PHYSICAL REVIEW E, VOLUME 64, 066107
Nonequilibrium dynamics of random field Ising spin chains: Exact results via real space renormalization group
Daniel S. Fisher Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138
Pierre Le Doussal CNRS-Laboratoire de Physique The´orique de l’Ecole, Normale Supe´rieure, 24 rue Lhomond, F-75231 Paris, France
Ce´cile Monthus Service de Physique The´orique, CEA Saclay, 91191 Gif-sur-Yvette, France ͑Received 18 December 2000; published 14 November 2001͒ The nonequilibrium dynamics of classical random Ising spin chains with nonconserved magnetization are studied using an asymptotically exact real space renormalization group ͑RSRG͒. We focus on random field Ising model ͑RFIM͒ spin chains with and without a uniform applied field, as well as on Ising spin glass chains in an applied field. For the RFIM we consider a universal regime where the random field and the temperature are both much smaller than the exchange coupling. In this regime, the Imry-Ma length that sets the scale of the equilibrium correlations is large and the coarsening of domains from random initial conditions ͑e.g., a quench from high temperature͒ occurs over a wide range of length scales. The two types of domain walls that occur diffuse in opposite random potentials, of the form studied by Sinai, and domain walls annihilate when they meet. Using the RSRG we compute many universal asymptotic properties of both the nonequilibrium dynamics and the equilibrium limit. We find that the configurations of the domain walls converge rapidly toward a set of system-specific time-dependent positions that are independent of the initial conditions. Thus the behavior of this nonequilibrium system is pseudodeterministic at long times because of the broad distributions of barriers that occur on the long length scales involved. Specifically, we obtain the time dependence of the energy, the magnetization, and the distribution of domain sizes ͑found to be statistically independent͒. The equilibrium limits agree with known exact results. We obtain the exact scaling form of the two-point equal time correlation
function ͗S0(t)Sx(t)͘ and the two-time autocorrelations ͗S0(tЈ)S0(t)͘. We also compute the persistence prop- erties of a single spin, of local magnetization, and of domains. The analogous quantities for the ϮJ Ising spin glass in an applied field are obtained from the RFIM via a gauge transformation. In addition to these we
compute the two-point two-time correlation function ͗S0(t)Sx(t)͗͘S0(tЈ)Sx(tЈ)͘ which can in principle be measured by experiments on spin-glass-like systems. The thermal fluctuations are studied and found to be dominated by rare events; in particular all moments of truncated equal time correlations are computed. Physical properties which are typically measured in aging experiments are also studied, focusing on the response to a
small magnetic field which is applied after waiting for the system to equilibrate for a time tw . The nonequi- Ϫ ϳ ␣ˆ ␣ˆ Ͻ librium fluctuation-dissipation ratio X(t,tw) is computed. We find that for (t tw) tw with 1, the ratio equal to its equilibrium value Xϭ1, although time translational invariance does not hold in this regime. For Ϫ ϳ ϭ Þ t tw tw the ratio exhibits an aging regime with a nontrivial X X(t/tw) 1, but the behavior is markedly different from mean field theory. Finally the distribution of the total magnetization and of the number of domains is computed for large finite size systems. General issues about convergence toward equilibrium and the possibilities of weakly history-dependent evolution in other random systems are discussed.
DOI: 10.1103/PhysRevE.64.066107 PACS number͑s͒: 05.50.ϩq
I. INTRODUCTION which it is qualitatively understood, to the large free energy barriers that must be overcome for order to be established on In many systems, the development of long range order is long length scales. Some of the best studied cases both ex- controlled by the dynamics of domain walls. The coarsening perimentally and theoretically are a variety of random mag- of domain structures evolving toward equilibrium has been netic systems, particularly spin glasses and random field studied extensively in pure systems ͓1,2͔, but little is known magnets ͓3–5͔. Both of these systems have engendered a quantitatively about domain growth in the presence of great deal of controversy about their equilibrium behavior, quenched disorder. In random systems, the nonequilibrium and the resolution of these controversies has been greatly dynamics plays an even more important role than in pure hampered by the inability of experiments to reach equilib- systems, and is most relevant for understanding experiments, rium. One might well argue, however, that the most interest- since many random systems become glassy at low tempera- ing properties of such random systems are, in fact, not their tures, with ultraslow dynamics which prevent full thermal equilibrium properties, but the nonequilibrium dynamics in- equilibrium from being established within the accessible volved in their Sisyphean struggle to reach equilibrium. time scales. This slow dynamics is due, at least in cases in Because of the dominance of the behavior of so many
1063-651X/2001/64͑6͒/066107͑41͒/$20.0064 066107-1 ©2001 The American Physical Society FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107 random systems by the interplay between equilibrium and which both the random field and the temperature are much nonequilibrium dynamic effects, it is important to find mod- smaller than the exchange coupling, the length and time els with quenched randomness for which solid results about scales are sufficiently long, and the random field dominates dynamics can be obtained. In particular, one would like to be the dynamics. With the random field being weak and the able to analyze the effects of activated dynamics caused by temperature low, the equilbrium correlation length is long large barriers, and compare results to predictions which and essentially the same as that at zero temperature—the come from either phenomenolgical scaling approaches to the Imry-Ma length. The coarsening of domains starting from dynamics—often known somewhat misleadingly as ‘‘droplet initial conditions with only short distance correlations—such models’’ ͓5͔, or from mean field approaches ͓6͔. Many of the as a quench from high temperature—will, after a rapid initial interesting phenomena go under the general name of aging— transient, be dominated by the randomness over a wide range the dependence of measurable properties such as correlations of time scales. and responses on the history of the system, in particular on The basic tool that we use in this paper is a real space how long it has equilibrated: its ‘‘age.’’ For example, one renormalization group ͑RSRG͒ method which we have de- would like to compute quantities which probe the violations veloped recently to obtain exact results for the nonequilib- of the fluctuation dissipation relations caused by nonequilib- rium dynamics of several 1D disordered systems ͓25͔. Most rium effects, and compare them with results obtained in of our previous results have been for the Sinai model that mean field models ͓6͔; this was done previously for coarsen- describes the diffusion of a random walker in a 1D random ing of pure models ͓7–9͔. static force field, which is equivalent to a random potential One of the simplest nontrivial random models, and one in that itself has the statistics of a 1D random walk ͓26͔.Itcan which coarsening occurs, is the random field Ising model readily be seen that individual domain walls in the classical ͑RFIM͒ in one dimension. Although this system does not RFIM diffuse in a random potential of exactly this Sinai have true long-range order, for weak randomness it exhibits a form, the complication being that they annihilate upon meet- wide range of scales over which the dynamics is qualitatively ing. As shown in Refs. ͓25,27͔ the RSRG model can also be like those of other random systems, especially those like applied to many-domain-wall problems such as that which two-dimensional random field magnets and two-dimensional corresponds to the RFIM. A few of the results of this paper spin glasses, which do not have phase transitions but exhibit were already presented in a short paper ͓25͔; the aim of the much dynamic behavior qualitatively similar to their three- present paper is to show in detail how the RSRG method dimensional counterparts which do have phase transitions. In applies to such disordered spin models, and to explore more particular, the weakly random one-dimensional ͑1D͒ RFIM of its consequences. Although we will give here a detailed has a wide range of length scales over which the typical size discussion of the RSRG method for the spin model, we will of ordered domains grows logarithmically with time at low rely on Ref. ͓28͔ for many results about the single particle temperatures. In one-dimensional the RFIM is equivalent to diffusion aspects of the problem; these we will only sketch, a spin glass in an applied magnetic field; this, or some analo- referring the reader to Ref. ͓28͔ for details. gous 1D systems, should be conducive to experimental in- As for the single particle problem, the RSRG method al- vestigation. For a recent review on the RFIM, see e.g. Ref. lows us to compute a great variety of physical quantities, ͓10͔. remarkably including even some which are not known for The equilibrium properties of the 1D RFIM and of the 1D the corresponding pure model ͑e.g., the domain persistence spin glass in a field were extensively studied ͓10–24͔. Sev- exponents ␦ and ). This provides another impetus for the eral thermodynamic quantities such as the energy, entropy, study of the random models. Using the RSRG method we and magnetization were computed exactly at low tempera- also obtain the equilibrium behavior which corresponds to a ture for a binary distribution of the randomness in Refs. ͓13– well defined scale at which the decimation is stopped. 15͔ and for continuous distributions in Refs. ͓22–24͔. Re- The RSRG method is closely related to that used to study sults are also available for distributions of bonds with disordered quantum spin chains ͓29–34͔. The crucial feature anomalous weight near the origin ͓16͔. The free energy dis- of the renormalization group ͑RG͒ is coarse graining the en- tribution was studied in Ref. ͓17͔. Equilibrium correlation ergy landscape in a way that preserves the long time dynam- functions are harder to obtain, and only a few explicit exact ics. Despite its approximate character, the RSRG method results exist. For the binary distribution some results are pre- yields asymptotically exact results for many quantities. As in sented in Ref. ͓20͔. Certain special limits have been solved, Ref. ͓31͔, it works because the width of the distribution of such as the infinite field strength limit which is related to barrier heights grows without bounds on long length scales, percolation ͓24͔, but for, e.g., higher cumulants of averages consistent with the rigorous results of Ref. ͓26͔. It is inter- of truncated correlations, only the general structure has been esting to note that an exact RG has also been applied to the discussed ͓21͔. problem of coarsening of the pure 1D soft-spin Ising ⌽4 In this paper we derive a host of exact results for the model at zero temperature, for which persistence exponents nonequilibrium dynamics of the RFIM and for the 1D Ising have been computed ͓35,36͔. Extensions to higher dimen- spin glass in a field, and, as a side benefit, also obtain many sions of the RSRG method for equilibrium quantum models equilibrium results, some to our knowledge new. Those of were recently obtained ͓37͔; these introduce hope that other our results for equilibrium quantities that can be compared dynamical models could be studied in higher dimensions as with previously known ones are found to agree with these. well. Here we focus on the universal regime of the RFIM in The outline of this paper is as follows. In Sec. II we define
066107-2 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107 the RFIM and spin glass models and their dynamics, and in lattice: on the bonds ͕(i,iϩ1)͖ will be indexed by their left- Sec. III we summarize some of the main results and the hand site i. A configuration of spins ͕Si͖ can be represented physics involved. In Sec. IV we explain how the RSRG as a series of ‘‘particles’’ A corresponding to domain walls method is applied to the RFIM, give the corresponding equa- of type (ϩ͉Ϫ) at positions a ,...,a , and ‘‘particles’’ B 1 NA tions and fixed point solutions, and discuss the properties of corresponding to domain walls of type (Ϫ͉ϩ) at positions the asymptotic large time state. In Sec. V we derive exact b1 ,...,bN ; these must of course occur in an alternating large time results for single time quantities such as the en- B sequence ABABAB ....͓The relation between the numbers ergy, the magnetization, and the domain size distribution. In N and N of domain walls depends on the boundary con- Sec. VI we obtain the time dependent single-time two-spin A B ditions: for instance, in a system with periodic boundary con- correlation functions both with and without an applied field. ditions, N ϭN , while in a system with free boundary con- In Sec. VII we compute the two time spin autocorrelation as A B ditions ͉N ϪN ͉р1.͔ well as the two-point two-time correlations. In Sec. VIII we A B It is very useful to introduce the potential ‘‘felt’’ by the study aging phenomena which necessitate considering rare domain walls, events which dominate all moments of the thermal ͑trun- cated͒ correlations which we obtain, as well as the response x ͒ϭϪ ͑ ͒ to a small uniform magnetic field applied after a waiting time V͑x 2͚ hi , 4 iϭ1 tw as in a typical aging experiment. We also compute the fluctuation dissipation ratio X(t,t ). In Sec. IX we compute w and rewrite the energy of a configuration ͕S ͖ in terms of the the persistence properties—the probabilities of no i positions of the set of domain walls as changes—of a single spin, of the local magnetization and of domains. Finally, in Sec. X we obtain the distribution of the NA NB HϭH ϩ ϩ ͒ϩ ͒Ϫ ͒ ͑ ͒ total magnetization and of the number of domains for large ref 2J͑NA NB ͚ V͑a␣ ͚ V͑b␣ , 5 finite size systems. A brief discussion of the possibilities for ␣ϭ1 ␣ϭ1 experimental tests of the predictions and speculation on the where H is the energy of the reference configuration applicability of the some of the general features to higher ref where all spins are (ϩ): dimensional systems are presented at the end of the paper. Various technical results are relegated to appendixes. In nϭN H ϭϪ Ϫ ͒Ϫ ͑ ͒ Appendix A the convergence towards the asymptotic state is ref J͑N 1 ͚ hi . 6 shown. In Appendix B the time dependent single-time two- iϭ1 spin correlation functions in an applied field is derived, while Each domain wall costs an energy 2J; the domain walls A in Appendix C, the two-point two-time correlations are com- feel the random potential ϩV(x); whereas the B walls feel puted in detail and in Appendix D some of the finite size the potential ϪV(x), i.e., the two types of domain walls feel properties are analyzed. opposite random potentials. Let us first recall some known features of the statics. In II. MODELS AND NOTATION the absence of an applied field (Hϭ0) the system is disor- ϭ A. Random field Ising model dered at T 0, and contains domains with typical size given by the Imry-Ma length 1. Statics 4J2 We consider the random field Ising chain consisting of N L Ϸ , ͑7͒ ϭϮ IM g spins ͕Si 1͖iϭ1,N with Hamiltonian
nϭNϪ1 iϭN obtained from the following simple argument: the creation of HϭϪ Ϫ ͑ ͒ a single pair of domain walls (A,B) a distance L apart costs J ͚ SiSiϩ1 ͚ hiSi , 1 iϭ1 iϭ1 exchange energy 4J independent of L. But the random po- ͉ Ϫ ͉ ϳͱ ͉ Ϫ ͉ tential has typical variations V(x) V(y) typ 4g x y , with independent random fields ͕͕hi͖͖ with identical distri- and thus the typical energy that the system can gain on a bution whose important moments are length L by using a favorable configuration of the random potential is of order ͱ4gL. The two energies become com- Hϵ¯h , ͑2͒ ϳ у i parable for L LIM and for L LIM , it becomes favorable for the system to create domain walls. Thus the ground state ϵ¯ 2Ϫ 2 ͑ ͒ g hi H , 3 will contain domains of typical size LIM . One should note the difference between the case of, e.g., bimodal distribu- so that H can be considered as a uniform applied field, and g tions ͑for which the ground state can be degenerate͒ and is the mean-square disorder strength; here and henceforth we continuous distributions for which it is nondegenerate; we denote averages over the quenched randomness—usually will generally consider continuous distributions for simplic- equivalent to averages over different parts of the system—by ity. overbars. At positive temperature without a random field, the ther- ϳ The statics and dynamics of the random field Ising model mal correlation length LT exp(2J/T) gives the typical size can be studied in terms of domain walls living on the dual of domains in the system in equilibrium at temperature T.
066107-3 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107
But in the presence of the random field the equilibrium state B. Ising spin glass in a magnetic field at finite temperature is dominated by the random fields and In this paper we also consider the 1D Ising spin glass in a still given by the Imry-Ma picture provided that uniform field h:
Ӷ Ӷ ͑ ͒ iϭNϪ1 N 1 LIM LT , 8 HϭϪ Ϫ ͑ ͒ ͚ Ji i iϩ1 ͚ h i . 16 which is the regime studied in the present paper. In the pres- iϭ1 iϭ1 Ͼ ence of a uniform applied field H 0, the system at low Ϯ temperature will contain domains of both orientations but As is well known, in the case of a bimodal ( J) distri- with different typical sizes leading to a finite magnetization bution with equal probabilities for either sign, 1D Ising spin Ͻ glasses in a field are equivalent via a gauge transformation to per spin m(g,H) 1. ϭ ⑀ random field ferromagnets. More precisely, setting Ji J i ϭϮ ϭ Ϫ ϭ 2. Nonequilibrium dynamics 1 for i 1,...,N 1, and defining 1 S1 and i ϭ⑀ •••⑀ ϭ 1 iϪ1Si for i 2,...,N, the gauge transformation In this paper we will study the magnetization non- gives the new Hamiltonian conserving dynamics of the RFIM model ͓Eq. ͑1͔͒ starting from a random initial condition at time tϭ0 corresponding iϭNϪ1 N HϭϪ Ϫ ͑ ͒ to a quench from a high temperature state. Although our ͚ JSiSiϩ1 ͚ hiSi , 17 results are independent of the details of the dynamics ͑pro- iϭ1 iϭ1 ͒ vided it satisfies detailed balance and is nonconserving , for ϭ ⑀ •••⑀ ⑀ definiteness let us consider the Glauber dynamics, where the where hi h 1 iϪ1. Since the i are independent random C C variables taking the values Ϯ1 with probability 1/2, the transition rate from a configuration to the configuration j , obtained from C by a flip of the spin j,is fields hi are also independent random variables taking the values Ϯh with probability 1/2. Hence the Hamiltonian de- Ϫ⌬ scribes a (Ϯh) random field Ising model with Hϭ0 and g e E 1 ⌬E ͑C→C ͒ϭ ϭ ͫ Ϫ ͩ ͪͬ ͑ ͒ ϭh2 in Eq. ͑1͒. W j ⌬ Ϫ⌬ 1 tanh , 9 e Eϩe E 2 2 The physical interpretation is as follows. The spin glass chain in zero field (hϭ0) has two ground states, given by Ϯ(0) (0)ϭ⑀ •••⑀ which satisfies the detailed balance condition and where i , where i 1 iϪ1, which correspond, via the ⌬Eϭ2JS (S Ϫ ϩS ϩ )ϩ2h S is the energy difference be- (0)ϭ j j 1 j 1 j j gauge tranformation, to the pure Ising ground states Si tween the two configurations, which takes the following pos- ϩ (0)ϭϪ Ͼ 1 and Si 1. In the presence of a field h 0, the sible values in terms of domain walls: ground state of the spin glass is made out of domains of either zero field ground states. Their typical size is thus given ⌬ ϭ Ϯ ͑ ͒ E ͕creation of two domain walls͖ 4J 2h j , 10 ϭ 2 2 by the Imry-Ma length LIM 4J /h . These domains corre- spond to the intervals between frustrated bonds since at the ⌬E ͕diffusion of one domain wall͖ϭϮ2h , ͑11͒ ϭ Ͻ j position of a domain wall one has Ji i iϩ1 JSiSiϩ1 0. Similarly each domain has magnetization: ⌬ ϭϪ Ϯ E ͕annihilation of two domain walls͖ 4J 2h j . ͑12͒ 1 1 ͯ ͚ (0)ͯϭ ͯ ͚ ͯϭ ͉ ͑ ͒Ϫ ͑ ͉͒ i hi V a␣ V b␣ , h 2h In this paper we focus on the regimes i domain i domain ͑18͒ ͕ ͖Ӷ ͑ ͒ hi J, 13 which is thus proportional to the absolute value of the corre- sponding barrier of the Sinai random potential, a property TӶJ, ͑14͒ which will be used below. Via the gauge transformation, the nonequilibrium dynam- in which Eq. ͑8͒ holds. The randomness will dominate on ics ͑e.g., the Glauber dynamics͒ of the spin glass in a field length scales that are sufficiently large so that the cumulative starting either from ͑i͒ random initial conditions ͑correspond- effect of the random field energies is greater than T, i.e., for ing to a quench from high temperature͒ or from ͑ii͒ the pure scales ferromagnetic state ͑obtained by applying a large magnetic field that is quickly reduced to be hӶJ) corresponds to the lӷ¯h2/T2. ͑15͒ nonequilibrium dynamics ͑e.g., Glauber͒ of the RFIM, also starting from random initial conditions. Thus in the regime So that there will be only one basic length and time scale, it hϳTӶJ, many of the universal results obtained for the ϳ is simplest to consider ͕hi͖ T. RFIM will hold directly for the spin glass in a field. Thus we There will be a wide range of time scales during which will study the two models in parallel in the present paper. the domains grow from initial sizes of O(1) to sizes of order An important complication in applying the results to 1D LIM by which time the system will be close to equilibrium. spin glasses is that the strengths of the exchange couplings Since the Imry-Ma length LIM is very large, we expect the will in general be random. This provides an extra random universality of the long time dynamics, as we indeed find. potential for the domain walls which has, however, only
066107-4 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107 short range correlations. The conditions for there to be a wall is pseudodeterministic: rescaled by the typical distance wide regime of validity of the universal coarsening behavior it has gone, ln2t, the wall’s position is asymptotically deter- found here is that the distribution of exchanges be narrow, ministic at long times. In a random field Ising chain, the two types of domain ͉ ͉ Ϫ͉ ͉ Ӷ͉ ͉ ͑ ͒ J max J min J min , 19 walls move in random potentials which are identical except ͉ ͉ Ϫ͉ ͉ ϳ for their sign. When walls meet, they annihilate. Not surpris- ideally with J max J min h or smaller. Note that the 2 ͉ ͉ ingly, since each wall can move a distance of order ln t in equilbrium correlation length will be set by J min for small h time t and they cannot pass through each other or occupy the ͉ ͉ as the domain walls can easily find positions with J same position, the density of domain walls remaining at time Х͉ ͉ J min that are near to extrema of the potential caused by t is simply of order 1/ln2t so that the correlation length start- the random fields. Having defined the models, before turning ing from random ͑or short-range correlated͒ initial conditions to the details of the calculations we now briefly decribe the grows as important physics and summarize our main results. ͑t͒ϳ⌫2/gϳln2t. ͑21͒ III. QUALITATIVE BEHAVIOR AND SUMMARY OF RESULTS This is in sharp contrast to the much faster power-law growth of the correlation length in most nonrandom systems, As for any Ising system, the static and dynamic properties for example the (t)ϳͱt for Ising systems with a noncon- of random spin chains can be fully described in terms of served order parameter. domain walls between ‘‘up’’ and ‘‘down’’ domains. The cru- The time-dependent correlation length can be more pre- cial simplification in one dimension is that the domain walls, ϵ ϩ͉Ϫ ϵ Ϫ͉ϩ cisely defined from the average nonequilibrium correlation which come in two types A ( ) and B ( ), are function. This, and most other nonequilibrium properties, are point objects. As discussed above, a random field induces a scaling functions of the ratio of lengths to powers of log- potential that the walls feel which has the statistics of a ran- times, as is characteristic of ‘‘activated dynamic scaling’’ dom walk so that its variations on a length scale L are of ͱ ͓39͔ which occurs in many random systems. We find that in order L. The A walls tend to minima of the potential, while zero applied field the average equal-time correlation function the B walls tend to maxima. If they meet they annihilate but, behaves as on the time scales of interest here, the probability that a pair Ϫ4J/T is spontaneously created, e can be ignored. x As shown by Sinai ͓26͔, the motion of a single domain 48ϩ64͑2nϩ1͒22g ϱ ⌫2 wall in such a random potential is completely dominated by ͒ ͒ ϭ ͗S0͑t Sx͑t ͘ ͚ the barriers that have to be surmounted to find low energy nϭϪϱ ͑2nϩ1͒44 extrema. Since the time to surmount a barrier of height b is b/T 2 2 2 of order e , and to find a minima a distance L away a ϫeϪ(2nϩ1) 2g(x/⌫ ) ͑22͒ barrier of order bϳhͱL will have to be overcome, the typi- cal distance a wall moves in time t is only of order l(t) whose Fourier transform is, at large time, ϳln2t. Although this motion is controlled by rare thermal ϩϱ fluctuations that take the wall over a large barrier, the posi- iqx͗ ͑ ͒ ͑ ͒͘ tion of a wall that started at a known point can, at long times, ͚ e S0 t Sx t xϭϪϱ be predicted with surprising accuracy. This is because the width of the distribution of barriers to go distances of order L 8 is as broad as the magnitude of the lowest barrier which ϳ Re͓tanh2 ͱiq͑t͒/2 ͔, ͑23͒ 2 2͑ ͒ „ … enabled the wall to move that distance. Thus at long times, q t when the barriers become very high, the probability of going first over other than the lowest surmountable barrier that de- with Re denoting the real part. These results obtain until the limits the region in which the wall is currently, is extremely nonequilibrium correlation length small. The position of a single wall at long times is thus 2 2 determined by its initial position and by the height of the T ln t ͑t͒ϭ ͑24͒ maximum barrier, which it could surmount up to that time; 2g2 this quantity, which we denote reaches the equilibrium correlation length ⌫ϵT ln t, ͑20͒ 8J2 2 yields a well defined region that the wall can explore up to ϭ ϭ ͑ ͒ eq LIM . 25 time t. The boundaries of the appropriate region that encom- 2g 2 passes the wall’s initial position are determined by ⌫(t). The wall will be in local equilbrium in this ‘‘valley’’ on time It is intriguing that the form of the nonequilibrium correla- scales of order t, and thus tend to spend most of its time near tions in the universal scaling limit are identical to those of the bottom of the valley. This behavior, as proved by the equilibrium correlations, the only difference being the Golosov ͓38͔, implies that the long time dynamics of a single correlation length.
066107-5 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107
A remarkable property of the coarsening in the RFIM are complications here due to the fact that a small applied chain was conjectured in Ref. ͓25͔: the positions of the set of field needs to be left on to provide the randomness, but the domain walls and hence all of the correlations in the non- basic physics is similar. equilibrium state are asymptotically deterministic at long The single spin autocorrelation function provides some times. This means that while the evolution of the domain information on how much memory the system has of its wall structure is only logarithmic in time, the domain wall earlier history. More information is provided by the two-spin configurations of two runs following quenches using the two-time correlation function ͗Sx(t)Sy(t)͗͘Sx(tЈ)Sy(tЈ)͘ same sample converge to each other much more rapidly as a which converges to a scaling function of (xϪy)/(t) and power of time. More precisely, the probability that the spin (tЈ)/(t), whose Fourier transform in (xϪy), D(q,t,tЈ) configurations of the two runs measured at the same time we compute exactly; to our knowledge, this is the first such after the quench differ substantially in a region of size of computation for a physical model pure or random. For an order (t) decays to zero as a nonuniversal power of time, ideal spin glass in which there are no correlations between becoming negligible well before the system is anywhere near the positions of positive and negative exchanges and the dis- equilibrium. ͓Strictly speaking, because in a rare region, as tribution of the exchange strengths is symmetric in J→ϪJ, discussed later, fluctuations of the position of an individual this correlation function D is the average over the or neighboring pair of domain walls will exist, one will need randomness—or equivalently a positional average over the to do a certain amount of time averaging for this pseudode- regions being probed by the scattering—of the product of the terminism to become most evident.͔ magnetic scattering intensity at time t and wave vector k and The asymptotic determinism will occur even if the initial that at time tЈ and wave vector qϪk. conditions were macroscopically distinct in the two runs: for Other properties associated with ‘‘aging’’ can also be example, if one was quenched from a high temperature state computed exactly; we study the thermal fluctuations around in zero applied field, and the other from a high temperature the configurations at two different times, and the dynamic state with a small net magnetization caused by a uniform linear response to a uniform magnetic field that is turned on applied field. This pseudodeterminism in a system with many after waiting for some time for the system to equilibrate. In degrees of freedom is a dramatic effect; it implies that the equilibrium, these are related by the fluctuation dissipation history dependence can, even under strongly nonequilibrium theorem but we find, as expected, that this relation generally conditions, be weak; one might thereby be fooled into think- fails except in the limit that the time difference is much less ing that such history independence implies equilibrium. than the waiting time. The behavior we find is, however, While equal-time correlations during coarsening converge rather different from that found in mean-field models ͓40͔, to an equilibriumlike form, two-time quantities show an in- and we discuss the contrasts between these results. teresting history dependence, generally depending on both In random systems controlled by zero temperature fixed times rather than just the time difference as they would do in points, such as is the case in the regime of scales studied equilibrium. There are typically three regimes: the later time here, thermal fluctuations and linear response functions are t, much longer than the earlier, time tЈ, the two times of the both dominated by rare spatially isolated regions of the same order, and the difference between the two times (t system—although which regions dominate depends on the ϪtЈ) much smaller than either time. The scaling variables in time scales and properties of interest. The study of these thus these three regimes are ln tЈ/ln t, tЈ/t, and ln(tϪtЈ)/ln t, re- involves the corrections to the deterministic approximation spectively. In the first and third of these, the condition for to the dynamics that led, as discussed above, to exact asymptopia, that the scaling variable is small, is very difficult asymptotic results for many other quantities. The dominant to attain in practice; thus knowing the full form of the scaling events are rare by a factor of, typically, 1/ln t, but neverthe- functions is essential for analyzing experimental or numeri- less still lead to universal results. cal data. Note that only in the last of these regimes should ‘‘Persistence’’ properties provide another probe of how one expect to find an equilibriumlike behavior characteristic much memory a system retains of its initial configuration, for of the local equilibrium that is being probed. example, what the probability is that a spin has never flipped. In this paper we compute a variety of correlations and Surprisingly, this and other related quantities—including response functions that illustrate some of the interesting non- some which are not known in pure systems—can also be equilibrium behaviors. The simplest of these is the temporal computed exactly for the RFIM chains. autocorrelations of a single spin. With no applied field, the average autocorrelation function is found to decay as a power of the ratio of the correlation lengths at two times, IV. REAL-SPACE RENORMALIZATION METHOD The coarsening process taking place in nonequilibrium ͑tЈ͒ dynamics of the RFIM starting from random initial condi- ͗S ͑t͒S ͑tЈ͒͘ϳͩ ͪ , ͑26͒ tions can be thought of as a reaction-diffusion process in a x x ͑t͒ 1D random environment ͑Sinai landscape͒ for the domain walls. Indeed, in the regime TӶJ considered in this paper, for tϾtЈ with the exponent ϭ1/2. In a three-dimensional the creation of pairs of domain walls is highly suppressed. spin glass, the autocorrelation function determines the non- The dynamics of the domain walls is thus dominated by the equilibrium decay of the magnetization after a large uniform random field as follows. The domains walls A quickly fall to ͓ ͔ ϵ field is turned off below the transition temperature 5 . There the local minima of the random potential V(i) Vi , whereas
066107-6 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107
FIG. 1. ͑a͒ Energy landscape in the Sinai model ͑b͒ decimation FIG. 2. The four possible cases of evolution under the decima- ϭ tion of the middle bond of the domain walls, given the constraint of method: the bond with the smallest barrier Fmin F2 is eliminated, resulting in three bonds being grouped into one. alternating sequence of domains. A and B denote the domain walls (ϩ͉Ϫ) and (Ϫ͉ϩ), respectively, л denotes no domain wall present at the top or bottom of the renormalized bond, while C the B walls quickly move to the local maxima of V . Then i represents either B or л and D represents either A or л. they slowly diffuse by going over barriers in opposite Sinai Ϯ potentials Vi . When a A and B walls meet, they annihilate This defines a renormalized landscape at scale ⌫, where all preserving the alternating ABAB . . . sequence. barriers smaller than ⌫ have been eliminated. We can thus use the real-space renormalisation procedure ͓ ͔ Since the distribution of barriers is found to become introduced in Ref. 28 to study the Sinai diffusion of a broader and broader ͓31͔ an Arrhenius argument implies that single particle, and extend it to take into account the annihi- the diffusion of a particle becomes better and better approxi- lation processes. In the single particle case this procedure mated at large time by the following ‘‘effective dynamics’’ was shown to be asymptotically exact at long length and ͓25,28͔. The position x(t) of a particle that started at x at time scales. Here it is also expected to be asymptotically 0 tϭ0 coincides with—or is at least very close to—the bottom exact at large time, since, as in the Sinai case, we find that of the renormalized bond at scale the the effective distribution of the random barrier heights become infinitely broad in the limit of large scales. Thus the ⌫ϭT ln t, ͑29͒ motion over the barriers becomes more and more determin- istic at long times. which contains x0. Note that we choose time units so as to set the microscopic ͑nonuniversal͒ inverse attempt frequency ͓ ͔ A. Definition of the real space renormalization procedure to unity 25,28 . This RG procedure is thus essentially deci- mation in time. Processes that are faster than a given time We briefly outline the renormalization procedure, detailed scale are decimated away and assumed to be in local equi- ͓ ͔ in Refs. 25,28 , for the diffusion of a particle in the Sinai librium. landscape Vi . Grouping segments with the same sign of the In the presence of domain walls of types A and B,we random field, one can start with no loss of generality from a must keep track of both the diffusion and the possible reac- ͑ ͒ ‘‘zigzag’’ potential Vi where each segment ‘‘bond’’ is tions of the domain walls that occur during the decimation. ϭ͉ Ϫ ͉ characterized by an energy barrier Fi Vi Viϩ1 and a Upon the decimation of bond 2 ͑see Fig. 1͒, there are four length li . From the independence of the random fields on possible cases, illustrated in Fig. 2, according to whether or each site, the pairs of bond variables (F,l) are independent not there is a type A wall at the bottom of bond ͑2͒, and from bond to bond and are chosen from a distribution P(F,l) whether or not there is a type B wall at the top of bond ͑2͒. normalized to unity. If ͑i͒ there are no type A or B walls, then one simply renor- The RG procedure which captures the long time behavior malizes the bond and nothing happens to the domain walls. If in a given energy landscape is illustrated in Fig. 1. It consists ͑ii͒ there is an A wall but no B wall, then the A wall goes to of the iterative decimation of the bond with the smallest the bottom of the new renormalized bond. If ͑iii͒ there is a B barrier, and hence the shortest time scale for domain walls to wall and no A wall then the B wall goes to the top of the new ͓ ͔ overcome it 25,28 . This smallest barrier, say, F2, together renormalized bond. And if ͑iv͒ there is both an A wall and a with its neighbors, the two bonds 1 and 3, are replaced by a B wall, then the two domain walls meet and annihilate upon single renormalized bond with barrier decimation. This annihilation will occur at a time of order e⌫/T determined by the barrier of the decimated bond: by FЈϭF ϪF ϩF ͑27͒ 1 2 3 assumption, ⌫ at this scale. The only truly new process compared to the single par- and length ticle diffusion is the case ͑iv͒. The above RG rule is consis- tent with the large time dynamics in this case for the follow- ϭ ϩ ϩ ͑ ͒ lЈ l1 l2 l3 . 28 ing reason. Since A and B domains diffuse with independent
066107-7 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107 thermal noises, the time it takes for them to meet is again T ln tϭF ͑accurate on a log scale͒. Indeed the probability that they meet at a point at potential V is ϳexp Ϫ(FϪV)/ Ϫ ϭ Ϫ ͓ „ T)exp( V/T) exp( F/T… as can be seen from considering the equivalent 2d diffusion problem ͑of the pair of wall po- sitions͒ with an absorbing wall at xϩyϭl͔. The second difference from the procedure in the case of FIG. 3. Full state in the renormalized landscape; as we show, the single particle, is that it must be stopped when the renor- this corresponds to the state at large time ͑see the text͒. The top and malisation scale ⌫ reaches the energy cost of creation of a bottom of the bonds are occupied by B and A domain walls, respec- pair of domain walls: tively. The correponding spin orientations are also indicated for the ⌫ ⌫ϭ⌫ ϭ ͑ ͒ RFIM. Decimation under an increase of corresponds to an anni- J 4J. 30 hilation of the domain with the smallest barrier. For the spin glass the position of the domain walls corresponds to the frustrated Indeed beyond this scale T ln tϾ⌫ one must take into ac- J bonds. count creation of pairs of domain walls. As will be shown below, the state at ⌫ϭ⌫ gives the final equilibrium state. Ϯ 2 2 J u⌫ ͑p͒ϭͱpϩ␦ coth͓⌫ͱpϩ␦ ͔ϯ␦
B. RG equations and statistical fixed point of the landscape ͱ ϩ␦2 Ϯ p ϯ␦⌫ U⌫ ͑p͒ϭ e . ͑37͒ Independently of whether the bonds contain domain walls sinh͓⌫ͱpϩ␦2͔ (A or B) or not, which is studied in Sec. V, one can study the evolution under RG of the landscape. Since the RG rules This exact knowledge of the renormalized landscape will be preserve the statistical independence of the variables (F,l) used below to extract physical quantities for the spin models. from bonds to bonds, it is possible to write closed RG equa- ϩ Ϫ C. Convergence toward ‘‘full’’ states: tions for the landscape, i.e., for P⌫ (ϭFϪ⌫,l) and P⌫ ( Asymptotic determinism ϭFϪ⌫,l) which denote the probabilities that a Ϯ renormal- ized bond at scale ⌫ has a barrier Even armed with the statistical properties of the renormal- ized landscape, we still have to determine the long time dis- Fϭ⌫ϩϾ⌫͑31͒ tribution of the occupation of the extrema of this landscape by A and B domain walls. At first sight this seems a difficult ͐ϱ ͐ϱ Ϯ ϭ and a length l, each normalized by 0 d 0 dlP⌫ ( ,l) 1, problem. Indeed, the positions of A and B walls can be cor- Ϯ ϯ Ϯ Ϯ related over many bonds of the renormalized landscape since ͒P ͑,l͒ϭP ͑0,.͒* P ͑.,.͒* P ͑.,.͒ ͑32͒ץϪ⌫ץ͑ ⌫ ⌫ l ⌫ ,l ⌫ there are a priori empty maxima and minima and the domain ϱ wall positions must respect the alternating constraint ϩ Ϯ͑ ͒ ͵ Ј Ϯ͑ Ј͒Ϫ ϯ͑ Ј͒ ͑ ͒ ABABAB. However, the RG analysis becomes simple if the P⌫ ,l dl „P⌫ 0,l P⌫ 0,l …, 33 0 system reaches at some stage a ‘‘full’’ state which has one A wall at each minimum and one B wall at each maximum of where *l denotes a convolution with respect to l only and the renormalized landscape, as illustrated in Fig. 3. It is easy *,l with respect to both and l with the variables to be ͓ ͔ to see that such a ‘‘full’’ state is preserved by the RG pro- convoluted denoted by dots. As discussed in Refs. 31,28 , cedure. Also note that this ‘‘full’’ state would be obtained the solutions of these RG equations depend on an assymetry ␦ from the beginning if, for instance, the initial condition were parameter defined as the nonvanishing root of the equation completly antiferromagnetic. Ϫ ␦ Generally, we consider random initial conditions and thus e 4 hϭ1, ͑34͒ the initial state is& not a ‘‘full’’ state. However, one can show which reduces in the limit of weak bias H to that the system converges exponentially in ⌫ toward a ‘‘full’’ state, as we now discuss. H The renormalization procedure for the coarsening process ␦Ӎ , ͑35͒ 2g of the RFIM has the following important property: the con- figuration for the spins at scale ⌫ depends only on the renor- with ␦ϭ0 in the absence of a uniform applied field. Our malized landscape at ⌫ and on the initial configuration of the results are valid for long times as long as spins ͑e.g., equilibrium at high temperature before the quench at tϭ0). In particular, it does not depend on the ␦TӶ1. ͑36͒ initial landscape or—except occasionally at early times be- fore the barrier distribution becomes very broad—on the ⌫ Ϯ For large , the Laplace transform of the distributions P⌫ whole history of the reaction-diffusion processes of domain take the following form, in the scaling regime of small ␦ and walls. 2 small p with ␦⌫ fixed and p⌫ fixed ͓31͔: Let us first consider one ascending bond with extremities
ϱ (x,y) of the renormalized landscape and assume that there Ϯ Ϫ Ϯ q Ϫ Ϯ ͵ ͑ ͒ qlϭ ͩ ͪ u⌫ (q/2g) were initially n domain walls in the interval (x,y). Neglect- dlP⌫ ,l e U⌫ e 0 2g ing for the time being the influence of the two neighboring
066107-8 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107 bonds, it is easy to see that the state at ⌫ is determined only 1/⌫ϳ1/ln2t—we restrict consideration in what follows to the by n and does not depend on the order in which the reactions analysis of full states. Note that our results for the biased between domain walls have occured: indeed, it is determined case in which there is a small uniform field applied in addi- by the parity of n and by the nature of the domain wall tion to the random field do yield a power law growth of the closest to the bottom end. There are several cases. correlation length with time; however, in the regime of va- ͑i͒ If nϭ0, then the final state of the occupation of the lidity of these results, the power law is very small, and again end points of the bond is (л,л). the effects of ‘‘missing’’ domain walls are negligible at long ͑ii͒ If n is odd, and if the domain wall closest to the times. bottom is of type A, then the final state is (A,л) ͑iii͒ If n is odd, and if the domain wall closest to the D. Convergence toward equilibrium bottom is of type B, then the final state is (л,B) As mentioned earlier, the RG procedure must be stopped ͑iv͒ If n is even with nу2 and the domain wall closest to at the bottom is of type A, then the final state is (A,B) ͑v͒ If n is even with nу2 and the domain wall closest to ⌫ϭ⌫ ϭ ͑ ͒ J 4J, 38 the bottom wall is of type B, then the final state is (л,л). Of course, to obtain the real occupation of the top and the since at this scale one must start to take into account the bottom ends of one renormalized bond, one also needs to energetic benfits of creation of pairs of domain walls. At a consider what happens on the two neighboring renormalized ⌫ϭ⌫ scale J the typical domain size is the Imry-Ma length bonds, and to compute the probabilities of the various states, ϭ 2 LIM 4J /g, and the energy cannot be lowered further by taking properly care of the alternating constraint ABAB. any process in the full state. Indeed, moving the domains This is done in Appendix A. It is found that the crucial walls without changing their number cannot lower the total feature is that in order for a bond not to have both ends energy, since domain walls already occupy all tops and bot- occupied by domain walls, either it or one of its neighboring toms of the renormalized potential. Decreasing the number bonds must have had no domain walls on it initially. Since of domain walls by two also cannot lower the energy, since the bonds tend to become progressively longer with time, the the gain is 4J while the loss due to the random field for walls chance of this occuring drops rapidly with increasing ⌫. This separated by a bond of barrier F is FϾ⌫ϭ4J. Similarly, to will be true even if the positions of the domain walls have add the two walls the cost is 4J and the gain is FϽ⌫ϭ4J, some local correlations, in particular for the case of an initial since the only positions they can occupy are by definition state that corresponds to a high temperature configuration in separated by a barrier Ͻ⌫ which has already been deci- a small magnetic field so that there is an initial magnetization ⌫ mated. Thus if the renormalization is stopped at J , in the and the typical distances between a neighboring A and B wall small field, low T scaling limit the configuration of the walls will depend on which of the two is on the left. corresponds precisely to the ground state and, up to negli- We thus find that the system converges towards the gible thermal fluctuations, to the thermal equilibrium state. ‘‘full’’ state of the renormalized landscape exponentially fast Thus we are able to compute equilibrium properties straight- ⌫ in , with a nonuniversal coefficient that depends on the forwardly from the renormalization group analysis. initial concentration of domain walls, on the inital magneti- Thus the RG approach allows one to study the approach zation, and on the strength of the randomness. This corre- Ϫ to equilibrium starting from any initial condition character- sponds to a power law decay in time, as t with nonuni- 2J/T0Ӷ ized by typical domain sizes L0e LIM , where T0 rep- versal, of the probability that a maximum or minimum of the resents the temperature before the quench. As explained renormalized landscape at time t is unoccupied. The posi- above, under these conditions the relaxation towards equilib- tions of the full set of domain walls at long times are thus rium always takes place by diffusion and annihilation of do- asymptotically deterministic and independent of the initial main walls before any domain creation can occur. conditions provided these have only short range correlations. Concretely, this asymptotic determinism can be character- E. Approach to equilibrium from more ordered initial ized by the typical mean distance between ‘‘missing’’ do- conditions main walls, i.e., deviations from the deterministic full state. The distance between these ‘‘errors’’ will grow exponen- A qualitatively different type of evolution towards equi- tially in ⌫ and hence as a power of time. The only exceptions librium takes place in the other limit ͑not studied here͒, ͑ ͒ to this are associated with fluctuations in the nominally full 2 states: because of the rare times or configurations in which 4J L ϳe2J/T0ӷL ϭ , ͑39͒ domain walls are either about to annihilate or are spending 0 IM g substantial fractions of their time in more than one valley; Ӷ these fluctuation induced ‘‘errors’’ we will study in detail still imposing h T0. This is a regime of initial temperatures later; they decay far more slowly with time than any persis- low enough that the initial density of domains is very small. tent initial condition induced differences between runs. In this case a very different relaxation process toward the Since all the universal quantities that we will study have same equilibrium state discussed above takes place: for ⌫ Ͻ⌫ ϭ length scales that grow much more slowly—typically J 4J well separated domain walls diffuse independently ⌫ ⌫ logarithmically—in time, or probabilites of occurring which with very rare annihilations. When reaches J many do- decay much more slowly with time—typically as powers of main walls are ‘‘suddenly’’—within a factor of 2 or so in
066107-9 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107 time—created, and the large initial domains break into many 4g smaller ones of size L . Thus, the relaxation is more abrupt ϭϪJϩ ͑2JϪT ln t͒, ͑47͒ IM 2 in this case than in the more interesting one studied in the ͑T ln t͒ present paper. where ͗F͘⌫ denotes the averaged barrier at scale ⌫. This formula holds up to time t ϳexp(4J/T) where the ground V. ENERGY, MAGNETIZATION, AND DOMAIN SIZE eq state energy at equilibrium is reached: DISTRIBUTION g E ӍϪJϪ . ͑48͒ From the knowledge of the fixed point for renormalized gs 2J landscape ͓Eq. ͑37͔͒ and the fact that the system reaches the ͑ ⌫͒ full state exponentially fast in where each top is occupied Since we consider the regime gӶJ, this result is expected to by a B domain and each bottom by a A domain, we can be exact to first order in g/J. It does indeed agrees with the immediately compute several simple quantities. We will exact result of Ref. ͓19͔ concerning the bimodal distribution, compare these results with the existing exact results known expanded to first order in g. To obtain higher orders in the in the statics. expansion in g/J one would need to compute within the RG higher orders in a 1/⌫ expansion. A. RFIM without applied field Note that the entropy per spin at Tϭ0 computed in Ref. ͓ ͔ Ϯ Specializing Eq. ͑37͒ to the case ␦ϭ0 in the absence of 19 for h distributions originates from degenerate con- an applied field, the fixed point of the RG equation is ͓30,31͔ figurations occuring from short scales and is thus nonuniver- sal. If the distribution is continuous we expect SϳT from ͱs short scales, also nonuniversal. ˜ *͑ ͒ϭ Ϫ1 ͩ Ϫͱs coth ͱs ͪ ͑ ͒ P , LTs→ ͱ e , 40 sinh s 3. Distribution of lengths of domains where we have introduced the dimensionless rescaled vari- Since the bond lengths in the renormalized landscape are ables for barriers, uncorrelated, we obtain the result that the lengths of the do- mains in the RFIM ͑both in the long time dynamics and at ϭ͑FϪ⌫͒/⌫, ͑41͒ equilibrium͒ are independent random variables. ͑Note that this is different from the exact result for the dynamics of the and for bond lengths, pure Ising chain obtained in Ref. ͓41͔.͒ Moreover, during the coarsening process, the probability l ϭ ⌫2ϭ 2 2 ϭ2g . ͑42͒ distribution of the rescaled length 2gl/ 2gl/T ln t is ⌫2 obtained as
1. Number of domain walls per unit length Ϫ 1 P*͑͒ϭLT 1 ͩ ͪ ͑49͒ p→ ͑ͱ ͒ Thus for large ⌫ϭT ln t, the average bond length ¯l ⌫ , cosh p equal to the average distance between two domain walls, ϱ behaves as 1 2 2 ϭ ͚ ͩ nϩ ͪ ͑Ϫ1͒neϪ [nϩ(1/2)] ϭϪϱ 2 1 1 n 2 2 2 ¯l ⌫ϭ ⌫ ϭ T ln t. ͑43͒ ϱ 4g 4g 1 1 ϭ ͚ ͑Ϫ1͒mͩ mϩ ͪ ͱ 3/2 ϭϪϱ The number of domain walls per unit length decays as m 2 ϫ Ϫ(1/)[mϩ(1/2)]2 ͑ ͒ 4g e . 50 n͑t͒ϭ , ͑44͒ T2 ln2 t The distribution of the length of domains at equilibrium is also given by P( ), where ϳ eq up to time teq exp(4J/T) at which equilibrium is reached, and 2gl l ϭ ϭ ͑ ͒ eq 2 . 51 1 ⌫ 2LIM ͒ϭ ϭ ͑ ͒ J n͑teq neq . 45 LIM B. Spin glass in a field 2. Energy density Using the gauge transformation described in Sec. II B, the The energy per spin as a function of time ͑i.e., of ⌫ above results readily apply also to the spin glass in a field. ϭT ln t) is simply given by Let us recall that a domain in the RFIM corresponds in the SG to an interval between two frustrated bonds. Using the ӍϪ ϩ ͑ Ϫ 1 ͗ ͘ ͒ ͑ ͒ E⌫ J n⌫ 2J 2 F ⌫ 46 above expressions for the zero applied field RFIM and re-
066107-10 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107 placing g→h2, we thus obtain the averaged size of these where ␥ϭ⌫␦ϭ␦T ln t and ⌫ϭT ln t. In the long time limit it domains from Eq. ͑43͒, their number per unit length from is these are Eqs. ͑44͒ and ͑45͒, and their distribution of lengths from Eqs. ͑49͒ and ͑51͒. Ϫ T ln t ¯l ⌫ Ϸ , ͑56͒ The distribution P⌫(M) of the magnetization M 2H ϭ͉͚ ͉ i domain i of each domain is obtained from the distri- bution of barriers as g ϩ TH/g ¯l ⌫ Ϸ t . ͑57͒ 2 1 MϪM ⌫ 2H ͑ ͒ϭ ͩ Ϫ ͪ ͑ Ϫ ͒ ͑ ͒ P⌫ M exp M M ⌫ , 52 M ⌫ M ⌫ Note that the fact that the length of domains with spins in the ⌫ opposite direction from the applied field still grow, on aver- M ⌫ϭ , ͑53͒ age, is simply due to the fact that the smallest ones ͑with 2h barriers smaller than T ln t) keep being eliminated. with ⌫ϭT ln t. In equilibrium the same result holds with The number of domain walls per unit length thus decays as M ⌫→2J/h. Note that since this variable is proportional to the barrier, there are no domains of magnetization smaller 2 ␦2 than M ⌫ . Similarly the joint distribution of magnetization n͑t͒Ӎ ϭ4g , ͑58͒ ϩ Ϫ 2 and length is given by Eq. ͑40͒ in Laplace transform and ¯l ⌫ ϩ¯l ⌫ sinh ͑␦T ln t͒ 2 2 rescaled variables ϭ(MϪM ⌫)/M ⌫ and ϭ2h l/⌫ , with again the property of statistical independence of domains. and the magnetization per spin grows as Finally, the energy per spin is given by expression ͑47͒, re- 2 ϩ Ϫ placing g→h . ¯l ⌫ Ϫ¯l ⌫ H ͑ ͒ϭ ϭMͫ ͬ ͑ ͒ Note that the results here are, strictly speaking, restricted m t ϩ Ϫ T ln t , 59 ¯l ϩ¯l 2g to the case in which the mapping to a random field Ising ⌫ ⌫ model is exact: the case in which all of the exchange inter- ␥ actions have the same magnitude and only differ in sign. M͓␥͔ϭcoth͑␥͒Ϫ . ͑60͒ Nevertheless, in the more general case with a distribution of sinh͑␥͒2 ͉J͉’s, the universal aspects of the nonequilibrium behavior M͓␥͔ M͓␥͔ϳ 2 ␥ ␥ will be the same as long as there is a nonzero lower bound to The function starts as 3 for small , and ͉ ͉ this distribution, J min . However, at times longer than goes exponentially to Mϭ1 for large ␥. ϭ͉ ͉ ϳ T ln t J min , domain walls will no longer necessarily be an- These results hold up to time teq exp(4J/T) at which nihilated; whether they are or not will depend on the local J equilibrium is reached. The number of domain walls per unit as well as on the renormalized potential. This will, of course, length in the equilibrium state is thus also affect the equilibrium positions of domain walls, but because there will always tend to be weak exchanges near to ␦2 the extrema of the potential caused by the random fields, the n Ӎ4g , ͑61͒ eq 2͑ ␦ ͒ changes in the positions of the walls will be negligible on the sinh 4 J scale of the correlation length. with averaged sizes
C. RFIM in an applied field Ϫ 2J ¯l Ϸ , ͑62͒ In a similar manner to the above analysis, we obtain re- eq H sults when a small uniform field H is applied. Solutions ͑37͒ of the RG equations now depend on the parameter ␦ defined ϩ g as the nonvanishing root of Eq. ͑34͒, equal to ␦ϭH/(2g)in ¯l Ϸ e4JH/g. ͑63͒ eq 2 the small field limit for which our results will be asymptoti- 2H cally exact. The equilibrium magnetization per spin meq is 1. Number of domain walls and magnetization 2JH The averaged sizes of domains (ϯ) ͑with spins oriented m ϭm͑t ͒ϭMͫ ͬ. ͑64͒ eq eq g respectively against and along the field͒ are found both to grow with time as 2. Energy per spin ͑␥͒ Ϫ 1 sinh Ϫ␥ The energy per spin as a function of time ͑i.e., of ⌫ ¯l ϭ⌫2 ͩ 1Ϫ e ͪ , ͑54͒ ⌫ 4␥g ␥ ϭT ln t) is simply
1 sinh͑␥͒ 1 ¯ϩϭ⌫2 ͩ ␥ Ϫ ͪ ͑ ͒ ϭϪ Ϫ ϩ ͩ Ϫ ͗ ͘ ͪ ͑ ͒ l e 1 , 55 E⌫ J H n⌫ 2J F Ϫ 65 ⌫ 4␥g ␥ 2
066107-11 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107
(n) where Q⌫ (x) is defined as the probability that point x be- longs to the bond n given that the point 0 belongs to the bond 0 of the renormalized landscape ͑see Fig. 4͒. We have ͚ϱ (n) ϭ ϭ the normalization nϭ0Q⌫ (x) 1. For n 0, the probabil- ity that x is on the same renormalized bond as 0 is FIG. 4. Renormalized landscape at scale ⌫ϭT ln t, indicating sites 0 and x of two spins and the bonds between them. ϱ l l (0) 1 0 0 Q ͑x͒ϭ ͵ dl P⌫͑l ͒ ͵ dy ͵ dy ␦ xϪ͑y Ϫy ͒ , ⌫ ¯ 0 0 1 2 „ 2 1 … ␦2 l ⌫ 0 0 y1 4 g 1 1 Ϫ ⌫␦ ͑ ͒ ϭϪJϪHϩ ͫ2JϪ ͩ ⌫ϩ ͑1Ϫe 2 ͒ͪͬ, 70 sinh2͑␦⌫͒ 2 2␦ ͑66͒ and thus, after rescaling the variables, we find where ͗F͘Ϫ denotes the averaged barrier of bonds against x ⌫ ⌫ ϭ (0)͑ ͒ϭ ͩ ϭ ͪ ͑ ͒ the field at scale . This obtains up to scale 4J,at Q⌫ x q X 2g , 71 eq 0 ⌫2 which equilibrium is reached with ground state energy where ␦g ϭϪ Ϫ Ϫ Ϫ Ϫ8J␦͒ ͑ ͒ Egs J H ͑1 e . 67 sinh2͑4J␦͒ ϱ ͑ ͒ϭ ͵ ͑ Ϫ ͒ ͑ ͒ ͑ ͒ q0 X 2 d 0 0 X P* 0 72 X This result is compatible with the result Eq. ͑80͒ of ͓ ͔ Derrida-Hilhorst 18 obtained from studying products of in terms of the fixed point solution P*() given in Eq. ͑49͒ random matrices, as can be checked with the correspondance for the distribution of rescaled lengths ϭ2gl/⌫2. For n ␣ ϭ ␦ DH 2 . In addition, here we obtain the explicit scaling у1, we have form in the small ␦ limit with ␦J fixed. We have checked that this scaling form is also consistent with the exact result x ͓Eq. ͑8͒ in Ref. ͓19͔͔ for the bimodal distribution at leading (n)͑ ͒ϭ ͩ ϭ ͪ ͑ ͒ Q⌫ x qn X 2g , 73 order in ␦. ⌫2
Á 3. Distribution of domain lengths P⌫ (l) with As in the zero field case, the lengths of the domains are independent random variables. Their probability distribu- ͑ ͒ϭ ͵ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ qn X 2 P* 0 P* 1 P* nϪ1 P* n Ͼ ••• tions can be obtained from Laplace inversion of Eq. ͑37͒. For y1 ,y2 , i 0 Ϯ domains they read ͑74͒
ϩϱ ␦ XϪ͑y ϩ ϩ ϩ ϩ Ϫ ϩy ͒ ͑ Ϫy ͒ Ϯ Ϯ Ϯ Ϫ Ϯ ␥ „ 1 1 2 ••• n 1 2 … 0 1 ͑ ͒ϭ ͑␥͒ ͑␥͒ lsn ( ) ͑ ͒ P l ͚ cn sn e , 68 ϭ ϫ Ϫ ͒ ͑ ͒ n 0 ͑ n y 2 . 75 ␥ϭ␦ Ϯ ␥ Ϯ ␥ where T ln t, and the functions cn ( ) and sn ( ) are The Laplace transforms read given in Eqs. ͑50͒–͑53͒ of Ref. ͓28͔. ϱ 2 s ͑ ͒ϭ ͵ ϪsX ͑ ͒ϭ ͩ ͑ ͒Ϫ ϩ ͪ ͑ ͒ VI. EQUAL-TIME TWO-SPIN CORRELATION FUNCTION q0 s dXe q0 X P* s 1 , 76 0 s2 2 ŠS0„T…Sx„T…‹ IN THE RFIM We can compute the disorder averaged two spin correla- ϱ ͑ ͒ϭ ͵ ϪsX ͑ ͒ qn s dXe qn X tion function by noting that, in a given environment, the 0 equal-time two-spin thermal correlation ͗S0(t)Sx(t)͘ is equal to ϩ1 if the points 0 and x are at scale ⌫ϭT ln t on 1ϪP*͑s͒ 2 ϭ2ͩ ͪ ͓P*͑s͔͒nϪ1 for nу1, ͑77͒ renormalized bonds of the same orientation ͑i.e. both ascend- s ing or both descending͒, and is equal to Ϫ1 otherwise. and thus A. Zero applied field ϱ 1 4 1ϪP*͑s͒ The average over the environments is, in zero field H ͚ ͑Ϫ ͒n ͑ ͒ϭ Ϫ ͩ ͪ ͑ ͒ 1 qn s . 78 ϭ0, nϭ0 s s2 1ϩP*͑s͒ ϱ ͱ ͒ ͒ ϭ Ϫ ͒n (n) ͒ ͑ ͒ Now using the explicit solution P*(s)ϭ1/cosh( s), we ͗S0͑t Sx͑t ͘ ͚ ͑ 1 Q⌫ ͑x , 69 nϭ0 finally obtain
066107-12 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107
ϩ 1 4 ͱs ¯l ⌫ ͗ ͑ ͒ ͑ ͒͘ϭ Ϫ1 Ϫ 2ͩ ͪ ϩϩ͑ ͒ϩ ϩϪ͑ ͒ϭ ͕ ͓ ͑ ͔͒ϭϩ ͖ϭ S t S t LT 2 ͫ tanh ͬ 0 x s→Xϭ2g ͉͑x͉/⌫ ͒ 2 Ct x Ct x Prob sgn S0 t 1 ϩ Ϫ , s s 2 ¯l ⌫ ϩ¯l ⌫ ͑79͒ ͑86͒
¯Ϫ ͉x͉ ϪϪ ϩϪ l ⌫ 2 2 ͑ ͒ϩ ͑ ͒ϭ ͕ ͓ ͑ ͔͒ϭϪ ͖ϭ ϱ 48ϩ64͑2nϩ1͒ g Ct x Ct x Prob sgn S0 t 1 ϩ Ϫ , ⌫2 ¯l ⌫ ϩ¯l ⌫ ϭ ͚ ͑87͒ nϭϪϱ ͑2nϩ1͒44
2 2 2 and we can thus write the correlation function as ϫeϪ(2nϩ1) 2g(͉x͉/⌫ ) ͑80͒ ͒ ͒ Ϫ ͒ ͒ ͗S0͑t Sx͑t ͘ ͗S0͑t ͘ ͗Sx͑t ͘ with ⌫ϭT ln t. The leading long-distance behavior of the Ϫx/(t) ¯ϩϪ¯Ϫ 2 equal time correlation function is proportional to xe ϩϪ l ⌫ l ⌫ ϭ1Ϫ4C ͑x͒Ϫͩ ͪ . ͑88͒ rather than a simple exponential. The Fourier transform of t ϩ Ϫ ¯l ϩ¯l the spin-spin correlation function is simply ⌫ ⌫
ϩϱ Performing an analysis similar to the case of zero field, we 8 obtain the Laplace transform iqx ͒ ͒ ϭ 2 ͱ ͒ ͚ e ͗S0͑t Sx͑t ͘ Re͓tanh iq ͑t /2 ͔, xϭϪϱ 2q2͑t͒ „ … ϱ Ϫ ϩ͑ ͒ Ϫ Ϫ͑ ͒ ͑ ͒ ϩϪ 1 P⌫ q 1 P⌫ q 81 ͵ dxeϪqxC ͑x͒ϭ „ …„ … . t ͑¯ϩϩ¯Ϫ͒ 2 Ϫ ϩ͑ ͒ Ϫ͑ ͒ 0 l ⌫ l ⌫ q „1 P⌫ q P⌫ q … with Re denoting the real part. As explained previously, the ͑89͒ ⌫Ӎ⌫ ϭ renormalization procedure has to be stopped at J 4J, ϳ ϳ The Laplace inversion can be performed as in Ref. ͓28͔. The i.e., t teq exp(4J/T) where the equilibrium state has been ͑ ͒ ⌫ϭ⌫ correlation decays as a sum of exponentials, and the term reached. Thus Eq. 80 with J gives the mean equilib- rium spin correlation function. The correlation length (t)is which dominates the asymptotic decay gives a correlation given by the decay of the (nϭ0) term which dominates at length large distances ⌫2 ͑t͒ϭ , ͑90͒ 2 2 ϩ͑␥͒ T ln t 2gs0 ͑t͒ϭ ͑82͒ 2 2g ⌫ϭ ␥ϭ␦ ϩ ␥ where T ln t, T ln t, and the function s0 ( ) is de- fined by Eqs. ͑50͒ and ͑52͒ in Ref. ͓28͔. In particular the ϭ⌫ ϭ up to scale T ln teq J 4J, and we obtain the correlation asymptotic behavior for large ␥ is length at equilibrium: g ͑ ͒ϳ¯ϩϳ HT/g ͑ ͒ 8J2 2 t l ⌫ t . 91 ϭ ͒ϭ ϭ ͑ ͒ 2H2 eq ͑teq LIM . 83 2g 2 ␥ϭ␦⌫ ϭ ␦ Note that even for the equilibrium case ( J 4 J), this This formula is in agreement with the limit hӶJ of the exact correlation length result for the equilibrium correlation of Ref. ͓20͔ in the case Ϯ g of a bimodal distribution ( h). Finally, note that the RFIM ϳ e4HJ/g ͑92͒ eq 2 two-point correlation function ͗Si(t)Siϩx(t)͘ also corre- 2H sponds for the spin-glass to the following correlation func- ϭ (0) does not seem to have been obtained previously; it is very tion involving the zero-field T 0 ground state i : different from the correlation length of the truncated (0) (0) correlations—i.e., that of the thermal fluctuations— ͗S ͑t͒S ϩ ͑t͒͘ϭ ͗ ͑t͒ ϩ ͑t͒͘. ͑84͒ i i x i iϩx i i x computed for bimodal distribution in ͓20͔ and discussed here in Sec. VIII A. B. Nonzero applied field In the presence of an applied field HϾ0, one defines VII. AGING AND TWO-TIME CORRELATIONS ⑀ ⑀ 1 , 2 ͓ ͔ϭ⑀ Ct (x) as the probability that sgn S0(t) 1 and Some of the most interesting properties of random sys- ϭ⑀ sgn͓Sx(t)͔ 2. One has tems involve ‘‘aging,’’ the dependence of measured quanti- ties on the history of the system, particularly on how long it ͗ ͑ ͒ ͑ ͒͘ϭ ϩϩ͑ ͒ϩ ϪϪ͑ ͒Ϫ ϩϪ͑ ͒ ͑ ͒ S0 t Sx t Ct x Ct x 2Ct x , 85 has been equilibrated for. In this section we study one of the fundamental properties which show the effects of aging: We know already that two-time nonequilibrium correlations. As before, the system
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is quenched from a random initial condition at time tϭ0, and together with initial conditions at ␣ϭ1, we study the aging dynamics at late times between time tЈ ϭt —the waiting time—and t, both tЈ and tϾtЈ being large. ϱ w ͵ dP*͑,͒ We first consider the autocorrelation function of a given 1 0 1 ϩϩ ͑͒ϭ ϭ ͑ ϩ ͒ Ϫ spin, from which one can extract the autocorrelation expo- P␣ϭ1 1 2 e , 2 ¯ 6 nent . Note that since this is a single site quantity it applies ͑ ͒ directly to both the random field and spin glass problems. 99 Ϫϩ ϭ and P␣ϭ1( ) 0. The solutions are A. Spin autocorrelations in zero applied field We consider the autocorrelations of the random field Ising Ϯϩ 1 Ϯϩ Ϯϩ Ϫ P␣ ͑͒ϭ ͑A␣ ϩB␣ ͒e , ͑100͒ model in zero applied field, Hϭ0: 2 ͒ϭ ͒ ͒ ͑ ͒ C͑t,tЈ ͗Si͑t Si͑tЈ ͘. 93 with
Except at short times the system is in the ‘‘full’’ state, and Ϯϩ 1 1 1 A ϭ Ϯ ϯ , ͑101͒ hence C(t,tЈ) is simply the probability that the site i—which ␣ 6 3␣ 6␣2 we take to be the origin—belongs at both time tЈ and time t
to renormalized bonds with the same orientation. Thus this Ϯϩ 1 1 B ϭ Ϯ , ͑102͒ quantity can in principle be obtained from the result in Ref. ␣ 3 3␣ ͓28͔ for the probability P(x,t;xЈtЈ͉0,0) that a particle diffus- ing in a Sinai landscape starting at 0 at time tϭ0isatxЈ at ͐ϱ ϩϩ which obey the normalization condition 0 d „P␣ ( ) tЈ and x at t. Indeed C(t,tЈ) ͑since it is computed in the full ϩ Ϫϩ ϭ 1 ϭ ϮϪ P␣ ( ) 2 . Since with H 0 we have P␣ ( ) state͒ is simply related to the probability that a particles pos- ϯϩ … ϭP␣ (), we obtain titions X(t) and X(tЈ) have the same sign. However, it can also be obtained through a much simpler direct computation, ϱ ͑ Ј͒ϭ ͵ ϩϩ͑͒ϩ ϩϪ͑͒Ϫ ϩϪ͑͒Ϫ Ϫϩ͑͒ which we now present. C t,t d P␣ P␣ P␣ P␣ ϩϩ Ϫϩ „ … We define P (). P () as the probability that the 0 ⌫,⌫Ј „ ⌫,⌫Ј … ͑103͒ origin is on a descending bond at ⌫Ј, and is on a descending ͑ascending͒ bond of strength at a later stage, ⌫. The RG ϩ ϩ Ϫϩ Ϫϩ 4 1 equations read ϭA2 ϩB2 ϪA ϪB ϭ Ϫ , ͑104͒ ␣ ␣ ␣ ␣ 3␣ 3␣2 Ϯϩ ϯ Ϯϩ ͒P ͑͒ϭϪ2P ͑0͒P ͑͒ץϪ⌫ץ͑ ⌫,⌫Ј ⌫ ⌫,⌫Ј and thus the autocorrelation function of the RFIM in zero ϯ Ϯ Ϯϩ applied field at large times tуtЈ is ϩ2P ͑0͒P ͑ ͒*P ͑ ͒ ⌫ ⌫ • ⌫,⌫Ј • Ϯ Ϯ ϯϩ Ј Ј 2 ϩP ͑ ͒*P ͑ ͒P ͑0͒ ͑94͒ 4 ln t 1 ln t ⌫ • ⌫ • ⌫,⌫Ј ͗S ͑t͒S ͑tЈ͒͘ϭ ͩ ͪ Ϫ ͩ ͪ . ͑105͒ i i 3 ln t 3 ln t together with the initial conditions In particular the asymptotic behavior for fixed tЈ is ϱ ϩϩ 1 lP⌫Ј͑,l͒ P ͑͒ϭ ͵ dl , ͑95͒ ⌫Ј,⌫Ј 2 ¯ ¯l͑tЈ͒ 0 l ⌫Ј ͗ ͑ ͒ ͑ Ј͒͘ϰͩ ͪ ͑ ͒ Si t Si t , 106 ¯l͑t͒ Ϫϩ ͑͒ϭ ͑ ͒ P⌫Ј,⌫Ј 0. 96 where ¯l (t)ϳln2t is the characteristic length of the coarsening We introduce the scaling variable in the RFIM, and where the autocorrelation exponent is ϭ 1 ͑ ͒ ⌫ ln t 2 . 107 ␣ϵ ϭ . ͑97͒ ⌫Ј ln tЈ The autocorrelation being invariant under gauge transforma- tions, for the spin glass we immediately obtain Since for large ⌫Ј, P⌫Ј has reached its fixed point value ͓Eq. ͑40͔͒, in terms of the rescaled variable ϭ/⌫ one has ¯l͑tЈ͒ 1 ͗ ͑ ͒ ͑ Ј͒͘ϭ͗ ͑ ͒ ͑ Ј͒͘ϳͩ ͪ ϭ i t i t Si t Si t with . Ϯϩ ¯l͑t͒ 2 ϩ1͒P␣ ͑͒ץϪ͑1ϩ͒␣ץ␣͑ ͑108͒ Ϫ(ϪЈ) Ϯϩ Ϫ ϯϩ ϭ2 ͵ dЈe P␣ ͑Ј͒ϩe P␣ ͑0͒, Note that this value of saturates the lower bound of d/2 in 0 contrast to the pure 1D Ising case which saturates the upper ͑98͒ bound of ϭd ͓5͔.
066107-14 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107
B. Autocorrelations for the RFIM in an applied field This result is valid, strictly, to leading order in ␦T. In general In the presence of an applied field HϾ0 the calculation is the exponent for the power law decay in time which occurs similar to the above; it is detailed in Appendix B. Using the here and for other quantities in the presence of a uniform ␦2 2 scaling variables applied field will have O( T ) corrections.
H ␥ϭ␦⌫Ϸ T ln t, ͑109͒ C. Two-point two-time correlation function for the spin glass 2g In order to characterize the spatial aspects of aging dy- H namics in the spin glass we have computed the correlation ␥Јϭ␦⌫ЈϷ T ln tЈ ͑110͒ function 2g x ⌫ for small H, and the magnetization per spin, ͗S ͑t͒S ͑t͒͗͘S ͑tЈ͒S ͑tЈ͒͘ϭFͩ ; ͪ , ͑114͒ 0 x 0 x ⌫Ј2 ⌫Ј ␥ ͗S ͑t͒͘ϭm͑t͒ϭM͑␥͒ϭcoth ␥Ϫ , ͑111͒ i sinh2␥ with tϾtЈ; this becomes, at long times, a scaling function of the result for the autocorrelation function is ͗ ͑ Ј͒ ͑ ͒͘Ϫ͗ ͑ ͒͘ ͗ ͑ Ј͒͘ x Si t Si t Si t Si t Xϵ ͑115͒ T2 ln2 tЈ 1 ϭ ͑2␥Ϫ␥Ј͒M͑␥Ј͒ϩ␥Ј coth ␥ЈϪ1 . sinh2␥ „ … and ͑112͒ The long time asymptotic behavior is ␣ϭln t/lntЈу1. ͑116͒
4HT ln t ͒ ͒ Ϫ ͒ ͒ ϭ ͒ We compute the scaling function F͓X,␣͔ in Appendix C. ͗Si͑tЈ Si͑t ͘ ͗Si͑t ͘ ͗Si͑tЈ ͘ m͑tЈ . gtHT/g For simplicity we have set 2gϭ1. In Laplace transform vari- ͑113͒ ables we have
ϱ 1 4 ͱp 2 ␣ͱp Fˆ ͓p,␣͔ϭ ͵ dXeϪpXF͑X;␣͒ϭ Ϫ tanh2ͩ ͪ ϩ coth2ͩ ͪ ͓8ϩ3p 0 p p2 2 ␣2p3 sinh2͑ͱp͒ 2
␣ͱp cothͩ ͪ 16 2 Ϫ16 cosh͑ͱp͒ϩ8ͱp sinh͑ͱp͒ϩ͑8ϩ5p͒cosh͑2ͱp͒Ϫ12ͱp sinh͑2ͱp͔͒Ϫ ␣2p2 ͱp
2 ͱp 16 ϫ 1Ϫͱp coth͑ͱp͒ ͩ tanhͩ ͪ Ϫ1 ͪ Ϫ 1Ϫͱp coth͑ͱp͒ 2. ͑117͒ „ … „ … ͱp 2 ␣ͱp ␣2p3 sinh2ͩ ͪ 2
Because the change of correlations between two points is ␣ϳ1, corresponding to ln tϷln tЈ, we have the following ex- caused only by passing of domain walls through one of the pansion to order O(␣Ϫ1): end points, the two-time correlation function decays at large x to the square of the autocorrelation function ͑105͒, i.e., ϱ 1 8 ͱp ˆ ͓ ␣͔ϭ ␣Ϫ 2 ␣4 ϪpX limpϪϾ0pF p, (4 1) /(9 ). The spatial decay to ͵ dXe F͑X;␣͒ϭ Ϫ͑␣Ϫ1͒ ͩ 1Ϫ ͪ . this constant value is determined by the closest poles to the 0 p p2 sinh͑ͱp͒ imaginary axis in the complex p plane. This yields an expo- ͑118͒ nential decay with a characteristic length which is the maxi- mum of (tЈ)ϭ(⌫Ј/)2 and (t)/4ϭ(⌫/2)2. In the regime Note that in experiments one could in principle measure
066107-15 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107
FIG. 5. Value at qϭ0 of the Fourier transform Hˆ (qϭ0,␣)(y axis͒ as a function of ␣ϭln t/ln tЈ (x axis͒. FIG. 6. Fourier transform Hˆ (q,␣)(y axis͒ of the scaling func- the Fourier transform which is related to the cross- tion for the two-point two-time spin glass correlation as a function of q (x axis͒ for four different values of ␣ϭ1.25, 2, 5, and 20. correlations of the scattering speckle patterns at two different times. It reads effective dynamics, which just places each wall at a specific ϩϱ location at each time. Indeed, in the effective dynamics, the ͒ ͒ ͒ ͒ iQx ͑ ͒ ͗ ͘ ͚ ͗S0͑t Sx͑t ͗͘S0͑tЈ Sx͑tЈ ͘e local magnetization i.e., the thermal average Sx(t) at a xϭϪϱ given point x is given by the orientation of the renormalized bond containing the point x at scale ⌫, and is thus either ϩ1 ln t Ϫ ϭ͑T ln tЈ͒2Hˆ ͩ qϭQ͑T ln tЈ͒2,␣ϭ ͪ , or 1. Thus truncated correlations are zero to leading order, ln tЈ and to estimate them one needs to consider the rare events in which a domain wall can be found with substantial probabili- ͑119͒ ties at two different positions. Such events occur with a prob- ability 1/⌫, and the two positions of the domain wall when ˆ ͑ ␣͒ϭ R ˆ ͓ ␣͔Ϫ͑ ␣Ϫ ͒2 ͑ ␣4 ͒ ͉ Þ H q, 2 „F p, 4 1 / 9 p … pϭiq, q 0, they do occur are typically separated by distance of order ⌫2. ͑120͒ For example, for the single point Edwards-Anderson order where R denotes the real part and the 1/p part has been parameter these lead to corrections to the zero temperature substracted to get rid of the ␦(q) part. The value of the value of unity of order Fourier transform as q→0is 1 1Ϫ ͗S ͑t͒͘ 2ϰ . ͑122͒ Ϫ6ϩ40␣Ϫ59␣2Ϫ20␣3ϩ45␣4 „ x … ln t Hˆ ͑qϭ0,␣͒ϭ , ͑121͒ 135␣4 In this section, to simplify the notation somewhat, we set which is plotted in Fig. 5. The Fourier transform is plotted in 2gϭ1. Fig. 6 for several values of ␣. Note the maximum at qϾ0 ␣ which develops for large and which is related to the non- A. Description of the important rare events monotonic behavior of the correlation as a function of x. The fact that the above correlation indeed reaches its limit by The rare events that are important for the RFIM turn out ͑ ͒ below can be seen as follows for large ␣. For xӶ⌫2, 0 and to be the same as far as the energy landscape is concerned x belong to the same domain, and thus the above two points as the ones that we considered in our previous study of the ͓ ͔ two time correlation is approximately equal to aging properties of the Sinai model 28 , with a slightly dif- ͗S (tЈ)S (tЈ)͘. However, this decays exponentially with ferent physical interpretation and different observables to be 0 x ͓ ͔ x/⌫Ј2 ͑and thus exponentially in ␣2 if one chooses x/⌫2 computed. There are two types 42 of such rare events that ⌫ ͑ ͒ ͑ ͒ ͓ ͔ fixed but very small͒ while the asymptotic value ͓Eq. ͑105͔͒ occur with probability 1/ , denoted a and c in Ref. 28 . decays only algebraically. We now describe them. Events (a) concern bonds which contain two almost de- generate extrema ͑see Fig. 7͒. In the following, we will use VIII. RARE EVENTS, TRUNCATED CORRELATIONS the probability D(a) (x) that two points 0 and xϾ0 belong AND RESPONSE TO A FIELD ⌫,⌫Ј to a renormalized bond at ⌫ which has two degenerate ex- In this section we compute time dependent thermal ͑trun- trema separated by a barrier smaller than ⌫Ј such that the cated͒ correlations as well as the response to a uniform field two points are both located between the two degenerate ex- ͓ ͔ applied at time tw . For this one needs to go beyond the trema. This is, using the calculations of Ref. 28 ,
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ϩϱ 4 1 2 2 2 ϭ ͚ ͑Ϫ1͒kϩ1 eϪ(x/⌫ )k . ⌫ 2 2 kϭ1 k ͑127͒
For events ͑c͒ associated with a barrier ⌫ϩ⑀, the probability pc that the two corresponding domain walls have not yet ⌫ ϭ Ϫ Ϫ⑀/T annihilated at is given by pc exp( e ). In the follow- ing, we will need
ϩϱdz ͑ ͒ϭ ͑ Ϫ ͒ nϭ n ͵ Ϫnz͑ Ϫ Ϫz͒n dn T 4pc 1 pc T 4 e 1 e „ … 0 z Ϫ n 1 1 ϭ 2n k ͩ ϩ ͪ ͑ ͒ T2 ͚ CnϪ1 ln 1 , 128 kϭ0 kϩn FIG. 7. Rare events ͑a͒ and ͑c͒ discussed in the text where we have used that the distribution in ⑀ is uniform around ⑀ϭ0. ⌫ (a) 4 Ј 1 y D ͑x͒ϭ ͵ P⌫͑lϪy͒͑yϪx͒ ͵ d⌫ rˆ ͩ ͪ ⌫,⌫Ј 0 ⌫2 Ͼ Ͼ ⌫4 ⌫2 B. Two point truncated equal time correlations l y,y x 0 0 0 ͑123͒ 1. Nonequilibrium behavior
4 1 y We consider the various moments of the truncated equal ϭ ͵ ͑yϪx͒ Gͩ ͪ time correlation function defined as 2 ⌫ yϾx ⌫Ј3 ⌫Ј2 ͒ϭ ͒ ͒ Ϫ ͒ ͒ n ͑ ͒ Cn͑x,t ͓͗S0͑t Sx͑t ͘ ͗S0͑t ͗͘Sx͑t ͔͘ ; 129 ⌫Ј ϩϱ 16 1 Ϫ ⌫Ј2 22 ϭ ͚ e (x/ )k , ͑124͒ C (x,t) is the sum of two contributions. The first contribu- ⌫2 ϭ 22 n k 1 k (a) ͑ ͒ tion Cn (x,t) originates from events a described above, ͗ ͘ϭϩ ͗ ͘ϭ͗ ͘ ˆ where we have S0(t)Sx(t) 1, while S0(t) Sx(t) where the functions r(Y) and G(Y) were defined in Eqs. ϭϮ(1Ϫ2p) because the domain wall fluctuates between the ͑E5-8͒ and ͑174͒ of Ref. ͓28͔. ͑ ͒ ͑ ͒ two extrema see Fig. 7 . We thus obtain the first contribu- In event a , the corresponding domain wall (A or B) will tion as fluctuate thermally between the two extrema ͑see Fig. 7͒.At equilibrium, the thermal probabilities of finding the domain (a) a C ͑x,t͒ϭc ͑T͒D⌫ ⌫͑x͒, ͑130͒ wall in the two positions are p and (1Ϫp) with pϭ1/(1 n n , ϩ Ϫu/T e ) and u representing the energy difference between with ⌫ϭT ln t. the two minima. The random variable u is distributed uni- (c) The second contribution Cn (x,t) originates from events formly around uϭ0. In the following, we will need ͑ ͒ ϭϩ c described above, where we have ͗S0(t)Sx(t)͘ 1 while ϭ ϭϮ Ϫ ͗S0(t)͘ ͗Sx(t)͘ (1 2pc). The second contribution is 1 ϩϱ znϪ1 ͑ ͒ϭ͑ ͑ Ϫ ͒͒nϭ n ͵ thus given by cn T 4p 1 p T 4 dz 2 0 ͑1ϩz͒2n (c)͑ ͒ϭ ͑ ͒ c ͑ ͒ ͑ ͒ Cn x,t dn T D⌫ x . 131 ⌫͓n͔ ϭTͱ , ͑125͒ 1 The final result for the moments of the truncated equal time ⌫ͫnϩ ͬ correlations is thus 2 ϩϱ 4 4c ͑T͒ϩ͑Ϫ1͒kϩ1d ͑T͒ the factor 1/2 arises as the integral is only ͓43͔ over uϾ0. ͒ϭ „ n n … Cn͑x,t ͚ Events (c) correspond to bonds about to be decimated, T ln t kϭ1 k22 (c) with a barrier ⌫ϩ⑀Ϸ⌫. The probability D⌫ (x) that the Ϫ͉ ͉ 22 2 segment ͓0,x͔, xϾ0 belongs at scale ⌫ to a bond of barrier ϫe x k /(T ln t) , ͑132͒ ⌫͑i.e., ϭ0): ͑ ͒ ͑ ͒ where cn(T) and dn(T) are given in Eqs. 125 and 128 . ⌫ 2 Note that to this order in 1/ the n dependence is only con- (c)͑ ͒ϭ ͵ ͑ϭ ͒͑ Ϫ ͒ D⌫ x P⌫ 0,l l x tained in the prefactor and in particular the correlation length ⌫2 lϾx ϭ 2 2 ͑ ͒ th(t) (T ln t) / extracted from Eq. 132 does not de- pend on n and is equal to result ͑82͒. For the spin glass, the 2 ϩϱ x ϭ ͵ dP*ͩ ϭ0,ϩ ͪ ͑126͒ above formula gives the even moments C2n(x,t). For the ⌫ 2 0 ⌫ following sections, we will need
066107-17 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107
xϭϩϱ 32 14 1 eϪ⑀/T t C ͑t͒ϭ ͚ C ͑x,t͒ϭͩ ϩ ln 2ͪ T2 ln t. ͑133͒ p͑t͒ϭ ϩ expͩ Ϫ͑1ϩeϪ⑀/T͒ ͪ , 1 1 Ϫ⑀/T Ϫ⑀/T xϭϪϱ 45 45 1ϩe 1ϩe teq ͑136͒ 2. Equilibrium truncated correlations Ϫ ⌫ Ϫ⌫ ϭ⌫ ϭ ( J )/T To compute the equilibrium truncated correlations the where ln teq J /T, and one has t/teq e . Integrat- ⑀ method is very similar. One must stop the RG at scale ⌫ ing over , one obtains that the crossover to equilibrium for J ϳ ϭ4J, and again consider events ͑a͒͑which give the same t teq is described by ⌫ ⌫ ͑ ͒ contribution as above replacing by J) and events c but ͑ ͒ϭ (a)͑ ͒ϩ (c)͑ ͒ ͑ ͒ with a different interpretation and result since there are now Cn x,t Cn x Cn x , 137 at equilibrium. Indeed in the renormalized landscape at scale ⌫ the only thermal fluctuations ͓apart from the ͑a͒ events͔ J (a) ͒ϭ ͒ (a) ͒ ͑ ͒ ⌫ϭ⌫ ϩ⑀Ϸ⌫ C ͑x cn͑T D⌫ ⌫ ͑x , 138 come from barriers J J . The barriers much n J , J ⌫ larger than J are occupied by a pair of domains with a ⌫ probability of almost 1, while the barriers well below J (c) ͒ϭ ͒ (c) ͒ ͑ ͒ C ͑x en͑T,t D⌫ ͑x , 139 ͑which have been decimated at previous stages͒ are occupied n J ⌫ϭ⌫ ϩ⑀ with a probability of almost 0. The barriers with J Ϸ⌫ ϭ ϩ Ϫ⑀/T J are occupied with a probability p 1/(1 e ). Thus with we now have the equilibrium truncated correlations:
e ͑T,t͒ϭ4p͑t͒͑1Ϫp͑t͒͒ (eq) a c n C ͑x͒ϭc ͑T͒ D⌫ ⌫ ͑x͒ϩD⌫ ͑x͒ n n „ J , J J … ϩϱ znϪ1 n ϩϱ kϩ1 ϭ4 T ͵ dz 1 ͑4ϩ͑Ϫ1͒ ͒ 2n ϭ ͒ Ϫ͉x͉k22/(4J)2 0 ͑1ϩz͒ cn͑T ͚ e . ϭ 2 2 J k 1 k Ϫ ϩ Ϫ ϩ ϫ͑1Ϫe (1 z)t/teq͒n͑1ϩze (1 z)t/teq͒n ͑134͒ ͑140͒
eqϭ 2 2 Here, as above, the correlation length th 16J / ex- n n ͑ ͒ ⌫ tracted from Eq. 134 to this order in J does not depend on ϭ n ͑Ϫ ͒k k p⌫͓ ϩ ͔ 4 T ͚ ͚ 1 CnCn n p n. Since it was argued in Ref. ͓21͔ that the correlation kϭ0 pϭ0 (eq) lengths of the Cn (x) generically depend on n, our results ⌫ t suggest that here this dependence is subleading in J . Our ϫUͩ nϩp,1ϩpϪn,͑kϩp͒ ͪ (eq) t result for the correlation length of C1 (t) coincides with the eq ͓ ͔ ͑ ͒ ϭ ͓ ͔ Ϫ ϩ result of Ref. 20 and with result 83 with 2g 1 44 . ϫ (k p)(t/teq) ͑ ͒ (eq) e , 141 However, the detailed form of the functions Cn (x) ob- tained here depends explicitly on n. ϭ where ln teq 4J/T. This expression crosses over from e (T,t/t Ӷ1)→d (T) and e (T,t/t ӷ1)→c (T). 3. Approach to equilibrium for truncated correlations n eq n n eq n ⌫Ͻ⌫ ϭ Since the equal time result for J 4J and the equi- C. Two-point two-time truncated correlations librium result differ only by substituting dn(T)bycn(T)in the ͑c͒ events, there should be a nontrivial crossover near We now consider the truncated two-point two-time corre- ⌫ϭ⌫ ͑ ͒ J toward equilibrium controlled by events c , which lations we now analyze. Let us consider a bond with barrier F ϭ⌫ ϩ⑀ ⌫ϭ ⌫ ͑ ͒ϭ͑͗ ͑ ͒ ͑ ͒͘Ϫ͗ ͑ ͒͗͘ ͑ ͒͒͘n J . When T ln t is close to J this bond can be Cn x,t,tw S0 t Sx tw S0 t Sx tw . either occupied ͓with probability p(t)͔ or empty ͓with prob- ͑142͒ ability 1Ϫp(t)͔. One has The calculation is very similar to the equal-time truncated ͑ ͒ dp 1 1 correlations. We first consider the events of type a , where ϭ Ϫ ͑ ͒ Ϫ ͑ ͒ ͑ ͒ we have now to keep track of the barrier ⌫ between the two „1 p t … p t , 135 0 dt 1 2 almost degenerate extrema. There is a nonvanishing contri- ⌫ ⌫ bution if the barrier 0 is smaller than w but larger than ⌫ ϭ J /T ⌫ˆ ϭ Ϫ where 1 e is the inverse rate of creation of a pair of T ln(t tw), so that equilibration cannot take place be- ͑ ϭϩ domain walls which immediately migrate to the end points tween tw and t. In that case we have ͗S0(tw)Sx(t)͘ 1 ⌫ ϩ⑀ ͒ ϭ ( J )/T ϭ ϭϮ Ϫ ϭ of the bond and 2 e is the inverse rate of annihi- while ͗S0(tw)͘ ͗Sx(t)͘ (1 2p), where p 1/(1 lation of the pair of domain walls located at the end points of ϩeϪu/T) as introduced above in the description of events of the bond. Thus, substituting tϭe⌫/T, one finds type ͑a͒. These events lead to the contribution
066107-18 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107
ӷ In the limit t tw these truncated correlations decay, for fixed (a) 4 2 n C ͑x,t,t ͒ϭc ͑T͒ ͵ P⌫͑lϪy͒͑yϪx͒ x/(T ln t ) ,asC (x,t,t )ϳ(t /t) . They decay to zero as n w n 2 w n w w ⌫ lϾy,yϾx there are no other contributions for later time t. In the fol-
⌫ lowing we will need w 1 x ϫ ͵ d⌫ rˆ ͩ ͪ ͑143͒ ϭϩϱ 0 x ⌫ˆ ⌫4 ⌫2 0 0 ͒ϭ ͒ ͑ ͒ C1͑t,tw ͚ C1͑x,t,tw 152 xϭϪϱ a a ϭc ͑T͒ D⌫ ⌫ ͑x͒ϪD ˆ ͑x͒ , n „ , w ⌫,⌫ … Ϫ ͒ 3 ͑144͒ 32 ln͑t tw 14 ϭ T2ͩ ln t Ϫ „ … ͪ ϩ T2 ln 2 ln t 45 w ͒2 45 w a ͑ln tw where c (T) and D⌫ ⌫ (x) are given above. Note that the n , w contribution of these events vanishes when ⌫ˆ becomes equal ln͑tϪt ͒ Ͻ w Ͻ ͑ ͒ to ⌫ . for 0 1 153 w ln tw We now consider events of type ͑c͒, which give different contributions and must be examined separately, in the scal- 14 t ln͑tϪt ͒ ϭ 2 ͩ ϩ w ͪ w ϳ ⌫ˆ ϭ Ϫ Ͻ ϭ⌫ T ln twln 1 for 1 ing regime T ln(t tw) T ln tw w and the scaling re- 45 t ln tw Ϫ ϳ ⌫ˆ ϭ⌫ ͑154͒ gime t tw tw , i.e., w . ⌫ˆ Ͻ⌫ We first consider w . In that regime the events of 14 t ln͑tϪt ͒ ͑ ͒ ⌫ ϭ 2 w w Ͼ ͑ ͒ type c should be considered at scale w where we have T ln tw for 1. 155 ϭϩ ϭ ϭϮ Ϫ 45 t ln tw ͗S0(tw)Sx(t)͘ 1 while ͗S0(tw)͘ ͗Sx(t)͘ (1 2p) which, together with the ͑a͒ events, gives the total contribu- tion D. Response to an applied field
͒ϭ (a) ͒ϩ ͒ c ͒ ͑ ͒ In order to compare with typical aging experiments, we Cn͑x,t,tw C ͑x,t,tw dn͑T D⌫ ͑x ; 145 n w will consider the following two histories for the system and compare them. ⌫ˆ ϭ Ϫ Ͻ ϭ⌫ ͑ ͒ Ͼ ϭ this formula holds for T ln(t tw) T ln tw w . i Apply H 0 starting from t 0: the magnetization per ͑ ͒ In the regime tϪt ϳt i.e., ⌫ˆ ϭ⌫ , the ͑c͒ events also spin m(t) will then grow in time as computed in 60 up to w w w ϳ 4J/T time teq e , where meq is reached. start to equilibriats, which we now study. Let pc(tw) Ϫ⑀ ͑ ͒ ϭ ϭ ϭexp(e /T) the probability that the domain walls separated ii Keep H 0 between t 0 and tw , and then apply H Ͼ Ͼ ⌫ ϩ⑀ 0 for t tw : in this case the magnetization per spin by the barrier w have not yet annihilated at tw . Let m(t,tw) remains 0 up to time tw , and then grows to again tϪt t reach meq in the large time limit. ͑ ͒ϭ ͑ ͒ ͩ Ϫ w Ϫ⑀/T ͪ ϭ ͩ Ϫ Ϫ⑀/T ͪ ͑ ͒ pc t pc tw exp e exp e We now estimate m(t,tw) in the case ii in the ‘‘small tw tw ϳ ⌫2 applied field regime,’’ where H 1/ w . It is convenient to be the probability that they also have not yet annihilated at t. define ͑ One has with x and 0 belonging to the bond being deci- ˆϵ Ϫ ͑ ͒ mated͒ t t tw , 156
͒ ϭ Ϫ ͒ ͑ ͒ and introduce the ratio between ͗Sx͑tw ͘ 1 2pc͑tw , 146 ⌫ˆ ϵ ˆϭ ͑ Ϫ ͒ ͑ ͒ ͒ ϭ Ϫ ͒ ͑ ͒ T ln t T ln t tw 157 ͗Sx͑t ͘ 1 2pc͑t , 147 and ⌫ ϭT ln t : ͒ ͒ ϭ Ϫ ͒ϩ ͒ ͑ ͒ w w ͗S0͑t Sx͑tw ͘ 1 2pc͑tw 2pc͑t , 148 ⌫ˆ Ϫ ͒ ln͑t tw ͗S ͑t͒S ͑t ͒͘Ϫ͗S ͑t͒͗͘S ͑t ͒͘ϭ4p ͑t͒ 1Ϫp ͑t ͒ . ␣ˆ ϭ ϭ . ͑158͒ 0 x w 0 x w c „ c w … ⌫ ln t ͑149͒ w w Ϫ ϳ We separately discuss the three regimes 0Ͻ␣ˆ Ͻ1, ␣ˆ ϳ1, and Thus in the regime t tw tw , one obtains the total contri- butions ␣ˆ Ͼ1.
c Ë␣ˆ Ë C ͑x,t,t ͒ϭd ͑T,t,t ͒D⌫ ͑x͒, ͑150͒ 1. Response at early times 0 1 from degenerate wells n w n w w ⌫ We first study the scaling limit of small H and large w ͑ ͒ ϭ ͑ ͒ Ϫ ͑ ͒ n ⌫2 Ͻ␣ˆ ϭ ˆ Ͻ dn T,t,tw ) „4pc t 1 pc tw … with H w fixed and 0 (lnt/ln tw) 1 fixed. In this re- ϩϱ gime the dominant contributions come from bonds with near dz Ϫ Ϫ ͑ ͒ ϭT 4n ͵ e nt/twz͑1Ϫe z͒n. ͑151͒ degenerate extrema, i.e., the rare events of type a described 0 z in Sec. VIII A.
066107-19 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107
Let us consider an ascending bond with a secondary mini- 2. Response at times ␣ˆ ¶ 1 ⌫ mum separated by a distance y and a barrier 0 at a potential Ϫ Ͼ When t tw is of order tw a second effect adds to the one u 0 above the minimum. Using the results and notations of ͑ ͒ ͓ ͔ computed above. It corresponds to the events of type c Ref. 28 we find that the probability that an ascending bond ⌫ ⌫ ͑ ͒ described in Sec. VIII A where the barrier of a bond at w is of this type at w before the field is turned on is ⌫ ϩ⑀ ⑀ϭ ͑ ͒ equal to w where O(1) of arbitrary sign .Inthe absence of the field the pair of domains at the endpoints of r⌫ ͑y,⌫ ͒d⌫ dyϭ2͑⌫ Ϫ⌫ ͒P⌫ ͑0,.͒* P⌫ ͑0,.͒d⌫ dy. w 0 0 w 0 0 y 0 0 ͑ ͒ the bond have not yet annihilated at time tw , with a prob- 159 ϭ Ϫ Ϫ⑀/T ability pc(tw) exp( e ). When adding the field at tw the ͑ ͒ Just before ⌫ , there is thermal equilibrium between the two barriers suddenly either increases for descending bonds or w ͑ ͒ extrema, and thus the probability that the primary extremum decreases ascending bonds by 2Hl where l is the length of ϭ ϩ Ϫu/T the bond. For tϾt ͓and such that tϪt ϳO(t )͔ the prob- is occupied by an A domain wall is peq(u) 1/(1 e ). w w w The thermally averaged total magnetization of the segment y ability pc(t) that the domain has not yet annihilated depends ͗ ͘ϭ Ϫ on H and is is then M y„1 2peq(u)…. One now turns the field on at ⌫ , and the thermally averaged magnetization of the seg- w tϪt ment remains the same ͑up to negligible probability͒ until the ͑ ͒ϭ ͑ ͒ ͫϪͩ w ͪ (Ϫ⑀ϯ2Hl)/Tͬ ͑ ͒ pc t pc tw exp e 164 ⌫ˆ ϭ⌫ ͑ tw time 0 when a new equilibrium is attained here we can ͒ ⌫ˆ ϭ⌫ ͑ ͒ neglect the equilibration time scale . Just after 0 the for descending and ascending bonds respectively. Events c occupation probability of the left minimum is now peq(uЈ) thus result in a difference in magnetization compared to the ϭ1/(1ϩeϪuЈ/T), where uЈϭuϪ2Hy. The new magnetiza- zero field case equal to ϭ Ϫ Ϫ tion is ͗M͘Ј y 1 2peq(u 2Hy) . Similarly, the contri- „ ͑ … ⌫ 2 ϱ ϩϱ bution of a descending bond with y, 0) is given by the (c)͑ ͒ϭ ͵ ͵ ⑀ ͑ ͒ ͑Ϫ Ϫ⑀/T͒ m t,tw dl d P* 0,l l exp e same formula with uϽ0. Finally the contribution of degen- ⌫2 Ϫϱ w 0 eracy of hills—fluctuations of the B domains–yields an over- Ϫ all factor of two. The total contribution of all these events to t tw ϫͩ expͩ Ϫ e(1/T)(Ϫ⑀Ϫ2Hl) ͪ the magnetization per spin is tw (a) Ϫ m͑t,t ͒ϭm ͑t,t ͒ t tw w w Ϫexpͩ Ϫ e(1/T)(Ϫ⑀ϩ2Hl) ͪͪ ͑165͒ tw 1 ⌫ˆ ϩϱ ϩϱ ϭ2 ͵ d⌫ ͵ dy͵ du r⌫ ͑y,⌫ ͒y ⌫2 0 w 0 2T ϱ w 0 0 0 ϭ ͵ ͑ϭ ͒ ⌫ d P* 0, ϫ Ϫ ͑ Ϫ ͒ϩ Ϫ ͑Ϫ Ϫ ͒ w 0 „1 2peq u 2Hy 1 2peq u 2Hy …. ⌫2 ϩ Ϫ ͒ 2H w/T ͑160͒ tw ͑t tw e ϫlnͩ ͪ . ͑166͒ Ϫ ⌫2 ϩ͑ Ϫ ͒ 2H w/T This gives tw t tw e
͒ϭ (a) ͒ The total magnetization is now m͑t,tw m ͑t,tw
⌫ˆ ϩϱ ϩϱ 32 1 1 m͑t,t ͒ϭm(a)͑t,t ͒ϩm(c)͑t,t ͒ϭ HT ln t ϩm(c)͑t,t ͒. ϭ4 ͵ dYYG͑Y ͒ ͵ duͩ w w w 45 w w ⌫ ⌫ (uϪ2H⌫ˆ 2Y)/T w w 0 0 1ϩe ͑167͒
1 Note that in the present regime the magnetization as a scal- Ϫ ͪ ͑161͒ 2 ϩ ⌫ˆ 2 ⌫ 1ϩe(u 2H Y)/T ing function of H w is complicated and nonlinear. ⌫2 In the limit where H w is small, one has ˆ ϩϱ 1 ⌫ Ϫ ϱ ϭ ͑ ⌫ˆ 2͒ 2 ͑ ͒ ͑ ͒ 8 t tw 8 H ͵ dYY G Y 162 m(c)͑t,t ͒ϭ H⌫2 ͵ dP*͑ϭ0,͒2 ⌫ ⌫ 0 w ⌫ w w w w t 0 ϩ ͑ ⌫2 ͒2 32 ⌫ˆ 3 32 ␣ˆ 3 O„ H w … ϭ H ϭ ͑H⌫2 ͒ . ͑163͒ 2 w ⌫ Ϫ 45 ⌫ 45 w 14 t tw w ϭ H⌫ ϩO ͑H⌫2 ͒2 , ͑168͒ 45 w t „ w … In this regime, the magnetization as a scaling function of ⌫2 ͐ϱ ϭ 2ϭ H w is thus exactly linear. It can be shown that a nonlinear where we have used 0 d P*( 0, ) 7/180. ⌫2 ͓ ͑ ͔͒ response in H w couples to the curvature of the distribution Although the above function Eq. 166 is complicated, Ϫ ϭ of difference of potential near zero and is of higher order in at the special time such that t tw tw it takes the simple ⌫ 1/ w . value
066107-20 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107
Ϯ 4 ϱ ϭ ˆ (c) 2 2 We write, to linear order P⌫ˆ (F,l) P⌫(F,l) m ͑2t ,t ͒ϭ H⌫ ͵ dP*͑ϭ0,͒ Ϯ w w ⌫ w ϩ w 0 2HQ⌫ˆ (F,l), with the initial condition in Laplace trans- ⌫ˆ ϭ⌫ 7 form variables at w , ϭ ⌫ ͑ ͒ H w . 169 45 ϩ Ϫu⌫ (p) U⌫ ͑p͒u⌫ ͑p͒e w ץQ⌫ ͑,p͒ϭϪ w p„ w w …
2 Ϫu⌫ (p) ϪU⌫Ј ͑0͒U⌫ ͑p͒ e w 3. Response at times ␣Ä␣ˆ Ì1 w w
Ϫu⌫ (p) Ϫ ӷӷ ␣ˆ ϩ2U⌫Ј ͑0͒U⌫ ͑p͒e w , ͑174͒ For time differences t tw tw , corresponding to w w ϭ␣Ͼ1, the response of the RFIM chain to an applied field Ϫ ϩ will be dominated by the effective dynamics described by the Q⌫ ͑,p͒ϭϪQ⌫ ͑,p͒. ͑175͒ RSRG procedure. When the field HϾ0 is turned on, the w w descending bonds with (F,l) become (Fϩ2Hl,l) and the To compute the magnetization we are interested only in Ϫ ⌫ Ͻ Ͻ⌫ ϩ Ϫ ascending bonds become (F 2Hl,l) except if w F w Ϫ ϭ ˆ ˆ P⌫ˆ (F,l) P⌫ˆ (F,l) 2HQ⌫(F,l), where Q⌫(F,l) ϩ2Hl, since in this case they must be immediately deci- ϩ Ϫ ϭ Ϫ mated. Technically, from the point of view of the landscape, Q⌫ˆ (F,l) Q⌫ˆ (F,l) satisifes the linearized RG equation it is more convenient to symmetrize the initial condition at ͒Q⌫ˆ ͑,p͒ϭϪQ⌫ˆ ͑0,p͒P⌫ˆ ͑.,p͒* P⌫ˆ ͑.,p͒ץϪ ˆ⌫ץ͑ ⌫ˆ ϭ⌫ l w which amounts to artificially reintroducing the de- ⌫ Ϫ Ͻ Ͻ⌫ ͑ ϩ scending bonds w 2Hl F w these bonds, being re- 2P⌫ˆ ͑0,p͒Q⌫ˆ ͑.,p͒*P⌫ˆ ͑.,p͒ decimated immediately, do not introduce any errors for ␣ ϩ ͑ ͒ ͑ ͒ ͑ ͒ Ͼ1). This corresponds to the following initial distributions 2Q⌫ˆ 0,0 P⌫ˆ ,p . 176 ⌫ˆ ϭ⌫ at w : The solution therefore has the form
Ϫu⌫ˆ (p) Q⌫ˆ ͑,p͒ϭ A⌫ˆ ͑p͒ϩB⌫ˆ ͑p͒ e , ͑177͒ ϩ „ … P⌫ ͑F,l͒ ץP⌫ ͑F,l͒ϭP⌫ ͑F,l͒Ϫ2Hl w w F w where the coefficients A⌫ˆ (p) and B⌫ˆ (p) satisfy the RG ϩ2HP⌫ * P⌫ * lP⌫ ͑0,l͒ equations w F,l w l„ w … ,A⌫ˆ ͑p͒ϭϪu⌫ˆ ͑p͒A⌫ˆ ͑p͒ϩB⌫ˆ ͑p͒ϩ2U⌫ˆ ͑p͒A⌫ˆ ͑0͒ ˆ⌫ץ ϱ Ϫ4HP⌫ ͑F,l͒ ͵ dlЈlЈP⌫ ͑0,lЈ͒, ͑170͒ ͑178͒ w 0 w
B⌫ˆ ͑p͒ϭϪu⌫ˆ ͑p͒B⌫ˆ ͑p͒, ͑179͒ ˆ⌫ץ
Ϫ : ⌫P⌫ ͑F,l͒ with initial condition at ⌫ˆ ϭ ץP⌫ ͑F,l͒ϭP⌫ ͑F,l͒ϩ2Hl w w F w w ͒ ϩ ͒ Ј ͒ ͒ ץ Ϫ ͒ϭϪ 2HP⌫ * P⌫ * lP⌫ ͑0,l͒ A⌫ ͑p 2 p U⌫ ͑p u⌫ ͑p 4U⌫ ͑p U⌫ ͑0 , w F,l w l„ w … w „ w w … w w ͑ ͒ ϱ 180 ϩ ͑ ͒ ͵ Ј Ј ͑ Ј͒ ͑ ͒ 4HP⌫ F,l dl l P⌫ 0,l . 171 2 w 0 w B⌫ ͑p͒ϭ2U⌫ ͑p͒u⌫ ͑p͒u⌫Ј ͑p͒Ϫ2U⌫ ͑p͒ U⌫Ј ͑p͒. w w w w w w ͑181͒
We now check that the magnetization corresponding to this The solutions are ␣ˆ ϭ initial condition is the one at the end of the 1 regime ͑⌫ ͱ ͒ ͓ →ϩϱ ͑ ͔͒ sinh w p i.e., the t limit of Eq. 168 . It is, to first order in H, B⌫ˆ ͑p͒ϭB⌫ ͑p͒ , ͑182͒ w sinh͑⌫ˆ ͱp͒
ϩϱ ϱ ͑⌫ ͱ ͒ ͵ ͵ ͓ ϩ ͑ ͒Ϫ Ϫ ͑ ͔͒ sinh w p dll d P⌫ ,l P⌫ ,l A⌫ˆ ͑p͒ϭ A⌫ ͑p͒ϩ͑⌫ˆ Ϫ⌫ ͒B⌫ ͑p͒ 0 0 w w „ w w w … ͑⌫ˆ ϭ⌫ ⌫ ͒ϭ sinh͑⌫ˆ ͱp͒ meff w , w ϩϱ ϱ ͓ ϩ ͑ ͒ϩ Ϫ ͑ ͔͒ ͵ dll͵ d P⌫ ,l P⌫ ,l ͱp 0 0 w w Ϫ ⌫ˆ Ϫ⌫ ͒ ͑ ͒ 2͑ w . 183 ͑172͒ sinh͑⌫ˆ ͱp͒
⌫ˆ ӷ⌫ In the limit w , we have 8H ϱ 14 Ӎ ͵ dllP⌫ ͑⌫ ,l͒ϭ H⌫ . ͱ ⌫2 w w 45 w p w 0 B⌫ˆ ͑p͒Ӎ , ͑184͒ ͑173͒ sinh͑⌫ˆ ͱp͒
066107-21 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107
ͱp E. Fluctuation-dissipation violation ratio A⌫ˆ ͑p͒ӍϪ⌫ˆ , ͑185͒ sinh͑⌫ˆ ͱp͒ Having computed truncated correlations and the response to an applied field, we now discuss the fluctuation- and recover the first order linearization in ␦ of the biased dissipation violation ratio, a measure of the nonequilibrium fixed point solutions ͓Eqs. ͑37͔͒: behavior of the system. For two observables A and B, the fluctuation-dissipation-theorem ͑FDT͒ violation ratio X is de- Ϯ Ϯ Ϯ Ϫu (p) fined as ͓6,40͔ P⌫ ͑,p͒ϭU⌫ ͑p͒e ⌫ ϭP⌫͑,p͒͑1ϯ␦⌫͒͑1Ϯ␦͒ C ͑t,t ͒, ͑195͒ ץ͒ TR ͑t,t ͒ϭX ͑t,t ͱp A,B w A,B w tw A,B w Ϫͱpcoth(⌫ͱp) ϭP⌫͑,p͒ϩ2͑␦Ϫ␦⌫͒ e . ͑⌫ͱ ͒ sinh p where CA,B(t,tw) represents the truncated correlation ͑186͒ ͒ϭ ͒ ͒ Ϫ ͒ ͒ ͑ ͒ CA,B͑t,tw ͗A͑t B͑tw ͘ ͗A͑t ͗͘B͑tw ͘, 196 The magnetization at first order in H is given by and RA,B(t,tw) represents the response in the observable A at ϩϱ ϱ time t to a field HB linearly coupled to the observable B in ͵ ͵ ͓ ϩ͑ ͒Ϫ Ϫ͑ ͔͒ Ϫ dll d P⌫ˆ ,l P⌫ˆ ,l the Hamiltonian through a term of the form HBB: 0 0 ͑⌫ˆ ͒ϭ ͑ ͒ meff ϩϱ ϱ 187 ␦ ͑ ͒ ϩ Ϫ ͗A t ͘ ͵ ͵ ͓ ͑ ͒ϩ ͑ ͔͒ R ͑t,t ͒ϭ ͯ . ͑197͒ dll d P⌫ˆ ,l P⌫ˆ ,l A,B w ␦ ͑ ͒ 0 0 HB tw ϭ HB 0
2H ϱ ϱ Here we have computed the magnetization resulting from Ӎ ͵ dll͵ dQ⌫ˆ ͑,l͒ an uniform magnetic field H, so that the observables A and B ⌫ˆ 2 0 0 ϭ ͚L ϭ͚L are given by A (1/L) iϭ1Si and B iϭ1Si , respectively. From the magnetization 2H A⌫ˆ ͑p͒ B⌫ˆ ͑p͒ ϩ ͪͬͯ ͑188͒ ͩ ץϭ ͫ Ϫ 2 p 2 t ⌫ˆ u⌫ˆ ͑p͒ u ˆ ͑p͒ 1 ⌫ ϭ ͑ ͒ϭ ͗ ͑ ͒͘ϭ͗ ͑ ͒͘ϭ ͵ ͑ ͒ p 0 m t,tw ͚ Si t S0 t H duR t,u L i tw ͑ ͒ ⌫ 3 198 2 16 w ϭH⌫ˆ ͫ Ϫ ͩ ͪ ͬ. ͑189͒ 3 45 ⌫ˆ and the truncated correlation
1 L L It grows from ͒ϭ ͒ ͒ Ϫ ͒ ͒ C1͑t,tw ͚ ͚ ͓͗Si͑t S j͑tw ͘ ͗Si͑t ͗͘S j͑tw ͔͘ L iϭ1 jϭ1 14 ⌫ˆ ϭ⌫ ͒ϭ ⌫ ͑ ͒ meff͑ w H w 190 45 ϭ ͒ ͒ Ϫ ͒ ͒ ͑ ͒ ͚ ͗S0͑t Sx͑tw ͘ ͗S0͑t ͗͘Sx͑tw ͘, 199 x ⌫ˆ ӷ⌫ to the asymptotic regime for large w , one obtains the fluctuation-dissipation ratio X(t,tw)as 2 m͑t,t ͒/H ץ ͒ ͑ ˆ⌫ ӷ⌫ ͒ϭ ˆ⌫͑ meff w H 191 tw w 3 X͑t,t ͒ϭϪT . ͑200͒ ͒ C ͑t,t ץ w tw 1 w that corresponds to the behavior of the magnetization of the biased case at first order in ␦ ͓Eq. ͑60͔͒. Note that we have used the infinite size limit to replace trans- We now summarize our results for the magnetization in lational averages by disorder averages. ͓ ͑ ͒ ͑ ͔͒ the regimes ␣ˆ Ͻ1, ␣ˆ Ӎ1, and ␣ˆ Ͼ1: We have computed C1(t,tw) Eqs. 153 – 155 and ͓ ͑ ͒ ͑ ͔͒ Ͻ␣ˆ Ͻ m(t,tw) Eqs. 192 – 194 in the three regimes 0 1, 32 ⌫ˆ ␣ˆ ϳ1, and ␣ˆ Ͼ1, and obtained the following expressions for ͑⌫ˆ ⌫ ͒ϭ ⌫ ␣ˆ 3 ␣ˆ ϭ Ͻ ͑ ͒ m , w H w for ⌫ 1, 192 the fluctuation dissipation violation ratio at large times t,t 45 w w ӷ1: Ϫ ⌫ˆ 32 14 t tw ln͑tϪt ͒ m͑⌫ˆ ,⌫ ͒ϭ H⌫ ϩ H⌫ for ␣ˆ ϭ Ӎ1, ͑ ͒ϭ Ͻ␣ϭ w Ͻ ͑ ͒ w 45 w 45 w t ⌫ X t,tw 1 for 0 ˆ 1, 201 w ln tw ͑193͒ tϩt tϪt ͑ ͒ϭ w w 2 16 32 ⌫ˆ X t,tw for a fixed number m͑⌫ˆ ,⌫ ͒ϭH⌫ˆ ͫ Ϫ ͬϩ H⌫ for ␣ˆ ϭ Ͼ1. t tw w 3 w ⌫ 3 45␣ˆ 45 w ͑194͒ ͑␣ˆ ϭ1͒, ͑202͒
066107-22 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107
2 regime of length scales studied here for a system—the 1D t 24 ln tw X͑t,t ͒ϭ ͩ 1ϩ ͪ for RFIM—with no phase transition. w 2 tw ln tw 7 ln t It is also interesting to compare our results with the fluctuation-dissipation ratio X in pure systems presenting do- ln͑tϪt ͒ ␣ϭ w Ͼ ͑ ͒ main growth, typically ferromagnets below their critical tem- ˆ 1. 203 Ͻ ͓ ͔ ϭ ln tw perature T Tc 7–9 or at T Tc . The ratio X is usually ϭ ϭ computed by considering the local observables A B S0 The behavior of X upon increasing the time difference t but to compare with the present study we will translate these Ϫ tw is as follows. First, of course, one expects, when t results for a spatially uniform applied field. Ϫ ϭ tw is a fixed number, a nonuniversal equilibrium regime Let us briefly recall why X 0 in the large time scaling ͑ ͒ Ͻ not studied here where time translational invariance and the regime for T Tc (X decays to 0 for large tw). When initial Ϫ ϳ ␣ˆ conditions ͑high temperature TϾT ) before the quench are FDT is obeyed. Increasing the time difference as t tw tw , c chosen to have zero magnetization ͓͚ ͗S (t)͘ϭ0͔ the spin in the regime 0Ͻ␣ˆ Ͻ1, we find that the FDT theorem is still x x autocorrelation coincides with the truncated correlation obeyed to leading order, which is quite interesting since time ͚ ͗S (t)͘ϭ0, and takes a simple scaling form ͓1͔ translational invariance does not hold in that regime. Next, in x x the regime ␣ˆ ϭ1, t/t fixed, X becomes a nontrivial scaling w L͑t ͒ Ϫ ϭ pure d w function of (t tw)/tw , which interpolates between X 2 for ͑ ͒ϭ ͗ ͑ ͒ ͑ ͒͘Ӎ ͑ ͒ ͩ ͪ C1 t,tw ͚ S0 t Sx tw L tw f 1 , Ϫ → ϭ Ϫ →ϩϱ x L͑t͒ (t tw)/tw 0 and X 1 for (t tw)/tw . In order to match the value Xϭ1, there is thus a nontrivial crossover ͑207͒ regime between the end of the quasiequilibrium regime ␣ˆ where L(t) is the typical size of domains at time t an d the Ͻ1 and the beginning of the ␣ˆ ϭ1 regime. This crossover Ϫ ϳ dimension of space. On the other hand, the magnetization occurs for (t tw) tw /lntw , and is given by when a uniform field has been applied between tw , and t ͓ ͔ ͑ ͒ϭ ͑ Ϫ ͒ ͑ ͒ behaves as 8,9,46 X t,tw F1„ t tw ln tw /tw…, 204 ͒ ϩ L͑tw 14y 96 ͑ ͒ϳ ͑ ͒dϪa ͩ ͪ ͑ ͒ F ͑y͒ϭ . ͑205͒ M auto t,tw L tw f 2 ͑ ͒ , 208 1 7yϩ96 L t ϭ ͓ ͔ ϭ Ϫ Finally, for very separated times, in the regime ␣ˆ Ͼ1, we with a 1 for Ising order parameter 7–9 and a d 2(d Ͼ ͓ ͔ find that X grows toward ϩϱ. This occurs again after a 2) for O(N) model 9 . Note the reduction of M with respect to C by a factor 1/L(t ) in ferromagnets, which crossover between the end of the ␣ˆ ϭ1 and the ␣ˆ Ͼ1 regime 1 w immediately implies Xϭ0 in the scaling regime L(t ) which occurs on time scale tϳt ln t , where X is given by w w w ϳL(t)(Xϭ0 as soon as aϾ0). As is usually argued, the 31 t origin of this factor can be seen by considering the system at ͒ϭ ϩ ͑ ͒ ϭ X͑t,tw 1 , 206 t tw and focusing on the immediate response to a small 7 tw ln tw applied pulse field ͓46͔͑here in dϭ1 for simplicity͒: each interface responds by a factor O(1) and since they occupy which matches both the required limits. only a portion 1/L(t ) of the system this yield a total re- We can now compare with the mean field models ͓45͔.As w sponse 1/L(t ). ln contrast, note that exactly at criticality, in the mean field ͓6,40͔, here we find an aging regime where w TϭT , there is a nontrivial XϭX(t/t ) which appears as a X is nontrivial, and for t/t fixed it is a function of this c w w dimensionless amplitude ratio, by a different mechanism ͓9͔. scaling variable. On the other hand, contrary to mean field For the RFIM case, the full correlation function has the models, the ratio X here is never a function of only C (t,t ), 1 w same scaling form as Eq. ͑207͓͒as can be seen by general- ␣ˆ ϭ in the regime 1 because of the extra power of ln t in izing the result for the autocorrelation computed in Eq. C1(t,tw), and in general because of the presence of both ͑105͔͒ but the truncated does not since we have obtained scales (t,tw) and (ln t,ln tw). In addition, X here is found to be ͓Eqs. ͑153͒–͑155͔͒ nonmonotonic, and the values taken by X are not within the ͓ ͔ ϩϱ interval 0,1 . In particular X tends to in the asymptotic ln͑tϪt ͒ ln͑tϪt ͒ ϳ ␣ ␣Ͼ ͑ ͒ϭ ͩ w ͪ Ͻ w Ͻ regime t tw with 1 since truncated correlations become C1 t,tw ln tw 1 for 0 1 very small compared to the response in that this regime. ln tw ln tw ͑209͒ Since the ratio X has an interpretation in some contexts as an ϭ ͓ ͔ inverse effective temperature X 1/Teff 40 , one would find → t ln͑tϪt ͒ here that Teff 0 at large time separations, in contrast to the ϭ ͩ w ͪ w у →ϩϱ ln tw 2 for 1, result that Teff in mean field models. Although this t ln tw might appear surprising at first sight, one should remember ͑210͒ that in finite dimensions many of the properties of the RFIM are controlled—in the the renormalization group sense—by a whereas for the RFIM magnetization we have found ͓Eqs. zero temperature fixed point; this includes the intermediate ͑192͒–͑194͔͒
066107-23 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107
Ϫ ͒ ⌫ˆ ⌸͑ ͒ϳ¯͑ ͒Ϫ ϭ ϭ ͑ ͒ ln͑t tw t l t with 2 1 1. 214 m͑⌫ˆ ,⌫ ͒ϭln t ͩ ͪ for ␣ˆ ϭ Ͻ1 w w 1 ln t ⌫ w w ͓ ͔ ͑211͒ This should be compared with the result 47 for the pure ϭ 3 ͓ Ising model, where 4 corresponding there to the charac- ϳͱ ͔ t ⌫ˆ teristic length l(t) t . This out of equilibrium behavior ͑⌫ˆ ⌫ ͒ϭ ͩ w ͪ ␣ˆ ϭ Ӎ ͑ ͒ holds up to time tϭt corresponding to renormalization m , w ln tw 2 for ⌫ 1 212 eq t w ⌫ϭ⌫ scale J at which equilibrium is reached. In the presence of an applied field HϾ0, we can use our ln t ⌫ˆ previous result ͓28͔ for the biased Sinai diffusion. We have ͑⌫ˆ ⌫ ͒ϭ ͩ ͪ ␣ˆ ϭ Ͼ ϩ Ϫ m , w ln tw 3 for ⌫ 1. ⌸ ͓⌸ ͔ ln tw w found that the probabilities 1 (t) 1 (t) that a given Si- ͑213͒ nai particle remains on the right ͑left͒ of its starting point up to time t, in the case of a drift in the ͑ϩ͒ direction, behave as These expressions are rather different from that for the fer- romagnet ͓Eq. ͑207͔͒, first because both t and ln t appear in ␦ ϩ 2 these scaling forms. The only domain growthlike nontrivial ⌸ ͑t͒Ϸ , ͑215͒ 1 Ϫ Ϫ2␦⌫ scaling regime with L(t)ϭT ln2t occurs for ␣Ͼ1. In this 1 e regime the magnetization has a form similar to Eq. ͑208͒ with a reduction factor 1/ͱL(t ) ͓rather than 1/L(t ) in the 2␦ w w ⌸Ϫ͑ ͒Ϸ ͑ ͒ pure system͔. Using the discussion of Sec. VIII D 3 one un- 1 t ␦⌫ . 216 e2 Ϫ1 derstands that the origin is quite different from the pure case. ͓ ͑ ͔͒ In the immediate response to an applied field Eq. 173 ⌸Ϯ ⌫ ϳ ͱ These give the probabilities (t) that a given spin in the only a small fraction of domains 1/ w 1/ L(tw) responds, RFIM keeps the value (Ϯ) up to time t, which in the limit of but their response is very large since the full domain, of ␦ ӷ ϳ large T ln t 1 behaves as length L(tw) flips. The truncated correlation on the other ͓ hand is very small and does not take the scaling form Eq. 2 ͑ ͔͒ ϭϩϱ ϩ H 207 . This yield a value X in this regime. ⌸ ͑t͒Ϸ , ͑217͒ Similarly, the origin of the nontrivial value of X in the g2 ϳ ␣ϭ regime t tw ( 1) is very different from the pure case. Both the response and correlations originiate from rare H2 ⌸Ϫ͑ ͒Ϸ Ϫ2HT/g ͑ ͒ events and take the form (1/ln tw)L(tw)f(t/tw), where now the 1 t t . 218 g2 factor 1/ln tw is the probability of the rare event and f (t/tw) its contribution to activated dynamics. Since they are of the same order this yield a nontrivial X. B. Persistence of the local time-averaged magnetization In addition to the persistence of a single spin, one can also IX. PERSISTENCE PROPERTIES OF THE RFIM obtain the statistics of the flips of the thermal average of the We now turn to a study of the persistence properties of local magnetization, i.e., ͗Sx(t)͘, at a given site x.Asex- the random field Ising model, which have received substan- plained in Ref. ͓28͔ in the context of the Sinai model, quan- tial attention for other systems evolving towards equilibrium. tities averaged over many runs behave very differently than Two of the primary properties of interest in this context are quantities for a single run; in particular while the spin of the time decay of the probability that a spin has never flipped interest in one run may flip many times, if the same spin is up to time t and the time decay of the probability that a examined in many runs with statistically similar initial con- domain has survived up to time t. The results for the single ditions, the average over the runs at a given time into the spin persistence for the RFIM ͑Secs IX A and IX B below͒ runs will flip far less often. are also valid for the spin glass. The large time limit of these The local magnetization will successively be equal to 1 quantities can be computed from the asymptotic full state and Ϫ1 with only very small probability, at large times, of ͑see Fig. 3͒ on which we focus below. being be equal to an intermadiate value. The sequence of flips is given by the sequence of changes of orientation of a A. Persistence of a single spin bond during the renormalisation procedure extensively stud- ied in Ref. ͓28͔. From that analysis, we obtain the following In zero applied field the probability ⌸(t) that a given spin results for the RFIM. at xϭ0 has never flipped up to time t in a single run, is equal In zero field Hϭ0, we denote by k the number of changes to the probability that neither the nearest domain wall on one of sign of ͗Sx(t)͘ at a given point x between 0 and t. The side, nor the nearest ͑opposite type͒ domain wall on the other distribution of the rescaled variable side have crossed xϭ0. In Ref. ͓28͔, we found that the prob- ⌸ ability 1(t) that a given Sinai particle does not cross its k k Ϫ ϭ ϭ ͑219͒ ⌸ ϳ¯ 1 starting point up to time t decays as 1(t) l (t) with ln ⌫ ln͑T ln t͒ ϭ ⌸ 1 1/2. We thus obtain that (t) in the zero field RFIM decays as is characterized by the asymptotic decay
066107-24 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107
¯ Prob͑͒ϳ¯l͑t͒Ϫ(), ͑220͒ where the generalized exponent ¯()is
5 3 1 5 ¯͑͒ϭ lnͫ 2ͩ ϩͱ2ϩ ͪͬϩ Ϫ Ϫ ͱ2ϩ . 2 4 4 2 2 4 ͑221͒
The exponent ¯() is a positive convex function : it decays ¯ ϭ ϭ Ϫͱ ¯ 1 ϭ from ( 0) (3 5)/4 to ( 3 ) 0, and then grows again for Ͼ1/3. This implies, in particular, that
k 1 FIG. 8. Temporal evolution of domains in the RFIM. For each ϭ → ͑with probability 1 at large time͒. ln͑T ln t͒ 3 surviving domain at time t, m denotes the number of ancestor do- ͑222͒ mains in the initial condition at tϭ0.
All of the moments of will be dominated by the typical and we thus recover that the number k of changes of ͗Sx(t)͘ behavior; i.e., ͗m͘ϵ3Ϫm for all m. The full dependence on grows as ln ⌫ϭln ln t and that the rescaled variable ϭ of the ¯() function describes the tails of the probability k/ln(T ln t) is equal to 1/3 with probability 1, as in Eq. ͑ ͒ distribution of , i.e., the large deviations. Note also that the 222 . These results can be extended to the RFIM in the presence probability that ͗Sx(t)͘ doesn’t flip up to time t decays as ¯ of an applied field HϾ0. The total number of flips in this ¯l (t)Ϫ with exponent case saturates to a finite value given by a scaling function of H and J identical to the one given in Sec. IV E in Ref. ͓28͔ 3Ϫͱ5 ¯ϭ¯͑ϭ0͒ϭ , ͑223͒ for the total number of returns to the origin in the Sinai 4 model. which is significantly smaller than the corresponding decay C. Domain persistence exponent ϭ1 in Eq. ͑214͒ for a single spin. Since the renormalization procedure has to be stopped at Persistence can also be defined for larger scale patterns. In ⌫ϭ⌫ ͑ ͒ J at which the equilibrium is reached, and that at later the RFIM the simplest pattern beyond a single spin is a times no more changes occur in the local magnetization we domain. When a domain of, e.g., consecutive ϩ spins disap- obtain that the total number of flips is pears, the two nearest domains of Ϫ spins merge. Thus do- mains can either grow by merging with other domains, or 1 die, and this leads to interesting survival properties, that were k → ln͑4J͒, ͑224͒ ͓ ͔ tot 3 studied by Krapivsky and Ben Naim 48 for the pure Ising model. Here we slightly change their definitions of the expo- the decay of the tails of the probability distribution of nents to adapt to the logarithmic characteristic length scale 2 ϭ ¯ ¯l (t)ϭ¯l ⌫ϳ(T ln t) of the RFIM. ktot /ln(4J), being described by the same function ( )as Of all the domains which still exist at time t, we define above in terms of the length LIM . Another result from Ref. ͓28͔ is the characterization of the Rm(t) as the number of domains which were formed out of ⌫ ϭ ⌫ ϭ m of the initial domains. This is illustrated in Fig. 8. Then full sequence of the times 1 T ln t1 ,..., k T ln tk where ͚ R (t)ϭN(t), the total number of domains at time t, and the local magnetization ͗Sx(t)͘ flips. The sequence of scales m m ⌫ the fraction of initial domains which have a descendent still ͕ k͖ is a multiplicative Markovian process constructed with ⌫ ϭ␣ ⌫ ␣ ‘‘alive’’ at t is S(t)ϭ͚ mR (t)ϭ͗m͘N(t). Following Ref. the simple rule kϩ1 k k , where ͕ k͖ are independent m m identically distributed random variables of probability distri- ͓48͔, one defines the exponents bution (␣): N͑t͒ϳ¯l͑t͒Ϫ1, ͑227͒ 1 1 3Ϯͱ5 ϪϪ Ϫϩ Ϫ ͑␣͒ϭ ͑␣ Ϫ␣ ͒ with Ϯϭ . S͑t͒ϳ¯l͑t͒ , ͑228͒ ␣ ϩϪϪ 2 ͑225͒ ͒ϳ¯ ͒Ϫ␦ ͑ ͒ R1͑t l͑t , 229 ⌫ ϭ␣ ␣ ␣ ⌫ As a consequence, k kϪ1 kϪ2••• 2 1 is simply the product of random variables; thus we obtain, using the cen- in terms of the mean domain length ¯l (t). Asymptotically, tral limit theorem, that Rm(t) has a scaling form ⌫ ln k m lim ͩ ͪ ϭ͗ln ␣͘ϭ3, ͑226͒ R ͑t͒Ϸ¯l͑t͒Ϫ2˜Rͩ ͪ . ͑230͒ m 1Ϫ k→ϱ k ¯l͑t͒
066107-25 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107
The scaling function behaves, in the pure case, as ˜R(z) X. FINITE SIZE PROPERTIES ϳ z for small z and as an exponential at large z. One finds The real-space renormalization procedure can also be the exponent relation ␦ϭϩ(Ϫ)(1ϩ). used analytically to study large finite size systems ͓32,28͔. Let us now study the RFIM using decimation. In the We will extensively use the analysis of the Sinai walker with asymptotic state ͑full state͒, domains coincide with bonds, either absorbing or reflecting boundary counditions ͓28͔. For and when a bond is decimated the domain on this bond dies, the random field Ising model, one may consider several and the two neighbors merge into a new domain ͑the new boundary conditions: ͑i͒ Fixed spins at the ends: this corre- bond͒. We thus associate with each bond an auxiliary vari- sponds for the Glauber dynamics of the RFIM to domain able m counting the number of initial domains which are walls which behave like Sinai walkers with reflecting bound- ancestors of the domain supported by the bond. Upon deci- ary conditions. ͑ii͒ Free boundary conditions: this corre- mation of bond 2, the rule for the variables m is simply sponds to absorbing boundary conditions for the domain walls. Here we give some results for ͑i͒, whose derivation is ϭ ϩ ͑ ͒ mЈ m1 m3 . 231 detailed in Appendix D, and discuss ͑ii͒ at the end.
It turns out that this rule coincides with the magnetization A. Fixed spins at both ends rule of the RFTIC ͓31͔ and thus one has the scaling m ⌽ Assume, for definiteness, that spins at both extremities are ϳ⌫ with ϭ(1ϩͱ5)/2, leading to ϭϩ ϭϪ fixed to the values S0 1 and SL 1, where L is the system size. There is thus an even number of domains in the 3Ϫͱ5 ϭ ϭ ͑ ͒ system, and one can describe its statistics at large time using 0.190983. 232 ⌫ 4 the following generating function. Let us define IL(k; )as probability that the system of size L at time t ͑i.e., at scale This should be compared with the value for the pure Ising ⌫ϭT ln t) contains exactly nϭ2kϩ2 domains, with k model, which has only been determined numerically ͓48͔ as ϭ0,1,2,.... One obtains ͑see Appendix D͒ the generating Ϸ0.252. function in Laplace transform with respect to the length L as Here it is interesting to note that we have found that ϱ ϱ ϭ¯, i.e., the general exact bound р¯ is saturated ͓27͔. ͵ ϪqLͩ k ͑ ⌫͒ͪ dLe ͚ z IL k; This bound for comes from the fact that a point that has 0 kϭ0 never been crossed by any domain wall up to time t for the effective dynamics, necessarily belongs to a domain that has sinh2͑⌫ͱpϩ␦2͒ ϭ , a descendant still living up to time t. In the effective dynam- p cosh2͑⌫ͱpϩ␦2͒ϩ␦2Ϫz͑pϩ␦2͒ ics in the ‘‘full’’ renormalized landscape, we have also the reciprocal property: a domain wall surviving between tЈ and ͑234͒ t necessarily contains points that have never been crossed by where pϭq/2g. Ј any domain wall between t and t. Note that the equality In the case of zero applied field Hϭ0(␦ϭ0) one easily ϭ also appears in the coarsening of domains for the 1D performs the Laplace inversion, and obtains Tϭ0 Landau-Ginzburg 4 soft-spin Ising model ͓35͔ for the ϱ same reason. The strict inequality Ͻ¯ requires the possi- sinh2͑⌫ͱp͒ ͚ k ͑ ⌫͒ϭ Ϫ1 ͩ ͪ bility that a domain wall can survive between tЈ and t even if z IL k; LTp→2gL kϭ0 p͑cosh2͑⌫ͱp͒Ϫz͒ all points inside it at tЈ get crossed by a domain wall between ϩϱ 2 2 tЈ and t ͑see Ref. ͓27͔ for examples͒. eϪ(2gL/⌫ )(␣ϩn) We now study the probability R (t) that a domain has ϭtan͑␣͒ ͚ , 1 ϭϪϱ ␣ϩn survived up to time t without merging with any other do- n main. This requires that the two domain walls A and B living ͑235͒ at the boundaries of this domain do not meet any other do- ␣ϭ ͱ main wall up to time t. In the asymptotic full state, these where arccos z (0, /2) for z (0,1). In particular we conditions imply that three consecutive bonds cannot be obtain the average and mean squared total number of do- decimated. Thus the domain in the middle cannot grow and mains in the system at time t: the probability that it survives upon decimation thus decays L 4 ⌫ 0 exponentially in . ͗n͘ϭ2͗k͘ ⌫ϩ2ϭ4g ϩ ϩo͑L ͒, ͑236͒ ⌫͑ L, ⌫2 3 We thus obtain that R1(t) decays exponentially in and thus with a nonuniversal power of time determined by the initial condition which determines the convergence towards 8gL 2 Ϫ͑ ͒2ϭ ϩ ͑ 0͒ ͑ ͒ ͒ ͗n ͘L,⌫ ͗n͘L,⌫ O L , 237 the asymptotic state . It thus corresponds to 3⌫2 ␦ϭϩϱ. ͑233͒ with ⌫ϭT ln t. Again, the same quantities at equilibrium are simply obtained by setting ⌫ϭ4J in the above formulas. As a consequence, the scaling function R(z), not computed In the presence of an applied field HϾ0(␦Ͼ0), one here, has an essential singularity at the origin. similarly obtains
066107-26 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107
␦2 1Ϫ␥ coth ␥ ϭ ϩ ϵ ϩ ϩ ϩ 0͒ ͗n͘ 2͗k͘L,⌫ 2 4gL 2 2 o͑L , sinh2␥ sinh2␥ ͑238͒
␦2 2͑1Ϫ␥ coth ␥͒ 2 2 ͗ ͘ ⌫Ϫ͑͗ ͘ ⌫͒ ϭ ͩ ϩ ͪ n L, n L, 8gL 1 sinh2␥ sinh2␥ ϩO͑L0͒, ͑239͒ with ␥ϭ␦T ln t. We have also obtained the probability distribution ⌫ FL(M; ) of the total magnetization of the system:
L iϭ2kϩ2 ϭ ϭ Ϫ ͒iϩ1 ͑ ͒ M L ͚ ͗S j͘ ͚ ͑ 1 li . 240 jϭ1 iϭ1
In a Laplace transform with respect to L and M,itis FIG. 9. RG rule near a free boundary, as explained in the text.
ϩϱ ϩL The same rule holds when the first renormalized bond is descending ͵ ϪqL ͵ ϪrM ͑ ⌫͒ by exchanging A and B. dLe dMe FL M; 0 ϪL ͑ Ϯ ϩ Ϫ and length and becomes the new bond 1). Let R (l) be the E ͑qϩr͒E ͑qϪr͒ ϭ¯ ⌫ ⌫ ͑ ͒ probability distribution of the length l of the absorbing zone. l ⌫ ϩ Ϫ , 241 1ϪP⌫ ͑qϩr͒P⌫ ͑qϪr͒ It satisfies the RG equation
Ϯ͑ ͒ϭ Ϯ͑ ͒ ϯ͑ ͒Ϫ Ϯ͑ ͒ ͵ Ј ϯ͑ Ј͒ ץ where ⌫R⌫ l P⌫ 0,• *lR⌫ • R⌫ l dl P⌫ 0,l . ␦ ϯ␦⌫ ͑ ͒ Ϯ e 244 E⌫ ͑q͒ϭ , ͑␦⌫͒ ͱ ϩ␦2 ͑⌫ͱ ϩ␦2͒ϯ␦ sinh „ p coth p … In the symmetric case the solution in Laplace transform is ͑242͒
ϯ␦⌫ 2 ⌫ͱp ͱpϩ␦2e ͑ ͒ϭ ͩ ͪ ͑ ͒ Ϯ R⌫ p tanh . 245 P⌫ ͑q͒ϭ , ⌫ͱ 2 ͑ͱ ϩ␦2⌫͒ ͱ ϩ␦2 ͑⌫ͱ ϩ␦2͒ϯ␦ p sinh p „ p coth p … ͑243͒ Interestingly the shape of the size distribution is the same as where pϭq/2g. Note that qϩr and qϪr are simply the for the Sinai particle with absorbing boundaries but with a Laplace variables associated with the total rescaled positive global length rescaling by a factor 1/4. A similar study can and negative lengths. be made in the biased case. Let us close by noting that the approach to equilibrium B. Free boundary conditions will be modified near a free boundary, as compared with the bulk. Indeed, near the free boundary it is possible to create a Free boundary conditions in the RFIM correspond to ab- single domain wall with energy cost 2J. Thus, for times such sorbing boundary conditions for the diffusing domain walls. that 2JϽ⌫ϭT ln tϽ4J one must consider different rules for However, the study is slightly different from the one carried the first renormalized bond. in Ref. ͓28͔ because now there are particles (A or B) both at the bottom and the top. XI. DISCUSSION AND FUTURE PROSPECTS The structure of the renormalized landscape at ⌫ and the full state near the boundary is now the following ͑see Fig. 9͒. In this paper we have shown how a powerful real space There is an absorbing zone of length l0. Then there is a first dynamical renormalization group method can be used to ͑ ͑ bond barrier F1 length l1) of arbitrary sign contrary to study the properties of the one dimensional random field Sinai’s case, where the first bond was always descending͒. Ising model and 1D spin glasses in a field. We have recov- The first particle at the common endpoint of the first bond ered many known results for the long time equilibrium be- and the second bond and is either an a A particle ͑descending havior, and obtained what we believe are some new ones. first bond͒ or a B particle ͑ascending first bond͒. The RG But the main advantages of this method is that it enables the procedure is unchanged in the bulk. The new case is when computation and physical understanding of many nonequi- bond 1 is decimated. Then the absorbing zone becomes of librium features of the coarsening process. Although the ϩ length l0 l1, while the domain wall (A or B) leaves the RSRG method is approximate, it retains all the information system ͑as the cluster formed by the absorbing zone and the needed to obtain exact results for universal long time, large previously first bond flips͒. Bond 2 keeps the same barrier distance quantities. As exemplified by the calculations of the
066107-27 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107 response and truncated correlations, it has the great advan- with a0 a microscopic scale. Thus the correlation length can tage over many more formal methods by virtue of providing be controlled by a combination of increasing time, increasing a clear physical interpretation of the origins of the processes T, and decreasing h. To obtain a wide range of (t), it is ͑e.g., particular types of rare events͒ that dominate many beneficial to start with h large enough that the shortest time properties of interest. In this last section we address two scales which can be measured still correspond to microscopic issues: the potential observability of some of the one- lengths and then decrease the field gradually. dimensional physics that we have studied in detail; and the While some of the nonequilibrium properties studied here crucial issue of which features of these one-dimensional sys- should be amenable to conventional experimental probes in tems might apply in higher dimensional random systems. magnetic systems, some of the most interesting predictions will probably not be: how the specific set of domain walls A. One-dimensional systems depends on equilibration time and how it varies—or does not Although random field Ising ferromagnets are not directly vary—from run to run on the same sample. For these one realizable experimentally, other systems in the same univer- needs either microscopic imaging probes—perhaps with sality class should be. For random antiferromagnets, an ap- magnetic atomic force or tunnelling microscopy—or scatter- plied magnetic field couples to the antiferromagnetic order ing data with enough resolution that speckle patterns from a parameter as a random field, most simply if the local mag- finite spot size can be measured. As mentioned earlier, the netic moments vary randomly; such systems have the obvi- spatial Fourier transform of the two-point two-time correla- ous advantage of the strength of the dominant randomness tion calculated here is, for spin glasses, related to the corre- being readily adjustable. Similarly, spin glasses, although lations between the speckle patterns at the two different unfrustrated in zero field, do exist in quasi-one-dimensional times. While measuring this with magnetic x-ray or neutron systems. In both these types of systems, rapid quenches scattering may not be possible, it should be feasible via light could be performed by decreasing the field at low tempera- scattering on systems in the same universality class in which tures down to a value of order TӶJ at which coarsening can the length scales are sufficiently long. take place. In practice, other types of randomness in, for One system on which light scattering measurements example, antiferromagnetic or frustrated two-leg spin-ladder might be possible is a nematic liquid crystal in a long thin systems in an applied field may have advantages. tubes with a square crossection and heterogeneities on the There is one physical phenomenon of which one must be surfaces which would couple randomly to the two possible wary: in many random systems—especially spin-glasses symmetry related orientations of the nematic director. ͓5͔—the equilibrium states toward which the system strives Whether this or other systems can be formed in a regime in depend hypersensitively on macroscopic parameters such as which the dynamics are sufficiently fast to allow a wide the temperature and the magnetic field; this is often referred range of length scales to equilibrate is a question whose an- to—somewhat confusingly—as ‘‘chaos’’ ͓49͔. If this were swer must rely on a quantitative analysis of the physics on the case here, then changing the temperature or the magnetic the scale of the domain wall structures that should occur. field would not correspond simply to speeding up the dynam- ics, but would instead drive the system toward a new equi- librium which might differ on the scales being probed from B. Higher-dimensional systems the original one. Indeed, this effect is the origin of much of Many of the qualitative features of the coarsening process the most interesting aging phenomena seen in three- in RFIM chains are also expected to obtain in higher dimen- dimensional spin glasses ͓5,3͔. Fortunately—although less sional random systems. These are particularly interesting in interestingly—this effect does not occur in random field systems in which, in contrast to the one-dimensional models, chains: although the effective local random fields could true long-range order should exist in equilibrium. Neverthe- change in a nonuniform way with temperature or with ap- less, the characteristics of the coarsening process in one- plied field, this will lead only to smooth modifications of the dimensional models with weak randomness should have large scale potential and thus, provided the changes are made much in common with such systems as long as consideration sufficiently slowly, will not change the universal aspects of is restricted, as we have primarily done, to time scales such the coarsening provided all lengths are scaled appropriately. that the full equilibrium correlation length cannot be at- The most obvious difficulty for any experiment is one that tained. is ubiquitous in random systems: how does one obtain a Features that are common to many higher dimensional reasonable range of length scales if the correlations are random systems—random exchange ferromagnets, random growing only logarithmically in time? The situation here is field magnets and, although more controversially, spin not quite as bad as it might seem as on macroscopic time glasses—include the growth of some kind of domain struc- ϳ 2 scales ln t is not all that short: with a microscopic time 0 ture by thermal activation over barriers that grow with length of the order of picoseconds, 1 min corresponds to coarsening scale and are broadly distributed and the existence of aging by a factor of 1000 in length scale. But for one-dimensional and other history dependent phenomena. From a renormal- random-field systems, one can do much better: the correla- ization group point of view, these features are general char- tion length is of order acteristics of systems controlled by random zero-temperature fixed points ͓5,39͔. The notions of local equilibrium within ͑ ͒ 2 T ln t/ 0 constraints caused by large barriers, and of domination of the ͑t͒ϳa ͩ ͪ , ͑246͒ o h dynamics at any given long time scale by rare regions of the
066107-28 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107 sample in which the rate for going over barriers is of order deep potential wells come, as in one dimension, from long- the frequency being probed, are both important. In general, wavelength slowly varying components; thus coarse graining these will, as in the one-dimensional models, result in three the potential, to yield approximate dynamics which are different regimes for two-time properties depending on asymptotically exact, should be possible. We should empha- whether the difference between the times is much smaller, of size, however, that it is by no means established even for this order of, or much larger than, the earlier time. simple model of a random walk in a smoothly varying One of the most intriguing questions concerns the pseudo- Gaussian random potential that the conclusion argued determinism found in the one-dimensional random models. above—that the dynamics will be asymptotically Will this exist in some higher dimensional systems in which deterministic—will be valid in high dimensions. What hap- the domain walls are lines or surfaces whose evolution in pens in three dimensional random magnetic systems in time will involve topological changes, rather than points which there are truly many degrees of freedom, we must which can simply move or annihilate? That is, will domain leave as an important open question. walls tend to lie, at long times, in similar sample-and-time- specific positions which are only weakly dependent on the ACKNOWLEDGMENTS initial conditions? Or will they evolve in different runs in One of us ͑D.S.F.͒, thanks the National Science Founda- very distinct ways–perhaps by retaining much more memory tion for support via DMR-9976621 and Harvard’s Materials of the initial conditions? Research Science and Engineering Center. Two examples of a single random walker diffusing in random potentials illustrate some of the difficulties of draw- ing any definitive conclusions. Consider, first, a random po- APPENDIX A: CONVERGENCE TOWARD THE FULL tential which is independently random at each site with a STATE long power law tail ot the distribution of the depth of the potential wells. The deep wells will give rise to a subdiffu- In this appendix we analyze the rapid convergence to- sive behavior; indeed, simple considerations of the time to wards the full state that was discussed in Sec. IV C. We find a deep trap and the time to escape it imply that the consider, for simplicity, random initial configurations of the typical distance the walker will diffuse in time t only grows spins, in which there is an independent probability w that as a power of ln t. But the statistics of the set of sites the each intersite position is occupied by a domain wall, and a Ϫ walker visits by the time it has made a given number of probability 1 w that it is not. Such random initial configu- jumps from one site to another will be identical to that of a rations describe, for example, initial equilibrium before a normal free random walker; it is just the time spent on each quench from a temperature T0 which is high enough that the site that causes the slow diffusion. Since a pair of random random fields are negligible and the system behaves like a ϭ  ϳ ӷ͕ ͖ walkers in dimensions higher than 4 have a nonzero prob- pure chain. This corresponds to T0 1/ 0 J hn and the Ϫ 2J Ϫ 2J ability of never visiting any sites in common, it is clear that choice wϭe 0 /(1ϩe 0 ). In particular, initial condi- the long time behavior is not deterministic in high dimen- tions where all spins are independent and take value Ϯ1 with  ϭ sions: two different runs starting from the same point will probability 1/2 corresponds to infinite temperature 0 0, have a probablity that vanishes at long times of the two and wϭ1/2. walkers being at the same site. In two dimensions, in con- For these type of initial configurations, the probability trast, the pair of walkers will be very likely to be trapped at that there exist nϭ0,...,l domain walls among l consecu- the same site at long times. ͑The three-dimensional case is tive lattice spacings is simply given by the binomial distri- more subtle but will be more like the high dimensional that bution the low-dimensional case.͒ A second type of random potential yields different behav- l! R͑n͉l͒ϭ wn͑1Ϫw͒lϪn. ͑A1͒ ior: a Gaussian random potential with mean zero and n!͑lϪn͒! ͓V(x)ϪV(y)͔2ϳ͉xϪy͉2 with Ͼ0. A pair of random walkers in such a potential will tend to be trapped in the In the renormalized landscape, the length l of descending same deep valley at long times. This can be argued by con- ϩ ͑ascending͒ bonds at scale ⌫ is distributed with P⌫ (l) sidering the borders of valleys which are surrounded by bar- Ϫ riers of at least a height ⌫. If these typically do not have a lot ͓P⌫ (l)͔, whose Laplace transforms are obtained after inte- of fine-scale structure, then they should act as effective traps gration over in Eq. ͑37͒. The probability that there had for all walkers in the vicinity. All the walkers will then tend been initially no domain walls in the interval occupied by a to eventualy leave such a valley over the same barrier, and descending ͑ascending͒ bond of the renormalized landscape hence end up in the same next-larger-scale valley as well. plays an important role; it is The behavior would then be asymptotically deterministic as in the one-dimensional Sinai model. ϱ ϱ Ϯ ͑⌫͒ϭ Ϯ͑ ͒ ͑ ͉ ͒ϭ Ϯ͑ ͒͑ Ϫ ͒l The crucial feature that distinguishes these two cases, rzero ͚ P⌫ l R 0 l ͚ P⌫ l 1 w . seems to be the smoothness of the potential: in the first ex- lϭ1 lϭ1 ͑ ͒ ample, the deep potential wells come from very short- A2 wavelength fluctuations, and there is thus no useful notion of a coarse-grained potential. But in the second example, the The probability that there had been an odd number n is
066107-29 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107
ϱ ϱ ϩ l 1Ϫ͑1Ϫ2w͒l r ͑⌫͒ Ϯ ͑⌫͒ϭ Ϯ͑ ͒ ͑ ͉ ͒ϭ Ϯ͑ ͒ desc͑ϩ ͕ Ϫ͖͒ϭ Ϫ Ϫ ͑⌫͒ ͩ Ϫ zero rodd ͚ P⌫ l ͚ R n l ͚ P⌫ l , V⌫ ; 2 1 rzero 1 lϭ1 n odd lϭ1 2 „ … 2 ͑A3͒ ϩ Ϫ r ͑⌫͒r ͑⌫͒ Ϫ zero zero ͪ and the probability that there had been a nonvanishing even 2 number is Ϫ r ͑⌫͒ 2 ϩ „ zero … ϩ ͑⌫͒ ͑ ͒ reven , A11 Ϯ ͑⌫͒ϭ Ϫ Ϯ ͑⌫͒Ϫ Ϯ ͑⌫͒ ͑ ͒ 2 reven 1 rodd rzero . A4 Ϫ r ͑⌫͒ desc desc zero To characterize the statistical properties of the spin con- V⌫ ͑ϩ;͕ϩϪ͖͒ϭV⌫ ͑ϩ;͕Ϫϩ͖͒ϭ figurations on the renormalized landscape, we focus on one 2 ⌫ Ϫ renormalized bond at scale —call it ‘‘2’’—and its local r ͑⌫͒ 2 environment which determines whether or not the maximum Ϫ „ zero … ϩ ͑⌫͒ϩ ϩ ͑⌫͒ „rzero reven …, and minimum at the ends of bond 2 are occupied by domain 2 desc ⑀ ⑀ ⑀ walls. We introduce the probabilities V⌫ ( 2 ;͕ 1 , 3͖) ͑A12͒ asc ⑀ ⑀ ⑀ ⑀ ϭϮ ͑ ͓V⌫ ( 2 ;͕ 1 , 3͖)͔, where i , that a descending resp. Ϫ ascending͒ bond of the renormalized landscape at scale ⌫ is r ͑⌫͒ 2 desc „ zero … ϩ ϩ ⑀ ϭϮ ⑀ V⌫ ͑ϩ;͕2ϩ͖͒ϭ r ͑⌫͒ϩr ͑⌫͒ , in phase 2 , with its left neighboring bond in phase 1 2 „ zero even … ⑀ and its right neighboring bond in phase 3. The normaliza- ͑A13͒ tion conditions are and similarly for ascending bonds by exchanging Ϯ→ϯ. The important property of these probabilities is that apart desc ⑀ ⑀ ⑀ ͒ϭ ͑ ͒ ͚ ͚ ͚ V⌫ ͑ 2 ;͕ 1 , 3͖ 1, A5 desc ϩ ͕ϪϪ͖ asc Ϫ ͕ϩϩ͖ ⑀ ϭϮ ⑀ ϭϮ ⑀ ϭϮ from V⌫ ( ; ) and V⌫ ( ; ), which corre- 1 2 3 spond to locally full configuration of the domain walls, all Ϯ ⌫ the allowed V have rzero( ) as a factor, i.e., to have a bond asc ⑀ ⑀ ⑀ ͒ϭ ͑ ͒ 2 that does not have domain walls at both its extremities ͚ ͚ ͚ V⌫ ͑ 2 ;͕ 1 , 3͖ 1. A6 ⑀ ϭϮ ⑀ ϭϮ ⑀ ϭϮ 1 2 3 requires that at least one of the three bonds ͑1,2, or 3͒ had exactly zero domain walls in the initial configuration. ⑀ ϭϪ ͓ ͑ ͔͒ If 2 on a descending bond, then it is not possible to Using the fixed point solution Eq. 37 , we find that ⑀ ϭϩ have 1 , since this would correspond to a domain wall ϱ 1 of type A at a maxima, and similarly it is not possible to have Ϯ ͑⌫͒Ӎ ͵ Ϯ͑ ͒͑ Ϫ ͒lϭ ˆ Ϯͩ ϭ ͩ ͪͪ ⑀ ϭϩ rzero dlP⌫ l 1 w P⌫ q ln , 3 since this would correspond to a domain wall of type 0 1Ϫw B at a minima; thus we have immediately that ͑A14͒
Vdesc͑Ϫ;͕ϩϪ͖͒ϭVdesc͑Ϫ;͕Ϫϩ͖͒ϭVdesc͑Ϫ;͕2ϩ͖͒ϭ0. and thus this and the probabilities of non-full bonds con- ⌫ ⌫ ⌫ ⌫ ͑A7͒ verges exponentially in to 0,
ͱ 2 Ϯ ͑⌫͒ϳ Ϫ⌫( pϩ␦ Ϯ␦) ͑ ͒ Similarly we have rzero e , A15
with pϭ(1/2g)ln͓1/(1Ϫw)͔. Vasc͑ϩ;͕ϩϪ͖͒ϭVasc͑Ϫ;͕Ϫϩ͖͒ϭVasc͑Ϫ;͕2Ϫ͖͒ϭ0. ⌫ ⌫ ⌫ Thus the system converges towards the ‘‘full’’ state of the ͑A8͒ renormalized landscape exponentially in ⌫, with a non uni- versal coefficient depending on the initial concentration of ⑀ ϭϩ ⑀ ϭϪ If, however, 2 on a descending bond ( 2 on a domain walls through the parameter w, and on the strength of ͒ ascending bond , then all four possible neighborhoods of this the disorder through g. For the symmetric case of no applied bond can occur. Since it is a bit lengthy to give the full field (␦ϭ0), the fraction of the extrema of the renormalized enumeration of all possible cases with their corresponding landscape at time t that are not occupied by a domain wall probabilities, we give here only the final results goes to zero as a power of time,
ϩ r ͑⌫͒ desc zero 1 V⌫ ͑Ϫ;͕ϪϪ͖͒ϭ , ͑A9͒ ͓ ͔ϳ ͑ ͒ Prob missing domain wall , A16 2 t e
Ϫ ͑⌫͒ rzero with Vasc͑ϩ;͕ϩϩ͖͒ϭ , ͑A10͒ ⌫ 2 ln 2 ϭ ͱ ͑ ͒ e T . A17 and 2¯h2
066107-30 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107
Note that in the initial state for the landscape, and in the ⌺ϩ ϭ ϩϩ ϩ Ϫϩ Introducing the sum ⌫,⌫Ј A⌫,⌫Ј A⌫,⌫Ј and the difference ϭ ϩ 2ϩ Ϫϩ symmetric case H 0, hn is positive or negative with prob- ⌬ ϭ Ϫ ⌫,⌫Ј A⌫,⌫Ј A⌫,⌫Ј , we obtain the decoupled equations ability 1/2, and thus after grouping the consecutive hn of the ϩ ϭ0, ͑B9͒ ⌬ 2ץ same sign to give the initial zig-zag landscape, we have, for an initial distribution of length, ⌫ ⌫,⌫Ј ͒ ͑ ⌺ϩ ϭ ϩ Ϫ⌺ϩ 2ץ 1 ⌫ ⌫,⌫Ј 2u⌫ u⌫ ⌫,⌫Ј , B10 P ͑l͒ϭ for lϭ1,2,...,ϱ. ͑A18͒ 0 l 2 together with the boundary conditions This corresponds to ϩ ϩ ϩ 1 ⌬ ϭ⌺ ϭA2 ϭ ͑␥Ј coth ␥ЈϪ1͒, 1Ϫw ⌫Ј,⌫Ј ⌫Ј,⌫Ј ⌫Ј,⌫Ј 2␦2 r ͑⌫ϭ0͒ϭ . ͑A19͒ ͑ ͒ zero 1ϩw B11 ϩ ϩ ͉ ⌺ ץϭ ͉ ⌬ ץ ϭ 1 ⌫ϭ ϭ 1 As an example, for w 2 , we have rzero( 0) 3 , and ⌫ ⌫,⌫Ј ⌫ϭ⌫Ј ⌫ ⌫,⌫Ј ⌫ϭ⌫Ј desc Ϫ ͕ϪϪ͖ ϭ 1 V⌫ϭ0( ; ) 6 . ϩϩ 1 ␥Ј ϭB ϭ ͩ coth ␥ЈϪ ͪ , ⌫Ј,⌫Ј ␦ 2␥Ј APPENDIX B: AUTOCORRELATIONS IN THE RFIM 2 sinh WITH AN APPLIED FIELD ͑B12͒ ͑ ͒ We have now to solve RG equations 94 given in the text where ␥Јϭ␦⌫Ј as usual. In terms of the function with the initial conditions
ϩ ͑␥͒ϭ␥ coth ␥Ϫ1, ͑B13͒ ϱ lP⌫ ͑,l͒ ϩϩ ͑͒ϭ ͵ Ј ͑ ͒ P⌫Ј,⌫Ј dl , B1 ␥ 0 ¯l ⌫Ј and its derivative with respect to , Ϫϩ ␥ P ͑͒ϭ0. ͑B2͒ ⌫Ј,⌫Ј M͑␥͒ϭcoth ␥Ϫ , ͑B14͒ sinh2␥ Since for large ⌫, PϮ have reach their fixed point values Ϯ ͓Eq. ͑37͔͒, we obtain ͓using the simplified notations u⌫ we finally obtain Ϯ Ϯ ϭu⌫ (pϭ0)ϭU⌫ (pϭ0)͔ Ϯϩ 1 Ϯϩ ϯ Ϯϩ A ϭ ͑␥͒Ϯ␥M͑␥Ј͒Ϯ͑␥Ј͒ϯ␥ЈM͑␥Ј͒ , … „ ͒ϭϪ ͑͒ ⌫,⌫Ј ␦2͑ ͒ ץϪ ץ͑ ⌫ P⌫,⌫Ј 2u⌫ P⌫,⌫Ј 4 ͑B15͒ ϱ ϯ Ϯ Ϫ Ϯ ϪЈ Ϯϩ ϩ ͵ Ј u⌫ ( ) ͑Ј͒ 2u⌫ u⌫ d e P⌫,⌫Ј 0 Ϯϩ 1 B ϭ M͑␥͒ϮM͑␥Ј͒ . ͑B16͒ ⌫,⌫Ј ␦ „ … Ϯ Ϫ Ϯ ϯϩ 4 ϩ͑ ͒2 u⌫ ͑ ͒ ͑ ͒ u⌫ e P⌫,⌫Ј 0 , B3 In the same way we obtain the solutions for the Ϫ initial together with the initial conditions condition,
ϩ 2ϩ 1 Ϫ ϩ Ϯ u⌫ ϮϪ 1 Ϫ Ϯ ϮϪ ϮϪ ⌫U⌫ ͑p͒ u ץP⌫ ⌫ ͑͒ϭ e Ј Ϫ Ј, Ј „ p P⌫ ⌫ ͑͒ϭ e u⌫ ͑A⌫ ⌫ ϩB⌫ ⌫ ͒ ͑B17͒ ¯l ⌫ , Ј , Ј , Ј ¯l ⌫ ϩ ϩ u ͑p͒ ͉ ϭ , ͑B4͒ ץϩU ͑p͒ ⌫ p ⌫ … p 0 ϮϪ ϭ ϯϩ ϮϪ ϭ ϯϩ with A⌫,⌫Ј A⌫,⌫Ј and B⌫,⌫Ј B⌫,⌫Ј . We may check the Ϫϩ ͑͒ϭ ͑ ͒ normalizations P⌫Ј,⌫Ј 0. B5 ϱ 1 The solutions are thus of the form ͵ ϩϮ ͑͒ϩ ϪϮ ͑͒ ϭ ϮM͑␥Ј͒ ͑ ͒ d P⌫,⌫Ј P⌫,⌫Ј 1 . B18 0 „ … 2 „ … Ϯϩ 1 Ϫ Ϯ Ϯ Ϯϩ Ϯϩ P ͑͒ϭ e u⌫ u ͑A ϩB ͒, ͑B6͒ ⌫,⌫Ј ⌫ ⌫,⌫Ј ⌫,⌫ The autocorrelation is ¯l ⌫ ͒ ͒ where the coefficients satisfy ͗Si͑tЈ Si͑t ͘
Ϯϩ Ϯϩ ϱ 2ϩ 2Ϫ ϩϪ Ϫϩ ͒ ͑ ץϭ B⌫,⌫Ј ⌫A⌫,⌫Ј , B7 ϭ ͵ ͑͒ϩ ͑͒Ϫ ͑͒Ϫ ͑͒ d P⌫,⌫Ј P⌫,⌫Ј P⌫,⌫Ј P⌫,⌫Ј 0 „ … ͒ ͑ ͒ Ϯϩ ϭ ϩ Ϫ͑ 2ϩ ϩ Ϫϩ 2ץ ⌫A⌫,⌫Ј u⌫ u⌫ A⌫,⌫Ј A⌫,⌫Ј . B8 ͑B19͒
066107-31 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107
⑀,⑀Ј 1 ϩ Ϫϩ 1 1 ϩ Ϫϩ In the end we are interested in the probabilities P (x) ϭ ͩ 2͑A2 ϪA ͒ϩͩ ϩ ͪ ͑B2 ϪB ͒ͪ ⌫,⌫Ј ⌫,⌫Ј ⌫,⌫Ј ϩ Ϫ ⌫,⌫Ј ⌫,⌫Ј ͗ Ј Ј ͘ϭ⑀Ј ͗ ͘ϭ⑀ ¯l ⌫ u⌫ u⌫ that S0(t )Sx(t ) and S0(t)Sx(t) . We have the ͑B20͒ normalization
ϩ ϩ Ϫ Ϫ Ϫ ϩ ϩ Ϫ 1 P , ͑x͒ϩP , ͑x͒ϩP , ͑x͒ϩP , ͑x͒ϭ1. ͑C7͒ ϭ͑coth ␥͒M͑␥Ј͒ϩ ␥M͑␥Ј͒ϩ͑␥Ј͒ ⌫,⌫Ј ⌫,⌫Ј ⌫,⌫Ј ⌫,⌫Ј sinh2␥ „ The correlation function is Ϫ␥ЈM͑␥Ј͒ ͑ ͒ …. B21 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ϭM ␥ ϭM ␥ ϭ ␥ ͗S0 t Sx t ͗͘S0 tЈ Sx tЈ ͘ Note that ͗Si(tЈ)͘ ( Ј) and ͗Si(t)͘ ( ) coth 2 Ϫ␥/sinh ␥. ϭ ϩ,ϩ ͑ ͒ϩ Ϫ,Ϫ ͑ ͒Ϫ Ϫ,ϩ ͑ ͒Ϫ ϩ,Ϫ ͑ ͒ P⌫,⌫Ј x P⌫,⌫Ј x P⌫,⌫Ј x P⌫,⌫Ј x . APPENDIX C: TWO-POINT TWO-TIME CORRELATIONS ͑C8͒ Ј Ј ŠS0„t…Sx„t…‹ŠS0„t …Sx„t …‹ In terms of the functions ⍀, we have 1. Definitions ϱ Two points at 0 and xϾ0 are given. To compute the two ϩ,ϩ ͑ ͒ϭ ͵ ⍀2n,ϩ P⌫ ⌫ x ͚ ⌫ ⌫ spin two time correlation we introduce the following quanti- , Ј ϭ , Ј ⍀2n,ϩ ϭ n 0 0 ,l0 ,... 2n ,l2n ties. Let ⌫,⌫Ј ( 0 ,l0 ; 1 ,l1 ,...; 2n ,l2n ;x), n 0,1,2 ..., 2n,Ϫ ϫ ͒ ͑ ͒ ͓ ⍀ ͔ ͑ 0 ,l0 ; 1 ,l1 ,...; 2n ,l2n ;x , C9 and, respectively, ⌫,⌫Ј ( 0 ,l0 ; 1 ,l1 ,...; 2n ,l2n ;x) be the probability that at ⌫Ј, x, and 0 are on bonds of ϱ same orientation ͑respectively different orientation͒, and ϩ,Ϫ ͑ ͒ϭ ͵ ⍀2n,Ϫ ⌫ P⌫ ⌫ x ͚ ⌫ ⌫ that at , x, and 0 are on bonds of same orientation, , Ј ϭ , Ј n 0 0 ,l0 ,... 2n ,l2n with a configuration ( 0 ,l0 ; 1 ,l1 ,...; 2n ,l2n). One simi- ⍀2nϩ1,ϩ ϫ͑ ͒ ͑ ͒ larly defines ⌫,⌫Ј ( 0 ,l0 ; 1 ,l1 ,...; 2nϩ1 ,l2nϩ1 ;x), n 0 ,l0 ; 1 ,l1 ,...; 2n ,l2n ;x , C10 ϭ ͓ ⍀2nϩ1,Ϫ 0,1,2 ..., respectively ⌫ ⌫ ( 0 ,l0 ; 1 ,l1 ,...; 2nϩ1 , , Ј ϱ l ϩ ;x)͔ be the probability that, at ⌫Ј, x and 0 are on bonds 2n 1 Ϫ,ϩ ͑ ͒ϭ ͵ of same orientation ͑different orientations͒ and that, at ⌫, x P⌫,⌫Ј x ͚ nϭ0 ,l ,... ,l and 0 are on bonds of different orientations, with a configu- 0 0 2nϩ1 2nϩ1 2nϩ1,ϩ ration ( ,l ; ,l ,...; ϩ ,l ϩ ). ϫ⍀ ͑ ͒ 0 0 1 1 2n 1 2n 1 ⌫,⌫Ј 0 ,l0 ; 1 ,l1 ,...; 2nϩ1 ,l2nϩ1 ;x , Initial condition at ⌫ϭ⌫Ј ͑for all nу0): ͑C11͒ ⍀2n,ϩ ͒ ͑ ͒ ⌫ ⌫ ͑ 0 ,l0 ; 1 ,l1 ,...; 2n ,l2n ;x , C1 Ј, Ј ϱ Ϫ,Ϫ ͑ ͒ϭ ͵ ϭP⌫ ͑ ,l ͒P⌫ ͑ ,l ͒ ...P⌫ ͑ ,l ͒W⌫ P⌫ ⌫ x ͚ Ј 0 0 Ј 1 1 Ј 2n 2n Ј , Ј ϭ n 0 0 ,l0 ,... 2nϩ1 ,l2nϩ1 2n ϫ⍀2nϩ1,Ϫ͑ ͒ ϫͩ Ϫ ͪ ͑ ͒ ⌫ ⌫ 0 ,l0 ; 1 ,l1 ,...; 2nϩ1 ,l2nϩ1 ;x . l0 ,l2n ,͚ li x , C2 , Ј iϭ0 ͑C12͒ ⍀2nϩ1,Ϫ͑ ͒ ͑ ͒ ⌫Ј,⌫Ј 0 ,l0 ; 1 ,l1 ,...; 2nϩ1 ,l2nϩ1 ;x , C3 2. RG equations ϭ ͑ ͒ ͑ ͒ ͑ ͒ P⌫Ј 0 ,l0 P⌫Ј 1 ,l1 ...P⌫Ј 2nϩ1 ,l2nϩ1 W⌫Ј m,⑀Ј The RG equations for ⍀⌫ ⌫ for ⑀ЈϭϮ and m ϩ , Ј 2n 1 ϭ0,1,2,... read ϫͩ Ϫ ͪ ͑ ͒ l0 ,l2nϩ1 , ͚ li x , C4 iϭ0 m ͒ ⍀m,⑀Ј͑ ͪ ץ Ϫ ץ ͩ ⌫ ͚ ⌫,⌫Ј 0 ,l0 ; 1 ,l1 ;... m ,lm ;x with the notation kϭ0 k
⑀Ј l l ϭϪ ͑ϭ ͒⍀m, ͑ ͒ 2 1 2 2P⌫ 0 ⌫,⌫Ј 0 ,l0 ; 1 ,l1 ;... m ,lm ;x W⌫͑l ,l ,L͒ϭ ͵ dy ͵ dy ␦͑LϪy Ϫy ͒ ͑C5͒ 1 2 2 1 2 1 2 ⌫ 0 0 ͑C13͒
2 m ϭ ͓ ͑ ͒Ϫ ͑ Ϫ ͔͓͒ ͑ ͒ ϩ ⑀Ј min l1 ,L max 0,L l2 min l1 ,L ϩ ⍀m 2, ⌫2 ͚ ͵ ⌫ ⌫ ϭ ϩ ϩ ϭ , Ј k 0 z,l lЈ lЉ lk Ϫ Ϫ ͒ ͑ ͒ max͑0,L l2 ͔. C6 ϫ ͑ 0 ,l0 ;...; kϪ1 ,lkϪ1 ;z,l;0,lЈ; k ⍀2n,Ϫ ϭ⍀2nϩ1,ϩϭ Ϫ Љ ͒ ͑ ͒ We have, of course, ⌫Ј,⌫Ј ⌫Ј,⌫Ј 0. z,l ;...; m ,lm ;x C14
066107-32 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107
3. Form of solutions ϩ ͵ ͑ Ϫ ͒⍀mϩ1,⑀Ј P⌫ 0 z,l ⌫ ⌫ ϩ Јϩ Јϭ , Ј For nу1 and ⑀ЈϭϮ,weset z,l l0 l1 l0 ϫ͑0,lЈ ;z,lЈ ; ,l ;...; ,l ;x͒ ͑C15͒ ⍀2n,⑀Ј͑ ͒ 0 1 1 1 m m ⌫,⌫Ј 0 ,l0 ;...; 2n ,l2n ;x 2n ϩ,⑀Ј mϩ1,⑀Ј ϭ ͩ Ϫ ͪ ϩ ͵ ⍀ E⌫,⌫Ј 0 ,l0 ; 2n ,l2n ;͚ li x ⌫ ⌫ ϭ ϩ Ј ϩ Ј ϭ , Ј i 0 z,l lm lmϩ1 lm ϫ ͑ ͒ ͑ ͒ ͑ ͒ ϫ͑ Ј Ј ͒ P⌫ 1 ,l1 ...P⌫ 2nϪ1 ,l2nϪ1 , C26 0 ,l0 ;...; mϪ1 ,lmϪ1 ;z,lm ;0,lmϩ1 ;x ϫ Ϫ ͒ ͑ ͒ 2nϩ1,⑀Ј P⌫͑ m z,l C16 ⍀ ͑ ͒ ⌫,⌫Ј 0 ,l0 ;...; 2nϩ1 ,l2nϩ1 ;x 2nϩ1 Ϫ ⑀Ј m,⑀Ј ϭ , ͩ Ϫ ͪ ϩ ͵ ͑ ͒ ͑ Ј͒⍀ ͑ E⌫ ⌫ 0 ,l0 ; 2nϩ1 ,l2nϩ1 ; ͚ li x P⌫ z,l P⌫ 0,l ⌫ ⌫ 0 , Ј ϩ Јϩ Јϭ , Ј iϭ0 z,l l l0 l0 ϫ ͑ ͒ ͑ ͒ ͑ ͒ Ϫ Ј ͒ ͑ ͒ P⌫ 1 ,l1 ...P⌫ 2n ,l2n , C27 z,l0 ; 1 ,l1 ;...; m ,lm ;x C17 and also ϩ ͵ ⍀m,⑀Ј͑ ⌫ ⌫ 0 ,l0 ;...; mϪ1 ,lmϪ1 ; 1,⑀Ј Ϫ,⑀Ј Ј ϩ Јϩ ϭ , Ј ⍀ ͑ ͒ϭ ͑ ϩ Ϫ ͒ z,lm l l lm ⌫,⌫Ј 0 ,l0 ; 1 ,l1 ;x E⌫,⌫Ј 0 ,l0 ; 1 ,l1 ;l0 l1 x , ͑C28͒ ϫ Ј ͒ ͑ Ј͒ ͑ Ϫ ͒ ͑ ͒ z,lm ;x P⌫ 0,l P⌫ p z,l . C18 where P⌫(,l) satisfy the bond equation, and where E satis- fies the RG equations In particular, for mϭ0, we have Ϯ,⑀Ј ͒E ͑ ,l ; ,l ;L͒ץ ϪץϪ⌫ץ͑ 1 2 ⌫,⌫Ј 1 1 2 2 0,⑀Ј ͒⍀ ͑ ,l ;x͒ Ϯ,⑀ЈץϪ⌫ץ͑ 0 ⌫,⌫Ј 0 0 ϭϪ ͑ϭ ͒ ͑ ͒ ͑ ͒ 2P⌫ 0 E⌫,⌫Ј 1 ,l1 ; 2 ,l2 ;L C29 0,⑀Ј ϭϪ2P⌫͑ϭ0͒⍀ ͑ ,l ;x͒ ͑C19͒ ⌫,⌫Ј 0 0 Ϯ,⑀Ј ϩP⌫͑0,.͒* P⌫͑.,.͒* E ͑.,.; ,l ;L͒ l1 1 ,l1 ⌫,⌫Ј 2 2 ͑C30͒ ϩ ͵ ⍀2,⑀Ј ͑ Ј Ϫ Љ ͒ ⌫ ⌫ z,l;0,l ; 0 z,l ;x ϩ ϩ ϭ , Ј Ϯ ⑀ z,l lЈ lЉ l0 , Ј ϩP⌫͑0,.͒* P⌫͑.,.͒* E⌫ ⌫ ͑ ,l ;.,.;L͒ ͑C20͒ l2 2 ,l2 , Ј 1 1 ͑C31͒
1,⑀Ј ϩ ͵ ͑ Ϫ ͒⍀ ͑ Ј Ј ͒ ϯ,⑀Ј P⌫ 0 z,l ⌫,⌫Ј 0,l0 ;z,l1 ;x ϩ ͵ ͑ Ј Ϫ ͒ z,lϩlЈϩlЈϭl E⌫,⌫Ј 0,l1 ; 2 ,l2 ;L l 0 1 0 lϩlЈϩlЈϭl ͑C21͒ 1 1 ϫP⌫͑.,l͒* P⌫͑.,lЈ͒ ͑C32͒ 1 ϩ ͵ ⍀1,⑀Ј ͑ Љ Ј ͒ ͑ Ϫ ͒ ⌫ ⌫ z,l ;0,l ;x P⌫ 0 z,l ϩ Љϩ Јϭ , Ј ϯ,⑀Ј z,l l l l0 ϩ ͵ ͑ Ј Ϫ ͒ E⌫ ⌫ 1 ,l1 ;0,l2 ;L l ͑ ͒ ϩ Јϩ Јϭ , Ј C22 l l l2 l2
ϫP⌫͑.,l͒* P⌫͑.,lЈ͒ ͑C33͒ 2 ϩ ͵ ͑ ͒ ͑ Ј͒⍀0,⑀Ј ͑ Ϫ Ј ͒ P⌫ z,l P⌫ 0,l ⌫ ⌫ 0 z,l0 ;x ϩ Јϩ Јϭ , Ј z,l l l0 l0 ͑ ͒ ϩ ͵ Ϯ,⑀Ј͑ Ј Ϫ Ϫ Ј͒ C23 E⌫ ⌫ .,l1 ; 2 ,l2 ;L l l * ϩ Јϩ Јϭ , Ј 1 l l l1 l1
ϫP⌫͑.,l͒P⌫͑0,lЈ͒ ͑C34͒ ϩ ͵ ⍀0,⑀Ј ͑ Ј ͒ ͑ Ј͒ ͑ Ϫ ͒ ⌫ ⌫ z,l0 ;x P⌫ 0,l P⌫ 0 z,l Јϩ Јϩ ϭ , Ј z,l0 l l l0 ͑C24͒ ϩ ͵ Ϯ,⑀Ј͑ Ј Ϫ Ϫ Ј͒ E⌫ ⌫ 1 ,l1 ;.,l2 ;L l l * ϩ Јϩ Јϭ , Ј 2 l l l2 l2 ⑀Ј ϩ ͵ ⍀0, ͑ Ј ͒ ͑ Ј͒ ͑ Ϫ ͒ ϫP⌫͑.,l͒P⌫͑0,lЈ͒, ͑C35͒ ⌫ ⌫ 0,l0 ;x P⌫ z,l P⌫ 0 z,l . Јϩ Јϩ ϭ , Ј z,l0 l l l0 ͑C25͒ with the initial conditions
066107-33 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107
Ϫ ϩ ⑀,⑀Ј , ͑ ͒ P⌫ ⌫ ͑x͒ ͑C48͒ E⌫Ј,⌫Ј 1 ,l1 ; 2 ,l2 ;L , Ј ϭ␦ ͒ ͒ ͒ ϱ ⑀,⑀ЈP⌫Ј͑ 1 ,l1 P⌫Ј͑ 2 ,l2 W⌫Ј͑l1 ,l2 ,L . ϭϩ ͵ ͵ Ϫ,ϩ ͚ E⌫ ⌫ ͑ ͒ ϭ , Ј C36 n 0 0 , 2nϩ1 L,l0 ,l1 ,l2nϩ1 ⑀Ј 2nϩ1 The RG equation for ⍀0, becomes ⌫,⌫Ј ϫ͑ ͒␦ͩ Ϫͩ Ϫ ͪͪ 0 ,l0 ; 2nϩ1 ,l2nϩ1 ;L L ͚ li x iϭ0 0,⑀Ј ͒⍀⌫ ⌫ ͑ ,l ;x͒ץϪ⌫ץ͑ 0 , Ј 0 0 ϫ ͑ ͒ ͑ ͒ ͑ ͒ P⌫ l1 •••P⌫ l2n C49 0,⑀Ј ϭϪ2P⌫͑ϭ0͒⍀ ͑ ,l ;x͒ ͑C37͒ Ϫ Ϫ ⌫,⌫Ј 0 0 , ͑ ͒ ͑ ͒ P⌫,⌫Ј x C50 ϩ ͵ ϩ,⑀Ј͑ Ϫ Љ Ϫ ͒ ͑ Ј͒ ϱ E⌫,⌫Ј z,l; 0 z,l ;l0 x P⌫ 0,l Ϫ,Ϫ z,lϩlЈϩlЉϭl ϭ ͵ ͵ 0 ͚ E⌫ ⌫ ϭ , Ј ͑C38͒ n 0 0 , 2nϩ1 L,l0 ,l1 ,l2nϩ1 ϫ ͒␦ ͑ 0 ,l0 ; 2nϩ1 ,l2nϩ1 ;L ϩ ͵ ͑ ͒ ⍀1,⑀Ј ͑ Ј Ј ͒ P⌫ .,l * ⌫,⌫Ј 0,l0 ;.,l1 ;x 2nϩ1 lϩlЈϩlЈϭl 0 0 1 0 ϫͩ Ϫͩ Ϫ ͪͪ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ L ͚ li x P⌫ l1 P⌫ l2n . C51 C39 iϭ0 •••
ϩ ͵ ⍀1,⑀Ј ͑ Ј Ј ͒ ͑ ͒ ⌫ ⌫ .,l0 ;0,l1 ;x * P⌫ .,l 4. Laplace transforms ϩ Јϩ Јϭ , Ј 0 l l0 l1 l0 ͑C40͒ It is convenient to introduce the Laplace transforms
⑀,⑀Ј 0,⑀Ј E ͑ ,p ; ,p ;p͒ ϩ2P⌫͑0,.͒* P⌫͑.,.͒* ⍀ ͑.,.;x͒ ͑C41͒ ⌫,⌫Ј 1 1 2 2 l0 0 ,l0 ⌫,⌫Ј ϱ ϱ ϱ Ϫ Ϫ Ϫ ⑀ ϭ ͵ ͵ ͵ p1l1 p2l2 pL 0, Ј dl1 dl2 dLe ϩP⌫͑.,.͒* P⌫͑.,.͒* ⍀ ͑0,.;x͒, ͑C42͒ 0 ,l0 l0 ⌫,⌫Ј 0 0 0 ϫ ͑ ͒ ͑ ͒ with the initial condition E⌫,⌫Ј 1 ,l1 ; 2 ,l2 ;L . C52
⍀0,⑀Ј ͑ ͒ϭ␦ ͑ ͒ ͑ Ϫ ͒ Using the fixed point solution ⌫Ј,⌫Ј 0 ,l0 ;x ⑀Ј,ϩ1P⌫Ј 0 ,l0 W⌫Ј l0 ,l0 ,l0 x . Ϫu⌫(p) ͑C43͒ P⌫͑,p͒ϭU⌫͑p͒e , ͑C53͒
Using the form of the solutions, we thus obtain the RG equation for E becomes
Ϯ,⑀Ј ͒ ͑ ͒ ץϪ ץϪ ץ͑ ϩ,ϩ 0,ϩ ͑ ͒ϭ ͵ ⍀ ͑ ͒ ͑ ͒ ⌫ E⌫,⌫Ј 1 ,p1 ; 2 ,p2 ;p P⌫ ⌫ x ⌫ ⌫ 0 ,l0 ;x C44 1 2 , Ј , Ј 0 ,l0 Ϯ,⑀Ј ϭϪ2U⌫͑0͒E ͑ ,p ; ,p ;p͒ ͑C54͒ ϱ ⌫,⌫Ј 1 1 2 2 ϩ ͵ ͵ ϩϩ ͑ ͚ E⌫ ⌫ 0 ,l0 ; 2n , ϭ , Ј 2 1 Ϫ Ϫ Ϯ,⑀Ј n 1 0 , 2n L,l0 ,l1 ...l2n ϩ ͑ ͒ ͵ ( 1 z)u⌫(p1) ͑ ͒ U⌫ p1 dze E⌫,⌫Ј z,p1 ; 2 ,p2 ;p 0 2n ͑ ͒ ϫ ͒␦ͩ Ϫͩ Ϫ ͪͪ C55 l2n ;L L ͚ li x ϭ i 0 2 Ϫ Ϫ Ϯ,⑀Ј ϩ 2 ͑ ͒ ͵ ( 2 z)u⌫(p2) ͑ ͒ U⌫ p dze E⌫ ⌫ ,p ;z,p ;p ϫP⌫͑l ͒ P⌫͑l Ϫ ͒ ͑C45͒ 2 , Ј 1 1 2 1 ••• 2n 1 0 ͑C56͒ ϩ,Ϫ 0,Ϫ ͑ ͒ϭ ͵ ⍀ ͑ ͒ ͑ ͒ Ϫ ϩ Ϫ P⌫ ⌫ x ⌫ ⌫ 0 ,l0 ;x C46 u⌫(p p) u⌫(p ) , Ј , Ј e 1 1 Ϫe 1 1 0 ,l0 ϩ ͑ ͒ ͑ ϩ ͒ U⌫ p1 U⌫ p1 p ͒Ϫ ϩ ͒ u⌫͑p1 u⌫͑p1 p ϱ ϩ ͵ ͵ ϩϪ ͑ ϫ ϯ,⑀Ј ͑ ͒ ͚ E⌫ ⌫ 0 ,l0 ; 2n , E ͑0,p ; ,p ;p͒ C57 ϭ , Ј ⌫,⌫Ј 1 2 2 n 1 0 , 2n L,l0 ,l1 ...l2n Ϫ ϩ Ϫ 2n e 2u⌫(p2 p)Ϫe 2u⌫(p2) ϩ ͑ ͒ ͑ ϩ ͒ ϫ ͒␦ͩ Ϫͩ Ϫ ͪͪ U⌫ p2 U⌫ p2 p l2n ;L L ͚ li x u⌫͑p ͒Ϫu⌫͑p ϩp͒ iϭ0 2 2 ϯ ⑀Ј ϫ ͑ ͒ ͑ ͒ ͑ ͒ ϫE , ͑ ,p ;0,p ;p͒ ͑C58͒ P⌫ l1 •••P⌫ l2nϪ1 C47 ⌫,⌫Ј 1 1 2
066107-34 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107
2nϭ0,⑀Ј 1 Ϫ Ϫ ϩ Ϯ,⑀Ј ⍀ ͑ ͒ ϩ 2 ͑ ϩ ͒ ͵ ( 1 z)u⌫(p1 p) ⌫Ј,⌫Ј 0 ,p0 ;p U⌫ p1 p dze E⌫,⌫Ј 0 2 Ϫ ϩ ϫ ͒ ͑ ͒ ϭ␦ ͓ ͑ ϩ ͒ 0u⌫Ј(p0 p) ͑z,p1 ; 2 ,p2 ;p C59 ⑀Ј,ϩ1 U⌫Ј p0 p e ⌫Ј2p2
Ϫ u⌫ (p ) 2 Ϫ Ϫ ϩ Ϯ,⑀Ј ϪU⌫ ͑p ͒e 0 Ј 0 ͑C70͒ ϩ 2 ͑ ϩ ͒ ͵ ( 2 z)u⌫(p2 p) Ј 0 U⌫ p2 p dze E⌫,⌫Ј 0 Ϫ ͒ ͑ (0u⌫Ј(p0 ͒ ץ Ϫ p p U⌫Ј͑p0 e ]. C71 ϫ ͒ ͑ ͒ 0„ … ͑ 1 ,p1 ;z,p2 ;p , C60 In Laplace terms with the initial conditions ϱ ⑀ ⑀Ј ⑀ ⑀Ј P , ͑q͒ϭ ͵ dxeϪqxP , ͑x͒ ͑C72͒ ⑀,⑀Ј 2 Ϫ ⌫,⌫Ј ⌫,⌫Ј ͑ ͒ϭ␦ ͓ ͑ ͒ 1u⌫Ј(p1) 0 E⌫Ј,⌫Ј 1 ,p1 ; 2 ,p2 ;p ⑀,⑀Ј U⌫Ј p1 e ⌫Ј2p2 Ϫ ϩ relations ͑C51͒ become Ϫ ϩ ͒ 1u⌫Ј(p1 p) U⌫Ј͑p1 p e ͔
͑C61͒ ϩ ϩ ϩ , ͑ ͒ϭ ͵ ⍀0, ͑ ͒ P⌫ ⌫ q ⌫ ⌫ 0,0;q , Ј , Ј Ϫ Ϫ ϩ 0 ͒ 2u⌫Ј(p2)Ϫ ϩ ͒ 2u⌫Ј(p2 p) ͓U⌫Ј͑p2 e U⌫Ј͑p2 p e ͔. ͑C62͒ P⌫͑q͒ ϩ ͵ ϩϩ ͑ Ϫ ͒ 2 E⌫,⌫Ј 1 ,q; 2 ,q; q , 1ϪP ͑q͒ , Also defining ⌫ 1 2 ͑C73͒ ⍀2nϭ0,⑀Ј͑ ͒ ⌫Ј,⌫Ј 0 ,p0 ;p ϩ,Ϫ ͑ ͒ϭ ͵ ⍀0,Ϫ ͑ ͒ ϱ ϱ P⌫ ⌫ q ⌫ ⌫ 0,0;q Ϫ Ϫ 2nϭ0,⑀Ј , Ј , Ј ϭ ͵ p0l0 ͵ px⍀ ͑ ͒ 0 dl0e dxe ⌫Ј,⌫Ј 0 ,l0 ;x , 0 0 P⌫͑q͒ ͑ ͒ ϩ ͵ ϩϪ ͑ Ϫ ͒ C63 E⌫ ⌫ 1 ,q; 2 ,q; q , 2 , Ј 1ϪP⌫͑q͒ 1 , 2 the RG equation becomes ͑C74͒
͒ ⍀0,⑀Ј ͑͒ ץϪ ץ͑ ⌫ ⌫,⌫Ј 0 ,p0 ;p 1 0 Ϫ,ϩ ͑ ͒ϭ ͵ Ϫϩ ͑ Ϫ ͒ P⌫ ⌫ q E⌫ ⌫ 1 ,q; 2 ,q; q , , Ј 2 , Ј 0,⑀Ј 1ϪP⌫͑q͒ 1 , 2 ϭϪ2u⌫⍀ ͑ ,p ;p͒ ͑C64͒ ⌫,⌫Ј 0 0 ͑C75͒
ϩ,⑀Ј ϩ ͵ ͑ ϩ Ϫ ϩ Ϫ ͒ ͑ ϩ ͒ Ϫ Ϫ 1 ϪϪ E⌫,⌫Ј z,p0 p; 0 z,p0 p; p P⌫ 0,p0 p , ͑ ͒ϭ ͵ ͑ Ϫ ͒ z P⌫,⌫Ј q 2 E⌫,⌫Ј 1 ,q; 2 ,q; q . 1ϪP ͑q͒ , ͑C65͒ ⌫ 1 2 ͑C76͒ 0 ⑀ Ϫ( Ϫz)u⌫(p ) 1, Ј Thus for the functions E, we need to solve only the case ϩU⌫͑p ͒ ͵ dze 0 0 ⍀ ͑0,p ;z,p ;p͒ 0 ⌫,⌫Ј 0 0 ϭ ϭ ϭϪ ϭ ϭ 0 p1 p2 q p, i.e., with the notation U⌫(p 0) u⌫(p ͑C66͒ ϭ0)ϭu⌫ ,
Ϯ ⑀Ј 0 , ͒E⌫ ⌫ ͑ ,q; ,q;Ϫq͒ץ ϪץϪ⌫ץ͑ Ϫ( Ϫz)u (p ) 1,⑀Ј ϩ ͑ ͒ ͵ 0 ⌫ 0 ⍀ ͑ ͒ 1 2 , Ј 1 2 U⌫ p0 dze ⌫,⌫Ј z,p0 ;0,p0 ;p 0 ͑ ͒ ϭϪ Ϯ,⑀Ј͑ Ϫ ͒ ͑ ͒ C67 2u⌫E⌫,⌫Ј 1 ,q; 2 ,q; q C77
0 2 Ϫ( Ϫz)u (p ) 0,⑀Ј 1 Ϯ ⑀ ϩ ͑ ͒ ͵ 0 ⌫ 0 ⍀ ͑ ͒ 2 Ϫ( Ϫz)u⌫(q) , Ј 2U⌫ p dze ⌫ ⌫ z,p ;p ϩ ͑ ͒ ͵ 1 ͑ Ϫ ͒ 0 , Ј 0 U⌫ q dze E⌫,⌫Ј z,q; 2 ,q; q 0 0 ͑C68͒ ͑C78͒
⑀ 2 Ϫ u⌫(p ) 0, Ј ϩU⌫͑p ͒ e 0 0 ⍀ ͑0,p ;p͒, ͑C69͒ 2 Ϫ Ϫ Ϯ,⑀Ј 0 0 ⌫,⌫Ј 0 ϩ 2 ͑ ͒ ͵ ( 2 z)u⌫(q) ͑ Ϫ ͒ U⌫ q dze E⌫,⌫Ј 1 ,q;z,q; q 0 with the initial condition ͑C79͒
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Ϫ Ϫ e 1u⌫Ϫe 1u⌫(q) 5. Solution for the functions E ϩ ͑ ͒ ϯ,⑀Ј͑ Ϫ ͒ U⌫ q u⌫ E⌫,⌫Ј 0,q; 2 ,q; q u⌫͑q͒Ϫu⌫ Using the symmetry in ( 1 , 2), we set ͑C80͒ ⑀,⑀Ј ͑ Ϫ ͒ Ϫ Ϫ E⌫,⌫Ј 1 ,q; 2 ,q; q e 2u⌫Ϫe 2u⌫(q) ϩ ͑ ͒ ϯ,⑀Ј͑ Ϫ ͒ U⌫ q u⌫ E⌫,⌫Ј 1 ,q;0,q; q u⌫͑q͒Ϫu⌫ 2 ⑀,⑀Ј Ϫ Ϫ ϭ A ͑q͒e 1u⌫(q)e 2u⌫(q) ͑C81͒ ⌫2 „ ⌫,⌫Ј
1 ⑀,⑀Ј Ϫ Ϫ 2 Ϫ Ϫ Ϯ,⑀Ј ϩ 1u⌫ 2u⌫(q) ͑ ͒ ϩ ͵ ( 1 z)u⌫ ͑ Ϫ ͒ B⌫ ⌫ ͑q͒e e C94 u⌫ dze E⌫,⌫Ј z,q; 2 ,q; q , Ј 0 ͑C82͒ ⑀,⑀Ј Ϫ Ϫ ⑀,⑀Ј Ϫ Ϫ ϩ ͑ ͒ 1u⌫(q) 2u⌫ϩ ͑ ͒ 1u⌫ 2u⌫ B⌫,⌫Ј q e e D⌫,⌫Ј q e e . … 2 Ϫ Ϫ Ϯ,⑀Ј ͑C95͒ ϩ 2 ͵ ( 2 z)u⌫ ͑ Ϫ ͒ u⌫ dze E⌫,⌫Ј 1 ,q;z,q; q , 0 ͑C83͒ It is in fact more convenient to use the following combi- nations: and with the initial conditions ⌫2 ⑀ ⑀Ј ⑀ ⑀Ј ⑀ ⑀Ј E , ͑ ,q; ,q;Ϫq͒ I , ͑q͒ϭ E , ͑ ϭ0,q; ϭ0,q;Ϫq͒ ͑C96͒ ⌫Ј,⌫Ј 1 2 ⌫,⌫Ј 2 ⌫Ј,⌫Ј 1 2 2 Ϫ u⌫ (q) Ϫ u⌫ ϭ␦⑀ ⑀ ͓U⌫ ͑q͒e 1 Ј Ϫu⌫ e 1 Ј͔ ⑀,⑀Ј ⑀,⑀Ј ⑀,⑀Ј , Ј 2 2 Ј Ј ϭ ͑ ͒ϩ ͑ ͒ϩ ͑ ͒ ⌫Ј q A⌫,⌫Ј q 2B⌫,⌫Ј q D⌫,⌫Ј q , ͑C97͒ ͑C84͒
Ϫ Ϫ ⌫2 ϱ ͒ 2u⌫Ј(q)Ϫ 2u⌫Ј ͑ ͒ ⑀ ⑀ ⑀ ⑀ ͓U⌫Ј͑q e u⌫Јe ͔. C85 , Ј ͑ ͒ϭ ͵ , Ј ͑ ϭ Ϫ ͒ J⌫,⌫Ј q d 2E⌫Ј,⌫Ј 1 0,q; 2 ,q; q 2 0 ϭ ϭ ϭ For n 0 we need to solve only the cases p0 p1 0 and ͑C98͒ pϭq, i.e., the equation ⑀ ⑀Ј ⑀Ј , 0, A⌫ ⌫ ͑q͒ 1 1 ͒⍀⌫ ⌫ ͑ ,0;q͒ , Ј ⑀,⑀ЈץϪ⌫ץ͑ 0 , Ј 0 ϭ ϩ ͑ ͒ͩ ϩ ͪ B⌫,⌫Ј q u⌫͑q͒ u⌫͑q͒ u⌫ 0,⑀Ј ϭϪ2u⌫⍀ ͑ ,0;q͒ ͑C86͒ ⌫,⌫Ј 0 ⑀,⑀Ј ͑ ͒ D⌫,⌫Ј q ϩ , ͑C99͒ ϩ ⑀ u⌫ ϩ ͑ ͒ ͵ , Ј͑ Ϫ Ϫ ͒ ͑ ͒ U⌫ q E⌫,⌫Ј z,q; 0 z,q; q C87 z ⌫2 ϱ ϱ ⑀,⑀Ј ⑀,⑀Ј ͑ ͒ϭ ͵ ͵ ͑ Ϫ ͒ K⌫ ⌫ q d d E⌫ ⌫ ,q; ,q; q 0 Ϫ Ϫ 1,⑀Ј , Ј 1 2 Ј, Ј 1 2 ϩ ͵ ( 0 z)u⌫⍀ ͑ ͒ ͑ ͒ 2 0 0 u⌫ dze ⌫,⌫Ј 0,0;z,0;q C88 0 ͑C100͒
0 Ϫ Ϫ 1,⑀Ј ⑀,⑀Ј ⑀,⑀Ј ⑀,⑀Ј ϩ ͵ ( 0 z)u⌫⍀ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ u⌫ dze ⌫,⌫Ј z,0;0,0;q C89 A⌫,⌫Ј q B⌫,⌫Ј q D⌫,⌫Ј q 0 ϭ ϩ2 ϩ . 2 ͑ ͒ 2 u⌫͑q͒ u⌫ q u⌫ u⌫ ͑C101͒ 0 Ϫ Ϫ 0,⑀Ј ϩ 2 ͵ ( 0 z)u⌫⍀ ͑ ͒ ͑ ͒ 2u⌫ dze ⌫,⌫Ј z,0;q C90 0 They satisfy the system of equations
Ϫ 0,⑀Ј ϩ 2 0u⌫⍀ ͑ ͒ ͑ ͒ u⌫ 0e ⌫,⌫Ј 0,0;q , C91 Ϯ,⑀Ј Ϯ,⑀Ј Ϯ,⑀Ј ,I ͑q͒ϭϪ2 u⌫ϩu⌫͑q͒ I ͑q͒ϩ2u⌫u⌫͑q͒J ͑q͒⌫ץ ⌫,⌫Ј „ … ⌫,⌫Ј ⌫,⌫Ј with the initial conditions ͑C102͒
⑀Ј ⍀0, ͑ ͒ 2 ͒ ⌫Ј,⌫Ј 0,0;q Ϯ ⑀ U⌫͑q Ϯ ⑀ Ϯ ⑀ ͒ Ј͑ ͒ϭͩ Ϫ ͑ ͒ͪ , Ј͑ ͒Ϫ , Ј͑ , ץ ⌫J⌫,⌫Ј q u⌫ q J⌫,⌫Ј q I⌫,⌫Ј q u⌫͑q͒ 2 Ϫ u⌫ (q) Ϫ u⌫ ϭ␦⑀ ϩ ͓U⌫ ͑q͒e 0 Ј ϪU⌫ e 0 Ј ͑C92͒ Ј, 1 2 2 Ј Ј U⌫͑q͒ ⌫Ј q ϩ ͑ ͒ Ϯ,⑀Ј͑ ͒ϩ ϯ,⑀Ј͑ ͒ u⌫u⌫ q K⌫,⌫Ј q I⌫,⌫Ј q , u⌫͑q͒ Ϫ u⌫ (p ) U⌫ ͑p ͒e 0 Ј 0 ͉͒ ϭ ]. ͑C93͒͑ ץϪp p0 Ј 0 p0 0 ͑C103͒
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2 ͑ ͒ Ϯ ⑀Ј U⌫ q Ϯ ⑀Ј Ϯ ⑀Ј 1 ϭͩ ϩ ͪ , ͑ ͒Ϫ , ͑ ͒ S,⑀Ј 2͒ ͑ , ץ ⌫K q 2 2u⌫ K q 2J q ͑ ͒ϭ ͓ ͑ ͒Ϫ ͔ ͑ ͒ ⌫,⌫Ј ͑ ͒ ⌫,⌫Ј ⌫,⌫Ј I⌫Ј,⌫Ј q U⌫Ј q u⌫Ј , C117 u⌫ q q2
U⌫͑q͒ ϯ ⑀Ј ϩ2 J , ͑q͒, ͑C104͒ ͑ ͒ ͑ ͒ ⌫,⌫Ј S,⑀Ј 1 U⌫Ј q u⌫ q J ͑q͒ϭ ͓U⌫ ͑q͒Ϫu⌫ ͔ͫ Ϫ1ͬ, ⌫Ј,⌫Ј 2 Ј Ј q u⌫Ј͑q͒ and the initial conditions ͑C118͒
1 ͑ ͒ 2 ⑀,⑀Ј 2 S,⑀Ј 1 U⌫Ј q I⌫ ⌫ ͑q͒ϭ␦⑀ ⑀ ͓U⌫ ͑q͒Ϫu⌫ ͔ , ͑C105͒ ͑ ͒ϭ ͫ Ϫ ͬ ͑ ͒ , Ј Ј Ј K q 1 ; C119 Ј, Ј 2 ⌫Ј,⌫Ј 2 q q u⌫Ј(q)
͑ ͒ thus the solutions are simply ⑀,⑀Ј 1 U⌫Ј q ͑ ͒ϭ␦⑀ ⑀ ͓ ⌫ ͑ ͒Ϫ ⌫ ͔ͫ Ϫ ͬ J⌫ ⌫ q , Ј U Ј q u Ј 1 , Ј, Ј 2 ͑ ͒ q u⌫Ј q 1 ͑ ͒ S,⑀Ј ͑ ͒ϭ ͓ ͑ ͒Ϫ ͔2 ͑ ͒ C106 I⌫,⌫Ј q U⌫ q u⌫ , C120 q2 ͑ ͒ 2 ⑀,⑀Ј 1 U⌫Ј q K ͑q͒ϭ␦⑀ ⑀ ͫ Ϫ1ͬ . ͑C107͒ ͑ ͒ ⌫Ј,⌫Ј , Ј 2 S,⑀Ј 1 U⌫ q q u⌫ ͑ ͒ϭ ͓ ͑ ͒Ϫ ͔ͫ Ϫ ͬ ͑ ͒ Ј(q) J⌫,⌫Ј q U⌫ q u⌫ 1 , C121 q2 u⌫͑q͒ It is convenient to introduce the following sums and dif- 2 ferences: 1 U⌫͑q͒ S,⑀Ј ͑ ͒ϭ ͫ Ϫ ͬ ͑ ͒ K⌫,⌫Ј q 1 . C122 q2 u⌫͑q͒ S,⑀Ј ͑ ͒ϭ ϩ,⑀Ј ͑ ͒ϩ Ϫ,⑀Ј ͑ ͒ ͑ ͒ I⌫Ј,⌫Ј q I⌫Ј,⌫Ј q I⌫Ј,⌫Ј q , C108 For ⑀Ј fixed, the three functions ID,⑀Ј, JD,⑀Ј, and KD,⑀Ј D,⑀Ј ͑ ͒ϭ ϩ,⑀Ј ͑ ͒Ϫ Ϫ,⑀Ј ͑ ͒ ͑ ͒ satisfy the system of equations I⌫Ј,⌫Ј q I⌫Ј,⌫Ј q I⌫Ј,⌫Ј q , C109
D,⑀Ј D,⑀Ј D,⑀Ј ͒ ͑ ͒ ͑ ϭϪ ϩ ͑ ͒ ͑ ͒ϩ͒ ͑ ץ Ј ϩ ⑀Ј Ϫ ⑀Ј⑀ S, ͑ ͒ϭ , ͑ ͒ϩ , ͑ ͒ ͑ ͒ ⌫I⌫,⌫Ј q 2 u⌫ u⌫ q I⌫,⌫Ј q 2u⌫u⌫ q J⌫,⌫Ј q , J⌫Ј,⌫Ј q J⌫Ј,⌫Ј q J⌫Ј,⌫Ј q , C110 „ … ͑C123͒
D,⑀Ј ϩ,⑀Ј Ϫ,⑀Ј 2 J⌫ ⌫ ͑q͒ϭJ⌫ ⌫ ͑q͒ϪJ⌫ ⌫ ͑q͒, ͑C111͒ U⌫͑q͒ ͒ D,⑀Ј͑ ͒ϭͩ Ϫ ͑ ͒ͪ D,⑀Ј͑ ͒Ϫ D,⑀Ј͑ ץ Ј, Ј Ј, Ј Ј, Ј ⌫J⌫,⌫Ј q u⌫ q J⌫,⌫Ј q I⌫,⌫Ј q u⌫͑q͒ S,⑀Ј ϩ,⑀Ј Ϫ,⑀Ј K⌫ ⌫ ͑q͒ϭK⌫ ⌫ ͑q͒ϩK⌫ ⌫ ͑q͒, ͑C112͒ Ј, Ј Ј, Ј Ј, Ј U⌫͑q͒ ϩ ͑ ͒ D,⑀Ј͑ ͒Ϫ D,⑀Ј͑ ͒ u⌫u⌫ q K⌫,⌫Ј q I⌫,⌫Ј q , u⌫͑q͒ D,⑀Ј ͑ ͒ϭ ϩ,⑀Ј ͑ ͒Ϫ Ϫ,⑀Ј ͑ ͒ ͑ ͒ K⌫Ј,⌫Ј q K⌫Ј,⌫Ј q K⌫Ј,⌫Ј q . C113 ͑C124͒ S,⑀Ј S,⑀Ј S,⑀Ј Then, for ⑀Ј fixed, the three functions I , J , and K 2 U⌫͑q͒ ͒ D,⑀Ј͑ ͒ϭͩ ϩ ͪ D,⑀Ј͑ ͒Ϫ D,⑀Ј͑ ץ satisfy the system of equations ⌫K⌫,⌫Ј q 2 2u⌫ K⌫,⌫Ј q 2J⌫,⌫Ј q u⌫͑q͒ ͒ ͑ S,⑀Ј ͑ ͒ϭϪ ϩ ͑ ͒ S,⑀Ј ͑ ͒ϩ ͑ ͒ S,⑀Ј ץ ⌫I⌫,⌫Ј q 2 u⌫ u⌫ q I⌫,⌫Ј q 2u⌫u⌫ q J⌫,⌫Ј q , U⌫͑q͒ „ … Ϫ D,⑀Ј͑ ͒ ͑ ͒ ͑ ͒ 2 J⌫,⌫Ј q , C125 C114 u⌫͑q͒ 2 U⌫͑q͒ S,⑀Ј ͑ ͒ϭͩ Ϫ ͑ ͒ͪ S,⑀Ј ͑ ͒Ϫ S,⑀Ј ͑ ͒ with the initial conditions ץ ⌫J⌫,⌫Ј q u⌫ q J⌫,⌫Ј q I⌫,⌫Ј q u⌫͑q͒ 1 D,⑀Ј ⑀Ј 2 ⑀Ј U⌫͑q͒ ⑀Ј I ͑q͒ϭ͑Ϫ1͒ ͓U⌫ ͑q͒Ϫu⌫ ͔ , ͑C126͒ ϩ ͑ ͒ S, ͑ ͒ϩ S, ͑ ͒ ⌫Ј,⌫Ј 2 Ј Ј u⌫u⌫ q K⌫,⌫Ј q I⌫,⌫Ј q , q u⌫͑q͒ ͑C115͒ ͑ ͒ D,⑀Ј ⑀Ј 1 U⌫Ј q ͑ ͒ϭ͑Ϫ ͒ ͓ ⌫ ͑ ͒Ϫ ⌫ ͔ͫ Ϫ ͬ J⌫Ј,⌫Ј q 1 U Ј q u Ј 1 , 2 2 ͑ ͒ U⌫͑q͒ q u⌫Ј q ͒ ͑ S,⑀Ј ͑ ͒ϭͩ ϩ ͪ S,⑀Ј ͑ ͒Ϫ S,⑀Ј ץ ⌫K⌫,⌫Ј q 2 2u⌫ K⌫,⌫Ј q 2J⌫,⌫Ј q ͑C127͒ u⌫͑q͒ ͒ 2 U⌫͑q͒ ⑀Ј 1 U⌫Ј͑q ϩ S,⑀Ј ͑ ͒ ͑ ͒ KD, ͑q͒ϭ͑Ϫ1͒⑀Ј ͫ Ϫ1ͬ . ͑C128͒ 2 ͑ ͒ J⌫,⌫Ј q , C116 ⌫Ј,⌫Ј 2 u⌫ q q u⌫Ј(q) with the initial conditions Three linearly independent solutions of the system read
066107-37 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107
D1 2 I⌫ ͑q͒ϭ͓U⌫͑q͒ϩu⌫͔ , ͑C129͒ It is useful to consider the matrix formed by these solu- tions,
U⌫͑q͒ D1͑ ͒ϭ͓ ͑ ͒ϩ ͔ͫ ϩ ͬ ͑ ͒ D1͑ ͒ D2͑ ͒ D3͑ ͒ J⌫ q U⌫ q u⌫ 1 , C130 I⌫ q I⌫ q I⌫ q u⌫͑q͒ ϭ D1͑ ͒ D2͑ ͒ D3͑ ͒ N⌫ ͩ J⌫ q J⌫ q J⌫ q ͪ , 2 U⌫͑q͒ D1͑ ͒ D2͑ ͒ D3͑ ͒ D1͑ ͒ϭͫ ϩ ͬ ͑ ͒ K⌫ q K⌫ q K⌫ q K⌫ q 1 , C131 u⌫(q) whose determinant gives the Wronskian W⌫ of the three lin- ear independent solutions: U⌫͑q͒ϩu⌫ D2͑ ͒ϭ ͑ ͒ I⌫ q 2u⌫ ͑ ͒ϩ ͑ ͒ , C132 U⌫ q u⌫ q 3 U⌫͑q͒ ⌫ϭ ͓ ⌫͔ϭϪ ͑ ͒ W det N 3 . C138 u⌫͑q͒ U⌫͑q͒ ϩ ͑ ͒ ͫ ϩ ͬ „2u⌫ U⌫ q … ͑ ͒ 1 D2 u⌫ q The solution satisfying the initial conditions we are inter- J⌫ ͑q͒ϭ , ͑C133͒ U⌫͑q͒ϩu⌫͑q͒ ested in will be obtained as a linear combination,
D,⑀Ј 1⑀Ј U⌫͑q͒ I ͑q͒ Di͑ ͒ ͑q͒ ͫ ϩ ͬ ⌫,⌫Ј I⌫ q ⌫Ј 1 3 u⌫(q) ⑀Ј ⑀Ј ⑀Ј KD2͑q͒ϭ2 , ͑C134͒ D, ͑ ͒ ϭ i ͑ ͒ JDi͑q͒ ϵ 2 ͑ ͒ ⌫ J⌫,⌫Ј q ͚ ⌫Ј q ͩ ⌫ ͪ N⌫ ⌫Ј q , ͑ ͒ u⌫ q ͩ ͪ iϭ1 ͩ ͪ ⑀ Di͑ ͒ ⑀ D, Ј͑ ͒ K⌫ q 3 Ј͑ ͒ K⌫,⌫Ј q ⌫Ј q 2 U⌫͑q͓͒2u⌫ϩ 2u⌫ϩU⌫͑q͒ u⌫͑q͒ϩU⌫͑q͒ ͔ D3 „ …„ … 1⑀Ј I⌫ ͑q͒ϭ , ͑ ͒ϩ ͑ ͒ ͑ ͒2Ϫ ͑ ͒2 where the coefficients ⌫Ј (q) are determined by the initial „u⌫ q U⌫ q …„u⌫ q U⌫ q … ͑C135͒ conditions
1⑀Ј͑ ͒ D,⑀Ј ͑ ͒ ͑ ͓͒ ϩ ͑ ͒ϩ ͑ ͔͒ ⌫Ј q I⌫Ј,⌫Ј q D3 U⌫ q 2u⌫ u⌫ q U⌫ q J⌫ ͑q͒ϭ , ͑C136͒ ͑ ͒ ͑ ͒2Ϫ ͑ ͒2 2⑀Ј Ϫ1 D,⑀Ј u⌫ q u⌫ q U⌫ q ) ͑q͒ ϭN J ͑q͒ . ͑C139͒ „ … ͩ ⌫Ј ͪ ⌫Ј ͩ ⌫Ј,⌫Ј ͪ 3⑀Ј͑ ͒ D,⑀Ј ͑ ͒ ͑ ͒ ⌫Ј q K⌫Ј,⌫Ј q D3 2U⌫ q K⌫ ͑q͒ϭ . ͑C137͒ ͑ ͒2 ͑ ͒Ϫ ͑ ͒ u⌫ q „u⌫ q U⌫ q … The inverse of the matrix N⌫ is
2 2 2u⌫͑q͒ 2u⌫͑q͒͑2u⌫u⌫͑q͒ϩU⌫͑q͓͒u⌫͑q͒ϩU⌫͑q͔͒͒ u⌫͑q͒ ͑2u u⌫͑q͒ϩU⌫͑q͓͒2u⌫ϩU⌫͑q͔͓͒u⌫͑q͒ϩU⌫͑q͔͒͒ Ϫ ⌫ 2 2 2 2 2 2 2 U⌫͑q͒ ͓u⌫͑q͒ϪU⌫͑q͔͒ U⌫͑q͒ ͑u⌫͑q͒ ϪU⌫͑q͒ ͒ U⌫͑q͒ ͓u⌫͑q͒ϩU⌫͑q͔͓͒u⌫͑q͒ ϪU⌫͑q͒ ͔
͑ ͑ ͒ϩ ͑ ͒͒ ͑ ͒͑ ϩ ͑ ͒͒ ͑ ͒2͓ ϩ ͑ ͔͒ Ϫ1 u⌫ q U⌫ q u⌫ q 2u⌫ U⌫ q u⌫u⌫ q u⌫ U⌫ q N⌫ ϭ Ϫ Ϫ , ͒2 ͒2 ͒2 ͒ϩ ͒ U⌫͑q U⌫͑q U⌫͑q ͓u⌫͑q U⌫͑q ͔ 2 2 2 ͓u⌫͑q͒ϩU⌫͑q͔͒ 2u⌫͑q͓͒u⌫ϩU⌫͑q͔͓͒u⌫͑q͒ϩU⌫͑q͔͒ u⌫͑q͒ ͓u⌫ϩU⌫͑q͔͒ Ϫ Ϫ 2 ͑ ͒ 2 U⌫͑q͒ U⌫ q U⌫͑q͒
͑C140͒ and, finally, we obtain the coefficients
͕Ϫ 2 ͑ ͒ϩ͓ ͑ ͒ϩ ͑ ͔͓͒Ϫ ͑ ͒2ϩ 2 ͑ ͒ϩ ͑ ͒Ϫ ͑ ͒ ͔͖ ⑀Ј 8u⌫Јu⌫Ј q u⌫Ј q U⌫Ј q 5u⌫Ј q U⌫Ј q 4u⌫Ј 3u⌫Ј q U⌫Ј q 1 ͑q͒ϭϪ͑Ϫ1͒⑀Ј „ … ͑C141͒ ⌫Ј 2 ͑ ͒Ϫ ͑ ͒ ͑ ͒ϩ ͑ ͒ 2 q „u⌫Ј q U⌫Ј q …„u⌫Ј q U⌫Ј q ) …
066107-38 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107
0,Ϫ 2⑀Ј ⍀ ͑ ͒ we do not have to compute the functions ⌫ ⌫ ( 0,0;q) ⌫Ј q , Ј separately. Indeed, we obtain 2 u⌫ЈϪu⌫Ј͑q͒ 2u⌫ЈϪu⌫Ј͑q͒ϪU⌫Ј͑q͒ ϭϪ͑Ϫ1͒⑀Ј „ …„ … 2 ͑ ͒ϩ ͑ ͒ q „u⌫Ј q U⌫Ј q … ϩ͑ ͒ϭ ͵ ⍀0,ϩ ͑ ͒ϩ ͵ ⍀0,Ϫ ͑ ͒ P⌫ q ⌫ ⌫ 0,0;q ⌫ ⌫ 0,0;q ͑C142͒ , Ј , Ј 0 0
2 4 u⌫ Ϫu⌫ ͑q͒ ͑ ͒ 3⑀Ј ⑀Ј „ Ј Ј … 2 P⌫ q S,ϩ ⌫ ͑q͒ϭϪ͑Ϫ1͒ . ͑C143͒ ϩ K ͑q͒, ͑C157͒ Ј 2 2 2 ⌫,⌫Ј q ⌫ 1ϪP⌫͑q͒ In the end we are interested in
͑ ͒ Ϫ 2 1 ϩ ϩ,ϩ 0,ϩ 2 P⌫ q ϩ,ϩ ϭ S, ͑ ͒ ͑ ͒ϭ ͵ ⍀ ͑ ͒ϩ ͑ ͒ P⌫ ͑q͒ K⌫ ⌫ ͑q͒, C158 P⌫ ⌫ q ⌫ ⌫ 0,0;q K⌫ ⌫ q , 2 2 , Ј , Ј , Ј 2 2 2 , Ј ⌫ 1ϪP ͑q͒ 0 ⌫ q 1ϪP⌫͑q͒ ⌫ ͑C144͒
ϩ Ϫ Ϫ 2 P⌫͑q͒ ϩ Ϫ 1 1 P , ͑q͒ϭ ͵ ⍀0, ͑ ,0;q͒ϩ K , ͑q͒, ϩ ͑ ͒ϭ ͵ ⍀0,ϩ ͑ ͒ϩ ⌫,⌫Ј ⌫,⌫Ј 0 2 ⌫,⌫Ј P⌫ q ⌫ ⌫ 0,0;q ⌫2 2 Ϫ ͑ ͒ Ј , Ј 2 2 0 q 1 P⌫ q 0 ⌫ 1ϪP⌫͑q͒ ͑C145͒ S,ϩ D,ϩ ϫ͓ 1ϩP⌫͑q͒ K ͑q͒Ϫ 1ϪP⌫͑q͒ K ͑q͔͒, „ … ⌫,⌫Ј „ … ⌫,⌫Ј Ϫ ϩ 2 1 Ϫ ϩ P , ͑q͒ϭ K , ͑q͒, ͑C146͒ ͑C159͒ ⌫,⌫Ј 2 2 2 ⌫,⌫Ј ⌫ q 1ϪP⌫͑q͒
Ϫ Ϫ 2 1 Ϫ Ϫ 1 1 P , ͑q͒ϭ K , ͑q͒. ͑C147͒ Ϫ ͑ ͒ϭ ͵ ⍀0,Ϫ ͑ ͒ϩ ⌫,⌫Ј 2 ⌫,⌫Ј P⌫ q ⌫ ⌫ 0,0;q ⌫2 2 Ϫ ͑ ͒ Ј , Ј 2 2 q 1 P⌫ q 0 ⌫ 1ϪP⌫͑q͒
S,ϩ Since we have the relations ϫ͓ 1ϩP⌫͑q͒ K ͑q͒ϩ 1ϪP⌫͑q͒ „ … ⌫,⌫Ј „ … S,ϩ ͑ ͒ϭ S,Ϫ ͑ ͒ ͑ ͒ ϫ D,ϩ ͑ ͔͒ ͑ ͒ K⌫,⌫Ј q K⌫,⌫Ј q , C148 K⌫,⌫Ј q . C160
D,ϩ ͑ ͒ϭϪ D,Ϫ ͑ ͒ ͑ ͒ K⌫,⌫Ј q K⌫,⌫Ј q , C149 S,ϩ ϭ The second equation is satisfied since K⌫,⌫Ј(q) (1/ ⑀,⑀Ј 2 ͓ Ϫ ͔2 we can express the four functions K⌫,⌫Ј(q) in terms of q ) 1 P⌫(q) . The three other are compatible, and give S,ϩ D,ϩ K⌫,⌫Ј(q) and K⌫,⌫Ј(q) only as
⑀,⑀Ј 1 S,ϩ ⑀ ⑀Ј D,ϩ Ϫ ͒ K ͑q͒ϭ K ͑q͒ϩ͑Ϫ1͒ ͑Ϫ1͒ K ͑q͒ ϩ 1 2 1 P⌫Ј͑q 1 ⌫,⌫Ј 2 „ ⌫,⌫Ј ⌫,⌫Ј … ͵ ⍀0, ͑ ͒ϭ Ϫ Ϫ ⌫ ⌫ 0,0;q ͑C150͒ , Ј q ⌫ 2 2 ϩ ͒ ⌫2 2 0 Ј q 1 P⌫Ј͑q q Since we have the constraints 1 1 ϫ Ϫ ͑ ͒ ϩ ϩ ϩ ϩ Ϫ 1 P⌫ q , ͑ ͒ϩ , ͑ ͒ϭ ϩ͑ ͒ ͑ ͒ „ … ⌫2 1ϩP⌫͑q͒ P⌫,⌫Ј q P⌫,⌫Ј q P⌫ q , C151 D,ϩ Ϫ ϩ Ϫ Ϫ ϫ ͑ ͒ ͑ ͒ , ͑ ͒ϩ , ͑ ͒ϭ Ϫ͑ ͒ ͑ ͒ K⌫,⌫Ј q , C161 P⌫,⌫Ј q P⌫,⌫Ј q P⌫ q , C152
ϩ,ϩ ͑ ͒ϩ Ϫ,ϩ ͑ ͒ϭ ϩ ͑ ͒ ͑ ͒ P⌫,⌫Ј q P⌫,⌫Ј q P⌫Ј q , C153 2 1ϪP⌫ ͑q͒ 1 ͵ ⍀0,Ϫ ͑ ͒ϭ Ј Ϫ Ϫ ͑ ͒ ϩ,Ϫ Ϫ,Ϫ Ϫ ⌫ ⌫ 0,0;q 1 P⌫ q ͑ ͒ϩ ͑ ͒ϭ ͑ ͒ ͑ ͒ , Ј ⌫ 2 2 ϩ ͒ ⌫2 2 „ … P⌫,⌫Ј q P⌫,⌫Ј q P⌫Ј q , C154 0 Ј q 1 P⌫Ј͑q q where 1 1 Ϫ D,ϩ ͑ ͒ ͑ ͒ K⌫,⌫Ј q . C162 ⌫2 1ϩP⌫͑q͒ Ϫ ͑ ͒ ϩ 1 2 1 P⌫ q P⌫ ͑q͒ϭ Ϫ , ͑C155͒ q ⌫2q2 1ϩP⌫͑q͒ 6. Final result 2 1ϪP⌫͑q͒ Ϫ The Laplace transform of the correlation function can P⌫ ͑q͒ϭ , ͑C156͒ ⌫2q2 1ϩP⌫͑q͒ now be obtained as
066107-39 FISHER, LE DOUSSAL, AND MONTHUS PHYSICAL REVIEW E 64 066107
ϱ APPENDIX D: FINITE SIZE PROPERTIES ͵ Ϫqx͗ ͑ ͒ ͑ ͒͗͘ ͑ Ј͒ ͑ Ј͒͘ dxe S0 t Sx t S0 t Sx t 0 In this appendix we sketch the derivation of the finite size results from the finite size measure for the RFIM. If spins at ϭ ϩ,ϩ ͑ ͒ϩ Ϫ,Ϫ ͑ ͒Ϫ Ϫ,ϩ ͑ ͒Ϫ ϩ,Ϫ ͑ ͒ P⌫,⌫Ј q P⌫,⌫Ј q P⌫,⌫Ј q P⌫,⌫Ј q both ends are fixed to have the same ͑opposite͒ value, there ͑ ͒ ͑C163͒ are an odd even number of bonds in the finite size measure. Assuming for definitness that spins at both extremities are ϭϩ ϭϪ fixed to the values S0 1 and SL 1. Then domains 1 4 1ϪP⌫Ј͑q͒ 4 1 ϩ closest to the boundaries are domain walls of type A which ϭ Ϫ ϩ KD, ͑q͒, q ⌫Ј2 2 ϩ ͑ ͒ ⌫2 Ϫ 2 ͑ ͒ ⌫,⌫Ј as Sinai walkers see reflecting boundary conditions. In this q 1 P⌫Ј q 1 P⌫ q ͓ ͔ ͑ ͒ case, as explained in Ref. 28 , the probability C164 2kϩ2 ⌫ N⌫,L (l1 ,l2 ,...,l2kϩ2) that the system at scale has (2k and thus the final explicit expression is ϩ2) bonds (kϭ0,1,...), with respective lengths (l1 ,...,l2kϩ2), is ϱ ͵ Ϫqx͗ ͑ ͒ ͑ ͒͗͘ ͑ Ј͒ ͑ Ј͒͘ dxe S0 t Sx t S0 t Sx t 2kϩ2͑ ͒ 0 N⌫,L l1 ,l2 ,...,l2kϩ2 ϭ ϩ͑ ͒ Ϫ͑ ͒ ϩ͑ ͒ ϩ͑ ͒ 1 4 ⌫Јͱq E⌫ l1 P⌫ l2 P⌫ l3 ...P⌫ l2kϩ1 ϭ Ϫ 2ͩ ͪ ͑ ͒ tanh C165 ϩ q ⌫Ј2q2 2 2k 2 ϫ Ϫ͑ ͒¯ ␦ͩ Ϫ ͪ ͑ ͒ E⌫ l2kϩ2 l ⌫ L ͚ li , D1 iϭ1 ⌫ͱq ͩ ͪ Ϯ cotanh where P (l) are the bulk length distributions ͓Eq. ͑68͔͒, ¯l ⌫ 4 ⌫ͱq 8 2 ⌫ ϩ 2ͩ ͪ 1ϩ͑ ͒ϩ ϭ¯ϩϩ¯Ϫ Ϯ cotanh ⌫Ј q l ⌫ l ⌫ , and E⌫ (l) the length distribution of boundary ⌫2 2 ⌫2 ͱq bonds ͑see Ref. ͓28͔͒. The normalization is
4 1 ϫ2ϩ͑ ͒ϩ 3ϩ͑ ͒ ͑ ͒ ϱ ⌫Ј q ⌫Ј q , C166 ⌫2 ⌫ͱ ͵ 2kϩ2͑ ͒ϭ ͑ ͒ q ͚ N⌫,L l1 ,l2 ,...,l2kϩ2 1. D2 2ͩ ͪ kϭ0 l ,...l ϩ q sinh 1 2k 2 2 The probability that the system has (2kϩ2) domains at where scale ⌫ is
1ϩ 1 ͑q͒ϭ ͓8ϩ3⌫Ј2q ϩ Ϫ ϩ ⌫Ј 2 3 2 ͑ ⌫͒ϭ ͵ ͑ ͒ ͑ ͒ ͑ ͒ 2⌫Ј q sinh ͑⌫Јͱq͒ IL k; E⌫ l1 P⌫ l2 •••P⌫ l2kϩ1 l1 ,...,l2kϩ2 Ϫ16 cosh͑⌫Јͱq͒ϩ8⌫Јͱq sinh͑⌫Јͱq͒ 2kϩ2 ϫ Ϫ͑ ͒ ␦ͩ Ϫ ͪ ͑ ͒ ͑ ͒ E⌫ l2kϩ2 l⌫ L ͚ li , D3 C167 iϭ1
ϩ͑8ϩ5⌫Ј2q͒cosh͑2⌫Јͱq͒ so that the Laplace transform with respect to the length of the generating function is Ϫ ⌫ ͱ ⌫ ͱ 12 Ј q sinh͑2 Ј q͒] ͑C168͒
ϱ ϱ ϩ Ϫ E⌫ ͑q/2g͒E⌫ ͑q/2g͒ 2 1 ͵ ϪqLͩ k ͑ ⌫͒ͪ ϭ¯ 2ϩ dLe ͚ z IL k; l ⌫ ϩ Ϫ ͑ ͒ϭϪ ͩ Ϫͱ ͑⌫Јͱ ͒ͪ kϭ0 Ϫ ͑ ͒ ͑ ͒ ⌫Ј q q coth q 0 1 zP⌫ q/2g P⌫ q/2g q2 ⌫Ј ϩ Ϫ 2g 1ϪP⌫ ͑q/2g͒P⌫ ͑q/2g͒ 2 ⌫Јͱq ϭ ϩ Ϫ , ϫͩ tanhͩ ͪ Ϫ1 ͪ ͑C169͒ q 1ϪzP ͑q/2g͒P ͑q/2g͒ ⌫Јͱq 2 ⌫ ⌫ ͑D4͒
2 ⑀ 4 1 ϭ 3 Ј͑ ͒ϭϪ ͩ Ϫͱ ͑⌫Јͱ ͒ͪ ͑ ͒ where p q/2g. This leads to the results given in the text. ⌫Ј q q coth q . C170 q2 ⌫Ј The magnetization can also be obtained. In the ‘‘full’’ renormalized landscape, the magnetization is given by M L ͑ ͒ ϭ͚iϭ2kϩ2 Ϫ iϩ1 This leads to the scaling form given in the text in Eqs. 114 iϭ1 ( 1) li . The probability that it has value M and ͑117͒. simply is
066107-40 NONEQUILIBRIUM DYNAMICS OF RANDOM FIELD . . . PHYSICAL REVIEW E 64 066107
kϭϩϱ 2kϩ2 iϭ2kϩ2 ͑ ⌫͒ϭ ͵ ϩ͑ ͒ Ϫ͑ ͒ ϩ͑ ͒ ¯ ␦ͩ Ϫ ͪ ␦ͩ Ϫ ͑Ϫ ͒iϩ1 ͪ ͑ ͒ FL M; ͚ E l1 P l2 P l3 l ⌫ L ͚ li M ͚ 1 li . D6 ϭ ••• ϭ ϭ k 0 l1 ,...,l2kϩ2 i 1 i 1 ϫ ϩ͑ ͒ Ϫ͑ ͒ ͑ ͒ P l2kϪ1 E l2kϩ2 D5 This leads to the result given in the text.
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