CHAOS VOLUME 8, NUMBER 2 JUNE 1998
Entropy balance, time reversibility, and mass transport in dynamical systems Wolfgang Breymann Institute of Physics, University of Basel, Klingelbergstr. 82, CH-4056 Basel, Switzerland and Olsen & Associates, Seefeldstrasse 233, CH-8008 Zu¨rich, Switzerland Tama´sTe´l Institute for Theoretical Physics, Eo¨tvo¨s University, H-1088 Budapest, Puskin u. 5-7, Hungary Ju¨rgen Vollmer Fachbereich Physik, Univ. GH Essen, D-45117 Essen, Germany ͑Received 15 December 1997; accepted for publication 6 April 1998͒ We review recent results concerning entropy balance in low-dimensional dynamical systems modeling mass ͑or charge͒ transport. The key ingredient for understanding entropy balance is the coarse graining of the local phase-space density. It mimics the fact that ever refining phase-space structures caused by chaotic dynamics can only be detected up to a finite resolution. In addition, we derive a new relation for the rate of irreversible entropy production in steady states of dynamical systems: It is proportional to the average growth rate of the local phase-space density. Previous results for the entropy production in steady states of thermostated systems without density gradients and of Hamiltonian systems with density gradients are recovered. As an extension we derive the entropy balance of dissipative systems with density gradients valid at any instant of time, not only in stationary states. We also find a condition for consistency with thermodynamics. A generalized multi-Baker map is used as an illustrative example. © 1998 American Institute of Physics. ͓S1054-1500͑98͒02602-0͔
Trajectories of chaotic dynamical systems closely ap- deterministic thermostats introduced to perform molecular proach the unstable manifold of an invariant set, namely, dynamics simulations of transport in spatially periodic a chaotic saddle for open systems, or a chaotic attractor systems.1,2 Since every ensemble of trajectories with smooth for closed and dissipative systems. Consequently, the initial distributions approaches a chaotic attractor in the Gibbs entropy of smooth initial distributions monotoni- phase space,3 it became clear soon that after an initial tran- cally decreases in time. We introduce here the concept of sient the Gibbs entropy decreases in time. It has a constant coarse-grained phase-space densities and the correspond- negative time derivative which coincides with the sum of all ing entropies, and point out that a local entropy-balance Lyapunov exponents on the attractor.1,4 It was tempting to equation with a non-negative irreversible entropy pro- identify the modulus of this quantity with the irreversible duction can always be derived under weak assumptions. entropy production. A qualitative explanation corroborating The structure of this microscopic balance equation is this view was based on the idea that the system together with similar to what is known from thermodynamics, but the the thermostat forms a larger closed system and, conse- contributing terms differ in general from those in the quently, the thermostat’s entropy should increase with at macroscopic case. We show that the macroscopic and the least the rate of decrease of the Gibbs entropy.2,5,6 microscopic balance equations are compatible only if it is Support of this picture is based on the observation that possible to take the limit of large systems while keeping the transport coefficients fixed. In addition, in the case of the negative sum of the Lyapunov exponents is the average a general driving due to an external field, a reversible phase-space contraction rate, which is expected to be non- dissipation mechanism is required, corresponding to a negative in physically relevant cases. It is then natural to say proper thermostating of the system. We also point out that the average phase-space contraction rate should be con- 4,6–13 that consistency with thermodynamic results can only be sidered as the irreversible entropy production. Rigorous achieved when considering the time evolution of the full mathematical statements have been proved concerning prop- phase-space density. Even though an entropy-balance erties of this entropy production.10–12 On the other hand, equation can also be established for the process projected there is a conceptual difficulty in these results because en- on the transport direction, this equation can no longer be tropy production is referred to without specifying the under- compatible with thermodynamics, except for the case of lying concept of entropy. unbiased mass transport „pure diffusion…. The cornerstone of nonequilibrium thermodynamics is an entropy which is ͑i͒ constant in a steady state ͑in contrast to the Gibbs entropy of thermostated systems͒, and ͑ii͒ has a I. INTRODUCTION time derivative which contains two contributions: the en- The concept of irreversible entropy production in dy- tropy flux deS/dt and the irreversible entropy production 14 namical systems first appeared in the context of dissipative diS/dtу0
1054-1500/98/8(2)/396/13/$15.00 396 © 1998 American Institute of Physics
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dS deS diS ϭ ϩ . ͑1͒ dt dt dt In any finite volume, the entropy density s, i.e., the entropy per unit volume, fulfills the balance equation
ds FIG. 1. General scheme of a map modeling mass transport. The mapping is ϭ⌽ϩ͑irr͒, ͑2͒ dt defined on a domain of N identical cells of phase-space volume ⌫ϭab where a is the width in the direction of an applied bias. Different boundary where ds/dt is the time derivative of entropy density, ⌽ de- conditions can be imposed by suitably defined action of the map on two notes the entropy flux into that volume, and the rate of irre- additional cells 0 and Nϩ1. versible entropy production (irr)у0 can be viewed as the source strength of entropy. In a steady state the entropy flux modeling this process is worked out for arbitrary nonstation- balances the irreversible entropy production, i.e., ⌽ (irr) ary states, and a general expression for the irreversible en- ϭϪ , so that ds/dtϭ0. tropy production is derived. It turns out that the entropy pro- The relation between the Gibbs entropy, which is con- duction stems from both phase-space contraction and tinually decreasing in a steady state, and the thermodynamic mixing. In other words, whenever mixing of phase-space entropy needs clarification. It is not a priori obvious why the volumes with different mass densities plays a role in the temporal change of the Gibbs entropy should be fully attrib- transport process, the irreversible entropy production does uted to the irreversible entropy production and not ͑at least not coincide with the average phase-space contraction rate. partially͒ to an entropy flux into an attached heat bath or We also show that the inclusion of dissipation is unavoidable particle reservoir. in order to be consistent with classical thermodynamics in To find a consistent description, the use of a coarse- this general case. The only choice consistent with this re- grained entropy has recently been proposed for dynamical 15–22 quirement is a time-reversible dissipation mechanism that systems. This concept emphasizes the importance of fi- models proper thermostating of the system. nite resolution in any observation, which is mimicked by In Sec. II we derive the entropy-balance equation for a coarse graining the density over phase-space cells of a fixed general dynamical system. Different levels of description size.23 The coarse-grained entropy computed with respect to 24 will be considered. We will find balance equations at all the coarse-grained density ͑or the ⑀-entropy ͒ differs quali- levels but an agreement with thermodynamics can be tatively from the Gibbs entropy. After a long time, the achieved only if the phase-space density is used to define the coarse-grained density approaches a stationary distribution entropy, and if the system size is large. In Sec. III we intro- localized around the unstable manifold of the chaotic set. duce the biased multi-Baker model. The microscopic balance This limiting distribution is a finite-precision approximation for this model is derived in Sec. IV, and its macroscopic of the natural measure on this manifold, which typically has limit is presented in Sec. V. In the conclusion we discuss the a fractal structure. Correspondingly, the coarse-grained en- role of coarse graining for the concept of entropy ͑Sec. VI͒. tropy eventually becomes independent of time. Then the ir- The paper is augmented by three appendices. For reference, reversible entropy production can be defined as the loss of in Appendix A relevant thermodynamic relations are derived information on the microscopic state of the system due to the in a form not relying on temperature. The discussion of the coarse-grained description. It is reflected in the growth of the entropy balance for the process projected on the transport difference between the coarse-grained and the Gibbs 15 direction is relegated to Appendix B. A generalization of the entropy. irreversible entropy-production formula to maps more gen- In open systems the concept of phase-space contraction eral than the multi-Baker is given as Appendix C. has to be generalized since there is an effective phase-space contraction even in Hamiltonian cases because trajectories can escape from the relevant part of the phase space. This II. ENTROPY BALANCE FOR MAPS MODELING MASS TRANSPORT leads to the observation that the escape rate has to be added to the previous result.10,15 We consider a system whose phase space is part of the To understand boundary driven problems, it turned out (x,p) plane and consists of N identical cells which are to be essential to use the balance equations in addition to the aligned along the direction of an external field. The cells are coarse-grained entropies. The first attempt to derive entropy coupled such that a trajectory can proceed from one cell to balance for a deterministic dynamical system is due to its neighbors. Since there are no temperature gradients in the Gaspard16 ͑in a noisy system to Nicolis and Daems17͒.To system, all cells can be taken equivalent as far as their ge- describe a one-dimensional diffusive current induced by flux ometry and dynamics is concerned. They are of linear size a boundary conditions, Gaspard considered a purely Hamil- with phase-space volume ⌫ϭab ͑Fig. 1͒. The time evolution tonian model ͑a multi-Baker map͒. The rate of entropy pro- is given by a mapping acting at integer multiples of the time duction he derived was consistent with the thermodynamic unit . Since we are interested in ensembles of trajectories, expression in the large-system limit. we consider the dynamics of phase-space densities. In order In the present paper we consider one-dimensional mass to model the effect of finite accuracy of observation, we will transport that is driven not only by density gradients but also consider coarse-grained densities where only the average by an external bias ͑a field͒. The entropy balance for maps phase-space density in the bins of a preselected fixed grid is
Downloaded 23 Feb 2010 to 10.0.105.87. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/chaos/copyright.jsp 398 Chaos, Vol. 8, No. 2, 1998 Breymann, Te´l, and Vollmer specified. For simplicity, in the present paper this grid is In meaningful models of transport, the entropy change taken to coincide with the cells depicted in Fig. 1.25 Thus the should always be related to particle motion contributing to mapping describing the time evolution of the coarse-grained transport. This means that for localized trajectories ͑as long densities will be considered as always acting on densities as they stay inside a given cell͒ phase-space volume should which are uniform within each cell. Such a density is also the be preserved. In view of this requirement we let the dynam- initial condition for the time evolution of the phase-space ics inside a cell be phase-space preserving ͑Hamiltonian͒, density. i.e., localized trajectories cannot contribute to the temporal Let %m(x,p) denote the phase-space density in cell m at change of a cell’s Gibbs entropy. Consequently, the change (G) some time t. For later reference we mention that the mass of ⌬Sm is entirely due to the entropy flux into the cell. The density m(x) in cell m is obtained by integrating the phase- entropy fluxes in the coarse grained and in the microscopic space density over the momentum coordinate description do not differ,
b ͑G͒ ⌬eSmϭ⌬Sm . ͑9͒ m͑x͒ϭ %m͑x,p͒dp. ͑3͒ ͵0 This equation states that the coarse-grained entropy flux is The difference between these quantities is also stressed by entirely due to a change of the Gibbs entropy. labeling them with different letters, and %. With this result Eq. ͑8͒ can also be written as (G) The Gibbs entropy Sm of cell m is defined as ͑G͒Ј ͑G͒ ͑G͒Ј ͑G͒ ⌬SmϭSm ϪSm ϩ͓͑SmЈϪSm ͒Ϫ͑SmϪSm ͔͒. ͑10͒ % ͑G͒ m͑x,p͒ In view of ͑7͒ and ͑9͒, Sm ϭϪ %m͑x,p͒ln à dx dp, ͑4͒ ͵cell m % ⌬Smϭ⌬eSmϩ⌬iSm , ͑11͒ where the Boltzmann constant kB has been suppressed, and where %à is a constant reference density which fixes the origin of ͑G͒Ј ͑G͒ the entropy scale. It does not depend on spatial coordinates, ⌬iSmϵ͑SmЈ ϪSm ͒Ϫ͑SmϪSm ͒. ͑12͒ time or the boundary conditions, and must not be confused Here, the expressions for ⌬S , ⌬ S , and ⌬ S take the with an equilibrium or steady-state density of the physical m e m i m form of discrete time derivatives, so that Eq. ͑11͒ has the system under consideration. structure of an entropy-balance equation ͓cf. ͑1͔͒. Conse- The coarse-grained or cell density % is the average den- m quently, ⌬ S / is a natural extension of the irreversible sity i m entropy production to dynamical systems. Being the change 1 in time of the difference between the coarse-grained and the %mϭ %m͑x,p͒dx dp ͑5͒ Gibbs entropy, it exactly measures the amount of informa- ⌫ ͵cell m tion lost per time unit because of coarse graining.15 in cell m. The corresponding entropy We remark that for the present choice of initial densities, which are uniform over full cells, the initial Gibbs and %m (G) S ϭϪ⌫% ln ͑6͒ coarse-grained entropies coincide: SmϭSm .In͑10͒–͑12͒ m m %à (G) the term SmϪSm was nevertheless written out explicitly, to is in general different from the Gibbs entropy. It will be make clear that ⌬iSm is indeed a discrete time derivative. (G)Ј called the coarse-grained entropy ͑coarse grained on the full Note that the Gibb’s entropy Sm contains more infor- cell͒. We argue below that it is a natural candidate for a mation about the phase-space structure than the coarse- generalization of the thermodynamic entropy to dynamical grained entropy SmЈ and, therefore, must be less than or equal systems. When the coarse-grained entropies are defined on a to SmЈ . This ensures that the irreversible entropy production grid with bins smaller than the full cell, the results depend on is always non-negative the form of the partitioning. In the macroscopic limit, how- ⌬iSmу0. ͑13͒ ever ͑cf. Sec. V͒, the chain consists of many cells (Nӷ1), and coarse graining over each cell corresponds to a rather The irreversible entropy production ⌬iSm vanishes only fine resolution. In this limit the time derivatives and the con- when, under the dynamics, the phase-space densities are no tributions to the entropy balance become independent of the longer developing fine structures. The absence of temporal details of the partitioning inside the cell, as corroborated in changes in the structure of the phase-space density can there- Ref. 22. fore be viewed as a signature of thermal equilibrium. To derive an entropy-balance equation, we compare In a nonequilibrium steady state we have SmЈ ϭSm , and quantities computed at two successive time steps, tϭt0 the rate of irreversible entropy production ϭn and tЈϭt0ϩϭ(nϩ1). The temporal variation of the ͑G͒ ⌬iSm ⌬Sm Gibbs entropy in cell m during one time step is ϭϪ ͑14͒ ͯ ss ⌬S͑G͒ϭS͑G͒ЈϪS͑G͒ , ͑7͒ m m m coincides with the negative time derivative of the Gibbs en- and, similarly, for the coarse-grained entropy we write tropy. Thus the rule that, after a long time, the irreversible entropy production is the negative of the Gibbs entropy’s ⌬SmϵSmЈ ϪSm . ͑8͒ time derivative appears as a natural consequence of the fun-
Downloaded 23 Feb 2010 to 10.0.105.87. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/chaos/copyright.jsp Chaos, Vol. 8, No. 2, 1998 Breymann, Te´l, and Vollmer 399 damental fact that any observation has finite precision. By that, the present approach also provides a foundation, within the framework of entropy balance, for the qualitative argu- ments mentioned in the Introduction. Although the structure of the microscopic balance equa- tion ͑11͒ is identical with that of the macroscopic equation ͑1͒, each term is yet a function of the microscopic parameters of the dynamics, e.g., of the cell size a, the time unit , and FIG. 2. Graphical illustration of the action of the multi-Baker map. Three typically also of additional parameters symbolically denoted vertical columns are squeezed and stretched to obtain horizontal strips of by . Therefore, the terms on the right hand side of Eq. ͑11͒ width a. Note the corresponding free spaces ͑white strips labeled R and S͒ ˜ ˜ will not coincide with the corresponding terms of the mac- where strips from the neighboring boxes are mapped to (rϩlϭrϩ l ). roscopic equation ͑1͒. In the macroscopic limit the microscopic parameters a, j2 and tend to zero in such a way that the transport coeffi- ͑irr͒ϭ , ͑19͒ cients stay finite. In that limit we expect that a finite entropy D density is obtained as the limit of Sm /a. Furthermore, we an expression quadratic in the total current density ͑16͒. also expect that the rates of entropy changes obtained from At this point general statements about the dependence Eqs. ͑9͒, ͑8͒, and ͑12͒ by dividing these equations by a are of the flux and entropy production appearing in ͑15͒ cannot also well-defined and finite. Thus ⌬Sm /(a) ds/dt(), be made. For the particular example of the next section there ⌬ S /(a) ⌽(), and ⌬ S /(a) (irr)(),→ so that the e m → i m → is one and only one set of parameters which corresponds to canonical form of entropy balance ͑2͒ is recovered the thermodynamic forms, namely, the one describing a ds time-reversible dissipation mechanism which models a ͑͒ϭ⌽͑͒ϩ͑irr͒͑͒. ͑15͒ proper thermostating of the system. dt We show in Appendix B that an entropy balance similar Note, however, that in general all terms still depend on the to ͑15͒ also holds for the transport process projected on the system parameter͑s͒. Only for special values of , these transport direction. However, this ‘‘projected’’ entropy bal- terms can coincide with the expressions known from irre- ance is in general inconsistent with thermodynamics. versible thermodynamics. To facilitate comparison with thermodynamics, the tem- III. THE MULTI-BAKER MODEL perature must be eliminated from the classical expressions for ds/dt, ⌽ and (irr). After all, in low-dimensional dy- Baker maps26–28 are archetype models for strongly cha- namical systems the concept of temperature is ill-defined. otic systems,26 and multi-Baker maps29–31,21,22 play an essen- Appendix A shows how this can be achieved. In the next tial role in understanding the connection between chaotic paragraphs we present temperature-independent expressions microscopic dynamics and macroscopic irreversibility, for the entropy flux ⌽ and the irreversible entropy produc- which has also been a subject investigated by means of sev- tion (irr). eral other approaches.32–39 According to irreversible thermodynamics, mass trans- The phase-space of the multi-Baker model to be treated port is characterized by the drift coefficient v and the diffu- here consists of a chain of identical cells of linear size a and sion coefficient D, and by two density distributions, namely, area ⌫ϭab ͑cf. Fig. 1͒. To define the dynamics every cell is the mass density (x) and the current density j(x). The cur- divided into three vertical columns ͑see Fig. 2͒: the rightmost x, ͑leftmost͒ column of width ra(la) of each cell is mappedץ rent density j is the sum of the diffusion current, ϪD and the drift current, v onto a strip of width a and height ˜rb(˜l b) in the square to the right ͑left͒. These columns are responsible for transport in . ͑16͒ ץjϭvϪD x one time step . The middle column of width sa stays inside The conservation of particles leads to a continuity equa- the cell, thus modeling the motion that does not contribute to tion which is a kind of transport ͑Fokker–Planck͒ equation transport during a single iteration. This column is mapped onto a strip of width a and height ˜sb. According to the ϭϪ jϭϪ ϩD 2 . 17 x ͑ ͒ argument of the previous section, the internal dynamicsץ xץx vץ tץ The entropy flux, which is the sum of a convective term should be area preserving, i.e., sϭ˜s. Motivated by other and of the heat flow, can be written as models of transport, global phase-space conservation is as- sumed, which implies the sum rules sϩlϩrϭ1ϭsϩ˜l ϩ˜r. v In order to characterize the local dissipation mechanism, we x͓ j͑cϩln %͔͒Ϫ j ͑18͒ץ⌽ϭ D introduce the parameter with c as a constant. The second term corresponds to the ˜rϪ˜l entropy flow related to the heat current into the thermostat ϵ , ͑20͒ rϪl ͑Joule’s heat, cf. Appendix A͒. The expression of the irreversible entropy production which measures the deviation from uniform phase-space formula is obtained as contraction on the strips R and L. For ϭ1 the Jacobians of
Downloaded 23 Feb 2010 to 10.0.105.87. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/chaos/copyright.jsp 400 Chaos, Vol. 8, No. 2, 1998 Breymann, Te´l, and Vollmer the strips R and L are identical, while for ϭϪ1 they are IV. ENTROPY BALANCE FOR THE MULTI-BAKER reciprocal to each other. Besides the microscopic units ͑, a, MODEL and b͒, the thermodynamic and transport properties of the In this section we calculate the different contributions model are characterized by three independent parameters: the 8 , 9 , 12 to the entropy balance 11 for the general transition probabilities to neighboring cells r and l, and the ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ multi-Baker map depicted in Fig. 2. As discussed above, we dissipation parameter . choose the initial density % (x,p) to be uniform in every The phase-space contraction rates ϭ(1/ )ln(l/˜l) and m L cell. Then the coarse-grained entropy of cell m at time t and ˜ Rϭ(1/)ln(r/r) on the respective strips L and R depend on tЈϭtϩ takes the form the parameter . According to their value, different transport dynamics can be modeled: %m S ϭϪ⌫% ln ϭS͑G͒ ͑23a͒ ͑i͒ Phase-space preserving ͑Hamiltonian͒ dynamics: m m %Ã m
LϭRϭ0 ͑21a͒ and
͑ϭ1 rϭ˜r, lϭ˜l ͒. %mЈ ⇒ SmЈ ϭϪ⌫%mЈ ln à , ͑23b͒ ͑ii͒ Dissipative dynamics with time-reversal symmetry: % 1 r respectively. In order to compute the Gibbs entropy at time RϭϪLϭ ln ͑21b͒ tЈ, we note that the densities on the three horizontal strips l depicted in Fig. 2 differ after having applied the map. On the ͑ϭϪ1 rϭ˜l, lϭ˜r͒. This choice mimics the effect of a strips R, S and L the density assumes the new values %RЈ ⇒ 21 ˜ ˜ thermostat. Indeed, in order to model the decelerating ef- ϭ%mϪ1r/r, %SЈϭ%m , and %LЈϭ%mϩ1l/ l , respectively. ͑The fect of the heat bath on particles accelerated by the external initial densities on these strips were %Rϭ%Sϭ%Lϭ%m .͒ (G)Ј field, a map modeling driven thermostated systems has to be Consequently, the Gibbs entropy Sm is area contracting if the trajectory moves in the direction of the %SЈ %RЈ %LЈ bias. Similarly, the map should be expanding for trajectories S͑G͒ЈϵϪ⌫ s%Ј ln ϩ˜r%Ј ln ϩ˜l %Ј ln ͑24a͒ moving against the bias, on which the heat bath has an ac- m ͫ S %à R %à L %Ãͬ celerating effect to compensate for the slowing down by the external field ͑cf. Refs. 2, 5, 9͒. In addition, the overall dis- %m %mϪ1 r ϭϪ⌫ ͑1ϪrϪl͒% ln ϩr% ln m à mϪ1 à sipation should vanish for closed trajectories which do not ͫ % ͩ % ˜r ͪ contribute to transport. Since this is also true for period-2 orbits, the overall contraction rates RϩL for making a %mϩ1 l ϩl% ln ͑24b͒ step to the right and a step to the left should add up to zero. mϩ1 à 40 ͩ % ˜l ͪͬ It is worth noting that the map is then time-reversible. ͑iii͒ General dissipative dynamics: %m %mϪ1r ϭϪ⌫ %Ј ln ϩr% ln ÞϪ Þ0 ͑21c͒ m à mϪ1 L R ͫ % ͩ % ˜r ͪ m (ÞϮ1). This case has similarities with the thermostating algorithm but does not fulfill the time-reversal symmetry.40 %mϩ1l ϩl%mϩ1 ln , ͑24c͒ We consider it as a model for improper thermostatting. ͩ % ˜l ͪͬ m Finally, we stress that also the coarse-grained densities where Eq. ͑22͒ has been used to arrive at the latter equality. depend, in general, on time. Due to the conservation of prob- (G)Ј ability, we have the following expression for the cell density It is clear from these relations that the Gibbs entropy Sm %Ј after one time step: coincides with the coarse-grained one, SmЈ , only if the map m does not generate any local inhomogeneities of the phase- %mЈ ϭ͑1ϪrϪl͒%mϩr%mϪ1ϩl%mϩ1 . ͑22͒ space density. Inserting Eqs. ͑23b͒ and ͑24͒ into the expressions for the This is a discrete-time master equation42 governing the dy- change ⌬Sm of the coarse-grained entropy ͓Eq. ͑8͔͒, the namics of the cell densities. In contrast to the full dynamics, change ⌬ S due to the flux ͓Eq. ͑9͔͒, and the irreversible it does not depend on the dissipation parameter . Because e m entropy change ⌬iSm ͓Eq. ͑12͔͒, one obtains the average mass density m of cell m is mϭb%m , Eq. ͑22͒ also holds for the average mass density. It provides a closed %mЈ %m ⌬S ϭϪ⌫ %Ј ln Ϫ% ln , ͑25a͒ set of equations for the time evolution of the coarse-grained m ͫ m %à m %à ͬ densities, which can rigorously be derived from the micro- scopic dynamics ͑cf. Refs. 31, 29͒. By this, modeling trans- %m %mϪ1 r ⌬ S ϭϪ⌫ ͑%ЈϪ% ͒ln ϩr% ln port by means of multi-Baker maps allows us to work out the e m m m à mϪ1 ͫ % ͩ %m ˜r ͪ relation between self-contained descriptions of transport be- havior on the microscopic level ͑phase-space densities and %mϩ1 l Gibbs entropy͒ and a macroscopic level ͑coarse-grained den- ϩl%mϩ1 ln , ͑25b͒ sities and coarse-grained entropy͒. ͩ %m ˜l ͪͬ
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is the mixing entropy based on neighboring cell densities, %mϪ1 r %mϩ1 l ⌬iSmϭ⌫ r%mϪ1 ln ϩl%mϩ1 ln and the contribution ͫ ͩ %m ˜r ͪ ͩ %m ˜l ͪ ͑con͒ ⌬Sm ϵ⌫͓%mϪ1ϩϩ%mϩ1Ϫ͔, ͑31͒ %Ј m accounts for an average contraction of phase space. The Ϫ%mЈ ln . ͑25c͒ %mͬ quantities
The sum of ⌬eSm and ⌬iSm is indeed the total change ⌬Sm 1 r 1 l of entropy. Consequently, the balance equation ͑11͒ is ful- ϩϭ r ln and Ϫϭ l ln ͑32͒ filled. In Appendix C we consider a map much more general ˜r ˜l than the multi-Baker model and show that, under very weak are the conditional phase-space contraction rates for the mo- assumptions, an expression structurally very similar to 25c ͑ ͒ tion in the positive and negative direction, respectively. ͑By follows for the irreversible change of entropy. conditional we mean that for a constant distribution of the Let us discuss expression ͑25c͒ for ⌬iSm in some more coarse-grained density these rates would be the average detail. First, we consider a steady state. Then the discrete phase-space contraction rates for the two directions.͒ The version of the rate of irreversible entropy production is third term of ͑29͒,
⌬iSm %mϪ1 %RЈ %mϩ1 %LЈ ϭ⌫% r ln ϩl ln . ͑26͒ %mЈ ͯ mͫ % % % % ͬ ⌬S͑evo͒ϭϪ⌫%Ј ln , ͑33͒ m R m L m m ss %m Notice that %RЈ/%R is the ratio of the local densities at time tЈ vanishes in a steady state. It accounts for irreversible entropy and t on strip R, and %LЈ/%L the corresponding quantity on production due to the temporal change of the cell densities, strip L. in addition to the contributions due to mixing and phase- The probability to be mapped onto these strips by time tЈ space contraction which are found in both stationary and is r%mϪ1 /%m ͑or l%mϩ1 /%m͒. Therefore, we can say that the nonstationary situations. In particular, the splitting of Eq. irreversible entropy production is related to the average of ͑29͒ shows that in a steady state with constant cell densities, the growth rates only the phase-space contraction contributes to the irrevers- 1 %Ј͑x,p͒ ible entropy production. In the other limiting case, namely, ͑x,p͒ϵ ln ͑27͒ for a steady state where the dynamics is everywhere Hamil- % %͑x,p͒ tonian, the mixing entropy provides the entire entropy pro- of the local ͑non coarse-grained͒ phase-space densities. In duction ͑see Ref. 16͒. other words, It is worth briefly mentioning the case of a homogenous temporal decay of the density towards an empty state, which ⌬iSm ⌬iSm ϭ ϭ͗ ͘ . ͑28͒ corresponds to the decay with the most stable eigenstate of a ⌫% ͯ a ͯ % ss m ss m ss system displaying transient chaos ͑escape rate formalism, cf. Refs. 5, 32, 31, 21͒. One can then write %Ј ϭexp Ϫ(t Thus the irreversible entropy production per particle ͑degree m ͓ 0 ˜ of freedom͒ is the average growth rate of the local phase- ϩ)͔%m , where is the decay rate ͑escape rate from a ˜ space density, where the average has to be taken with respect saddle͒, and %m is the conditional cell density which does to the steady-state distribution. In the special case when the not depend on time. An analogous form holds for the phase- ˜ coarse-grained density is constant along the chain ͑%mϪ1 space density %mЈ (x,p)ϭexp͓Ϫ(t0ϩ)͔%m(x,p), where ϭ% for every m͒, the local density growth rates reduce to ˜ m %m(x,p) is the so-called conditional phase-space density. In the phase-space contraction rates (x,p) and we recover the ˜ contrast to its cell average, %m(x,p) is time-dependent, ex- relation (⌬iSm /(am))ssϭ͗͘ss valid for thermostated sys- 1,4,6–13 pressing the approach towards the chaotic saddle’s unstable tems with periodic boundary conditions. We believe manifold. From ͑25c͒ we obtain for such a decaying state ͑28͒ to be a proper generalization that holds in every steady ͑ds͒ state of a dynamical systems where mixing ͑i.e., intertwining of different neighboring average cell densities͒ plays a role. ⌬iSm ϭϩ͗˜͘ , ͑34͒ To underline this observation, it is useful to separate the a exp͑Ϫ͒ͯ % ds m contributions stemming from mixing and phase-space con- ds (evo) traction. To this end we split the full entropy change ͑25c͒ where stems from ⌬Sm , and into a term with only % dependence under the logarithms, a ˜ ˜ ˜ ˜ term which accounts for the change of phase-space volumes, e %mϪ1 %RЈ %mϩ1 %LЈ ͗%˜ ͘dsϭ r ln ϩl ln ͑35͒ and a time-dependent rest: ͫ %˜ %˜ %˜ %˜ ͬ m R m L ͑mix͒ ͑con͒ ͑evo͒ ⌬iSmϭ⌬Sm ϩ⌬Sm ϩ⌬Sm . ͑29͒ is the average growth rate of the conditional phase-space Here, density evaluated under the condition that particles have not ˜ ˜ yet escaped. Note that %˜ ϭ(1/)ln(%Ј/%) appearing here is %mϪ1 %mϩ1 the direct generalization of ͓cf. Eq. ͑27͔͒ for the condi- ⌬S͑mix͒ϵ⌫ r% ln ϩl% ln ͑30͒ % m ͫ mϪ1 % mϩ1 % ͬ ˜ m m tional density %m(x,p).
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Equation ͑34͒ is an extension of the results of Ref. 15, on how the transition probabilities r and l behave. In order to which are recovered for two special cases: a homogeneous find a sufficient condition for the existence of this limit, we cell density distribution, %mϭ%mϩ1 , or averaging over the go back to the microscopic model. The cell-to-cell dynamics whole system. If the deviation from a homogeneous cell den- of the multi-Baker model is equivalent to a random walk sity distribution is important ͑or averaging takes place over, with fixed step length a.31,42,43 Such random walks are char- e.g., individual cells͒, there is, according to ͑34͒, ͑35͒, also a acterized by well defined drift (v) and diffusion (D) coeffi- contribution due to the mixing entropy, just as in a steady cients, which can be expressed in terms of r and l as (r state. Ϫl)aϭv and (rϩl)a2ϭ2D, respectively.44 Thus, D av D av rϭ 1ϩ , lϭ 1Ϫ . ͑39͒ V. THE MACROSCOPIC LIMIT a2 ͩ 2Dͪ a2 ͩ 2Dͪ As a first step towards a meaningful macroscopic limit, If v and D are fixed, these relations determine the scaling we assume that the system consists of a large number N behavior of the transition probabilities r and l as a function ӷ1 of cells. This implies that the total length LϵNa is of the length unit a and the time unit . much larger than the cell size a. We are interested in the Via the conditional contraction rates Ϯ , Eqs. ͑38a͒ and large-system limit corresponding to aӶL, i.e., to a 0at ͑38b͒ contain the parameters ˜r and ˜l . It is convenient to fixed length L. → express ˜r and ˜l in a similar way as r and l, which, however, The spatial variation of the average phase-space density now also contain the dissipation parameter %m , i.e., of the cell density, follows the one of the mass density in the large system limit (b% ϭ ). Thus the D av D av m m m ˜rϭ 1ϩ , ˜l ϭ 1Ϫ . ͑40͒ cell densities for the neighbors of cell m ͓i.e., at positions a2 ͩ 2Dͪ a2 ͩ 2Dͪ xϩ␦xϭ(mϮ1)a͔ can be expressed through spatial deriva- Since is also a parameter which is relevant for the tives of the mass density distribution (x)atxϭma.Upto second order in a, macroscopic behavior we define the macroscopic limit as 2 a 0, 0, with v,D, fixed ͑41͒ a 2 → → ͑x͒. ͑36͒ 2 2 ץ ͑x͒ϩ ץb% ϭ͑x͒Ϯa mϮ1 x 2 x ͑or aӶL,D/v; ӶL /D,D/v ͒. It is in this limit that the dynamics ͑37͒ of the cell ͑or mass͒ densities reduces to the In the same spirit, the temporal variation of the cell density macroscopic mass transport equation ͑17͒. However, this re- reads quirement only ensures that the transport process can be de- a2 scribed by a one-dimensional Fokker–Planck equation. It 2. ͑37͒ץϩ ͑rϩl͒ ץb͑%Ј Ϫ% ͒ϭϪa͑rϪl͒ m m x 2 x does not yet imply that the model also has a meaningful thermodynamics. Substituting these relations for the phase-space densities In the macroscopic limit one immediately finds in ͑25c͒ yields for the irreversible entropy production in the 2 2 large system limit v ͑1Ϫ͒ v͑1Ϫ͒ ϩϩϪϭ , ϩϪϪϭ . ͑42͒ 2 4D a ⌬iSm a ץ͒ 2 Ϫa͑ Ϫץ ϭ͑ ϩ ͒ ϩ a ϩ Ϫ ͩ 2 x ͪ ϩ Ϫ x Inserting these relations, as well as ͑39͒ and ͑40͒, into Eq. ͑38a͒, we obtain the asymptotic expression a2 2 x͒ 2ץ͑ 2 xץ ϩ ͓͑rϩl͒Ϫ͑rϪl͒ ͔ . ͑38a͒ v͑1Ϫ͒ 2 ͑irr͒͑͒ϭ ϪD . ͑43͒ D ͩ 2 ͪ The first two terms are consequences of the phase-space con- traction, while the third one arises from ⌬S(mix) and ⌬S(evo). In the same way we find for the entropy flux Eq. ͑38b͒ A similar rearrangement of ͑25b͒ yields for the entropy flux v2͑1Ϫ͒2 Ϫ , ͑44͒ ץj 1ϩln Ϫv ץ⌽͑͒ϭ ⌬ S a2 xͫ ͩ à ͪͬ x 4D e m 2 xץ ϭϪ͑ϩϩϪ͒ ϩ a ͩ 2 ͪ where the particle current density j is given by ͑16͒. A com- parison with the thermodynamic entropy balance also shows rϪl à à x that the limit of Ϫb%m ln(%m /% )ϭϪm ln(m / ) goes overץ ϩa ϩϪϪϪ ͫ ͬ into the macroscopic expression 2 rϪl a rϩl ͑x͒ x 1ϩln à , s͑x͒ϭϪ͑x͒ln ͑45͒ץ x a Ϫץϩ ͫͩ 2 ͪͩ ͪͬ à ͑38b͒ of the entropy density given in terms of the mass density. à à à -xj.Weemץ(( %/%)where ϵb% denotes the reference mass density. Thus for every we find ds/dtϭ(1ϩln The fact that these equations are expressed solely by phasize again that the balance equation and ͑43͒, ͑44͒ are spatial derivatives of the mass density distribution does not valid at every instant of time, not only in steady states. yet ensure the existence of a well defined macroscopic limit It is remarkable that the asymptotic macroscopic expres- for (irr) and ⌽. We see that the limit 0 strongly depends sions for (irr)() and ⌽͑͒ are finite for arbitrary values of →
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However, there exists only one whether or not the system is close to thermal equilibrium. value for which the thermodynamic results ͑18͒, ͑19͒ are Therefore, we expect this form to hold beyond linear- recovered, namely ϭϪ1. ͓The constant c of Eq. ͑18͒ cor- response theory. responds to 1Ϫln à and can be set zero by the choice of ͑vi͒ It was shown that besides for steady states of peri- à xϭ0, the rate of irreversible entropyץ ϭe.͔ We thus conclude that ͑a͒ although transport caused odic systems where by an external field ͑bias͒ is considered to be a typical many- production (irr) contains contributions not only from phase- body phenomenon, it can faithfully be modeled by discrete- space contraction but also due to spatial variations of the time dynamical systems with few degrees of freedom. The density. To our knowledge, these contributions can only be heat flow accompanying transport ͑Joule’s heat͒ is accounted derived from dynamical system theory via coarse graining. for by a reversible dissipation mechanism. ͑b͒ The thermo- stating condition is fulfilled in one point of the parameter space only. VI. DISCUSSION Several comments are in order. We conclude this paper with comments on the role of ͑i͒ For the derivation of Eqs. ͑43͒ and ͑44͒ we did not coarse graining when defining entropies. Dividing the phase specify boundary conditions, i.e., the macroscopic results space into elementary cells of area ⌬p⌬qϵ⌬⌫ is essential hold irrespective of their choice. In particular, ͑43͒ with ( for the foundation of equilibrium statistical mechanics.43 It is ϭϪ1) implies that even for periodic boundary conditions well-known that the equilibrium entropy is defined only up there is a non-negligible mixing entropy contribution ͑the to an additive constant whose value depends on the partition ,x͒ before the steady state is reached. size ⌬⌫. By the uncertainty principle of quantum mechanicsץ term proportional to It is only in the long time limit of such periodic cases that ⌬⌫ is bounded from below by Planck’s constant h. How- (irr) 2 / equals v /D and coincides with the negative sum of ever, the value of ⌬⌫ does not affect thermodynamic observ- the average Lyapunov exponents. ables. Indeed, the entropy itself is not an observable, and its ͑ii͒ The multi-Baker model is an example for which the derivatives, the observables, are not affected by the additive laws of thermodynamics can be derived without using the constant depending on ⌬⌫. concept of temperature and heat. Instead, irreversible entropy The same reasoning applies to coarse-grained entropies production was seen to arise from the mixing of phase-space of dynamical systems. The entropies themselves depend on regions due to a strongly chaotic dynamics. This mechanisms the way of coarse graining, but their temporal derivatives, is by no means restricted to low-dimensional systems. We which appear in the balance equation, are much less sensi- expect it to be the dynamical reason for irreversible entropy tive. In the macroscopic limit the dependence on the coarse production and for the possibility to find an entropy balance graining disappears completely.16,22 in systems with many degrees of freedom, too. In this ap- In the bulk of the paper, we have not discussed the ap- proach the concepts of heat and temperature appear as con- proach to thermal equilibrium in a closed system, i.e., in a venient tools to characterize certain macroscopically relevant Hamiltonian system with a closed phase space. Owing to effects of the mixing process. Liouville’s theorem, the Gibbs entropy does not depend on ͑iii͒ Entropy-balance equations have been established for time in that case, its value is preserved. Also in the context arbitrary values of the external control parameter .We of closed systems, one way to find an increase of entropy would like to point out that although a macroscopic limit when starting from nonequilibrium initial conditions is to exists in all these cases, the expressions obtained for the introduce a coarse-grained entropy. It was already Gibbs fluxes and for the irreversible entropy production do, in gen- who showed that the coarse-grained density approaches eral, not fulfill the thermodynamic relations between ͑ther- some finite limit in the long time limit.45 He also claimed modynamic͒ forces, induced currents and entropy produc- that the coarse-grained entropy based on this density can tion. Only for parameter values corresponding to a properly increase in time ͑for polemic discussions, see also Refs. 47– thermostated system ͑including time reversibility͒, the ther- 49͒. modynamic relations are recovered. This emphasizes the ne- It is interesting to compare the time evolution of the cessity of both the existence of a meaningful macroscopic difference ⌬iS(t) between the coarse-grained and the Gibbs limit and proper thermostating. entropy in this Hamiltonian case with the time evolution of ͑iv͒ It is worth pointing out where the chaoticity of the the same difference for a transport process approaching a underlying dynamics plays an essential role. It is the mixing steady state ͑cf. Fig. 3͒. For both situations, the Gibbs en- property ͑in the sense of ergodic theory5,26͒ of the multi- tropy S(G) is compared at any time with the entropy S com- Baker map without which an analogy with a random walk, puted, for the same initial conditions, with densities coarse and hence a transport equation, cannot be established. The grained on a fixed grid. In the case of a closed system, the mixing property leads to the appearance of a diffusion coef- Gibbs entropy remains unchanged, but S(t) initially grows ficient D which, in our approach, is of fully dynamical origin. with time ͓Fig. 3͑a͔͒. Since closed systems eventually ap- In view of point ͑ii͒, we also see how accurate Gibbs’ anal- proach equilibrium, the coarse-grained entropy saturates. ogy of irreversible processes with the mixing of a nondiffu- Consequently, the time derivative of ⌬iS(t), which can sive dye in a colorless solvent45 was, since the latter is of again be considered as irreversible entropy production, first course a chaotic process with mixing properties.46 grows but vanishes in the long-time limit. The basic differ- ͑v͒ The macroscopic expression (irr)ϭ j2/(D) was de- ence between a thermodynamically closed and an open sys- rived here ͑for ϭϪ1͒ without any assumption about tem relaxing towards a nonequilibrium state is that in the
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graining. This is why one recovers the thermodynamic re- sults, as we did in the present paper, even with the most straight-forward choice of coarse-graining size ͑namely, the full cell size͒ and the choice of a constant initial density in every cell.
ACKNOWLEDGMENTS This work was started during a workshop at the ESI in FIG. 3. Schematic time evolution of the coarse-grained density S and the Vienna. We are grateful for enlightening discussions with H. Gibbs entropy S(G) for ͑a͒ a closed system where the densities relax to their van Beijeren, E. G. D. Cohen, J. R. Dorfman, P. Gaspard, T. equilibrium values, and ͑b͒ an open system approaching a steady state that Geszti, W. G. Hoover, R. Klages, L. Ma´tya´s, G. Nicolis, L. carries a nonvanishing mass current. In both cases, the initial condition is ´ ´ smooth and distributed over a region of the phase space much bigger than Rondoni, L. Sasvari, P. Szepfalusy, and S. Tasaki. JV grate- the cell size used for coarse graining. The arrow indicate the effect of coarse fully acknowledges financial support by the Deutsche Fors- graining on the Gibbs entropy. chungsgemeinschaft, and hospitality of the Max-Planck In- stitute for polymer research ͑Mainz͒ where he was a guest scientist upon completion of this work. TT was supported by latter case the Gibbs entropy decreases forever ͓Fig. 3͑b͔͒. the Hungarian Science Foundation ͑OTKA T17493, A jump from the phase-space density to its coarse- T19483͒. grained analog causes the Gibbs entropy to jump up to the curve of S. Because the third column ͑S͒ is missing in their APPENDIX A: NONEQUILIBRIUM THERMODYNAMICS model, they can neither find consistency with a Fokker– OF MASS TRANSPORT Planck equation nor with the thermodynamical form of the irreversible entropy production. When such a coarse graining In this section we summarize the thermodynamic rela- tions describing transport due to the simultaneous presence is performed after the closed system has relaxed to equilib- 14 rium, S and S(G) remain identical ͓Fig. 3͑a͔͒. This is a partial of drift and diffusion currents. We restrict ourselves to extension of the approach of Ref. 22, where only two differ- cases where the external field and the density gradient are ent refined partitions were investigated but results for arbi- both parallel to the x-axis so that the diffusion and the con- trary non-stationary states could be obtained. Even though a ductivity are scalar quantities. The temperature T of the sys- jump from the phase-space density to its coarse-grained ana- tem will be kept constant throughout the system, i.e., we will log causes the Gibbs entropy to jump up to the curve of S, not consider heat conduction. For the derivation we use the the coarse-grained and Gibbs entropy always start to differ laws of nonequilibrium thermodynamics and linear response. again after the coarse graining. The central point of the We stress, however, that in spite of this assumption the ex- present article was to call attention to this qualitatively dif- pressions can very well be of more general validity. ferent behavior of closed and open systems, which leads to a The conductivity is the transport coefficient which strictly positive irreversible entropy production in nonequi- quantifies the amount of a current density j in response to an librium systems. The ever-lasting decrease of the Gibbs en- applied thermodynamic force X. In the presence of a gradi- ent of the chemical potential ͑͒ and an external force (E) tropy is a striking consequence of the fact that nonequilib- 14 rium steady states are associated with chaotic systems whose this leads to x͒. ͑A1͒ץsteady-state distributions are supported by fractal subsets of jϭTXϭ͑EϪ the phase space. Note added: After submitting this paper we became Next, we rewrite the current in a convenient form. Sub- aware of a recent preprint by Gilbert and Dorfman50 dealing stituting, in Eq. ͑A1͒, the thermodynamic expression for the with closely related problems. In a multi-Baker model, de- diffusion coefficient ץ fined with two columns only, they discuss properties of the Dϭ ͑A2͒ ͯ ץ entropy in stationary states of the Hamiltonian and of the properly thermostated version ͑the cases ϭϮ1 in our no- T x. By introducing the drift velocityץtation͒. By carrying out coarse graining on arbitrarily fine we find that jϭEϪD Markov-partitions, they derive an expression for the irrevers- E ible entropy production in steady states. This is an extension vϵ0Eϵ , ͑A3͒ of the results of Ref. 22, where only two different refined partitions were investigated, but for arbitrary nonstationary ͑Ohm’s law͒ where 0 is the mobility of the particles, we states. The authors of Ref. 50 show by explicit numerical find the full current in the form of ͑16͒. calculation that in the limit of large system size the irrevers- The entropy flux contains14 a convective contribution ible entropy production becomes independent of the parti- j cϩln ͒ ͑A4͒ ץ⌽͑conv͒ϭ tioning everywhere, except for a narrow boundary layer x͓ ͑ ͔ around the two ends of the chain. Based on the results of characterizing the entropy carried by the flow of particles Refs. 22 and 50 we firmly believe that in the bulk of every ͑here c is a constant which is typically scaled out to zero͒.In macroscopically large system none of the terms in the the presence of a thermostat there must be another contribu- entropy-balance equation depends on the details of coarse tion, which accounts for the entropy flux due to heat which
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FIG. 4. Chain of intervals along the x axis on which the projected dynamics is defined. ͑cf. the text͒.
dissipates the work done by the external field into a thermo- stat ͑Joule’s heat͒. This contribution will be called ⌽(heat), and it takes the form Ej v ⌽͑heat͒ϭϪ ϭϪ j. ͑A5͒ T D Here, we used the definition of the drift ͑A3͒ and Einstein’s relation DϭT to arrive at the last equality. The total flux is the sum: ⌽ϭ⌽(conv)ϩ⌽(heat). Finally, the rate of irreversible entropy production (irr) is the product of the current density j and the thermodynamic driving force X14 ͑irr͒ϭ jX. ͑A6͒ FIG. 5. Piecewise linear map governing the projected dynamics of the multi-Baker model. In view of Eq. ͑A1͒, this leads to j2 j2 ͑irr͒ϭ ϭ , ͑A7͒ this coincidence, however, the balance equation for the pro- T D jected entropy differs profoundly from its phase-space ana- where again Einstein’s relation has been used. log. For the projected entropy, the derivation of a balance equation is more formal than in the full phase-space ap- APPENDIX B: ENTROPY BALANCE FOR THE proach, because the projected dynamics is no longer invert- PROCESS PROJECTED ON THE DIRECTION OF ible. For states with constant mass densities in every interval, TRANSPORT the projected Gibbs entropy coincides thus with the coarse- grained entropy although the entropy production might still 1. Entropy balance be strictly positive. What can be done in any case is to iden- (p) Since the thermodynamic expressions only contain the tity a flux ⌬eSm of the projected entropy into interval m due mass density distribution (x) along the transport axis, the to the motion of one-dimensional volumes across the bound- question arises in which sense the orthogonal direction is aries of this interval during one time unit. This quantity is relevant. To clarify this, we discuss an entropy balance based typically different from the full change ⌬Sm of the entropy. on the projected density, i.e., on the mass density distribution Therefore, we can define the irreversible projected entropy inside the cells. change via a balance equation For a general model of the type of Fig. 1, the dynamics ⌬S͑p͒ϭ⌬S ϭ⌬ S͑p͒ϩ⌬ S͑p͒ , ͑B2͒ of the projected density is generated by an extended one- m m e m i m dimensional map containing identical intervals of size a ͑see i.e., as the difference Fig. 4͒. We define the ‘‘projected’’ ͑Gibbs͒ entropy as ͑p͒ ͑p͒ ⌬iSm ϵ⌬SmϪ⌬eSm . ͑B3͒ ͑p͒ m͑x͒ Sm ϭϪ m͑x͒ln à dx, ͑B1͒ Thus, in this approach the existence of an entropy-balance ͵interval m equation is necessary for the definition of entropy produc- which contains information about the inhomogeneities of the tion. projected distribution along the x axis. Here, again, we used Ãϭb%à as the reference density for the definition of the entropy. We also define a coarse-grained projected entropy 2. Entropy balance in the projected multi-Baker map computed with the average mass density m in each interval. The dynamics projected onto the x axis is governed by In view of Eq. ͑5͒, mϭb%m . Therefore, with this choice of the one-dimensional map depicted in Fig. 5. Since the 1D à (p) the coarse-grained projected entropy Sm coincides with map is piecewise-linear, the density m is a step function the coarse-grained entropy Sm . Thus we can drop the super- which is constant within every cell at any instant of time, if (p) script of the coarse-grained entropy: Sm ϭSm . In spite of the initial density is of this type. Again, we find mϭb%m .
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(p) The entropy flux ⌬eSm into interval m during one time step corresponds to the flow of Sm through the boundaries of the interval, i.e., it is given by the difference between the respective in- and outgoing fluxes
mϪ1 mϩ1 S͑p,in͒ϭϪa r ln ϩl ln , ͑B4a͒ m ͩ mϪ1 Ã mϩ1 Ã ͪ
m S͑p,out͒ϭϪa͑rϩl͒ ln . ͑B4b͒ m m à The total flux is obtained, after using ͑22͒ for the mass den- sity, as ͑p͒ ͑p,in͒ ͑p,out͒ ⌬eSm ϵSm ϪSm m mϪ1 FIG. 6. Notations used for the discussion of entropy production for general ϭϪa ͑mЈϪm͒ln à ϩrmϪ1 ln ͫ maps ͑cf. the text͒. m
mϩ1 ϩl ln . ͑B5͒ mϩ1 ͬ m tion about the phase-space contraction along the p axis, Note that this is exactly the same as Eq. ͑25b͒ except that which is related to the heat current into a thermostat. The here, the effect of phase-space contraction is missing. projected density faithfully reflects the entropy changes re- In view of ͑B3͒ the irreversible change of the projected lated to the interaction with particle reservoirs, but it cannot entropy reads account for the effects related to a heat bath.
mϪ1 mϩ1 mЈ ⌬ S͑p͒ϵa r ln ϩl ln ϪЈ ln . APPENDIX C: IRREVERSIBLE ENTROPY i m ͫ mϪ1 mϩ1 m ͬ m m m PRODUCTION FOR GENERAL MAPS ͑B6͒ We work out the irreversible entropy production Eq. ˜ ˜ As expected, it is independent of the parameters r and l ͑12͒ per time step in cell m for a general map modeling characterizing the phase-space contraction in the full baker mass transport on a grid of cells as schematically depicted in dynamics. Fig. 6. In order to describe the time evolution of the system on the level of coarse-grained densities, we introduce the fol- 3. Macroscopic limit lowing notations: The present approach corresponds to the Hamiltonian n: label of the neighbors of cell m. choice ϭ1 of the previous results ͓cf. Eq. ͑25͔͒. Therefore, % j : the coarse-grained density in cell j. in the macroscopic limit when m tends to (x) we find ⌫: the ͑possibly high dimensional͒ volume of each cell. w : the transition probability from cell j to k. j,k ͑p͒ s : volume fraction of cell j mapped into cell k. s de- x ͑B7͒ j,k m,mץx j 1ϩln à Ϫvץ⌽ ϭ ͫ ͩ ͪͬ notes the volume fraction which does not leave cell m and in one time step. The s j,k’s are subjected to the sum 2 rule x͒ץ͑ ͑p,irr͒ϭD . ͑B8͒ ͚ sm, jϭ1. ͑C1͒ j n,m Note that the same result is also obtained by directly consid- ͕ ͖ ˜ ering the large-system limit of the projected entropy in ͑B1͒. s j,k : volume fraction of cell k which contains all points with In this limit, m(x) approaches the thermodynamic mass preimages in cell j. We are only interested in globally density (x), and the Fokker–Planck equation ͑17͒ can be phase-space preserving flows implying used to directly evaluate the time derivative of the projected ˜ entropy. In this limit, Sm /a approaches s, and one obtains an ͚ s j,mϭ1. ͑C2͒ entropy-balance equation of the form j͕n,m͖ ˜ -sϭ⌽͑p͒ϩ͑p,irr͒, ͑B9͒ Note that s j, jϭs j, j for every j because of the assump ץ t tion that the internal dynamics in every cell is phase- (p) (p,irr) where ⌽ and are again given by Eqs. ͑B7͒ and space preserving. ͑B8͒. Although this is a well-defined macroscopic entropy- Taking the cell density to be constant because of coarse balance equation with a non-negative entropy production, it graining, the transition probability from cell j to k is the is markedly different from the thermodynamic form in the volume fraction mapped into cell k case of a nonvanishing drift. This is consistent with the ob- servation that the projected density cannot contain informa- w j,kϭs j,k . ͑C3͒
Downloaded 23 Feb 2010 to 10.0.105.87. Redistribution subject to AIP license or copyright; see http://chaos.aip.org/chaos/copyright.jsp Chaos, Vol. 8, No. 2, 1998 Breymann, Te´l, and Vollmer 407
˜ 5 J. R. Dorfman, An Introduction to Chaos in Non-Equilibrium Statistical The volume fraction sm,m of points not leaving cell m in one time step is obtained as Physics, Lecture Notes ͑University of Maryland Press, Maryland, 1998͒; Phys. Rep. ͑to appear 1998͒. 6 N. I. Chernov, G. L. Eyink, J. L. Lebowitz, and Ya. G. Sinai, Phys. Rev. ˜ sm,mϭsm,mϭ1Ϫ͚ wm,n. ͑C4͒ Lett. 70, 2209 ͑1993͒; Commun. Math. Phys. 154, 569 ͑1993͒. n 7 D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys. Rev. Lett. 71, 2401 ͑1993͒. Conservation of probability leads to a master equation 8 G. Gallavotti and E. G. D. Cohen, Phys. Rev. Lett. 74, 2694 ͑1995͒;J. 42 expressing the cell density %mЈ after one time step Stat. Phys. 80, 931 ͑1995͒; F. Bonetto, G. Gallavotti, and P. Garrido, Physica D 105, 226 ͑1997͒. 9 E. G. D. Cohen, Physica A 213, 293 ͑1995͒; 240,43͑1997͒; also cf. E. G. %mЈ ϭ%mϪ ͑wm,n%mϪwn,m%n͒ D. Cohen and L. Rondoni, Chaos 8, 357 1998 . ͚n ͑ ͒ 10 D. Ruelle, J. Stat. Phys. 85,1͑1996͒. 11 G. Gallavotti, J. Stat. Phys. 85, 899 ͑1996͒; Phys. Rev. Lett. 77, 4334 ϭs % ϩ s % . ͑C5͒ ͑1996͒. m,m m ͚ n,m n 12 n G. Gallavotti and D. Ruelle, Commun. Math. Phys. 190, 279 ͑1997͒. 13 L. Rondoni and E. G. D. Cohen, Orbital Measures in Nonequilibrium At time tЈ the coarse-grained entropy is Statistical Mechanics ͑preprint, 1997͒. 14 S. R. de Groot, and P. Mazur, Nonequilibrium Thermodynamics ͑Elsevier, %mЈ Amsterdam, 1962͒. SЈ ϭϪ⌫%Ј ln . ͑C6͒ 15 m m %à W. Breymann, T. Te´l, and J. Vollmer, Phys. Rev. Lett. 77, 2945 ͑1996͒. 16 P. Gaspard, Physica A 240,54͑1997͒; J. Stat. Phys. 88, 1215 ͑1997͒. In order to compute the Gibbs entropy at the same time, we 17 G. Nicolis and D. Daems, J. Phys. Chem. 100, 19187 ͑1996͒; D. Daems observe that the number of particles entering cell m from the and G. Nicolis, ‘‘Entropy production and phase space volume contrac- % tion’’ ͑preprint, 1998͒; Chaos 8, 311 ͑1998͒. volume sn,m⌫ of cell n is nsn,m⌫. Therefore, the new den- 18 L. Rondoni and G. P. Morriss, Phys. Rev. E 53, 2143 ͑1996͒. ˜ 19 sity %nЈ,m on volume sn,m⌫ of cell m is %nЈ,m J. R. Dorfman and H. van Beijeren, Physica A 240,12͑1997͒. 20 ϭ% s /˜s . The density in that part of cell m which is S. Lepri, R. Livi, and A. Politi, Phys. Rev. Lett. 78, 1896 ͑1997͒. n n,m n,m 21 J. Vollmer, T. Te´l, and W. Breymann, Phys. Rev. Lett. 79, 2759 ͑1997͒. mapped into the same cell does not change. The Gibbs en- 22 J. Vollmer, T. Te´l, and W. Breymann, ‘‘Entropy balance in the presence tropy is thus of drift and diffusion currents: an elementary chaotic map approach,’’ Phys. Rev. E ͑to be published͒. %Ј 23 16 ͑G͒Ј ˜ j,m It was pointed out by Gaspard that the integrals appearing in the defini- Sm ϵϪ ͚ ⌫s j,m%Јj,m ln à tion of, e.g., the Gibbs entropy are not even well defined in the long time j n,m ͩ % ͪ ͕ ͖ limit because of strictly formal reasons, unless the singular densities de- scribing steady states of nonequilibrium systems are regularized by a % j s j,m ϭϪ ⌫s % ln coarse-graining procedure as provided by the formalism of ⑀-entropies.24 ͚ j,m j à 24 j͕n,m͖ ͩ % ˜s ͪ P. Gaspard and X-J. Wang, Phys. Rep. 235, 321 ͑1993͒. j,m 25 In a more elaborate setting our results can immediately be extended to %n sn,m cases where the bins are smaller than the cells ͑cf. Ref. 22͒. In fact, the ϭ⌫ Ϫ%mЈ ln %mϪ͚ sn,m%n ln , formalism of ⑀-entropies is ideally suited to carry out this program, and ͫ n ͩ %m ˜s ͪͬ subsequently take the limit of vanishing bin size ͑cf. Ref. 24͒. n,m 26 E. Ott, Chaos in Dynamical Systems ͑Cambridge Univ. 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the Lyapunov exponents of the inverted dynamics coincide with those of 44 Using the more general relations (rϪl)aϭv and (rϩl)a2ϭ2Dϩ2v2 the forward dynamics. This differs slightly from the irreversibility concept only modifies corrections to the macroscopic results, see Ref. 22. of other authors41 who demand that the time-reversed dynamics is identi- 45 J. W. Gibbs, Elementary Principles in Statistical Mechanics ͑Yale Uni- cal to the original one up to a geometrical transformation. For multi-Baker versity Press, New Haven, CT, 1902͒, Chap. XII. 46 maps with a dissipative dynamics with time reversal symmetry ͑21b͒ the J. M. Ottino, The Kinematics of Transport and Mixing: Stretching, Chaos composition with a reflection on the diagonal of every cell complies with and Transport ͑Cambridge University Press, Cambridge, 1989͒. 47 P. and T. Ehrenfest, The Conceptual Foundations of the Statistical Ap- the latter requirement. proach in Mechanics ͑Cornell University Press, Ithaca, 1959͒, Sec. 23; 41 J. A. G. Roberts and G. R. W. Quispel, Phys. Rep. 216,63͑1992͒; as well translation of the German original in the Encyklopa¨die der Mathematis- as contributions to the proceedings volume Physica D 112, No. 1–2 ͑Jan chen Wissenschaften ͑Teubner, Leipzig, 1912͒. 15, 1998͒. 48 R. C. Tolman, The Principles of Statistical Mechanics ͑Oxford University 42 W. Feller, An Introduction to Probability Theory and its Applications Press, Oxford, 1938͒. ͑Wiley, New York, 1978͒. 49 D. ter Haar, Rev. Mod. Phys. 27, 289 ͑1955͒. 43 F. Reif, Fundamentals of Statistical and Thermal Physics ͑McGraw-Hill, 50 T. Gilbert and J. R. Dorfman, Entropy Production: From Open Volume New York, 1965͒. For random walks, see Section 1.9. Preserving to Dissipative Systems ͑preprint, 1998͒.
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