CHAOS VOLUME 8, NUMBER 2 JUNE 1998
An analytical construction of the SRB measures for Baker-type maps S. Tasaki Department of Physics, Nara Women’s University, Nara 630, Japan and Institute for Fundamental Chemistry, 34-4 Takano Nishihiraki-cho, Sakyo-ku, Kyoto 606, Japan Thomas Gilbert and J. R. Dorfman Institute for Physical Science and Technology, and Department of Physics, University of Maryland, College Park, Maryland 20742-3511 ͑Received 4 December 1997; accepted for publication 6 April 1998͒ For a class of dynamical systems, called the axiom-A systems, Sinai, Ruelle and Bowen showed the existence of an invariant measure ͑SRB measure͒ weakly attracting the temporal average of any initial distribution that is absolutely continuous with respect to the Lebesgue measure. Recently, the SRB measures were found to be related to the nonequilibrium stationary state distribution functions for thermostated or open systems. Inspite of the importance of these SRB measures, it is difficult to handle them analytically because they are often singular functions. In this article, for three kinds of Baker-type maps, the SRB measures are analytically constructed with the aid of a functional equation, which was proposed by de Rham in order to deal with a class of singular functions. We first briefly review the properties of singular functions including those of de Rham. Then, the Baker-type maps are described, one of which is nonconservative but time reversible, the second has a Cantor-like invariant set, and the third is a model of a simple chemical reaction R I P. For the second example, the cases with and without escape are considered. For the last↔ example,↔ we consider the reaction processes in a closed system and in an open system under a flux boundary condition. In all cases, we show that the evolution equation of the distribution functions partially integrated over the unstable direction is very similar to de Rham’s functional equation and, employing this analogy, we explicitly construct the SRB measures. © 1998 American Institute of Physics. ͓S1054-1500͑98͒02702-5͔
Characterization of nonequilibrium stationary states in I. STATISTICAL MECHANICS AND SRB MEASURES terms of dynamical ensembles is one of the main ques- One of the main questions in statistical mechanics is to tions in statistical mechanics. Recently, the so-called characterize nonequilibrium stationary states in terms of dy- Sinai-Ruelle-Bowen „SRB… measures, which had been namical ensembles ͑cf., e.g., Refs. 1–3͒. Recently, for ther- studied in dynamical systems theory, were found to be mostated or open systems, stationary nonequilibrium en- related to nonequilibrium stationary ensembles for ther- sembles were found to be related to the so-called Sinai- mostated or open systems. The SRB measures fully de- Ruelle-Bowen ͑SRB͒ measures,4–15 which had been scribe transport properties of the corresponding non- investigated in dynamical systems theory.16–19 In the thermo- equilibrium stationary states. Also they would provide an stated systems,4–12 a fictitious damping force mimicking a important insight about the emergence of irreversibility heat reservoir is introduced to avoid an uncontrolled growth in reversible dynamical systems, since they do not have of the kinetic energy due to an external driving force. The time-reversal invariance even when the dynamics is re- damping force is chosen so as to make the dynamics dissi- versible. It is therefore illustrative to know exact forms of pative while it preserves time-reversibility. As a result of the the SRB measures, but it is difficult to handle them ex- dissipation, there exists an attractor of information dimen- actly because they are often singular functions. In this sion less than the dimension of phase space and the nonequi- paper, we study three examples of Baker-type maps, librium stationary state is described by an asymptotic mea- which illustrate some aspects of the thermostated and/or sure, which is an SRB measure. The SRB measure fully open systems: One is nonconservative but time revers- characterizes the transport properties, such as the transport ible, the second has a Cantor-like invariant set, and the law, transport coefficients and their fluctuations. For ex- third is a model of a simple chemical reaction such as an ample, for the driven thermostated Lorentz gas,6 the conduc- isomerization R I P. For those maps, we analytically ↔ ↔ tivity tensor was calculated, and Ohm’s law and Einstein’s construct SRB measures with the aid of a new method, relation were verified by comparing the averaged current where the weak convergence of measures is converted to with respect to the SRB measure to the external electric field. the strong convergence of partially integrated distribu- On the other hand, open chaotic Hamiltonian systems with a tion functions „PIDFs… and the evolution equations for flux boundary condition13–15 admit a nonequilibrium station- the PIDFs are solved employing the analogy between ary state obeying Fick’s law that is described by a kind of them and de Rham’s functional equations. SRB measure. This measure again characterizes the transport properties. Even for an open Hamiltonian system with an
1054-1500/98/8(2)/424/20/$15.00 424 © 1998 American Institute of Physics
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absorbing boundary condition, an SRB-like measure may de- other mϪms directions grow more rapidly than an exponen- scribe the long-time behavior of averaged dynamical func- tial function et (tу0 and the same ). An invariant set ⌳ is tions. Such a system has a fractal repeller, that controls the said to be hyperbolic when ͑1͒ for any x⌳, the orbit Stx 12,20–26 chaotic scattering. The unstable manifold of the fractal has the ms stable and mϪms unstable directions as explained repeller supports a conditionally invariant measure, which above, ͑2͒ the stable and unstable directions depend continu- provides the long time limits of averaged dynamical ously on x and ͑3͒ the stable and unstable directions for the functions23,24 and may be of SRB-type ͑cf. Sec. IV B͒.We point x are mapped by St to the corresponding directions for note that the interrelation between the thermostated systems the point Stx. approach and the open systems approach has been discussed A point x is said to be nonwandering if the orbit Stx by Breymann, Te´l and Vollmer.29 In this article, we present returns indefinitely often to any neighborhood of its initial an analytical construction of SRB measures for three ex- point x. If the set ⍀ of all nonwandering points is hyperbolic amples of Baker-type maps, which illustrate some aspects of and the set of periodic points is dense in ⍀, S is called an the thermostated and/or open systems mentioned above. axiom-A diffeomorphism. In particular, if the whole phase Now we start with the general arguments on the SRB mea- space M is hyperbolic, S is called an Anosov diffeomor- sure. phism. The Arnold cat map is an example of an Anosov The long-term behavior of a dynamical system is char- diffeomorphism. acterized by an invariant measure on an invariant set A, Invariant measures that are smooth along the unstable which describes how frequently various parts of A are visited directions are called SRB measures. Sinai, Ruelle and Bo- by a given orbit x(t) ͑with t the time͒. The invariant measure wen showed that, for axiom-A diffeomorphisms, the SRB is said to be ergodic if it cannot be decomposed into different measure is the unique physical measure describing the invariant measures. Such an ergodic invariant measure sat- time averages ͑2͒ of observables of motion with initial data x isfies the ergodic theorem.1–3,16 In the case of a map S,it taken at random with respect to the Lebesgue measure 16–18 asserts that, for any continuous function (x), we have 0. For more details on axiom-A systems and SRB mea- T sures, see Refs. 16, 17, and 19. 1 ͐A͑xЈ͒͑dxЈ͒ lim ͚ ͑Stx͒ϭ In Gibbs’ picture of statistical mechanics, a macroscopic T ϩϱ Ttϭ0 ͐A͑dxЈ͒ state for an isolated system is described by a phase-space → distribution function, and a macroscopic observable by an x A E with E ϭ0 . 1 ͑ \ ͑ ͒ ͒ ͑ ͒ averaged dynamical function with respect to the distribution. A dynamical system typically admits uncountably many dis- Suppose that the dynamics satisfies the mixing condition tinct ergodic measures and not all of them are physically with respect to the microcanonical distribution mc observable. One criterion of choosing a physical measure is that describes the time averages of observables on mo- x dx t ͐M͑ ͒mc͑ ͒ tions with initial data x randomly sampled with respect to the lim ͵ ͑S x͒0͑x͒mc͑dx͒ϭ , ͑3͒ 16,17 t ϩϱ M ͐Mmc͑dx͒ Lebesgue measure 0 → where M stands for the whole phase space, (x) is a con- T 1 ͐A͑xЈ͒͑dxЈ͒ tinuous dynamical function and (x) is a normalized initial lim Stx͒ϭ 0 ͚ ͑ distribution function. Then, the system exhibits time evolu- T ϩϱ Ttϭ0 ͐A͑dxЈ͒ → tion as expected from statistical thermodynamics. Namely, x⌺ E with ⌺ʛA, the distribution function weakly approaches an equilibrium \ . ͑2͒ ͩ0͑⌺͒Ͼ0 and 0͑E͒ϭ0ͪ microcanonical distribution and the averaged dynamical functions approach well defined equilibrium values.1 From Sinai, Ruelle and Bowen showed that a class of dynamical this point of view, instead of Eq. ͑2͒, it is enough to consider systems, called axiom-A systems, uniquely admit such a the following condition as a criterion of choosing a physical physical invariant measure ͑SRB measure͒.16–18 Thus an measure : SRB measure is one for which the ergodic theorem is true for almost every point, x, with respect to the Lebesgue measure ͐A͑x͒͑dx͒ lim ͑Stx͒ ͑x͒ ͑dx͒ϭ , ͑4͒ . ͵ 0 0 0 t ϩϱ M ͐A͑dx͒ Axiom-A systems are characterized by the hyperbolic → structure, i.e., the existence of exponentially stable and ex- where 0 stands for the Lebesgue measure, M is the whole ponentially unstable directions which intersect transversally phase space, A the attractor and (x) and 0(x) are, respec- with each other. In case of bijective differentiable maps ͑i.e., tively, a continuous dynamical function and a normalized diffeomorphisms͒, the hyperbolic structure is defined as fol- initial distribution function. Sinai, Ruelle and Bowen18 t lows: Consider a given orbit S x (tϭ0,Ϯ1,Ϯ2,•••) and showed that the SRB measures for the axiom-A systems sat- small deviations ␦x of the initial value x. Note that there are isfy Eq. ͑4͒ as well. Hence, the measure satisfying Eq. ͑4͒ m independent possible directions of ␦x when the phase will be referred to as a physical measure. Since the left hand space dimension is m. Assume that the deviations along ms side of Eq. ͑4͒ is the average ͗͘t of at time t, Eq. ͑4͒ can directions decrease more rapidly than an exponential func- be generalized to define a physical measure for systems tion eϪt (Ͼ0 and tу0) and that the deviations along the with escape23,24
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t ͐ M͑S x͒0͑x͒0͑dx͒ plain for three Baker-type maps, the weak convergence of lim ͗͘tϭ lim t measures is converted to the usual convergence ͑technically t ϩϱ t ϩϱ͐MM͑S x͒0͑x͒0͑dx͒ → → speaking, the strong convergence͒ of partially integrated dis- tribution functions. Contrary to the evolution equation of a ͐AЈ͑x͒͑dx͒ ϭ , ͑5͒ distribution function ͑the Frobenius-Perron equation͒, the ͐ ͑dx͒ AЈ evolution equation of a partially integrated distribution is where M stands for the characteristic function of the phase contractive and possesses a well-defined long-time limit. space M and AЈ is the support of . The denominator of the This equation is similar to de Rham’s functional equation,39 middle expression is necessary as the total probability is not which was introduced to deal with singular functions such as preserved. When there is no escape, the physical measure continuous functions with zero derivatives almost every- defined by Eq. ͑4͒ is supported by the attractor and, thus, is where, and its contraction rate gives the convergence rate of invariant. On the other hand, when there is escape, the physi- the averaged dynamical function ͗͘t . In Sec. II, we review cal measure defined by Eq. ͑5͒ is, in general, not an invariant some properties of singular functions including those of de measure. However, since the ratio ͐AЈ(x)(dx)/͐AЈ(dx) Rham’s functional equation. In Sec. III, an SRB measure is does not evolve in time, such a measure is called condition- constructed for a nonconservative reversible Baker map with ally invariant.23,24 We remark that, when they are of axiom-A the aid of de Rham’s equation. The model illustrates the two type, the systems with escape also possess ‘‘natural’’ invari- fundamental features of the thermostated systems, namely, ant measures supported by the fractal repeller, which are the phase space contraction and time reversibility, and we specified, e.g., by a variational principle.1,12,22–28 It is the discuss the interrelation between the two features. In Sec. IV, natural invariant measure that characterizes the ergodic prop- a Baker map with a Cantor-like invariant set is studied. erties such as the Lyapunov exponents, but not the condition- When there is no escape, the map possesses a strange attrac- ally invariant physical measure defined by Eq. ͑5͒. tor, which is a direct product of the unit interval and a Cantor Now we revisit thermostated and open systems. As ex- set. On the other hand, when there is escape, the map has an plained before, a thermostated system is dissipative because invariant set, which is the direct product of two Cantor sets, of the fictitious damping force. Then, a nonequilibrium sta- and is a simple example of an open system with an absorbing tionary state is described by an asymptotic SRB measure boundary condition. Physical measures for the map with and defined by Eq. ͑2͒.4–12 For open chaotic Hamiltonian systems without escape are constructed with the aid of de Rham’s with a flux boundary condition, a nonequilibrium stationary equation and the natural invariant measure for the map with state obeying Fick’s law is described by a measure with a escape is derived. In Sec. V, we investigate the properties of fractal structure along the contracting direction.13–15 Since a Baker map with a flux boundary condition, which mimics a the measure is smooth along the expanding direction and can chemical-reaction dynamics with a flux boundary condition be defined by Eq. ͑4͒, it is an SRB measure.13 In both cases, ͑cf. Ref. 40͒. We show that an SRB-type stationary distribu- those SRB measures fully characterize the transport proper- tion describes the reaction dynamics and that the slowest ties. However, it should be noticed that the two cases are relaxation to it is characterized by a decay mode ͑i.e., the different because the invariant set of an open system is a Pollicott-Ruelle resonance͒, which is a conditionally invari- fractal repeller which is fractal in both the stable and un- ant measure. Section VI is devoted to concluding remarks. stable directions and, thus, does not support an SRB mea- Technical details of the construction of measures are pre- sure, while the invariant set of a thermostated system is an sented in Appendixes. attractor which does support an SRB measure. By applying Eq. ͑4͒ for an initial constant density, one II. SINGULAR FUNCTIONS AND de RHAM EQUATION obtains a method of constructing an SRB measure for a Basic tools of our analytical construction of SRB mea- 30 map: ͑1͒ Approximate the measure by iterating an initial sures are singular functions and de Rham’s functional equa- constant density finite times; ͑2͒ calculate the average with tion. Singular functions such as the Weierstrass function or its result and ͑3͒ take the limit of infinite iterations. Several the Takagi function were originally introduced as pathologi- methods are also proposed where unstable periodic orbits or cal counter examples to the intuitive picture of functions. trajectory segments are used to write down an SRB measure These singular functions play an important role in chaotic and averages with respect to it ͑cf. Refs. 5, 7, 8, 10, 21, 22, dynamics. A step towards the analytical treatment of singular and 31 and references therein͒. However, it is not easy to functions was given by de Rham,39 who showed many of evaluate the convergence rate of the limits in Eqs. ͑4͒ and them can be characterized as a unique fixed point of a con- ͑5͒, particularly for nonexpanding maps, and to extract ex- traction mapping in a space of functions. In this section, we ponentially decaying terms from an averaged dynamical briefly review the properties of some singular functions and function ͗͘t , which are the Pollicott-Ruelle the relation between them and chaotic dynamical systems 32–34,14,35–38 resonances. One of the reasons is that the long- ͑cf. Refs. 47 and 51͒. time limit of the measure can be defined only via the en- The first example of a nowhere differentiable continuous semble average as shown in Eqs. ͑4͒ and ͑5͒. For nonexpand- function was given by Weierstrass in 187241 ͑Fig. 1͒: ing maps, the distribution function itself does not have a well-defined long-time limit. In other words, an asymptotic ϱ SRB measure is the weak limit of an initial measure. In an n n Wa,b͑x͒ϭ a cos͑b x͒, ͑6͒ analytical construction of SRB measures, which we shall ex- n͚ϭ0
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FIG. 1. Weierstrass function for aϭ2/3 and bϭ2.
FIG. 3. Cantor function. where 0ϽaϽ1, b is a positive number and abу1. Moser42 used the Weierstrass function to construct a nowhere differ- entiable torus for an analytic Anosov system. Yamaguti and functions.48,49 One typical example is the Cantor function 43 Hata used it as a generating function of orbits for the lo- ͑Fig. 3͒. A more interesting example f ␣(x)(0Ͻ␣Ͻ1,␣ gistic map and discussed some generalizations. Also Weier- Þ1/2) is the unique function satisfying strass functions are eigenfunctions of the Frobenius-Perron ␣ f ␣͑2x͒, x͓0,1/2͔ operator for the Renyi map Sxϭrx ͑mod 1͒ with r a positive f ͑x͒ϭ ͑8͒ 35,44 ␣ integer. ͭ ͑1Ϫ␣͒f ␣͑2xϪ1͒ϩ␣, x͓1/2,1͔, In 1903, Takagi gave a simpler example of a nowhere which is strictly increasing and continuous, but has zero de- 45 differentiable continuous function ͑Fig. 2͒: rivatives almost everywhere with respect to the Lebesgue 49 ϱ 1 measure ͑Fig. 4͒. Note that such functions do not satisfy n 48 T͑x͒ϭ n ͑2 x͒, ͑7͒ the fundamental theorem of calculus n͚ϭ0 2 1 f 1 Ϫ f 0 ϭ1ϭ0ϭ f Ј x dx. where (x)ϭ͉xϪ͓xϩ1/2͔͉ and ͓y͔ stands for the maximum ␣͑ ͒ ␣͑ ͒ ” ͵ ␣͑ ͒ integer which does not exceed y. In 1930, van der Waerden 0
gave a similar function which is obtained from Eq. ͑7͒ by The function f ␣(x) with real ␣ represents a cumulative dis- replacing 2n to 10n.46 In 1957, the Takagi function was re- tribution function of an ergodic measure for the dyadic map discovered by de Rham.39 Some generalizations of the Sxϭ2x ͑mod 1͒͑cf. examples given on p. 626 of Ref. 16 Takagi function were discussed by Hata and Yamaguti.43,47 and on p. 36 of Ref. 50͒. The eigenfunctionals of the The function T(x)Ϫ1/2 is the eigenfunction of the Frobenius-Perron operator for the multi-Bernoulli map and Frobenius-Perron operator for the map Sxϭ2x ͑mod 1͒ with the multi-Baker map can be represented as the Riemann- 36,37 eigenvalue 1/2. Also, the Takagi function and related func- Stieltjes integrals with respect to f ␣(x) with complex ␣. tions were found to describe the nonequilibrium stationary In Ref. 38, it was shown that, for a class of piecewise linear state obeying Fick’s law for the multi-Baker map.13 maps, the left eigenfunctionals of the Frobenius-Perron op- In the theory of the Lebesgue integral, there appear non- erator admit a representation in terms of singular functions 43,47 constant functions with zero derivatives almost everywhere, similar to f ␣(x). As pointed out by Hata and Yamaguti, which are sometimes referred to as Lebesgue’s singular f ␣(x) is analytic with respect to the parameter ␣ though it is
FIG. 2. Takagi function. FIG. 4. Lebesgue’s singular function for ␣ϭ0.75.
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a singular function of x, and there exists an interesting rela- tion between the parameter derivative of f ␣(x) and the Takagi function d f ͑x͉͒ ϭ2T͑x͒. ͑9͒ d␣ ␣ ␣ϭ1/2 In 1957, de Rham found that the Weierstrass function, the Takagi function and Lebesgue’s singular function as well as other singular functions are fully characterized as a unique FIG. 5. Schematic representation of the nonconservative reversible Baker solution f of a functional equation map. The shaded rectangle is expanded and the rest is contracted.
f ͑x͒ϭ f ͑x͒ϩg͑x͒, ͑10͒ a class of functions obeying de Rham’s equation, it is not the F case. As an example, consider the Fourier-Laplace transfor- where g(x) is a given function and is a contraction map- mation of f (x) ping with 0ϭ0 defined in the spaceF of bounded functions ␣ F 39 on the unit interval ͓0,1͔. Then, he generalized the func- 1 ix tional equation ͑10͒ to describe fractal continuous curves I͑͒ϵ ͵ e df␣͑x͒. such as the ones by Koch or Le´vy. The contraction mapping 0 means that the inequality By dividing the integral into the ones over ͓0,1/2͔ and ͓1/2,1͔ and changing the variable x to x/2 and (xϩ1)/2, respec- ʈ f 1Ϫ f 2ʈрʈf1Ϫf2ʈ tively, we have the recursion relation F F holds for some constant 0ϽϽ1 and for any functions f 1 1 1 ix/2 i͑xϩ1͒/2 and f , where the function norm ʈ ʈ is defined as the supre- I͑͒ϭ e df␣͑x/2͒ϩ e df␣͑͑xϩ1͒/2͒ 2 • ͵0 ͵0 mum: ʈ f ʈϵsup[0,1]͉f (x)͉. The existence of a unique solution of de Rham’s functional equation ͑10͒ immediately follows 1 1 ix/2 i/2 ix/2 49 ϭ␣ e df␣͑x͒ϩ͑1Ϫ␣͒e e df␣͑x͒ from Banach’s contraction mapping theorem and the fact ͵0 ͵0 that the mapping f fϩg is contraction. Let the mapping be →F ϭ͕␣ϩ͑1Ϫ␣͒ei/2͖I͑/2͒, F␣, where the de Rham equation ͑10͒ is used in the second ␣ f ͑2x͒, x͓0,1/2͔ 1 f ͑x͒ϵ ͑11͒ equality. Because I(0)ϭ͐0 df␣(x)ϭf␣(1)ϭ1, the above F␣, ͭ  f ͑2xϪ1͒, x͑1/2,1͔ recursion relation gives38,47,51,52 then the Weierstrass function W , the Takagi function T(x) ϱ a,2 1 n eix df x ϭI ϭ ␣ϩ 1Ϫ␣ ei/2 . ͑13͒ and Lebesgue’s singular function f ␣(x) are the unique solu- ͵ ␣͑ ͒ ͑ ͒ ͟ ͕ ͑ ͒ ͖ tion of ͑10͒, respectively, for ␣ϭϭa, g(x)ϭcosx; for 0 nϭ1 ␣ϭϭ1/2, g(x)ϭ͉xϪ͓xϩ1/2͔͉; and for ϭ1Ϫ␣, g(x) Note that, for ␣ϭ1/2, I(2m)ϭI(2)(ϭ0) (mϭ0,1, ) 39 ” ” ••• ϭ␣(xϪ1/2) with the step function. For more informa- and, hence, the Fourier-Laplace transformation of f ␣ does tion on singular functions, see Refs. 47 and 51. not satisfy the Riemann-Lebesgue lemma: lim ϱI()ϭ” 0, → Before closing this section, we remark on Riemann- that again implies the singular nature of f ␣ . Stieltjes integrals with respect to singular functions, which Further we notice that the formula ͑13͒ and the Hata- will appear in the next section. Suppose f (x) is of bounded Yamaguti relation ͑9͒ relate the Lebesgue’s singular function variation and (x) is continuous. Then, the Riemann- and the Takagi function to the Weierstrass functions Stieltjes integral of with respect to f is defined by the limit Im I͑2s͒ f ͑x͒ϭ␣Ϫ W ͑2sx͒ n ␣ ͚ 1/2,2 b sϾ0:odd s ͑x͒df͑x͒ϵ lim ͚ ͑k͕͒f ͑xk͒ ͵a kϭ1 Re I͑2s͒Ϫ1 max͑xkϪxkϪ1͒ 0 ˜ → ϩ W1/2,2͑2sx͒, sϾ͚0:odd s Ϫ f ͑xkϪ1͖͒, ͑12͒ 1 2 where ͕xk͖ is a partition of ͓a,b͔: aϭx0Ͻx1Ͻ•••Ͻxnϭb, 48,49 T͑x͒ϭ Ϫ ͚ 2 2 W1/2,2͑2sx͒, and xkϪ1рkрxk . Since the formula for the integration 2 sϾ0:odd s by parts where the sums run over positive odd integers, W1/2,2(x)is ˜ b b the Weierstrass function ͑6͒ and W1/2,2(x) is a singular func- ͑x͒df͑x͒ϩ f͑x͒d͑x͒ϭ͑b͒f͑b͒Ϫ͑a͒f͑a͒, tion obtained from ͑6͒ by replacing cos to sin. ͵a ͵a holds in general, the Riemann-Stieltjes integral of a function III. SRB MEASURE FOR A NONCONSERVATIVE of bounded variation with respect to a continuous function REVERSIBLE BAKER MAP can be defined as above. At first sight, the evaluation of the In thermostated systems, dynamics is nonconservative Riemann-Stieltjes integral ͑12͒ seems to be difficult. But, for due to the damping force mimicking a heat reservoir and
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thus, the forward time evolution is different from the back- distribution function, the symbol dy stands for the Riemann- 4–12 53 ward time evolution. However, it is time reversible. It is Stieltjes integral of Pt only with respect to the variable y therefore interesting to see how these two features are com- and the integration by parts is used in the second equality. patible, that we shall study with a simple map. One of the The evolution equation for Pt can be obtained easily from simplest nonconservative systems which have time reversal Eq. ͑15͒ symmetry is given by ͑Fig. 5͒ lPt͑lx,y/r͒, y͓0,r͔ P ͑x,y͒ϭ ͑x/l,ry͒, x͓0,l͔ tϩ1 ϩ Ϫ ϩ ⌽͑x,y͒ϭ ͑14͒ ͭ rPt͑rx l,͑y r͒/l͒ lPt͑lx,1͒, y͑r,1͔. ͭ ͑͑xϪl͒/r,lyϩr͒, x͑l,1͔, ͑17͒ where lϩrϭ1 and 0ϽlϽ1. The map is nonconservative Partial integration of the distribution function changes the since its Jacobian takes r/l for x͓0,l͔ and l/r for x prefactors from r/l and l/r, respectively, to r and l, which (l,1͔, both of which are different from 1. But the map has are strictly less than unity. Because of this, the evolution a time reversal symmetry represented by an involution equation ͑17͒ is similar to de Rham’s functional equation I(x,y)ϵ(1Ϫy,1Ϫx): I⌽Iϭ⌽Ϫ1. The Frobenius-Perron ͑10͒. equation governing the time evolution of distribution func- The next step in our construction of the singular measure tions ͑with respect to the Lebesgue measure͒ is given by is to calculate the long time limit of Pt . Putting yϭ1 in Eq. ͑17͒, we obtain tϩ1͑x,y͒ϭUt͑x,y͒ Ptϩ1͑x,1͒ϭrPt͑rxϩl,1͒ϩlPt͑lx,1͒. ͑18͒ ϵ ͵ dxЈ dyЈ ␦͓͑x,y͒Ϫ⌽͑xЈ,yЈ͔͒t͑xЈ,yЈ͒ Note that this is nothing but the Frobenius-Perron equation for a one-dimensional chaotic map ͑strictly speaking, an ex- l act map cf. Ref. 54͒ Sxϭx/l ͑for x͓0,l͔) and Sxϭ(x ͑lx,y/r͒, y͓0,r͔ r t Ϫl)/r ͑for x(l,1͔), which admits the Lebesgue measure as ϭ ͑15͒ r the invariant measure. Therefore, the normalization integral ͑rxϩl,͑yϪr͒/l͒, y͑r,1͔, ͐1 dx P (x,1) is invariant ͭ l t 0 t 1 1 where U stands for the Frobenius-Perron operator defined by ͵ dx Pt͑x,1͒ϭ ͵ dx PtϪ1͑x,1͒ the second equality and ␦͓•͔ is the two-dimensional delta 0 0 function. Since the map ⌽ is not conservative, the numerical 1 factors r/l and l/r different from 1 appear in the last expres- ϭ•••ϭ dx P0͑x,1͒, sion. ͵0 and is equal to the long time limit of the partially integrated A. An SRB measure for the forward time evolution 54 distribution function Pt(x,1). As will be shown in Appen- First we explain our algorithm to construct SRB mea- dix A, the convergence rate is ϵmax(l,r) sures and apply it to the forward time evolution. Our goal is 1 t to show that an expectation value of the dynamical function Pt͑x,1͒ϭ dxЈ P0͑xЈ,1͒ϩO͑ ͒ ͵0 (x,y) with respect to t converges for t ϩϱ to an expec- tation with respect to a singular measure→ given below, when t the initial distribution function is continuously differen- ϭ 0͑xЈ,yЈ͒dxЈ dyЈϩO͑ ͒. ͑19͒ 0 ͵ 2 tiable in x, and a dynamical function is continuously dif- [0,1] ferentiable in y and continuous in x. We remark that the In order to proceed with the calculation, one needs the convergence rate is controlled by the smoothness of the ini- following lemma. ͑For its proof, see Appendix A.͒ tial distribution function and the dynamical function, and the Lemma: Let be a linear contraction mapping with the F (0) t (1) condition given above is sufficient for the exponential con- contraction constant 0ϽϽ1. And let gtϵg ϩ g (2) vergence. ϩgt be a given function where is a constant satisfying (2) t The first step in our explicit construction of the singular Ͻ͉͉р1, and gt ϭO( ). Then, the solution of the func- measure is to express the expectation value by the partially tional equation integrated distribution function, i.e., f ϭ f ϩg , ͑20͒ tϩ1 F t t is given by ͑x,y͒t͑x,y͒dx dy ͵[0,1]2 ͑0͒ t ͑1͒ t f tϭ f ϱ ϩ f ϱ ϩO͑t ͒, ͑21͒ ϭ ͑x,y͒dy Pt͑x,y͒dx where f (0) and f (1) are the unique solutions of the following ͵[0,1]2 ϱ ϱ fixed point equations: 1 ͑0͒ ͑0͒ ͑0͒ y͑x,y͒Pt͑x,y͒dx dy, f ϱ ϭ f ϱ ϩg , ͑22͒ץ ϭ ͑x,1͒Pt͑x,1͒dxϪ ͵0 ͵[0,1]2 F ͑16͒ ͑1͒ ͑1͒ 1 ͑1͒ g y f ϱ ϭ f ϱ ϩ . ͑23͒ where Pt(x,y)ϵ͐0 dyЈ t(x,yЈ) is the partially integrated F
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Now we go back to the Eq. ͑17͒ for Pt(x,y), which can be rewritten as P ϭ P ϩg , ͑24͒ tϩ1 F t t where the contraction mapping and a function gt are, re- spectively, given by F lP͑lx,y/r͒, y͓0,r͔ P͑x,y͒ϭ ͑25͒ F ͭ rP͑rxϩl,͑yϪr͒/l͒, y͑r,1͔ and 0, y͓0,r͔ gt͑x,y͒ϵ ͭ lPt͑lx,1͒, y͑r,1͔
FIG. 6. Partially integrated distribution of the physical measure along y ¯ ͑0͒ t ph ϭg ͑y͒ 0͑xЈ,yЈ͒dxЈ dyЈϩO͑ ͒, ͑26͒ for the nonconservative reversible map with lϭ0.3. ͵[0,1]2 with Clearly it is absolutely continuous with respect to the 0, y͓0,r͔ Lebesgue measure along the expanding x-direction and, thus ¯g͑0͒͑y͒ϭ ͑27͒ ͭ l, y͑r,1͔. is an SRB measure. As studied in Ref. 38, the function Fl is nondecreasing, has zero derivatives almost every- The contraction constant of the mapping is ϭmax(l,r) where except for rϭlϭ1/2 with respect to the Lebesgue and Eq. ͑19͒ is used to derive the left-handF side of Eq. ͑26͒. measure and is Ho¨lder continuous with exponent ␦ As the Eqs. ͑24͒, ͑25͒, ͑26͒ and ͑27͒ satisfy the condition of ϭϪln max(l,r)/ln min(lϪ1,rϪ1)ϭ1 ͑i.e., ͉F (x)ϪF (y)͉ the lemma and the contraction mapping given by ͑25͒ is l l рA xϪy ). The graph of F is a fractal ͑Fig. 6͒, but its linear, the lemma implies F ͉ ͉ l fractal dimension is Dϭ1 as a result of the Besicovich- 56 t Ursell inequality: 1рDр2Ϫ␦ϭ1. Moreover, the one- Pt͑x,y͒ϭFl͑y͒ 0͑xЈ,yЈ͒dxЈ dyЈϩO͑t ͒, ͑28͒ ͵[0,1]2 dimensional measure defined by Fl is a multifractal two- scale Cantor measure, the dimension spectrum Dq ͑Ϫϱ where F (y) is the unique solution of de Rham’s functional 38,55 l ϽqϽϩϱ͒ of which is given as the solution of equation lqr͑1Ϫq͒Dqϩrql͑1Ϫq͒Dqϭ1. ¯ ͑0͒ Fl͑y͒ϭ Fl͑y͒ϩg ͑y͒ F There exists an interesting relation between Fl and Leb- lFl͑y/r͒, y͓0,r͔ esgue’s singular function f ␣ ϭ ͑29͒ Ϫ1 ͭ rFl͑͑yϪr͒/l͒ϩl, y͑r,1͔. ,Fl͑y͒ϭ f lؠ f r ͑y͒ By substituting Eq. ͑28͒ into Eq. ͑16͒ and employing the which immediately follows from the fact that the right hand integration by parts, we have side obeys the same functional equation as Fl . Note that, Ϫ1 since f r is continuous and strictly increasing, the inverse f r ͑x,y͒t͑x,y͒dx dy exists and is also strictly increasing. As a composite function ͵ 2 [0,1] Ϫ1 of two strictly increasing functions f l and f r , Fl is also strictly increasing. Because of these singular properties of ϭ ͑x,y͒dx dFl͑y͒ ͵[0,1]2 Fl , the physical measure ph is singular along the contract- ing y-direction. t The physical measure ph is mixing with respect to the ϫ 0͑xЈ,yЈ͒dxЈ dyЈϩO͑t ͒. ͑30͒ ͵ 2 [0,1] map ⌽. Indeed, by considering Pt(x,y) y y We remind the reader that Eq. ͑30͒ holds for any continuous ϭ͐0t(x,yЈ)dFl(yЈ) instead of Pt(x,y)ϭ͐0t(x,yЈ)dyЈ and following exactly the same arguments as above, one ob- function (x,y) and any integrable function 0(x,y) which tains are continuously differentiable, respectively, in y and x.If0 is normalized with respect to the Lebesgue measure, Eq. ͑30͒ gives lim ͑x,y͒t͑x,y͒dx dFl͑y͒ ͵ 2 t ϱ [0,1] →
lim ͑x,y͒t͑x,y͒dx dyϭ ͑x,y͒dx dFl͑y͒. ͵ 2 ͵ 2 t ϱ [0,1] [0,1] ϭ ͑x,y͒dx dFl͑y͒, ͑33͒ → ͵ 2 ͑31͒ [0,1] provided that (x,y) and (x,y) are continuously differen- This shows that the physical measure of the system is 0 ph tiable, respectively, in y and x and that is normalized: given by 0 ͐0(x,y)dx dFl(y)ϭ1. By using this fact, the Lyapunov ph͓͑0,x͒ϫ͓0,y͒͒ϭxFl͑y͒. ͑32͒ exponents can be analytically calculated as follows: The
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Jacobian matrix for ⌽ is diagonal, and the logarithm of the component along the expanding x direction, the local ex- ͑x,y͒t͑x,y͒dx dy ͵[0,1]2 panding rate ⌳x(x,y), is ¯ Ϫln l, x͓0,l͔ ϭ ͑x,y͒dFr͑x͒dy 0͑xЈ,yЈ͒dxЈ dyЈ ⌳ ͑x,y͒ϭ ͵[0,1]2 ͵[0,1]2 x ͭϪln r, x͑l,1͔. ϩO͉͑t͉͉t͉͒, ͑36͒ The Lyapunov exponent along the x direction ͑the positive where tϭ0,Ϫ1,Ϫ2, and a singular function ¯F is given Lyapunov exponent͒ is defined as the average of ⌳x(x,y) ••• r ¯ over an orbit starting from some initial point (x,y). Since the by Fr(x)ϭ1ϪFl(1Ϫx). As before, Eq. ͑36͒ implies that the ¯ ¯ system is ergodic, the Lyapunov exponent can be obtained asymptotic physical measure ph is given by ph(͓0,x) from Eq. 2 using the measure ¯ ¯ ͑ ͒ ph ϫ͓0,y))ϵFr(x)y. The measure ph is then absolutely con- tinuous with respect to the Lebesgue measure along y direc- 1 T t tion and singular along x-direction. This corresponds to the x͑⌽,ph͒ϵ lim ͚ ⌳x͑⌽ ͑x,y͒͒ T ϩϱTtϭ0 fact that the expanding and contracting directions are inter- → ¯ changed for the backward motion. Actually, the measure ph ϭ ⌳x͑x,y͒dx dFl͑y͒ is precisely the one obtained from ph via the time reversal ͵[0,1]2 ¯ ¯ operation I: phϭIph . The measure ph is again mixing ϭϪl ln lϪr ln r. ͑34͒ with respect to the backward time evolution ⌽Ϫt (tϭ0,1,•••). And the Lyapunov exponents are calculated as Similarly, the Lyapunov exponent along the y-direction ͑the the phase space averages of the local scaling rates for the negative Lyapunov exponent͒ is Ϫ1 ¯ inverse map ⌽ with respect to ph . For example, the posi- ¯ y͑⌽,ph͒ϭl ln rϩr ln l. ͑35͒ tive Lyapunov exponent is the ph-average of the local ex- ¯ The sum of the two Lyapunov exponents is negative panding rate: ⌳y(x,y)ϭϪ(rϪy)ln rϪ(yϪr)ln l and is equal to that for the original map ⌽ l ͑⌽, ͒ϩ ͑⌽, ͒ϭ͑rϪl͒ln Ͻ0. x ph y ph ͩ rͪ ͑⌽Ϫ1,¯ ͒ϭ ⌳¯ ͑x,y͒dF¯ ͑x͒dy y ph y r ͵[0,1]2 Hence, areas are contracted on average by the map ⌽ and the ϭϪr ln rϪl ln lϭ ⌽, ͒. ͑37͒ map possesses an attractor A. From Eq. ͑31͒, one finds that x͑ ph the attractor A is the support of the SRB measure ph . When The negative Lyapunov exponents of the two maps are also lϭ” 1/2, the two-dimensional Lebesgue measure of the attrac- the same tor A is zero since the function F has zero derivatives almost l Ϫ1 ¯ everywhere with respect to the Lebesgue measure. More- x͑⌽ ,ph͒ϭy͑⌽,ph͒. ͑38͒ 57 over, according to Young’s formula ͑which is the Kaplan- The equality of Lyapunov exponents for (⌽, ) and 16,17,30,58 ph Yorke formula for two-dimensional ergodic sys- Ϫ1 ¯ (⌽ ,ph) is a general consequence of the time reversal tems͒, the information dimension of A is given by symmetry of the system. ¯ Ϫ1 x͑⌽,ph͒ l ln lϩr ln r We notice that the natural invariant measure ph of ⌽ D ϭ1ϩ ϭ1ϩ Ͻ2, I ͑⌽, ͒ ͯl ln rϩr ln lͯ is also invariant under ⌽ as follows straightforwardly from ͉ y ph ͉ the reversibility of ⌽ which is less than two. Therefore, A is a fractal set. On the ¯ Ϫ1 ¯ other hand, since the function Fl is strictly increasing, for ph͑⌽ ͕͓0,x͒ϫ͓0,y͖͒͒ϭph͓͑0,x͒ϫ͓0,y͒͒, any nonempty rectangle ͓x0 ,x0ϩ⑀)ϫ͓y 0 ,y 0ϩ⑀Ј)(⑀Ͼ0 is equivalent to and ⑀ЈϾ0), we have ¯ ¯ ph͓͑0,x͒ϫ͓0,y͒͒ϭph͑⌽͕͓0,x͒ϫ͓0,y͖͒͒. ͑39͒ ph͓͑x0,x0ϩ⑀͒ϫ͓y0,y0ϩ⑀Ј͒͒ That is, we may think of ¯ as a repelling measure for ⌽,in ϭ⑀͕F ͑y ϩ⑀Ј͒ϪF ͑y ͖͒Ͼ0, ph l 0 l 0 the sense that, while it is indeed invariant, any slight devia- which implies that Aപ͓x0 ,x0ϩ⑀)ϫ͓y 0 ,y 0ϩ⑀Ј)ϭ” л and, tions from this measure, if they are absolutely continuous hence, the attractor A is dense in the whole phase space with respect to the Lebesgue measure, will evolve toward the 2 ͓0,1) . This phase-space structure is in contrast to the one of measure ph for the attractor for ⌽͓cf. Eq. ͑31͔͒. In short, a dissipative system usually studied30 ͑see also the next sec- we find, for a nonconservative reversible map ⌽, different ¯ tion͒, where an attractor is a Cantor-like set. SRB measures ph and ph for the forward and backward B. An SRB measure for the backward time evolution time evolutions, respectively. And for each time evolution, one plays a role of an attracting measure and the other a role Now we consider the backward time evolution. In the of a repelling measure in the sense just explained. This ob- same way as the forward evolution, we find that another servation is a key element of the compatibility between dy- ¯ partially integrated distribution function Pt(x,y) namical reversibility and irreversible behavior of statistical ϭ͐x dxЈ (xЈ,y) converges for t Ϫϱ and we have ensembles. Indeed, when the dynamics is reversible and sta- 0 t →
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tistical ensembles irreversibly approach a stationary en- semble ph for t ϩϱ, there should exist another stationary ¯ → ensemble ph which is obtained from ph by the time rever- ¯ sal operation. However, the new stationary ensemble ph is repelling and, thus its existence is compatible with the irre- versible behavior of statistical ensembles. ¯ Further, the measure ph is mixing with respect to the t forward time evolution ⌽ (tϭ0,1,•••). Then it is interest- ing to investigate the relation between Lyapunov exponents FIG. 7. Schematic representation of the Baker map with a Cantor-like in- ¯ and the Kolmogorov-Sinai ͑KS͒ entropy for (⌽,ph). The variant set. The points in the black rectangles are removed. The other shaded ¯ rectangles are mapped onto the corresponding ones. Lyapunov exponents for (⌽,ph) are easily found to be
¯ x͑⌽,ph͒ϭϪr ln lϪl ln r, ͑40͒ IV. SRB MEASURE FOR A BAKER MAP WITH A CANTOR-LIKE INVARIANT SET and Transport properties are also studied for open Hamil- tonian systems with a flux boundary condition13–15 or with ¯ 12,20–26 y͑⌽,ph͒ϭr ln rϩl ln l. ͑41͒ an absorbing boundary condition. Breymann, Te´l and Vollmer used an open nonconservative system to study an It is useful to note that the positive ͑negative͒ Lyapunov interrelation between the thermostated systems approach and ¯ 29 exponent of ph can be found by changing the sign of the the open systems approach. Further, a model used by Kauf- 59 negative ͑positive͒ Lyapunov exponent of ph . This fact is mann, Lustfeld, Ne´meth and Sze´pfalusky to investigate de- widely used in the analysis of the Lyapunov spectrum of terministic transient diffusion is a nonconservative open sys- thermostated many particle systems.4,7,8 In our case this re- tem. One of the important features of those open systems is sult follows from the observations that the existence of escape. So we investigate how escape affects the physical measure, by using a Baker map with a Cantor- like invariant set.10,30,58 The map is defined on the unit ͑⌽,¯ ͒ϭϪ ͑⌽, ͒ ͑42͒ x ph y ph square ͓0,1͔2 ͑Fig. 7͒:
and ͑x/l,⌳1y͒, x͓0,l͔ ⌿͑x,y͒ϭ ͑47͒ ͭ ͑͑xϪa͒/r,⌳2yϩb͒, x͓a,aϩr͔, ¯ y͑⌽,ph͒ϭϪx͑⌽,ph͒. ͑43͒ where 0Ͻlрa,0Ͻrр1Ϫa,0Ͻ⌳1рband 0Ͻ⌳2р1Ϫb. Here we introduce an escape for points x(l,a)ഫ(aϩr,1͔ Thus the Lyapunov spectrum changes sign under the ex- and an inhomogeneity. For ⌳ Ͻl and ⌳ Ͼr,orfor⌳Ͼl ¯ 1 2 1 change of ph and ph . Now we turn to a calculation of the and ⌳2Ͻr, the map ⌿ is partially attractive and partially KS-entropy. First we note that the KS-entropy of (⌽,ph)is repelling as discussed in the previous section and, for ⌳1 ϭr, ⌳2ϭl, and lϩrϭ1, it is reduced to the previous model. hKS͑⌽,ph͒ϭx͑⌽,ph͒, ͑44͒ It is everywhere attractive for ⌳1Ͻl and ⌳2Ͻr, is conser- 16 vative for ⌳1ϭl and ⌳2ϭr and is everywhere repelling for which follows from the Pesin’s identity, since ph is the SRB measure for the map ⌽, and which can also be com- ⌳1Ͼl and ⌳2Ͼr. Note that the last case is possible only puted directly from the entropy of the generating partition when there exists an escape: lϩrϽ1. formed by the two elements 0рxрl and lрxр1. From the We show that when the initial distribution function 0 is continuously differentiable in x, and a dynamical function same partition, it is readily seen that the entropy of (⌽,¯ ) ph is continuously differentiable in y and continuous in x,an is also expectation value of the dynamical function (x,y) with re- ¯ spect to t decays exponentially hKS͑⌽,ph͒ϭx͑⌽,ph͒. ͑45͒
Then the difference between the positive Lyapunov exponent ͑x,y͒t͑x,y͒dx dy ͵[0,1]2 ¯ and the KS-entropy for (⌽,ph)is ϭt ͑x,y͒dx dG͑y͒ ¯ ¯ ͵ 2 x͑⌽,ph͒ϪhKS͑⌽,ph͒ϭϪy͑⌽,ph͒Ϫx͑⌽,ph͒у0, [0,1] ͑46͒ t ϫ 0͑x,y͒dH͑x͒dyϩO͑tЈ ͒, ͑48͒ which is strictly positive for lϭ” 1/2 since the right hand side ͵[0,1]2 is just the phase space contraction rate. Therefore, the mixing where the decay rate is equal to the remainder volume per ¯ system (⌽,ph) violates Pesin’s identity, but satisfies iteration: ϵlϩr, Јϭmax(l,r)(р). The functions G and 16 ¯ Ruelle’s inequality, as it should be since the measure ph is H are ͑possibly͒ singular functions defined as the unique not an SRB measure for ⌽. solutions of de Rham equations
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l y G , y͓0,⌳ ͔ lϩr ͩ ⌳ ͪ 1 1 l , y͑⌳1 ,b͒ G͑y͒ϭ lϩr ͑49͒ r yϪb l G ϩ , y͓b,⌳ ϩb͔ lϩr ͩ ⌳ ͪ lϩr 2 2 Ά 1, y͑⌳2ϩb,1͔ and l x H , x͓0,l͔ lϩr ͩ l ͪ l , x͑l,a͒ FIG. 8. Partially integrated distribution of the physical measure ph along x H͑x͒ϭ lϩr ͑50͒ for the Baker map with a Cantor-like invariant set. The parameters are ⌳1 r xϪa l ϭ0.4, ⌳2ϭ0.3, bϭ0.5 and l/rϭ0.8. H ϩ , x͓a,aϩr͔ lϩr ͩ r ͪ lϩr Ά 1, x͑aϩr,1͔. First we consider the physical measure. The formula 48 holds even when lϩrϽ1. By setting 1in 48 , the The derivation of Eq. ͑48͒ is outlined in Appendix B. Now ͑ ͒ ϵ ͑ ͒ renormalization factor (x,y)dx dy is found to be we investigate the implications of Eq. ͑48͒ in cases without ͐t and with escape separately. t t͑x,y͒dx dyϭ 0͑x,y͒dH͑x͒dy ͵[0,1]2 ͵[0,1]2 A. Baker map without escape ϩO͑tЈt͒. Consider first the case where there is no escape (lϩr Hence, the expectation value of at time t is ϭ1). The unit square is then contracted onto a set which is a direct product of a Cantor set in the y-direction and the unit ͐[0,1]2͑x,y͒t͑x,y͒dx dy ͗͘tϵ interval, ͓0,1͔,inthex-direction. This direct product set is ͐[0,1]2t͑x,y͒dx dy nothing but the strange attractor of ⌿, whose Hausdorff di- Ј t mension 1ϽHDϽ2. In this case, H(x)ϭx and ͑48͒ reduces ϭ ͑x,y͒dx dG͑y͒ϩO t . ͑53͒ to an expression similar to ͑36͒ for the previous example. ͵[0,1]2 ͩ ͭ ͮ ͪ And the function G defines an invariant mixing measure ph This implies that the physical measure defined by Eq. ͑5͒ is identical to that for the Baker map without escape ph͓͑0,x͒ϫ͓0,y͒͒ϵxG͑y͒. ͑51͒ 0,x ϫ 0,y ϭxG y , 54 The measure ph is the one which provides long-term aver- ph͓͑ ͒ ͓ ͒͒ ͑ ͒ ͑ ͒ ages of dynamical functions and thus, is a physical measure. provided that the ratio l/r is common in two cases. As in the Indeed, when ͐ (x,y)dx dyϭ1, ͑48͒ gives 0 previous case, the physical measure ph is singular and is supported by a direct product of a Cantor set C1 along y and ͗͘tϵ ͑x,y͒t͑x,y͒dx dy the unit interval along x: ͓0,1͔ϫC1, which is the unstable ͵[0,1]2 manifold of the fractal repeller. Since it is smooth along the expanding x-direction, it is an SRB-like measure ͑in the ϭ ͑x,y͒dx dG͑y͒ϩO͑t͕Ј͖t͒. ͑52͒ ͵[0,1]2 sense that, although not invariant, it is smooth along the expanding direction͒. Since is smooth ͑more precisely absolutely continuous ph We observe that the support of is the unstable mani- with respect to the Lebesgue measure͒ along the expanding ph fold of the repeller and is not an invariant set with respect to coordinate x, it is an SRB measure. As shown in Fig. 8, the ⌿. Accordingly, the physical measure is not an invariant graph of G(y) is a typical devil’s staircase and the measure ph measure. Indeed, the measure ph satisfies ph is singular. Note that the support of ph is the strange Ϫ1 attractor of ⌿. ph͑⌿ ͕͓0,x͒ϫ͓0,y͖͒͒ϭ͑lϩr͒ph͓͑0,x͒ϫ͓0,y͒͒, ͑55͒
B. Baker map with escape which implies that ph shrinks as time goes on. This can be seen immediately from the functional equation for G. For When lϩrϽ1, almost all the points escape the unit example, when y͓0,⌳ ͔, square and there appears a fractal repeller, which is singular 1 Ϫ1 both in expanding and contracting directions. Then, the in- ph͑⌿ ͕͓0,x͒ϫ͓0,y͖͒͒ variant measure supported by the fractal repeller is different ϭ ͓͑0,lx͒ϫ͓0,y/⌳ ͒͒ϭlxG͑y/⌳ ͒ from the physical measure defined by Eq. ͑5͒, which is not ph 1 1 23,24 invariant but conditionally invariant under ⌿. ϭ͑lϩr͒xG͑y͒ϭ͑lϩr͒ph͓͑0,x͒ϫ͓0,y͒͒.
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Then, since the measure ph satisfies each trajectory starting from a reactant state to a product Ϫ1 state represents an individual reaction R P ͑cf., Ref. 60, ph͑⌿ E͒ → ͑E͒ϭ , and references therein͒. A connection between chemical re- ph Ϫ1 2 ph͑⌿ ͓0,1͒ ͒ actions and the escape-rate formalism was investigated by Dorfman and Gaspard22 and a Baker-type model of a chemi- for any Borel set Eʚ͓0,1)2, it is conditionally invariant.23,24 cal reaction was studied by Elskens and Kapral.40 In this Now we turn to an invariant measure on the repeller, in section, we introduce a Baker-type model of a reaction which is defined by R I P and study its statistical properties for two cases: In ↔ ↔ in͓͑0,x͒ϫ͓0,y͒͒ϭH͑x͒G͑y͒. ͑56͒ one case, the system is closed, and in the other case, a flux boundary condition is imposed. Now we begin with the phe- The invariance can be seen straightforwardly from the func- nomenology. tional equations for G and H. For example, when y ͓b,⌳2ϩb͔, one has ⌿Ϫ1 0,x͒ϫ 0,y͒ ͒ in͑ ͕͓ ͓ ͖ A. Phenomenology ϭin͓͑0,lx͒ϫ͓0,1͒ഫ͓a,rxϩa͒ϫ͓0,͑yϪb͒/⌳2͒͒ For a chemical reaction R I P, the discrete-time ver- sion of the phenomenological↔ rate↔ equation is ϭin͓͑0,lx͒ϫ͓0,1͒͒ϩin͓͑a,rxϩa͒ϫ͓0,͑yϪb͒/⌳2͒͒
Rtϩ1ϭkRRRtϩkRIIt , ϭH͑lx͒G͑1͒ϩ͕H͑rxϩa͒ϪH͑a͖͒G͑͑yϪb͒/⌳2͒) l r Itϩ1ϭkIRRtϩkIIItϩkIPPt , ͑57͒ ϭH͑x͒ ϩ H͑x͒G͑͑yϪb͒/⌳2͒) lϩr lϩr Ptϩ1ϭk PIItϩkPPPt ,
ϭH͑x͒G͑y͒ϭin͓͑0,x͒ϫ͓0,y͒͒. where Rt , It , and Pt are concentrations of the reactant R, intermediate I, and product P, respectively, and k (A,B Since H(x) is a fractal function similar to those discussed in AB ϭR,I,orP) are rate coefficients. Since the sum RtϩIt Sec. II, the invariant measure is singular both along the con- ϩP is preserved, the rate coefficients satisfy a sum rule tracting and expanding directions. This can be easily under- t stood as follows: Since the map ⌿ eventually transforms the kRRϩkIRϭ1, unit square into the unstable manifold of the repeller, which kRIϩkIIϩkPIϭ1, ͑58͒ is a direct product of a Cantor set C1 along y and the unit interval along x: ͓0,1͔ϫC1, the measure becomes singular kIPϩkPPϭ1. along y. On the other hand, only the orbits starting from the The stationary state solution of Eq. ͑57͒ is then given by stable manifold of the repeller remain in the unit square and the stable manifold is a direct product set of the unit interval Rst kRI Pst k PI along y and a Cantor set C along x: C ϫ 0,1 . Thus the ϭ , ϭ . ͑59͒ 2 2 ͓ ͔ Ist kIR Ist kIP invariant set is a subset of C2ϫ͓0,1͔. As a result, the invari- ant measure becomes singular also along x. Actually, the Now we consider a stationary solution of Eq. ͑57͒ under a flux boundary condition, where the concentrations of the direct product of the two Cantor sets C2ϫC1 is the fractal repeller of ⌿. reactant R and the product P are fixed to given values Rex In a similar argument to the derivation of ͑48͒, one can and Pex , respectively. This may be realized, e.g., by intro- ducing source terms to the equations for the reactant and show that the invariant measure in is mixing with respect to product and by adjusting them so as to keep the values of Rt ⌿. As shown in Appendix C, the invariant measure in is a Gibbs measure.1,12,22–28 and Pt constant. Equation ͑57͒ is then reduced to
Itϩ1ϭkIRRexϩkIIItϩkIPPex , ͑60͒ V. A BAKER-TYPE MAP UNDER A FLUX BOUNDARY which has a stationary solution CONDITION kIRRexϩkIPPex I ϭ . ͑61͒ To illustrate a stationary state for an open system with a fl 1Ϫk flux boundary condition, we study a simple model of a II chemical reaction. In simple reactions such as R I P, the The deviation ␦ItϵItϪIfl of the intermediate concentration reactant R, the intermediate I and the product P↔consist↔ of from the stationary value decays exponentially the same atoms, and they can be specified by configurations ␦I ϭk ␦I ϭ ϭkt ␦I . ͑62͒ of atoms, or points in the atomic-configuration space. An t II tϪ1 ••• II 0 example of the reaction R I P is an isomerization, or a The stationary state admits a nonvanishing concentration change in the conformation↔ ↔ of a molecule, such as the flow from the reactant to the product ‘‘chair’’ to ‘‘boat’’ transformation of cyclohexane, where R k PIkIR kRIkIP is the chair-shaped isomer, P is the boat-shaped isomer and I J ϭk I Ϫk P ϭ R Ϫ P , ͑63͒ R P PI fl IP ex 1Ϫk ex 1Ϫk ex is an intermediate unstable isomer, all of which consist of six → II II carbon atoms and twelve hydrogen atoms. Hence, the reac- and, hence, can be regarded as a stationary state under a flux tion process can be regarded as a dynamical process where boundary condition.
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B. A Baker-type model and a stationary state for a Area preserving asymmetric Baker maps are used to describe closed system the dynamics of the reactant and product states, and a Baker We introduce a Baker-type model of the reaction process map similar to the one discussed in the previous section is R I P. Microscopic dynamical states of each species R, used for the intermediate state dynamics. The model is then ↔ ↔ 2 I,orPare represented by the points in a unit square (0,1͔ . defined by ͑Fig. 9͒
x I: ,⌳ y , x͑0,l͔ ͩ l 1 ͪ xϪl R: ,͑bϪ⌳ ͒y , x͑l,a͔ ͩ aϪl 1 ͪ ⌿Ј͑I:x,y͒ϭ ͑64͒ xϪa I: ,⌳ yϩb , x͑a,aϩr͔ ͩ r 2 ͪ xϪaϪr P: ,͑1ϪbϪ⌳ ͒y , x͑aϩr,1͔, Ά ͩ 1ϪaϪr 2 ͪ
x I: ,͑bϪ⌳ ͒yϩ⌳ , x͑0,bϪ⌳ ͔ ͩ bϪ⌳ 1 1 ͪ 1 1 ⌿Ј͑R:x,y͒ϭ ͑65͒ xϪbϩ⌳1 R: ,͑1Ϫbϩ⌳ ͒yϩbϪ⌳ , x͑bϪ⌳ ,1͔, ͭ ͩ 1Ϫbϩ⌳ 1 1 ͪ 1 1
x I: ,͑1ϪbϪ⌳ ͒yϩbϩ⌳ , x͑0,1ϪbϪ⌳ ͔ ͩ 1ϪbϪ⌳ 2 2 ͪ 2 2 ⌿Ј͑P:x,y͒ϭ ͑66͒ xϪ1ϩbϩ⌳2 P: ,͑bϩ⌳ ͒yϩ1ϪbϪ⌳ , x͑1ϪbϪ⌳ ,1͔, ͭ ͩ bϩ⌳ 2 2 ͪ 2 2
where (x,y) denotes a point in a unit square and ␣ (␣ Q0(␣:x,y) does not depend on x. Then, as seen from the ϭR,I,orP) distinguishes different species. expressions of ¯ and ¯R , the partially integrated function F t As before, we study the time evolution of a measure Qt(␣:x,y) at time t is also independent of x. Particularly, its starting from an initial measure which is absolutely continu- value Qt(␣:x,yϭ1)ϵQt(␣)atyϭ1 obeys ous with respect to the two-dimensional Lebesgue measure. The Frobenius-Perron equation for the density function Qtϩ1͑R͒ϭ͑1Ϫbϩ⌳1͒Qt͑R͒ϩ͑aϪl͒Qt͑I͒, t(␣:x,y)(␣ϭR,I,orP) is obtained from Qtϩ1͑I͒ϭ͑bϪ⌳1͒Qt͑R͒ϩ͑lϩr͒Qt͑I͒
ϩ͑1ϪbϪ⌳2͒Qt͑P͒, ͑68͒ tϩ1͑␣:x,y͒ϭ dx dy ␦͑͑␣:x,y͒ ϭ͚R,I,P ͵[0,1]2 Qtϩ1͑P͒ϭ͑1ϪaϪr͒Qt͑I͒ϩ͑bϩ⌳2͒Qt͑P͒. Ϫ⌿Ј͑:xЈ,yЈ͒͒t͑:xЈ,yЈ͒, Since the distribution is uniform along the x direction, Qt(␣) where the delta function ␦((␣:x,y)Ϫ(:xЈ,yЈ)) stands for is equal to the total probability of finding the system in a the product ␦␣,␦(xϪxЈ)␦(yϪyЈ). By integrating the species ␣: Qt(␣)ϭ͐ dx dy t(␣:x,y). Therefore, Eq. ͑68͒ Frobenius-Perron equation with respect to y, one obtains the has exactly the same form as Eq. ͑57͒ where the correspond- evolution equation for the partially integrated distribution ing rate coefficients are given by y function Qt(␣:x,y)ϭ͐0 dyЈ t(␣:x,yЈ): kRRϭ1Ϫbϩ⌳1 , kRIϭaϪl, ¯ ¯ Qtϩ1͑␣:x,y͒ϭ Qt͑␣:x,y͒ϩRt͑␣:x,y͒, ͑67͒ F kIRϭbϪ⌳1 , kIIϭlϩr, kIPϭ1ϪbϪ⌳2, ͑69͒ ¯ ¯ where a contraction mapping and a functional Rt of F k PIϭ1ϪaϪr, kPPϭbϩ⌳2. Qt(␣:x,1) are given in Appendix D ͓cf., Eqs. ͑D2͒, ͑D3͒, ͑D4͒ and Eqs. ͑D5͒, ͑D6͒, ͑D7͒, respectively͔. Note that the rate coefficients given above satisfy the sum Now we consider the case where the initial measure is rule Eq. ͑58͒ and hence, the sum Qt(R)ϩQt(I)ϩQt(P)is uniform along the expanding x direction and thus, constant in time. We also remark that the rate coefficients
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from the intermediate states to the reactant states and, thus, its Lebesgue area (aϪl) corresponds to the transition prob- ability kRI from I to R, or we have the second equation of ͑69͒. The other rate coefficients can be obtained in the same way. According to the discussion given in the previous sub- section, the stationary solution of ͑68͒ is aϪl 1ϪaϪr Qst͑R͒ϭ Qst͑I͒, Qst͑P͒ϭ Qst͑I͒, bϪ⌳1 1ϪbϪ⌳2 ͑70͒ and the corresponding distribution is given by FIG. 9. Schematic representation of the Baker-type map describing a chemi- cal reaction process R I P. Three unit squares describe the dynamical ↔ ↔ aϪl states of the reactant, intermediate and product, respectively. The shaded Qst͑R:x,y͒ϭ Qst͑I͒qst͑R:y͒, rectangles are mapped onto the corresponding ones. bϪ⌳1
Qst͑I:x,y͒ϭQst͑I͒qst͑I:y͒, ͑71͒ 1ϪaϪr ͑69͒ admit a simple geometrical interpretation. As an ex- Qst͑P:x,y͒ϭ Qst͑I͒qst͑P:y͒, 1ϪbϪ⌳2 ample, we consider kRI , which is the transition probability from the intermediate I to the reactant R. According to the where the singular functions qst(␣:y)(␣ϭR,I, and P) are definition of the map ⌿Ј, a rectangle (l,a͔ϫ(0,1͔ moves the unique solutions of de Rham equations
y lq I: , y͑0,⌳ ͔ stͩ ⌳ ͪ 1 1
yϪ⌳1 lϩ͑aϪl͒q R: , y͑⌳ ,b͔ stͩ bϪ⌳ ͪ 1 1 q ͑I:y͒ϭ ͑72͒ st yϪb aϩrq I: , y͑b,⌳ ϩb͔ stͩ ⌳ ͪ 2 2
yϪbϪ⌳2 aϩrϩ͑1ϪaϪr͒qst P: , y͑⌳2ϩb,1͔, Ά ͩ 1ϪbϪ⌳2 ͪ
y ͑bϪ⌳ ͒q I: , y͑0,bϪ⌳ ͔ 1 stͩ bϪ⌳ ͪ 1 1 qst͑R:y͒ϭ ͑73͒ yϪbϩ⌳1 bϪ⌳ ϩ͑1Ϫbϩ⌳ ͒q R: , y͑bϪ⌳ ,1͔, ͭ 1 1 stͩ 1Ϫbϩ⌳ ͪ 1 1
y ͑1ϪbϪ⌳ ͒q I: , y͑0,1ϪbϪ⌳ ͔ 2 stͩ 1ϪbϪ⌳ ͪ 2 2 qst͑P:y͒ϭ ͑74͒ yϪ1ϩbϩ⌳2 1ϪbϪ⌳ ϩ͑bϩ⌳ ͒q P: , y͑1ϪbϪ⌳ ,1͔. ͭ 2 2 stͩ bϩ⌳ ͪ 2 2
One can show, by exactly the same method as before, that C. A stationary state under a flux boundary condition
the partially integrated function Qt(␣:x,y) approaches the stationary state Qst(␣:x,y) provided that the initial function As shown in Sec. V A, one has a flux from the reactant Q0(␣:x,y) is continuously differentiable with respect to x. to the product when their concentrations are fixed to the val- Then since the asymptotic measure Qst(␣:x,y) is absolutely ues different from the equilibrium ones. This can be realized continuous with respect to the Lebesgue measure along the in the Baker-type model by fixing the measures of the unit expanding x direction, it is the SRB measure. Note that, in squares corresponding to the reactant and the product to uni- this case, the quantity Qst(I) is a functional of the initial form Lebesgue measures with different densities. So we set distribution Q0(␣:x,y). Qt(R:x,y)ϭRexy and Qt(P:x,y)ϭPexy (tϭ0,1,•••). Note
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that the same procedure was used to achieve the flux bound- ary condition for the finite multi-Baker chain.13 Then, we have only one time-dependent variable Qt(I:x,y), which will be abbreviated as Qt(I:x,y)ϵQt(x,y). Then the equa- tion of motion ͑67͒ for the partially integrated distribution function Qt becomes
Q ͑x,y͒ϭ ЈQ ͑x,y͒ϩR ͑x,y͒ϩS͑y͒, ͑75͒ tϩ1 F t t where the contraction mapping Ј and Q (x,1)-dependent F t part Rt(x,y) are given by Eqs. ͑B3͒ and ͑B4͒ of Appendix B, respectively, and the source term S(y) is due to the reactant and product states
FIG. 10. Partially integrated distribution of the stationary measure along 0, y͓0,⌳1͔ fl x for the reaction model under a flux boundary condition. The parameters Rex͑yϪ⌳1͒, y͑⌳1 ,b͒ are chosen as ⌳ ϭ0.4, ⌳ ϭ0.3, bϭ0.5, lϭ0.4, rϭ0.5, R ϭ0, and P S y ϭ 1 2 ex ex ͑ ͒ ϭ(1ϪlϪr)/(1ϪbϪ⌳2). Rex͑bϪ⌳1͒, y͓b,⌳2ϩb͔
Ά Rex͑bϪ⌳1͒ϩPex͑yϪbϪ⌳2͒, y͑⌳2ϩb,1͔. ͑76͒ bϪ⌳1 ͓͑0,x͒ϫ͓0,y͒͒ϭ R x ͑y͒ By exactly the same argument as before, Eqs. ͑75͒ and ͑76͒ fl 1ϪlϪr ex R are found to have a solution 1ϪbϪ⌳2 ϩ Pexx P͑y͒. ͑80͒ bϪ⌳1 1ϪbϪ⌳2 1ϪlϪr Q ͑x,y͒ϭ R ͑y͒ϩ P ͑y͒ t 1ϪlϪr ex R 1ϪlϪr ex P As it is smooth ͑i.e., absolutely continuous with respect to 1 bϪ the Lebesgue measure͒ along the expanding coordinate x,it t ⌳1 ϩ dH͑xЈ͒ Q0͑xЈ,1͒Ϫ Rex is an SRB measure. However, the measure is different ͵0 ͭ 1ϪlϪr fl from the SRB measures obtained before since it is absolutely 1ϪbϪ⌳2 continuous with respect to y. It is only weakly singular in the Ϫ P G͑y͒ϩO͑tЈt͒, ͑77͒ 1ϪlϪr exͮ sense that the density is discontinuous on a Cantor-like set ͑cf., Fig. 10͒. Actually, such stationary measures become 13,15 where G(y) and H(x) are singular functions introduced in truly singular only for infinite systems. It is interesting to observe that the conditionally invariant measure (͓0,x) Sec. IV ͓cf. Eqs. ͑49͒ and ͑50͔͒, and the functions R(y) and ph ϫ͓0,y))ϭxG(y) appears in the time evolution of the mea- P(y) are the unique solutions of de Rham equations sure ͑77͒ as the decay mode, i.e., as the Pollicott-Ruelle reso- y nance. lR , y͓0,⌳1͔ Now we investigate macroscopic aspects. The total mea- ͩ ⌳ ͪ 1 1 2 sure t((0,1͔ )ϵ͐0 dx Qt(x,1) corresponding to the con- 1ϪlϪr centration of the intermediate I is 1Ϫrϩ ͑yϪb͒, y͑⌳1 ,b͒ ͑y͒ϭ bϪ⌳1 ͑78͒ R bϪ 1ϪbϪ 2 ⌳1 ⌳2 t yϪb t͑͑0,1͔ ͒ϭ Rexϩ Pexϩ ␦0 r ϩ1Ϫr, y͓b,⌳ ϩb͔ 1ϪlϪr 1ϪlϪr Rͩ ⌳ ͪ 2 2 ϩO͑tЈt͒, ͑81͒ Ά 1, y͑⌳2ϩb,1͔ where ␦0 stands for the deviation of the initial measure y from the stationary state fl l , y͓0,⌳ ͔ Pͩ ⌳ ͪ 1 1 1 bϪ⌳1 l, y͑⌳1 ,b͒ ␦0ϵ dH͑xЈ͒ Q0͑xЈ,1͒Ϫ Rex ͵0 ͭ 1ϪlϪr P͑y͒ϭ yϪb r P ϩl, y͓b,⌳2ϩb͔ 1ϪbϪ⌳2 ͩ ⌳2 ͪ Ϫ P . 1ϪlϪr exͮ 1ϪlϪr 1ϩ ͑yϪ1͒, y͑⌳2ϩb,1͔. 1ϪbϪ⌳ The stationary state fl admits a flux JR P from the reactant Ά 2 → ͑79͒ state to the product state. Indeed, the measure fl((aϩr,1͔ ϫ(0,1͔) is transferred to the product state and the measure Therefore the stationary measure fl for this open system is (1ϪbϪ⌳2)Pex is transferred from there. This implies the given by existence of the flux JR P given by →
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JR Pϭfl͑͑aϩr,1͔ϫ͑0,1͔͒Ϫ͑1ϪbϪ⌳2͒Pex condition, the physical measure is singular and condi- → tionally invariant. And for an open system under a flux ͑1ϪaϪr͒͑bϪ⌳1͒ ϭ R boundary condition, the physical measure is invariant, 1ϪlϪr ex but absolutely continuous with respect to the two- dimensional Lebesgue measure. It is only weakly sin- ͑aϪl͒͑1ϪbϪ⌳ ͒ 2 gular in the sense that the density is discontinuous on a Ϫ Pex . ͑82͒ 1ϪlϪr Cantor-like set. With the specification ͑69͒ of the rate coefficients, the ex- It is interesting to note that the conditionally invariant pressions ͑81͒ of the stationary measure and the decay mode measures appear as the decay modes ͑i.e., the Pollicott- as well as ͑82͒ of the flux agree with their phenomenological Ruelle resonances͒ for an open system under a flux counterparts ͑61͒, ͑62͒ and ͑63͒, respectively. In short, the boundary condition ͑cf. Sec. V C͒. Such a relation ex- measure fl describes a nonequlibrium stationary state which ists more generally. The Pollicott-Ruelle resonances has a nonvanishing flux JR P . are defined as the functionals ⌫ acting on a dynamical → variable ,14,32–38 which satisfy, in case of a map, VI. CONCLUSIONS , ؠ ⌽t͘ϭ t͗⌫,͘,⌫͗ We have explicitly constructed SRB measures for three Baker-type maps employing the similarity between the de where ⌽ is a map and ( Ͻ1) is a decay rate. When Rham equation and the evolution equation of partially inte- ͉ ͉ the characteristic function of any Borel set A is in grated distribution functions. Now we give a few more re- A the domain of the functional and , ϭ0 with M marks. ⌫ ͗⌫ M͘ ” the whole phase space, then the ͑possibly complex͒ ͑a͒ In our examples discussed in Secs. III and IV, we have measure defined by encountered different types of contracting dynamics that lead to invariant states all of which have singular ͑A͒ϭ͗⌫,A͘, measures supported on fractal sets. In the first case we considered a nonconservative reversible Baker map on is conditionally invariant since (⌽Ϫ1A)/(⌽Ϫ1M) the unit square which globally preserves the area of the ϭ(A). Such examples are the hydrodynamic modes square but forms an attractor-repeller pair due to the for open Hamiltonian systems.14,36,37 local contraction and expansion properties of the map. ͑d͒ For a nonconservative reversible map ⌽, we find dif- The invariant set is an attractor and is fractal since it ferent SRB measures and ¯ for the forward and has the information dimension less than 2, but it is ph ph backward time evolutions, respectively. For each time dense in the unit square. In the second example of the evolution, one plays a role of an attracting measure and Baker map with a Cantor-like invariant set, we consid- the other of a repelling measure. This is a key element ered two different cases—global contraction onto a of the compatibility between dynamical reversibility fractal set with and without escape of points from the unit square. When there is no escape, the invariant set and irreversible behavior of statistical ensembles. In- is a fractal attractor which is a nondense subset of the deed, when the dynamics is reversible and statistical unit square. If we add the possibility of escape, then the ensembles irreversibly approach a stationary ensemble invariant set is fractal in both x and y directions. ph for t ϩϱ, there should exist another stationary →¯ ͑b͒ It is worth mentioning that one can convert a physical ensemble ph which is obtained from ph by the time measure for a system with escape to the proper invari- reversal operation. Since the new stationary ensemble ¯ ant measure on the repeller by incorporating into the ph is repelling, its existence is compatible with the averaging process described in Eq. ͑5͒ both the charac- irreversible behavior of statistical ensembles. A similar teristic function on the region of the phase space from situation was observed for open Hamiltonian systems which points escape, as well as a ‘‘survival’’ function under a flux boundary condition.13,14 Such systems ad- which is unity if the phase point is in the region for the mit two different stationary states, one is the time re- time interval 0ϽϽT, with TϾt, and zero otherwise. versed state of the other. In this case, an attracting sta- Then by taking the limit T ϱ, one recovers the in- tionary state for the forward time evolution obeys variant measure on the repeller.→ Such a procedure was Fick’s law and a repelling state obeys anti-Fick’s law. used by van Beijeren and Dorfman in order to correctly ͑e͒ As mentioned in Introduction, an SRB measure for a compute the Lyapunov exponents on the repeller for a map may be constructed as follows:30 ͑1͒ Approximate Lorentz gas.27 This procedure is closely related to one the measure by iterating an initial constant density fi- described by Hunt, Ott, and Yorke28 to obtain natural nite times, ͑2͒ calculate the average with its result and measures on invariant sets. ͑3͒ take the limit of infinite iterations. On the other ͑c͒ We have encountered three different physical mea- hand, in our construction of an SRB measure, a func- sures, which are characterized by the smoothness along tional equation for the partially integrated distribution the expanding direction. For closed nonconservative function similar to the de Rham equation is derived systems, the physical measures are singular and invari- from the evolution equation for measures. Note that the ant. For an open system under an absorbing boundary functional equation is a direct consequence of the self-
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similarity of the measure. An iterative method of solv- ant measure. As a consequence, the integral of Pt(x,1) over ing the functional equation is similar to the procedure the unit interval ͓0,1͔ is invariant. Indeed, by integrating Eq. explained above. However, our method has some ad- ͑A1͒, we have
vantages. First, the de Rham-type functional equations 1 1 are fixed point equations for contraction mappings and, dx Ptϩ1͑x,1͒ϭ dx Pt͑x,1͒ as a result, the iterative solution strongly converges to ͵0 ͵0 the limit. On the contrary, the iterative approximation 1 of the density function does not converge by itself. ϭ•••ϭ͵ dx P0͑x,1͒. Therefore, one can obtain a good approximation to the 0
cumulative distribution function of the SRB measure On the other hand, since 0 and, hence, P0 is continu- by less iterations than by the conventional method. ously differentiable in x, Eq. ͑A1͒ implies that Pt(x,1) Second, in the functional equation approach, one can (tϭ1,2,•••) are also continuously differentiable in x. And xPt(x,1) satisfies the equationץ systematically extract exponentially decaying terms the derivative which are typically higher order derivatives of singular P x,1 ϭr2 P rxϩl,1 ϩl2 P lx,1 , ͒ x t͑ץ ͒ x t͑ץ ͒ x tϩ1͑ץ ,functions in the sense of Schwartz’ distributions. Third although the measure is defined as the solution of a which leads to an inequality functional equation, the average values of certain dy- P ͑ ,1͒ʈ ץP ͑ ,1͒ʈр͑r2ϩl2͒ʈ ץʈ namical functions can be calculated analytically as il- x tϩ1 • x t • tϩ1 ,xP0͑•,1͒ʈץxPt͑•,1͒ʈр•••р ʈץlustrated in Sec. II. So far, the functional equation рʈ method was mainly applied to piecewise linear one- (P ( ,1 ץ where the function norm is defined by dimensional maps and Baker-type maps,13,37,38 but we ʈ x t • ʈ -P (x,1) and ϭmax(l,r)Ͻ1. Since the func ץ ϵsup believe that the method can also be applied to other x[0,1]͉ x t ͉ tion P (x,1) can be represented as systems if the expanding and contracting directions can t be well-separated. 1 1 xЈPt͑xЈ,1͒ץPt͑x,1͒Ϫ dx ЈPt͑xЈ,1͒ϭ dx ЈxЈ ͵0 ͵0
ACKNOWLEDGMENTS 1 ,xЈPt͑xЈ,1͒ץϪ dx Ј ST and JRD thank Professor T. Te´l and Professor I. ͵x Kondor for having invited them to an exciting Summer we finally obtain School/Workshop: ‘‘Chaos and Irreversibility’’ ͑Etovo¨s Uni- versity, Budapest, 31 August–6 September, 1997͒ and for 1 their warm hospitality. The authors are grateful to Professor Pt͑x,1͒Ϫ dxЈ P0͑xЈ,1͒ ͯ ͵0 ͯ P. Gaspard for fruitful discussions and helpful comments, particularly on the relation between the physical measure and 1 1 xPt͑•,1͒ʈץр dxЈ xЈϩ dxЈ ʈ ´ the conditionally invariant measure, and to Professor T. Tel, ͭ ͵0 ͵x ͮ Professor J. Vollmer and Professor W. Breymann for fruitful discussions, particularly on their work about multi-Baker 3 3 ,P ͑ ,1͒ʈ ץP ͑ ,1͒ʈр tʈ ץр ʈ maps. This work is a part of the project of Institute for Fun- 2 x t • 2 x 0 • damental Chemistry, supported by Japan Society for the Pro- or Eq. ͑19͒. motion of Science–Research for the Future Program ͑JSPS- RFTF96P00206͒. Also this work is partially supported by a Grant-in-Aid for Scientific Research and a grant under the 2. Proof of the lemma International Scientific Research Program both from Minis- Since the statement of the lemma given in the text is not try of Education, Science and Culture of Japan. JRD also technically complete, we first give the precise statement and wishes to thank Professor Henk van Beijeren, and Dr. Arnulf then prove it. Latz for many helpful discussions as well as the National Lemma: Let be a linear contraction mapping with the Science Foundation for support under Grant PHY-96-00428. contraction constantF 0ϽϽ1 defined on a (Banach) space of bounded functions equipped with the supremum norm ʈ•ʈ (0) t (1) (2) APPENDIX A: PROOFS OF EQ. 19 AND THE LEMMA (i.e., ʈ f ʈрʈfʈ). And let gtϵg ϩ g ϩgt be a given „ … F (0) (1) (2) function where g ,g and gt are bounded functions, is 1. Derivation of Eq. „19… (2) t a constant satisfying Ͻ͉͉р1, and ʈgt ʈрK with some As explained in Sec. III, the evolution equation ͑18͒ for constant KϾ0. Then, the solution of the functional equation the partially averaged distribution function Pt(x,1) 1 f tϩ1ϭ f tϩgt , ͑A2͒ ϵ͐0 dyЈ t(x,yЈ) F is given by Ptϩ1͑x,1͒ϭrPt͑rxϩl,1͒ϩlPt͑lx,1͒, ͑A1͒ f ϭ f ͑0͒ϩt f ͑1͒ϩO͑tt͒, ͑A3͒ is the Frobenius-Perron equation for a one-dimensional exact t ϱ ϱ (0) (1) map Sxϭx/l ͑for x͓0,l͔) and Sxϭ(xϪl)/r ͑for x where f ϱ and f ϱ are the unique solutions of the following (l,1͔), which admits the Lebesgue measure as the invari- fixed point equations
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͑0͒ ͑0͒ 0 f ϭ f ϩg͑ ͒, ͑A4͒ and Rt(x,y) is a function of Qt(x,1) ϱ F ϱ 1 g͑1͒ 0, y͓0,⌳1͔ f ͑1͒ϭ f ͑1͒ϩ . ͑A5͒ ϱ F ϱ lQt͑lx,1͒, y͑⌳1 ,b͒ Rt͑x,y͒ϭ The proof of the lemma is straightforward: From Eq. lQt͑lx,1͒, y͓b,⌳2ϩb͔ ͑A2͒, one obtains Ά lQt͑lx,1͒ϩrQt͑rxϩa,1͒, y͑⌳2ϩb,1͔. tϪ1 ͑B4͒ t s f tϭ f 0ϩ ͚ gtϪ1Ϫs . In terms of Qt(x,y), the expectation value of a dynami- F sϭ0 F cal variable (x,y) is expressed as By rewriting the right hand side in terms of g(0), g(1) and (2) gt , this relation leads to ͑x,y͒t͑x,y͒dx dyϭ ͑x,y͒dx dyQt͑x,y͒, ͵[0,1]2 ͵[0,1]2 ϱ ϱ ͑B5͒ s ͑0͒ t ϪsϪ1 s ͑1͒ f tϭ g ϩ g s͚ϭ0 F s͚ϭ0 F where dy stands for the Riemann-Stieltjes integral of Qt with respect to y.53 tϪ1 ϱ To solve Eqs. B2 , B3 and B4 , we first investigate t s ͑2͒ s ͑0͒ ͑ ͒ ͑ ͒ ͑ ͒ ϩ f 0ϩ ͚ gtϪ1ϪsϪ͚ g the equation of motion of Q (x,1) ͭ F sϭ0 F sϭt F t ϱ Qtϩ1͑x,1͒ϭlQt͑lx,1͒ϩrQt͑rxϩa,1͒. ͑B6͒ t ϪsϪ1 s ͑1͒ Ϫ ͚ g . We observe that, in terms of a function H(x) defined by a sϭt F ͮ de Rham equation ͑50͒, one has ϱ s (0) (0) The first sum ͚sϭ0 g in the right-hand side is f ϱ and F (1) 1 1 the second sum ͚ ϱϪsϪ1 sg(1) is f , which satisfy sϭ0 ϱ dH͑xЈ͒Qt͑xЈ,1͒ϭ dH͑xЈ͒QtϪ1͑xЈ,1͒••• Eqs. ͑A4͒ and ͑A5͒, respectively.F By repeatedly using the ͵0 ͵0
property of , we have 1 F t ͑0͒ ͑1͒ ϭ dH͑xЈ͒Q0͑xЈ,1͒, ͑B7͒ ʈg ʈ ʈg ʈ ͵0 ʈ f Ϫ f ͑0͒Ϫt f ͑1͒ʈрt ʈf ʈϩ ϩ ϩKt , t ϱ ϱ ͭ 0 1Ϫ ͉͉Ϫ ͮ 1 t which is O(t ) and implies the desired result ͑A3͒. Qt͑x,1͒Ϫ dH͑xЈ͒Qt͑xЈ,1͒ ͵0
APPENDIX B: PHYSICAL MEASURE FOR A BAKER 1 xЈQt͑xЈ,1͒, ͑B8͒ץMAP WITH A CANTOR-LIKE INVARIANT SET ϭ dxЈ͕H͑xЈ͒Ϫ͑xЈϪx͖͒ ͵0 In this Appendix, we outline the derivation of Eq. ͑48͒, where ϭlϩr(р1) and Јϭmax(l,r)(Ͻ1). On the other which is quite similar to that of Eq. ͑30͒. hand, Eq. ͑B6͒ leads to an inequality From the definition ͑47͒ of the map ⌿, one finds that the t ,xQ0͑•,1͒ʈץxQtϪ1͑•,1͒ʈр•••рЈ ʈץxQt͑•,1͒ʈрЈʈץFrobenius-Perron equation governing the time evolution of ʈ the distribution function t(x,y) is given by xQt(x,1)͉. Then, one obtainsץ͉[xQt(•,1)ʈϵsupx[0,1ץwith ʈ l y from Eqs. ͑B7͒ and ͑B8͒ lx, , y͓0,⌳ ͔ ⌳ tͩ ⌳ ͪ 1 1 1 1 t t Qt͑x,1͒ϭ ͵ dH͑xЈ͒Q0͑xЈ,1͒ϩO͑Ј ͒, 0, y͑⌳1 ,b͒ 0 tϩ1͑x,y͒ϭ ͑B1͒ r yϪb and, thus rxϩa, , y͓b,⌳ ϩb͔ ⌳ tͩ ⌳ ͪ 2 2 2 1 t ͑1͒ t 0, y ͑⌳ ϩb,1͔ Rt͑x,y͒ϭ dH͑xЈ͒Q0͑xЈ,1͒g ͑y͒ϩO͑Ј ͒, ͑B9͒ Ά 2 ͵0 which leads to a contractive time evolution of the partially y 0, y͓0,⌳1͔ integrated distribution function Qt(x,y)ϵ͐0 dyЈ t(x,yЈ): ͑1͒ g ͑y͒ϭ l, y͑⌳1 ,⌳2ϩb͔ ͑B10͒ Qtϩ1͑x,y͒ϭ ЈQt͑x,y͒ϩRt͑x,y͒, ͑B2͒ F ͭ lϩr, y͑⌳ ϩb,1͔. 2 where Ј stands for a contraction mapping F Since ϭlϩrϾmax(l,r)ϭЈ, the lemma of Sec. III can y be applied to the evolution equations ͑B2͒, ͑B3͒ and ͑B4͒ lQ lx, , , y͓0,⌳ ͔ tͩ ⌳ ͪ 1 and one obtains 1 0, y͑⌳ ,b͒ 1 1 t t ЈQt͑x,y͒ϵ ͑B3͒ Qt͑x,y͒ϭ dH͑xЈ͒Q0͑xЈ,1͒G͑y͒ϩO͑tЈ ͒. ͑B11͒ F yϪb ͵0 rQt rxϩa, , y͓b,⌳2ϩb͔ ͩ ⌳2 ͪ where G(y) is the solution of a de Rham equation ͑49͒. The Ά 0, y͑⌳2ϩb,1͔ desired result ͑48͒ immediately follows from Eq. ͑B11͒.
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APPENDIX C: ON AN INVARIANT MEASURE FOR [1] [2] in in͑H2 പV3 ͒ A BAKER MAP WITH ESCAPE ϭ͕H͑lrϩla͒ϪH͑la͖͕͒G͑⌳2ϩb͒ϪG͑b͖͒ In this Appendix, we show that the invariant measure in on the fractal repeller considered in Sec. IV is a Gibbs mea- l r r ϭ ͕H͑1͒ϪH͑0͖͒ ͕G͑1͒ϪG͑0͖͒ sure. lϩr lϩr lϩr To show that, we first observe that the image ⌿m͓0,1͔2 m m 2 of the unit square by the map ⌿ consists of 2 horizontal lr exp͑Ϫu2,3͑2,1͒͒ ϭ ϭ , [m] [m] [m] lϩr 3 3 strips, which will be referred to as H1 ,H2 ,•••H2m ; and ͑ ͒ ͑lϩr͒ that the preimage of ⌿Ϫn͓0,1͔2 of the unit square by the map ⌿n consists of 2n vertical strips, which will be referred which is ͑C2͒. [n] [n] [n] [m] [n] Now we go to the proof of Eq. ͑C2͒. Since one can write to as V ,V , V n . Then, boxes H പV generated 1 2 ••• 2 i j H[m]ϭ 0,1 ϫ ␣[m] ,[m] and V[n]ϭ ␥[n] ,␦[n] ϫ 0,1 , by the overlap procedure provide a generating partition of i ͓ ͔ ͓ i i ͔ j ͓ j j ͔ ͓ ͔ [m] [n] and thus the repeller. As easily seen, for each box Hi പV j ,a stretching factor for a time interval ͓Ϫm,nϪ1͔ [m] [n] [m] [m] [n] [n] in͑Hi പV j ͒ϭ͕G͑i ͒ϪG͑␣i ͖͕͒H͑␦j ͒ϪH͑␥j ͖͒ [m] [n] nϪ1 ϭin͑Hi ͒in͑V j ͒, t uij͑n,m͒ϵ x͑⌿ ͑x,y͒͒, ͑C1͒ tϭϪ͚m it is enough to show
Ϫu j͑n͒ [m] [n] [n] e does not depend on the initial point (x,y)Hi പV j , in͑V j ͒ϭ , ͑C3͒ ͑lϩr͒n where x(x,y) is the local expanding rate defined by x(x,y)ϭϪln l ͑if x͓0,l͔) and x(x,y)ϭϪlnr ͑if x [m] [n] and ͓a,aϩr͔). Note that, when (x,y)Hi പV j , the preim- Ϫk age ⌿ (x,y) is unique for kϭ1,2,•••m. We show that the eϪui͑m͒ [m] [n] [m] -measure of the box H പV is given by in͑Hi ͒ϭ , ͑C4͒ in i j ͑lϩr͒m Ϫ Ϫu n,m͒ n 1 t e ij͑ where u j(n)ϭ͚tϭ0 x(⌿ (x,y)) and ui(m) [m] [n] Ϫ1 t in͑Hi പV j ͒ϭ , ͑C2͒ ϭ͚ x(⌿ (x,y)) are stretching factors for a vertical Ϫuij͑n,m͒ tϭϪm ͚i,je [n] [m] strip V j and a horizontal strip Hi , respectively. As be- fore, the stretching factors are constant on each strip. The which implies that the measure in is a Gibbs relations ͑C3͒ and ͑C4͒ are proved by induction with respect 1,18,19,24 measure. Note that, since the numerator of Eq. ͑C2͒ is to n and m, with the aid of the functional equations for H(x) s nϩmϪs a product l r of (nϩm) factors (sϭ0,1,2,•••nϩm) and G(y), respectively. Since the proofs of the two relations and there are (nϩm)!/͕s!(nϩmϪs)!͖ boxes with this are similar, we show ͑C3͒ only. value, the sum of the numerators, or the normalization factor, It is easy to see that ͑C3͒ holds for nϭ1. Now we sup- is pose that Eq. ͑C3͒ is valid for n. As easily seen, a vertical [nϩ1] [n] strip V j is expressed by some vertical strip V j as nϩm Ј ͑nϩm͒! [nϩ1] Ϫuij͑n,m͒ s nϩmϪs Ϫ1 [n] e ϭ l r V j ϭRപ⌿ Vj , ͚i, j s͚ϭ0 s!͑nϩmϪs͒! Ј
nϩm Ϫ͑nϩm͒ where ϭ0 or 1 with R0ϭ͓0,l͔ϫ͓0,1͔ and R1ϭ͓a,aϩr͔ ϭ͑lϩr͒ ϭe , [n] [n] ϫ͓0,1͔. In terms of ␥ j and ␦ j , we have
24 where ϭϪln(lϩr) is the escape rate. [n] [n] ͓l␥j ,l␦j ͔ϫ͓0,1͔, ϭ0 Before proving Eq. ͑C2͒, we verify it for a simple case V[nϩ1]ϭ [2] 2 jЈ [n] [n] ͓ͭr␥ j ϩa,r␦ j ϩa͔ϫ͓0,1͔, ϭ1. nϭ2 and mϭ1. In this case, we have V1 ϭ͓0,l ͔ϫ͓0,1͔, [2] [2] [2] V2 ϭ͓a,rlϩa͔ϫ͓0,1͔, V3 ϭ͓la,lrϩla͔ϫ͓0,1͔, V4 2 [1] When ϭ0, from the functional equation for H(x), one ob- ϭ͓raϩa,r ϩraϩa͔ϫ͓0,1͔; H1 ϭ͓0,1͔ϫ͓0,⌳1͔ and [1] tains H2 ϭ͓0,1͔ϫ͓b,⌳2ϩb͔. As an example, we consider a box [1] [2] [nϩ1] [n] [n] H2 പV3 ϭ͓la,lrϩla͔ϫ͓b,⌳2ϩb͔. For any point (x,y) in͑V ͒ϭH͑l␦j ͒ϪH͑l␥j ͒ [1] [2] Ϫ1 jЈ H2 പV3 , we have 0рxрl and ⌿(x,y),⌿ (x,y) ͓a,aϩr͔ϫ͓0,1͔ and, thus l [n] [n] ϭ ͕H͑␦ ͒ϪH͑␥ ͖͒ lϩr j j 1 t l exp͑Ϫu ͑n͒ϩln l͒ u2,3͑2,1͒ϵ x͑⌿ ͑x,y͒͒ϭϪ2lnrϪln l. [n] j tϭϪ͚1 ϭ in͑V j ͒ϭ . ͑C5͒ lϩr ͑lϩr͒nϩ1 [nϩ1] On the other hand, from the functional equations of H(x) Then, for (x,y)V j , one has (x,y)R0 and ⌿(x,y) [n] Ј and G(x), we have V j . Therefore, x(x,y)ϭϪln l and
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nϪ1 y ϭ͐0dyЈt(␣:x,yЈ)(␣ϭR,I,orP) for a chemical reaction t (u j͑n͒Ϫln lϭ ͚ x͑⌿ ؠ⌿͑x,y͒͒ϩx͑x,y͒ model introduced in Sec. V. The density function t(␣:x,y tϭ0 (␣ϭR,I,orP) at time t is given by n t ϭ͚ x͑⌿ ͑x,y͒͒ϭujЈ͑nϩ1͒. ͑C6͒ t͑␣:x,y͒ϭ dx dy ␦͑͑␣:x,y͒ tϭ0 ϭ͚R,I,P ͵[0,1]2 Similarly, one can verify Eq. ͑C6͒ when ϭ1. Hence, t Ϫ⌿Ј ͑:xЈ,yЈ͒͒0͑:xЈ,yЈ͒,
[nϩ1] exp͑Ϫu jЈ͑nϩ1͒͒ where 0 is the density function of the initial measure, ⌿Ј is in͑V j ͒ϭ , the map introduced in Sec. V ͓cf. Eqs. ͑64͒, ͑65͒, and ͑66͔͒ Ј ͑lϩr͒nϩ1 and the delta function ␦((␣:x,y)Ϫ(:xЈ,yЈ)) stands for the or Eq. ͑C3͒ holds for nϩ1 and, by induction, it is valid for product ␦␣,␦(xϪxЈ)␦(yϪyЈ). all positive integer n. By integrating the Frobenius-Perron equation for the density t(␣:x,y), one obtains the evolution equation for Qt APPENDIX D: THE EVOLUTION EQUATION OF MEASURES FOR A REACTION MODEL Q ͑␣:x,y͒ϭ ¯ Q ͑␣:x,y͒ϩ¯R ͑␣:x,y͒, ͑D1͒ tϩ1 F t t In this Appendix, we write down the evolution equation where ␣ϭR,I,orP, a linear contraction mapping ¯ is de- F of the partially integrated distribution function Qt(␣:x,y) fined by
y lQ I:lx, , y͑0,⌳ ͔ tͩ ⌳ ͪ 1 1
yϪ⌳1 ͑bϪ⌳ ͒Q R:͑bϪ⌳ ͒x, , y͑⌳ ,b͔ 1 tͩ 1 bϪ⌳ ͪ 1 1 ¯ Q ͑I:x,y͒ϭ ͑D2͒ F t yϪb rQ I:rxϩa, , y͑b,⌳ ϩb͔ tͩ ⌳ ͪ 2 2
yϪbϪ⌳2 ͑1ϪbϪ⌳2͒Qt P:͑1ϪbϪ⌳2͒x, , y͑⌳2ϩb,1͔ Ά ͩ 1ϪbϪ⌳2ͪ y ͑aϪl͒Q I : ͑aϪl͒xϩl, , y͑0,bϪ⌳ ͔ tͩ bϪ⌳ ͪ 1 ¯ 1 Qt͑R : x,y͒ϭ ͑D3͒ F yϪbϩ⌳1 ͑1Ϫbϩ⌳ ͒Q R : ͑1Ϫbϩ⌳ ͒xϩbϪ⌳ , , y͑bϪ⌳ ,1͔, ͭ 1 tͩ 1 1 1Ϫbϩ⌳ ͪ 1 1
y ͑1ϪaϪr͒Q I : ͑1ϪaϪr͒xϩaϩr, , y͑0,1ϪbϪ⌳ ͔ tͩ 1ϪbϪ⌳ ͪ 2 ¯ 2 Qt͑P : x,y͒ϭ ͑D4͒ F yϪ1ϩbϩ⌳2 ͑bϩ⌳ ͒Q P : ͑bϩ⌳ ͒xϩ1ϪbϪ⌳ , , y͑1ϪbϪ⌳ ,1͔, ͭ 2 tͩ 2 2 bϩ⌳ ͪ 2 2 ¯ and Rt(␣:x,y) is a functional of Qt(␣:x,1):
0, y͑0,⌳1͔ lQ ͑I:lx,1͒, y͑⌳ ,b͔ ¯ t 1 Rt͑I:x,y͒ϭ ͑D5͒ lQt͑I:lx,1͒ϩ͑bϪ⌳1͒Qt͑R:͑bϪ⌳1͒x,1͒, y͑b,⌳2ϩb͔
Ά lQt͑I:lx,1͒ϩ͑bϪ⌳1͒Qt͑R:͑bϪ⌳1͒x,1͒ϩrQt͑I:rxϩa,1͒, y͑⌳2ϩb,1͔ 0, y͑0,bϪ⌳ ͔ ¯ 1 Rt͑R:x,y͒ϭ ͑D6͒ ͭ ͑aϪl͒Qt͑I:͑aϪl͒xϩl,1͒, y͑bϪ⌳1,1͔,
0, y͑0,1ϪbϪ⌳ ͔ ¯ 2 Rt͑P:x,y͒ϭ ͑D7͒ ͭ ͑1ϪaϪr͒Qt͑I:͑1ϪaϪr͒xϩaϩr,1͒ y͑1ϪbϪ⌳2,1͔.
These are the desired results. Note that the contraction constant ¯ of the mapping ¯ is given by F ¯ ϭmax͑l,r,aϪl,1ϪaϪr,bϪ⌳1 ,bϩ⌳2,1Ϫbϩ⌳1 ,1ϪbϪ⌳2͒͑Ͻ1͒.
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