The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Entropy Production in Open Volume{Preserving Systems Pierre Gaspard Vienna, Preprint ESI 382 (1996) September 23, 1996 Supported by Federal Ministry of Science and Research, Austria Available via

2 Entropy Production in Open Volume-Preserving Systems Pierre Gaspard Center for Nonlinear Phenomena and Complex Systems, Universite Libre de Bruxelles, Campus Plaine, Code Postal 231, B-1050 Bruxelles, Belgium (October 1, 1996) Abstract We describe a mechanism leading to positive entropy production in volumepreserving systems under nonequilibrium conditions. We consider volumepreserving systems sustaining a diusion process like the multibaker map or the Lorentz gas. A continuous ux of particles is imposed across the system resulting in a stationary gradient of concentration. In the limit where such ux boundary conditions are imposed at arbitrarily separated boundaries for a xed gradient, the invariant measure becomes singular. For instance, in the multibaker map, the limit invariant measure has a cumulative function given in terms of the nondierentiable Takagi function. Because of this singularity of the invariant measure, the entropy must be dened as an -entropy instead of the Gibbs entropy which would require the existence of a regular measure with a density. The -entropy production is then shown to be asymptotically positive and, moreover, given by the entropy production expected from irreversible thermodynamics. PACS numbers: j, 05.20Dd, b 1 Typeset using REVTEX

3 I. INTRODUCTION The probem of entropy production is one of the oldest problems of nonequilibrium statistical mechanics. It originates from the confrontation between the thermodynamics of irreversible processes and the classical and later the quantum mechanics which are supposed to describe the motion of atoms and molecules in matter. Classical mechanics is reversible and preserves volumes in phase space (cf. Liouville's theorem). Both properties have remained for long in apparent contradiction with irreversible thermodynamics. Only recently, works in dynamical systems theory and on chaos have shown that we can understand exponential relaxations toward thermodynamic equilibrium in the theory of the time evolution of statistical ensembles of Newtonian trajectories, shortly called Liouvillian dynamics. The study of models like the Lorentz gas and the multibaker map has led to the explicit construction, within the context of the Liouvillian dynamics, of several relations and results typical of the irreversible thermodynamics like: (1) The dispersion relation of diusion has been obtained in terms of the so-called Pollicott-Ruelle resonances [1] which are generalized eigenvalues of the Frobenius-Perron operator [2]. The Green-Kubo formula is then given by the second derivative of this generalized eigenvalue with respect to the wavenumber [2]. (2) The hydrodynamic modes of diusion are constructed as the generalized eigenstates associated with the Pollicott-Ruelle resonances [2]. (3) In large open systems of scattering type or with absorbing boundaries the dynamics of escape of particles has been shown to be controlled by diusion and the geometry of the scatterer [3{8]. The set of trajectories which always remain strictly inside the absorbing boundaries forms a chaotic repeller which is fractal in both the stable and the unstable directions. It supports an invariant measure characterized by a nonvanishing escape rate, so that almost all trajectories reach the absorbing boundaries. A relation has been obtained between the diusion coecient and the characteristic quantities of chaos on the repeller like the Lyapunov exponent and the Kolmogorov-Sinai entropy, or the partial Hausdor 2

4 dimensions [3,6,7]. (4) The nonequilibrium steady states corresponding to gradients of concentration have also been constructed for ux boundary conditions [2,9]. In nite systems, such a nonequilibrium steady state is given by an invariant measure which is dierent from the Liouville equilibrium invariant measure but which is absolutely continuous with respect to the Liouville measure. In the limit of large systems where the concentration gradient is maintained to a xed value, the nonequilibrium invariant measure becomes singular with respect to the Liouville measure. For instance, in the multibaker map which is a model of deterministic diusion, Tasaki and Gaspard have shown that the invariant measure corresponding to a gradient of concentration is given in terms of the nondierentiable Takagi function in the limit where the gradient is imposed at borders which are more and more separated while keeping constant the gradient [9]. The convergence of the invariant measure { which remains absolutely continuous as long as the borders are nitely separated { to the singular measure is very rapid in the large system limit because it is determined by the Lyapunov exponential instability. In this way, the absolute continuity disappears exponentially fast below tiny scales in phase space. This result is of crucial importance for the following arguments. The previous results have been obtained for volume-preserving systems where the sum of all the positive and negative Lyapunov exponents vanishes. Similar results have been obtained for systems coupled to a thermostat, in which the kinetic energy is a constant of motion instead of the total energy [10,11]. The interest of thermostatted systems is that they allow to conceive the interaction of a few-body system with an external force in a simplied and deterministic way, which is convenient for certain numerical simulations [10]. However, the thermostatted dynamical systems do not obey to the Liouville theorem so that their invariant measure has in general an information dimension which is smaller than the phasespace dimension. The thermostatted systems present two distinct regimes [12]: (1) The Anosov regime at low external forces where the support of the invariant measure remains the full phase space; (2) The regime beyond a critical force where the invariant measure is supported by a fractal attractor. In both cases, the invariant measure remains absolutely 3

5 continuous with respect to the Lebesgue measure along the unstable directions but shows multifractality along the stable directions (in contrast to the case of fractal repellers). For thermostatted systems, relations have also been obtained between the transport coecients and the characteristic quantities of chaos. Moreover, the measures of nonequilibrium steady states have also been constructed and characterized in thermostatted systems [13]. The aforementioned properties allow us to understand the approach toward the thermodynamic equilibrium in terms of statistical properties like the Pollicott-Ruelle resonances. These properties are intimately related to the formulation of the approach to equilibrium in terms of the mixing property introduced by Gibbs. The mixing property assumes the decrease of the two-time correlation functions of the physical observables and can thus be formulated directly in the context of the Liouvillian dynamics of statistical ensembles of trajectories. In this formulation of the approach to equilibrium, the concept of entropy is not required and the connection to the 2nd law of thermodynamics can be eluded: the time evolution toward equilibrium is only formulated in terms of statistical averages of physical observables: hai t = R A(X)f t (X)dX! hai eq for t! 1. Recently, several works have been devoted to the problem of the possible connections between the aforementioned results on chaotic dynamical systems and the time evolution of a quantity of the type of the thermodynamic entropy and, thus, to the problem of our understanding of the entropy production and of the 2nd law in the context of dynamical systems theory [13{17]. It is a very old and dicult problem because it is well know that an entropy dened like the Gibbs entropy as S G (t) =? Z? dx f t (X) ln f t (X) ; (1) remains constant during the time evolution of the probability density f t (X) in closed volumepreserving systems. However, if the invariant measure becomes singular in some limit as it is the case in open volume-preserving systems, we are no longer allowed to use the Gibbs entropy. Indeed, the Gibbs entropy is given in terms of the probability density which exists only if the associated measure is absolutely continuous with respect to the Liouville measure: 4

6 f t (X) = d t =dx. (Let us recall here that the Liouville measure dx is invariant in volumepreserving systems.) Therefore, the constancy of the entropy is in question in large open volume-preserving systems. The purpose of the present paper is to reconsider the problem of entropy production in the light of our new result showing the singularity of the invariant measure of nonequilibrium states in open volume-preserving systems of large spatial extension. If the invariant measure becomes singular in some limit, the Gibbs entropy should be replaced by a Boltzmann entropy or -entropy which is essentially? P p log p where the p's are the probabilities for the trajectories to visit cells of size in phase space. Indeed, the work by Kolmogorov and Tikhomirov has shown that an -entropy diverges for! 0 in a way which is characteristic of the type of singularities of the measure [18]. If the type of the measure happens to change in some limit such an -entropy is thus required, especially, if we want to keep the operational interpretation of entropy as a measure of disorder. The plan of the paper is as follows. In Sec. II, we describe the problem of entropy production by going back to the original denition of entropy production in the thermodynamics of irreversible processes. The recent works based on the Gibbs entropy are then discussed in order to show the diculties arising in the dierent schemes proposed. In Sec. III, we introduce the open volume-preserving systems and we discuss the choice of appropriate boundary conditions. A probability measure is dened in such open systems with innitely many particles in terms of a Poisson suspension over the dynamical system. The time evolution is then introduced in this formulation. In Sec. IV, we dene the -entropy we use in the following and we show that the denition is consistent with the standard equilibrium entropy per unit volume. The time evolution of this -entropy and the corresponding -entropy production are then dened. In Sec. V, we apply our denitions to diusion in the multibaker map. We show that the -entropy production is determined by the nondierentiable Takagi function in the large system limit and gives precisely the entropy production expected from irreversible thermodynamics. Conclusions are drawn in Sec. VI. 5

7 II. IRREVERSIBLE THERMODYNAMICS AND THE PROBLEM OF ENTROPY PRODUCTION The concept of entropy production is introduced in the thermodynamics of irreversible processes [19{22]. In order to identify in deterministic dynamical systems a quantity like the entropy production, we shall rst present the phenomenological entropy production and discuss its properties which should be recovered in the deterministic approach. We only discuss here the case of diusion which is the process observed in the Lorentz gas and the multibaker map. A rst remark is that the thermodynamics of irreversible processes is a macroscopic theory where the quantities are dened as averages over volumes of size larger than the mean free path of the uid particles. In the case of the diusion of tracer particles in a uid, the density evolves in time according to the phenomenological t = D r 2 ; (2) where r r denotes the gradient with respect to the physical positions r = (x; y; z) and where D is the diusion coecient which is here supposed to be constant in space. This phenomenological equation is obtained from the conservation law of tracer t + r j = 0, where the tracer current is given by Fick's law, j =?Dr. If the tracer concentration is not too high we may suppose that the tracer particles and the uid form an ideal solution so that the entropy S is given as the integral of the entropy per unit volume or entropy density s: S = Z V s dr ; with s = ln 0 ; (3) where 0 is a constant dening a reference density for which s( 0 ) = 0. An evolution equation for the entropy density can be derived from its denition (3) and the diusion equation (2) t s + r J s = s ; (4) 6

8 with the following entropy current and entropy source J s = j ln 0 e =? Dr ln 0 e ; (5) s = D (r)2 0 : (6) The balance equation (4) for the entropy density can be expressed for the global entropy S as ds dt =? da J s + Z V s dr = d es dt + d is dt : (7) d es is the ow of entropy at the of the system and d is dt dt is the so-called entropy production inside the system due to the irreversible process of diusion. This entropy production is always nonnegative according to the 2nd law of thermodynamics: d i S dt = Z V s dr 0 : (8) The entropy production vanishes at equilibrium and is positive away from equilibrium. In contrast, the ow of entropy may take positive or negative values depending on the gradient of concentration imposed at borders. We now compare irreversible thermodynamics with the deterministic dynamics for the motion of atoms and molecules in the uid. We suppose that X denotes the positions and momenta of these particles, which dene the phase space? of the dynamical system. The motion is supposed to be governed by a set of dierential equations of rst order in time, given by a vector eld F(X) in phase space _X = F(X) : (9) We consider a statistical ensemble of copies of the system which is dened by a probability measure t, we suppose for the moment to be absolutely continuous with respect to the Lebesgue measure so that the corresponding probability density exists f t (X) = d t =dx. This probability density evolves in time according to the (generalized) Liouville equation which expresses the conservation of probability in phase space: 7

9 @ t f(x) + r[f(x)f(x)] = 0 ; (10) where r X denotes the gradient with respect to all the phase-space variables X. This equation is a partial dierential equation. Its resolution requires boundary conditions on the probability density f(x) at the of the phase space. Such boundary conditions allows us to express the nonequilibrium constraints in this formulation: for instance, if the temperature varies along the walls of the container of the uid the density on the corresponding boundary of phase space, must be taken as a Maxwellian distribution with a varying temperature parameter for the velocities of the incoming particles. The time evolution of the probability density induces a time evolution for the Gibbs entropy density s G =? f ln f : (11) The balance equation for the local evolution of this density is given t s G + r J sg = sg ; (12) with the following Gibbs entropy current and entropy source J sg =? f ln f F = s G F ; (13) sg = f r F : (14) Therefore, the variation of the Gibbs entropy is obtained as ds G dt =? da J sg + Z? dx sg : (15) The rst term is the ow of entropy at the boundary of the phase space and may be identied with the entropy ow in Eq. (7). The second term is the average value of the divergence of the vector eld (9). In a conservative system, this divergence vanishes, r F = 0, and sg = 0. Therefore, the second term vanishes in Eq. (15) so that the time variation of the 8

10 Gibbs entropy ds G =dt is only due to boundary conditions. At a steady state, the variation of entropy vanishes so that the ow of Gibbs entropy is also zero. These properties of Gibbs entropy are in contradiction with the properties expected for an entropy in view of the thermodynamics of irreversible processes. If we had to identify the Gibbs entropy (1) with the thermodynamic entropy (3) as it is the case in equilibrium statistical mechanics the famous problem would arise for the class of conservative systems where the source and thus the production of Gibbs entropy vanishes. Mackey has suggested that a positive entropy production should have its origin in the property of exactness of dynamical systems [23]. Exact dynamical systems are dened as discrete-time systems, X t+1 = (X t ), which are expansive j@ X j 1. The expansivity is compatible with a nite phase space if the mapping sends several dierent points X onto the same point (X). If entropy is conceived as a measure of disorder in phase space we understand that there is a loss of information and thus disorder production in such systems. In ows, the property of exactness should be expressed by the assumption that r F 0 which means that the ow is expansive. According to Eqs. (14)-(15), the entropy production would then be positive. However, Hamiltonian systems are not expansive. In thermostatted systems, the trajectories are attracted toward phase-space regions of contractivity where r F 0 on average [10{13]. In such systems, there is a negative entropy production. Indeed, since trajectories converge to a strange attractor which has an information dimension lower than the total phase-space dimension the probability distribution f t (X) is more disordered at the initial time than at following times. As a consequence, the Gibbs entropy { which is a measure of disorder { decreases! To overcome this problem, a hypothetical mechanism of entropy conservation between the system and the thermostat has been proposed [13]: S G; total = S G; system + S G; thermostat = constant. If we now consider the so-dened entropy of the thermostat there is a change of sign in its time variation and the entropy production of the thermostat should thus be positive. Although useful for the computations of thermodynamic quantities, this reasoning is unsatisfactory as an explanation of the origin of entropy production. 9

11 III. OPEN VOLUME-PRESERVING SYSTEMS We shall here below consider volume-preserving systems which are open. The openness of the system is very important if we want to conceive a process of the kind of those described by irreversible thermodynamics. As an example, we consider a nite Lorentz gas composed of a nite number of xed disks forming a nite lattice of size L. The set of these disks can be considered as a scatterer as studied in collision theory [3,5,6]. As L! 1, the lattice occupies the whole plane and becomes periodic. The dynamical system of the Lorentz gas is formed by a point particle in elastic collisions on the disks. This mechanical system has two degrees of freedom, is volume preserving, and conserves energy. If the lattice has a nite horizon (the horizon is the largest possible free ight for the point particle) the Lorentz gas is known to have a positive and nite diusion coecient [24]. A statistical ensemble of such systems is introduced which corresponds to lling space with a gas of innitely many particles (which are independent of each other). The time evolution of the statistical ensemble is governed by the Liouville equation (10) with extra conditions to describe the elastic collisions on the disks. Alternatively, the time evolution of the ensemble can be described by a Frobenius-Perron operator as shown elsewhere [2]. Dierent boundary conditions can be considered to solve the Liouvillian dynamics in such systems: (1) Absorbing boundary conditions [3{8]. We suppose here that the particle density is zero at the borders of the nite lattice for all times. This condition is equivalent to the escape of trajectories in free ight to innity outside the scatterer. In this case, the number N t of particles inside the scatterer decreases exponentially to zero: N t ' N 0 exp(?t) ; (16) in the double limit where the initial number of particles and the time become innite: N 0! 1 and t! 1. Eq. (16) denes the so-called escape rate. From the viewpoint of thermodynamics, the preceding situation translates as follows. At the level of the phe- 10

12 nomenological equation (2), the density at boundaries should be zero for all times: t = 0. The diusion equation is solved with this boundary condition to get where ' is the eigenfunction of t ' ' exp(?t) (t! 1) ; (17) D r 2 ' =? ' ; (18) associated with the smallest eigenvalue 0 and satisfying: = 0. If we replace this solution into the phenomenological entropy source (6) we get s = D (r)2 ' exp(?t) D (r')2 '! 0 ; (19) which vanishes when t! 1. As a consequence, this situation does not allow us to identify the thermodynamic entropy production because the entropy source vanishes together with the density itself due to the escape of all the particles. In order to properly identify the thermodynamic entropy production, we consider the following boundary conditions: (2) Flux boundary conditions [4,9]. We suppose that the Lorentz-type scatterer is submitted to a continuous ux of particles (see Fig. 1). For instance, a ux of density? of incoming particles reaches the left-hand side of the scatterer while a ux of density + >? reaches the right-hand side. The particles evolve according to the laws of mechanics. If the ux is continuous in time an invariant measure will established itself after some time at the level of the statistical ensemble. In such open and innite systems, we have thus to dene a measure t at time t which gives the local density of particles in phase space such that t (B) is the number of particles in the phase-space region B at time t. We notice that this measure is no longer normalizable because there is an innity of particles in the whole system: t (?) = 1. Thanks to the ux boundary conditions, a stationary gradient of concentration can be maintained in these open systems, which is a favorable situation for a possible identication of an entropy production. Indeed, the phenomenological entropy source (6) and entropy production (8) are now positive and constant in time. 11

13 We proceed with the construction of a probability measure t for the Poisson suspension over the dynamical system of measure t [25]. This Poisson suspension is a dynamical system on a phase space which is a direct product of innitely many copies of the original phase space?: M = 1? i=1 i. A point in this phase space M denes an ensemble of copies of the system: Y = fx i g 1 i=1 2 M. We can dene subsets of the phase space such that the number of copies X i inside the region B is xed to the integer k C Bk = fy 2 M : Card(Y \ B) = kg : (20) The probability measure of the Poisson suspension corresponding to the measure dened on? is dened by [25] (C Bk ) = [(B)]k k! exp[?(b)] ; (C B1 k 1 \ C B2 k 2 ) = (C B1 k 1 ) (C B2 k 2 ) if B 1 \ B 2 = ; : (21) and To show that this measure is a probability, we consider a partition fb i g of some subset A of the phase space: A = [ i B i? with B i \ B j = ; for i 6= j. Each cell B i of the partition contains a certain number of points of Y given by Card(Y \ B i ) = k i 2 f0; 1; 2; 3; :::g. The partition induced by fb i g in the phase space M is given by [ fki g C fbi k i g with C fbi k i g = C B1 k 1 \ C B2 k 2 \ C B3 k 3 \ \ C Bmkm ; (22) where fk i g denotes a conguration in which the number of particles in each cell B i is equal to a given integer k i. The measure of one element of the induced partition, i.e., the measure of a given conguration is given by applying the denition (21). Summing over all the congurations fk i g, we get that X fk i g (C fbi k i g) = my i=1 1X k=0 [(B i )] k k! exp[?(b i )] = 1 ; (23) so that the measure is normalized to unity and is thus a probability measure in this sense. The time evolution under a specic dynamical system (with discrete or continuous time) is given by 12

14 X t = t X 0 : (24) Accordingly, the measure t evolves in time as t+1 (B) = t (?1 B) ; (25) which induces a corresponding evolution for the probability measure t of the Poisson suspension. Let us remark here that the ux boundary conditions break the time-reversal invariance of the steady-state measure 1 under nonequilibrium conditions, in contrast to the Liouville equilibrium measure which is time-reversal invariant. IV. THE -ENTROPY Thanks to the probability measure of the Poisson suspension, we are now able to dene an -entropy for open systems with an innite number of particles. The entropy of the measure corresponding to the partition fb i g of A in the original phase space? is dened in terms of the probabilities of the elements of the induced partition in M. It is therefore an entropy of the Boltzmann type which characterizes the disorder of the probability measure. When all the cells B i have the same given size we shall speak of an -entropy S. Our denition is thus S = S(fB i g) =? X fk i g (C fbi k i g) ln (C fbi k i g) : (26) Remark. More systematic denitions of -entropy may be given as for instance S = Inf diambi S(fB i g) ; or S = Sup diambi S(fB ig) ; (27) or with -nets in the domain A [18] but, for simplicity, we shall use the denition (26) where all the cells are identical. Using the denitions (21) and (22), Eq. (26) becomes S =? X i 1X k=0 (C Bi k) ln (C Bi k) ; (28) 13

15 and S = X i e (B i ) ln (B i ) + R() ; (29) where the rest R() = P i O[(B i ) 2 ] = O() is important only if (B i ) 1 but is negligible for! 0 even if the measure is singular. This rest plays no role in the following argument and may be considered negligible but we shall keep it for rigor. In the case where the measure is absolutely continuous with respect to the Liouville measure, the associated density exists: f(x) = d=dx. The measures of the cells are given by (B i ) = f(x i )X where X i is a point inside B i according to the mean theorem of Riemann integration theory and where X is the Liouville measure of the cells B i. The -entropy is given by S =X = ln e X! Z A f(x) dx? Z A dx f(x) ln f(x) + O(X) + R(X) : (30) The second term is nothing else than the Gibbs entropy (1). The rst term diverges as = X! 0. Since R? fdx = 1 for a closed system with a normalized measure the rst term remains constant in time and may be disregarded. This term xes the famous constant of entropy according to the third law of thermodynamics. This term is very important to establish the correspondence with the entropy of quantum statistical mechanics where should be xed to X = f q f p = (2h) f. Thanks to the previous denition (26), we recover the usual expression of the equilibrium entropy per unit volume for instance in an ideal gas where S (vol) = ln e 5=2 2 3=2 + O( 3 q 3 p) : (31) 3 q 3 p m The previous denition is therefore entirely consistent with standard equilibrium statistical mechanics [26]. The -entropy (29) of a domain A evolves in time and we are interested in its time variation, i.e., in the dierence between its values between two successive instants of time separated for instance by a unit time: 14

16 S = S (t + 1; A)? S (t; A) X " = t (?1 e B i ) ln t (?1 B i )? e t(b i ) ln t (B i ) B i A # + R() : (32) If the measure t is absolutely continuous with respect to the Liouville measure we obtain S =X = + Z dx f t (X) Z! ZAin Aout e dx f t (X) ln f t (X)X + O(X) + R(X) ; (33) where we used the identity Z?1 A? Z A = Z A in? Z A out ; (34) that the dierence between the integrals over the preimage of A and over A itself is equal to the dierence between the integrals over the domain A in which enters A and the domain A out which exits A. In a volume-preserving system, the Jacobian of the mapping is equal to unity so that there is no term in (33) which could be identied with the entropy production. However, this holds as long as the density exists so that the terms O(X) may be neglected. This is no longer the case for singular measures. Let us now proceed with the separation of the -entropy variation into an -entropy ow and an -entropy production in analogy with Eq. (7). The -entropy ow can be naturally dened as the dierence between the -entropies of the domains ingoing and outgoing A e S = S (t; A in )? S (t; A out ) = S (t;?1 A)? S (t; A) ; (35) where the last identity follows from Eq. (34). The -entropy production can now be dened as i S = S? e S : (36) In the next section, we shall apply the previous denitions to a simple model of diusion. 15

17 V. ENTROPY PRODUCTION IN THE MULTIBAKER MAP The multibaker map is a model of deterministic diusion which can be seen as a caricature of the collision dynamics of the Lorentz gas [4,5,9]. Indeed, the dynamics of collisions from disk to disk is given by a Birkho map which governs the coordinates of the successive impact points and velocity angles at collisions. The Birkho map is area-preserving and of hyperbolic character. This map can be modeled by transformations of baker type between several squares which represents the dierent disks of the Lorentz gas. Points are mapped from square to square like particles undergoing collisions from disk to disk, which results in a deterministic motion of diusion. The multibaker can also be viewed as a deterministic realization of a symmetric random walk. Since we consider nite scatterers we suppose that the transformation is of baker type only on a nite number of squares forming a chain of length L + 1. At both ends of the chain, particles may exit or enter the chain in free motion with velocities +1 or?1. This is realized by a simple composition of translations to the left or the right in the half squares extending from both ends to innity (see Fig. 2). The phase space is therefore given by (n; x; y) where 0 x; y 1 and?1 < n < +1 is an integer labeling the square where the particle currently lies. The multibaker map is thus [9] (n; x; y) = 8 >< n? 1; 2x; y 2 n + 1; 2x? 1; y+1 2 ; 0 x < 1=2 ; +1 n L + 1 ; ; 1=2 x 1 ;?1 n L? 1 ; (n? 1; x; y) ; 0 x < 1=2 ; n 0 or L + 2 n ; >: (n + 1; x; y) ; 1=2 x 1 ; n?2 or L n : (37) A ux of particles is supposed to ow continuously across the chain. The particles in the half squares, 1=2 x 1 with n?1, arriving from innity on the left-hand end are assumed to be uniformly distributed with the density?, while those in the half squares, 0 x < 1=2 with L + 1 n, arriving on the right-hand end have a density +. As a consequence of the chaotic time evolution inside the chain, the measure is very complicated on the half squares which exit the chain. Indeed, there is a fractal repeller in 16

18 the squares 0 n L, which has the partial Hausdor dimension [9] d H = ln 2 ln cos L + 2 = 1? 1 4 ln 2 L O(L?4 ) : (38) Therefore, there are three types of orbits which exit the scatterer (i.e. for n?1 and L + 1 n): (1) The orbits which entered at the left-hand end: The density is? in their vicinity. (2) The orbits which entered at the right-hand end: The density is + in their vicinity. (3) The orbits of the unstable manifolds of the repeller. The unstable manifolds are segments of horizontal lines which separate the regions of density + from those of density?. Since the repeller is fractal, it is also the case for its unstable manifolds so that the measure on the exiting half squares is very complicated. However, the density always exists because it is equal to either + or?, except on the set of the unstable manifolds which is of zero Lebesgue measure. In the limit L! 1, the partial Hausdor dimension (38) of the unstable manifolds becomes equal to unity so that we can understand that the invariant measure becomes singular because the regions of densities alternate then everywhere. Since we expect singular measures in some limit we dene the cumulative distribution function associated with the measure t as [9] G t (n; x; y) = t n; [0; x[[0; y[ ; (39) which exists as a function even when the density does not exist. With this denition, Tasaki and Gaspard have shown that the invariant measure corresponding to the gradient of concentration r = +?? L + 2 ; (40) is given by the cumulative function: G 1 (n; x; y) = x [ n y + (r)t n (y)] x g n (y) ; (41) 17

19 where n = (r)(n + 1) +? ; (42) is the average density in the n th square and where ft n (y)g are the incomplete Takagi functions dened by the iterations T n (y) = 8 < : 1 2 T n?1(2y) + y ; 0 y < 1=2 ; 1 2 T n?1(2y? 1) + 1? y ; 1=2 y 1 ; with the boundary conditions, T?1 (y) = T L+1 (y) = 0 [9]. The incomplete Takagi functions are dierentiable almost everywhere because it is the case for the cumulative function (41) since the corresponding density exists according to the above reasoning. Let us mention here the property that T n (0) = T n (1) = 0. However, in the limit where L! 1 and ( +?? )! 1 keeping constant the gradient (40), the incomplete Takagi functions converge to the Takagi function dened by the following iteration [27] T (y) = 8 < : The convergence is exponentially fast like [9] 1 2 T (2y) + y ; 0 y < 1=2 ; (43) 1 T (2y? 1) + 1? y ; 1=2 y 1 (44) : 2 Sup 0y1 jt n (y)? T (y)j min(n;l?n) ; (45) so that T n (y) = T (y) + O(2?L=2 ) in the middle of the chain at n = [L=2]. The Takagi function is nondierentiable almost everywhere because its derivative with respect to y is formally given by [2] dt dy (y) = 1X t=0 [ t (y)] ; (46) where (y) = 2y (mod. 1) is the Bernoulli map of the interval and (y) = 1 if y < 1=2 or y > 1=2. The derivative of the Takagi funtion is thus given by a sum of plus and minus ones corresponding to the jumps of the particle to the right ( = +) or to the left ( =?) under the inverse multibaker mapping?1. Fig. 3 shows the rst few iterations of (44) where we 18

20 observe that, at the t th iteration, the function has already converged to its limit value at the points y = m=2 t with m = 0; 1; 2; :::; 2 t. These points can be assigned to symbolic sequences with symbols! t = 0 or 1 whether t = +1 or?1 according to y!1! t =! 1 2 +! ! t 2 t : (47) Using the construction of Fig. 3, we deduce the following properties of the Takagi function: 2 T? T y!1!t + 1? T (y 2 t!1! t ) = 1 2 ; (48) t and y!1! t t+1 " X T y!1!t + 1! 1! t 2 t Both properties will be used in the following. # 2? T (y!1! t ) = t 2 t : (49) In the limit L! 1, we moreover observe that the invariant measure (41) remains absolutely continuous with respect to the Lebesgue measure in the unstable direction x. This is because an initially regular measure is stretched in the unstable direction under time evolution and converges thus to a measure which is absolutely continuous in this direction. The measure is moreover uniform in the unstable direction if the stretching is uniform. In contrast, the measure (41) becomes singular in the stable direction because of the dierence of ingoing densities imposed at both ends. Formally, the density in the middle of an arbitrarily long chain is given by f 1 (n; x; y) y G 1 (n; x; y) = n + (r) 1X t=0 [ t (y)] ; (50) where the last term is a discrete form of the integral over the velocity of the diusive particle: R?1 0 v( t X)dt [2]. In the form (50), we recognize a steady-state measure of the type introduced by Lebowitz and McLennan [28,29]. The presence of this singular term is at the origin of a positive -entropy production, as shown below. Let us consider the time variation of the -entropy (32) in the n th square of the chain, we denote by A. We take the cells B i of size x in the unstable direction x and of size 19

21 y = 1 2 t ; (51) in the stable direction y to take advantage of the symbolic decomposition (47). In this case, the role of is played by (x; y). According to Eq. (32), we must consider the cells B i of size (x; y) in the square A = (n; [0; 1] [0; 1]) as well as their preimages?1 B i of size (x=2; 2y) which belongs to the half squares?1 A = (n + 1; [0; 1=2[[0; 1]) [ (n? 1; [1=2; 1] [0; 1]) (see Fig. 4). The time variation of the -entropy at the steady state is thus given by S = x 2 ; 2y?entropy of?1 A? x; y)?entropy of A : (52) The number of cells (x; y) in the unit square A is equal to 1=x in the unstable direction x and to 1=y = 2 t in the stable direction y. From Eq. (41), we infer that the measure of the cell B i = [x; x + x[[y!1! t ; y!1! t + y[ ; (53) is (B i ) = x g n (! 1! t ) with g n (! 1! t ) = g n (y!1! t + y)? g n (y!1! t ) : (54) Substituting in Eq. (52), the entropy variation becomes explicitly S = X " 1 g 2 n+1(! 1! t?1 ) ln! 1! t?1 xg n+1 (! 1! t?1 ) # + 1 g 2e 2 n?1(! 1! t?1 ) ln xg n?1 (! 1! t?1 )? X e g n (! 1! t ) ln! 1! t xg n (! 1! t ) 2e + O(x) : (55) We notice that the factors e=x in the logarithms can be eliminated using the properties that X! 1! t g n (! 1! t ) = g n (1)? g n (0) = n ; (56) 20

22 and that (1=2) n+1 + (1=2) n?1? n = 0. On the other hand, the -entropy ow (35) is given by e S = (x; y)?entropy of?1 A? (x; y)?entropy of A : (57) As a consequence, we obtain the -entropy production (36) as i S = x 2 ; 2y?entropy of?1 A? (x; y)?entropy of?1 A ; (58) or, by using the stationarity that implies S = 0 in Eq. (52), we get i S = (x; y)?entropy of A? 2x; y 2 The -entropy production of the multibaker is then explicitly given by?entropy of A : (59) X " 2g i S = g n (! 1 n (! 1! t 0)! t 0) ln! 1! t g n (! 1! t 0) + g n (! 1! t 1) # 2g + g n (! 1 n (! 1! t 1)! t 1) ln g n (! 1! t 0) + g n (! 1! t 1) ; (60) where we have taken the limit x! 0 to eliminate the term of O(x) which plays no role in our argument because the measure remains regular in the unstable direction x. The rst remarkable property of this -entropy production is its positivity i S 0 ; (61) which follows from the concavity of the function z lnz [i.e., (d=dz) 2 (z ln z) 0] so that X a ln 2a a + b + b ln 2b a + b! 0 : (62) We have numerically calculated the expression (60) for a multibaker chain of length L = 50. Fig. 5 shows i S as a function of t = log 2 (1=y). We observe that i S is approximately constant with respect to t. Therefore, the -entropy production displays a plateau at a positive value which depends on the position n along the chain. In the case of Fig. 5, we have taken? = 1 and r = 1 so that the mean density in the n th cell is n = n + 2. Fig. 6 shows i S as a function of the position n, for dierent values of. We 21

23 then compare with the behaviour expected from the phenomenological entropy production which is i S phenom = D (r)2 n = 1 2(n + 2) ; (63) for a square of unit length because the diusion coecient is D = 1=2 for the multibaker. We observe a remarkable agreement between both curves except at the ends of the chain. The decrease of the -entropy production at the ends is explained by the fact that the density is there constant to the values over large parts of the square so that i S tends to zero more rapidly at the ends than in the middle of the chain as! 0. The critical value of y below which i S tends to zero depends on the position n along the chain in a way which is determined by Eq. (45) as y c 2?min(n;L?n) : (64) Hence, the critical scale decreases exponentially fast as L; n! 1 due to the Lyapunov instability of the dynamics. For a xed value of y, we should thus observe the vanishing of the -entropy production only in some boundary layers of the order of the inverse Lyapunov distance: n c ln 2. Away from these small boundary layers, the -entropy production reaches a positive value we shall now calculate. We suppose that we are in the middle of the chain at values of y above the critical value (64) so that the incomplete Takagi functions in (41) can be replaced by the limiting Takagi function. In Eqs. (60) or (62), we set a = m + =2 and b = m? =2 and we expand in Taylor series of =m to get with i S = X m " 2m # + + O ; (65) 6 2m m 6 m = n y 2 + (r) T a + T b 2 ; and = (r)(t a? T b ) ; (66) where 22

24 T a T b = T y!1! t t+1 = T y!1!t t? T (y!1! t ) ;? T y!1! t t+1 : (67) We can here use the properties (48) and (49) of the Takagi function which imply that T a? T b = 1 2 t = y ; X (Ta + T b ) 2 = t 2 t = y log 2 1 y ; (68) and, moreover, the property that P (T a + T b ) = T (1)? T (0) = 0. Expanding in series of (r)= n, we nally obtain i S = (r)2 2 n + (r) n 6 + log 2 1 y + O (r) 6 5 n ; (69) for y > y c. The remarkable result is that the leading term is precisely the entropy production expected from irreversible thermodynamics. The next term is a correction which is small like (r) 4 and which slowly increases as y! y c. This behaviour is observed in Figs. 5 and 6: Near the left-hand end of the chain where n is small enough, we observe a slow linear increase of i S versus log 2 (1=y) in Fig. 5. This increase becomes negligible where n is larger. We emphasize that these results are entirely due to the Tagaki function and to its nondierentiability, which therefore controls the entropy production. VI. CONCLUSIONS In this paper, we have revisited the problem of entropy production in volume-preserving system under steady nonequilibrium conditions. We have supposed that the system sustains a diusion process and is submitted to ux boundary conditions. With such boundary conditions, the invariant measure of the Liouvillian dynamics is no longer the uniform Liouville measure. Indeed, the invariant measure at a phase-space point X 2? has the density of the boundary point X b from which the point X is issued under time evolution: X = t X b. In the limit where xed nonequilibrium gradients are imposed at arbitrarily large distances, arbitrarily small subsets 23

25 of phase space contain points coming from almost every point of the so that the density varies innitely fast and the measure becomes singular. These asymptotic invariant measures corresponding to nonequilibrium steady states have been known since works by Lebowitz and McLennan [28,29]. They are given as Zubarev local integrals of motion [2,30]. In the example of the multibaker map, the cumulative function of these measures is expressed in terms of the nondierentiable Takagi function [9]. The singular character of these steadystate measures forces us to use a Boltzmann entropy or -entropy instead of the Gibbs entropy which only applies to regular measure. We have then showed that the -entropy production has precisely the behaviour expected from irreversible thermodynamics for diusion in the multibaker map. Our result may be formulated as lim!0 lim L!1 lim (r)=!0 (r) 2 is = D 0 ; (70) where the limits are not commutative. Because the limit L! 1 of a large chain has to be taken before the ne-grained limit! 0 we should understand the entropy production as an emerging property appearing in the limit of large systems. We remark that the -entropy is the entropy obtained after a coarse graining of the phase space with an -partition. However, the previous result is independent of the particular coarse graining because the entropy production dened in (70) does not vanish in the ne-grained limit. This is in contrast to the usual coarse-graining considerations which depend on the particular partition. The nontrivial ne-grained limit is here due to the singular character of the invariant measure, which is the new result. We notice that the singular character of the invariant measure under nonequilibrium conditions appears very rapidly in large systems because of the convergence property (45). We can translate this rapid convergence for a uid of particles of diameter d as follows. >From the analogy with a Lorentz gas, the transition from one square of the multibaker to neighbouring squares corresponds to the free ight of a particle from a collision to the next one, i.e., to a mean free path ` = 1=() where is the particle density and = d 2 the collision cross-section. At each collision, a perturbation on a velocity angle is amplied 24

26 like! (2`=d). At a distance z = m` of m mean free paths from the wall, about m collisions have occurred so that the critical scale below which the absolute continuity is hidden is c (d=2`) m. Expressed in terms of the distance z from the wall, the critical scale would be c exp(?z) where (1=`) ln(2`=d) is the Lyapunov exponent per unit distance. The width of the boundary layer where the -entropy production should be smaller than its bulk thermodynamic value should thus be z c `= ln(`=d). It is only beyond this boundary layer that we may expect the -entropy production to reach its thermodynamic value. We notice that the scale where the absolute continuity of the nonequilibrium invariant measure is hidden becomes exponentially small as z z c toward the bulk of the uid because of the Lyapunov dynamical instability. Therefore, the bulk behaviour of the nonequilibrium steady state becomes essentially determined by the nondierentiability of the Tagaki function or, equivalently, by the singularity of the Lebowitz-McLennan steady-state measures. We may conclude that we have here identied the appropriate mechanism at the origin of the thermodynamic entropy production in volume-preserving systems. Acknowledgements This paper is dedicated to the 60 th anniversary of Bill Hoover to whom so much is owed for his pioneering works at this frontier of nonequilibrium statistical mechanics. The author would like to thank Prof. G. Nicolis for support and encouragement in this research. He expresses his gratitude to Prof. Ph. Choquard for the invitation to the Semester on `Hyperbolic Systems with Singularities' held at the Erwin Schrodinger International Institute for Mathematical Physics (ESI, Vienna) in September 1996, where this paper was written. He would also like to thank Profs. J. R. Dorfman, S. Tasaki, T. Tel, and H. van Beijeren for fruitful discussions. The author is nancially supported by the National Fund for Scientic Research (FNRS Belgium). 25

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30 FIGURE CAPTIONS Fig. 1. Schematic representation of a nite Lorentz gas under ux boundary conditions. The nite Lorentz gas is composed of a slab of width L cut out of the innite Lorentz gas. This slab of disks forms a scatterer for a gas of independent particles arriving at the left-hand border with a density? and at the right-hand wall with a density +, creating a gradient r = (1=L)( +?? )e x of concentration. Fig. 2. Representation of the action of the open multibaker map in its phase space which is composed of an innity of squares. The map acts like a baker transformation on the chain of the squares 0 n L and by left or right translations outside the chain up to innity. Fig. 3. Construction of the Takagi function by successive iterations according to Eq. (44): lim t!1 T (t) (y) = T (y), the seed function being zero. Fig. 4. Action of the multibaker map on three successive squares of the chain and on the cells fb i g of a (x; y)-partition of the n th square taken as the domain A in Eq. (58). Fig. 5. The -entropy production (60) calculated numerically for a multibaker chain of length L = 50, a unit gradient r = 1, and? = 1. The -entropy production is depicted as a function of t = log 2 (1=y), for dierent positions n along the chain. Fig. 6. The same -entropy production as in Fig. 50 for a multibaker chain of length L = 50 but depicted here as a function of the position n, for dierent values of y. The dashed line represents the phenomenological entropy production (63). 29