Chaos, Spatial Extension and Non-Equilibrium ... - CiteSeerX

Anosov systems are oversimplifications like square clouds or spherical chickens. Hoover [77, p. 219] Palmstr¨om denkt die Alpen sich als Kubus.. Und besteigt sie so mit seinem Tubus. Morgenstern [118]

Chaos, Spatial Extension and Non-Equilibrium Thermodynamics∗ J¨ urgen Vollmer Department of Theoretical Physics, University of Essen, 45117 Essen, Germany. Max-Planck-Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany. http://www.mpip-mainz.mpg.de/~vollmer November 3, 2000

Abstract The connection between the statistical physics of transport phenomena and a microscopic description of the underlying chaotic motion has recently received new attention due to the convergence of ongoing developments in the theory of deterministic chaotic systems, in the foundation of non-equilibrium statistical physics and of non-equilibrium molecular dynamics simulations. I give an overview of this developments with an emphasis on explicit calculations on exactly solvable models, which may serve as paradigms for the improved understanding.

∗

Contribution to Phys. Rep. Habilitation Thesis, Univ.-GH Essen (2000)

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Contents 1 Introduction 1.1 The Markovian Postulate in Statistical Physics . . . . . . . . . . 1.2 Chaos and the Chaotic Hypothesis . . . . . . . . . . . . . . . . 1.3 Dissipation and phase-space contraction . . . . . . . . . . . . . 1.4 A Gedankenexperiment: Particle transport through a wire . . . . 1.4.1 The Thermodynamic Entropy Balance . . . . . . . . . . 1.4.2 Steady-state entropy production according to Boltzmann 1.5 Objectives and Outline . . . . . . . . . . . . . . . . . . . . . .

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2 The 2.1 2.2 2.3 2.4 2.5 2.6

Galton Board: Coarse Graining and Markovian Dynamics Encoding and the probability distribution . . . . . . . . . . . . A one-dimensional map mimicking the Galton-board dynamics Time evolution of the probability density and Markov chains . Separation of scales and the Fokker-Planck equation . . . . . Many-Particle Systems and Irreversibility . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 The 3.1 3.2 3.3 3.4 3.5

MultiBernoulli Map: Hydrodynamic Modes and Escape Rate Formalism The dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absorbing boundary conditions and the escape rate formalism . . . . . . . . Parameter dependence of the diffusion coefficient . . . . . . . . . . . . . . . Symbolic dynamics and complex Markov chains . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 The 4.1 4.2 4.3 4.4

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Field-Free Lorentz Gas: Mixing, Relaxation, and Time-Reversibility The Poincar´e map for the Lorentz gas . . . . . . . . . . . . . . . . . . . Relaxation to equilibrium and time reversibility . . . . . . . . . . . . . . . The spatially-extended Lorentz gas . . . . . . . . . . . . . . . . . . . . . Implementation of boundary conditions . . . . . . . . . . . . . . . . . . . 4.4.1 Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . 4.4.2 Reflecting boundary conditions . . . . . . . . . . . . . . . . . . . 4.4.3 Absorbing boundary conditions . . . . . . . . . . . . . . . . . . . 4.4.4 Flux boundary conditions . . . . . . . . . . . . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 The Driven Lorentz Gas: Phase-Space Contraction and Time-Reversible Dissipation 5.1 The Poincar´e map for the Lorentz gas with an external force . . . . . . . . . . 5.2 Time-reversible thermostating at collisions . . . . . . . . . . . . . . . . . . . . 5.3 Time-reversible thermostating in the bulk . . . . . . . . . . . . . . . . . . . . 5.3.1 The Gaussian thermostat for the Lorentz gas . . . . . . . . . . . . . . 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 6 The 6.1 6.2 6.3 6.4 6.5 6.6

Isothermal MultiBaker Map: Particle Flow and Phase-Space Structures The multibaker map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time evolution of the coarse-grained particle density . . . . . . . . . . . . . Time Reversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Invariant densities and Takagi functions . . . . . . . . . . . . . . . . . . . . The physics underlying the structure of Takagi functions . . . . . . . . . . .

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Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 MultiBaker Maps for Particle and Heat Flow 7.1 Heat conduction and phase-space densities . . . . . . . . . 7.2 Time evolution of the coarse-grained energy density . . . . 7.3 Time evolution of the coarse-grained kinetic-energy density 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 The Information-Theoretic Entropy Balance for Dynamical Systems 8.1 Information entropy and statistical physics . . . . . . . . . . . . . . . . . . . .

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8.5

The Gibbs and the thermodynamic entropy . . . . . . . . . . . . . . . 8.2.1 The coarse-grained entropy . . . . . . . . . . . . . . . . . . . . 8.2.2 The Gibbs entropy . . . . . . . . . . . . . . . . . . . . . . . . . Information gain according to Kullback . . . . . . . . . . . . . . . . . . 8.3.1 Positivity of the information gain . . . . . . . . . . . . . . . . . 8.3.2 The steady-state entropy production as the average growth rate relative phase-space density . . . . . . . . . . . . . . . . . . . . Time evolution of the entropies . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Evolution of the entropy . . . . . . . . . . . . . . . . . . . . . 8.4.2 Relaxation to equilibrium . . . . . . . . . . . . . . . . . . . . . 8.4.3 Relaxation to a non-equilibrium steady state . . . . . . . . . . . A local entropy balance . . . . . . . . . . . . . . . . . . . . . . . . . .

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Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9 The Thermodynamic Entropy Balance for MultiBaker Maps 9.1 The discrete entropy balance . . . . . . . . . . . . . . . . . 9.1.1 Irreversible entropy production . . . . . . . . . . . . 9.1.2 Entropy flux . . . . . . . . . . . . . . . . . . . . . . 9.2 Entropy balance in the macroscopic limit . . . . . . . . . . . 9.2.1 Irreversible entropy production . . . . . . . . . . . .

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9.2.2 Entropy flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 The influence of the choice of spatial resolution . . . . . . . . . Steady-state entropy production according to Gaspard-Gilbert-Dorfman Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 10 Global vs. Local Entropy Production 10.1 Thermostating . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 An isothermal multibaker chain for field driven transport 10.1.2 Entropy production in thermostated systems . . . . . . . 10.2 Entropy Production in the Escape Rate Formalism . . . . . . . . 10.2.1 Bulk contribution to the irreversible entropy production . 10.2.2 The contributions from the boundaries . . . . . . . . . . 10.3 Thermoelectric cross effects . . . . . . . . . . . . . . . . . . . . 10.3.1 The Peltier effect . . . . . . . . . . . . . . . . . . . . . 10.3.2 The Seebeck effect . . . . . . . . . . . . . . . . . . . . 10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Fluctuations, Dissipation and Green-Kubo Relations 11.1 Isothermal systems with periodic boundary conditions . . . . . 11.1.1 Fluctuations of the phase-space contraction rate . . . . 11.1.2 The statistics of fluctuations . . . . . . . . . . . . . . 11.1.3 Einstein and Green-Kubo relations for the conductivity 11.2 The entropy production of a tagged particle . . . . . . . . . . 11.2.1 Fluctuations of the entropy-production rate . . . . . . 11.2.2 The fluctuation relation for the entropy production . . 11.2.3 The global fluctuation relation for a tagged particle . . 11.2.4 The generalized Einstein relation . . . . . . . . . . . . 11.3 Bounds to the linear-response regime . . . . . . . . . . . . . . 11.3.1 The Demoivre-Laplace theorem . . . . . . . . . . . . . 11.3.2 A sharper bound for multibaker maps . . . . . . . . . 11.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12 Outcomes, Outlook, Open Questions 101 12.1 Chaos and non-equilibrium thermodynamics . . . . . . . . . . . . . . . . . . . 102 12.1.1 The origin and modeling of irreversible behavior . . . . . . . . . . . . . 102 12.1.2 Coarse graining and balance equations . . . . . . . . . . . . . . . . . . 103 12.1.3 The macroscopic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 12.1.4 Necessary generalizations with respect to conventional dynamical systems 104 12.2 Applicability of non-equilibrium thermodynamics and linear response . . . . . . 105 12.2.1 Particles vs. thermodynamic fields . . . . . . . . . . . . . . . . . . . . 105 12.2.2 Independence on coarse graining . . . . . . . . . . . . . . . . . . . . . 105 12.2.3 Reversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 12.2.4 Validity of linear response . . . . . . . . . . . . . . . . . . . . . . . . . 106 12.3 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 12.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Acknowledgements

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5 A Elements of Non-Equilibrium Thermodynamics A.1 Thermodynamic forces and currents . . . . . A.2 Identifying transport coefficients . . . . . . . A.3 Relating transport and diffusion coefficients . A.4 Time evolution of the “multibaker gas” . . .

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B List of Symbols

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C Translation of German citations

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References

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Bring vor, was wahr ist; schreib’ so, daß klar ist und verficht’s, bis es dir gar ist Boltzmann [9]

1

Introduction

Boltzmann [9] put these words at the beginning of his book Vorlesungen u ¨ber die Principe 1 der Mechanik. They characterize his involvement with explaining the facts that (i) regular physical processes are described by continuous variables that emerge at the macroscopic level of everyday life from the extremely complicated motion of a vast number of particles; (ii) the passage to the macroscopic description is accompanied by a break of time-reversal symmetry. It leads from reversible microscopic motion to irreversible macroscopic phenomena. Establishing the link (i) between microscopic and macroscopic evolution equations lead to the rise of (equilibrium) statistical physics in the 20th century. Item (ii) touches upon the second law of thermodynamics, which states that the entropy can never decrease. How this behavior can emerge from a reversible microscopic dynamics is often referred to as the paradox of irreversibility. In establishing this link Boltzmann founded kinetic theory, which for a long time was the only analytical approach to evaluate thermodynamic response functions and transport coefficients. One of its early successes was Lorentz’s [108, 109] treatment of the Drude model [33] for electric and heat conduction in metals. Based on the assumption of independent electrons that propagate in free motion in between collisions after each free time τ the model gives an astonishingly detailed description of transport. According to Ashcroft and Mermin [3, chapter I] “it is still used today as a quick practical way to form simple pictures and rough estimates of properties.” In the 1980s and 90s the availability of powerful workstations for simulation and visualization of the properties of many-particle systems lead to the revival of interest in simple deterministic models for transport. In this context also deterministic variants of the Drude model initially considered by Lorentz [109] came back into the focus of research. The new emphasis lies on tracing relations between characteristics of the microscopic chaotic motion of the particles and transport coefficients. In the present review we describe these developments with an emphasis on explicit calculations on analytically solvable models. One of the objectives is to place the results into context with classical work on the foundations of statistical physics.

1

A translation due to Brush [15] is known under the title Lectures on Gas Theory.

1.1 The Markovian Postulate in Statistical Physics

1.1

7

The Markovian Postulate in Statistical Physics

A widely accepted2 resolution of the paradox of irreversibility lies in the statistical interpretation of the macroscopic laws, as it was lucidly suggested by Maxwell [114, chapter XXII]. In this spirit one recognizes that the second law should be seen as an educated guess predicting what the overwhelming majority of systems will do in a certain situation, rather than a physical law without exceptions. In the formulation of Gibbs [66] and Boltzmann [9] it comprises that “the impossibility of an uncompensated decrease of entropy seems to be reduced to an improbability.” This entails (cf. for instance van Kampen [166]) that there cannot be a rigorous derivation of transport equations without making use of additional information or assumptions. Often the ergodic hypothesis is evoked to that end, which has recently been rigorously underpinned for hard-ball systems (Sinai [149], Sz´asz [152]). This hypothesis focuses, however, on equilibrium properties of the system and has little to say about non-equilibrium processes and fluctuation phenomena (cf. Tolman [161], Penrose [127, 128], Ruelle [145]). A more general approach lies in adapting the Markovian postulate 3 thoroughly discussed by Penrose [127]. In this approach an observation on a physical system is idealized as an instantaneous measurement of a particular set of dynamical variables at discrete and equally spaced instances of time, and it is assumed that successive observational states of the macroscopic system constitute a Markov chain. This approach enables one to deal with properties of individual systems as well as of ensembles of systems, and it allows to treat equilibrium and non-equilibrium phenomena on equal footing. We introduce these notions in more detail in Sect. 2–5 by explicitly working out two model systems.

1.2

Chaos and the Chaotic Hypothesis

Remarkably, it was already pointed out in the mid 1970s that rather than as a technical assumption needed to derive thermodynamics, the Markovian postulate should be viewed as a consequence of a more fundamental requirement on the microscopic motion of the systems (Vstovsky [173]): In order to make the macroscopic description of the dynamical system possible, the latter must have the property of “exponential decay of correlations” or even be a “Y (or C) system” in the sense of Anosov. However, this is too strong a requirement, since real systems are usually extremely unstable with respect to the initial state (dynamically unstable). This conclusion is remarkably analogous to the starting point of very recent developments aiming at relating transport properties to characteristics of a chaotic microscopic motion. Many of these studies are based on (or at least compatible with) the chaotic hypothesis of Gallavotti and Cohen [51] which states that 2

An overview of mathematical subtleties, which are not yet settles is given in Lieb [104]. Alternative point of views were recently discussed at a round-table discussion of Klein, Lebowitz [96], Ruelle [145] and Prigogine [131] on StatPhys 1998 in Paris, and in a book by Hoover [77]. 3 According to Penrose [127] the postulate implicitly appears already in early work of Ehrenfest and Ehrenfest [34], of Onsager [124], and of Kirkwood [88], while the first explicit proposal is due to Green [71].

8

1 Introduction

B

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Figure 1: Action of the baker map on the unit square. The region 0.5 < x < 1 is uniformly shaded and marked by the letter B to illustrate the compression of vertical and expansion of horizontal distance due to the action of the map. The symbols in the region 0 < x < 0.5 indicate the position of 2500 randomly chosen initial conditions that are followed over the iterations. The 1st, 2nd, 5th, and 8th iteration, and the recovered initial condition after eightfold application of the inverse map are shown.

A reversible many-particle system in a stationary state can be regarded as a transitive Anosov system for the purpose of computing the macroscopic properties of the system. From the point of view of a physicist the Anosov property might be considered as requiring a strongly chaotic dynamics that acts bijectively on the underlying phase space. This requirement facilitates a rigorous mapping of the dynamics of suitably chosen state variables to a Markov chain. Reversibility is lost in the reduction of the reversible microscopic motion to the coarse-grained states of the Markov process, thus resolving the irreversibility paradox. The idea that strong chaos entails might be taken as a starting hypothesis to build statistical physics is actually much older. It was already described by Hopf [81] in the 1930s. He introduced to that end a map, now known as baker map, that acts on points on the unit square as follows ( y
2x, for 0 ≤ x < 0.5 2 B : (x, y) 7→ (1) y−1
2x − 1, 1 + 2 for 0.5 ≤ x < 1 The action of the map is illustrated in Fig. 1. It is one of the simplest illustrations of how a stretching (in the vertical direction) can work together with compression and folding (in the vertical direction) to produce deterministic chaos. The action of the baker map can also be represented as an action of the binary representation of the x and y coordinates. Let these coordinates be ∞ ∞ X X x= si 2−(i+1) , y= s−i 2−i i=0

i=1

where the si take the values 0 or 1. Then one easily verifies by induction that the nth image (x(n) , y (n) ) of (x, y) takes the coordinates

1.3 Dissipation and phase-space contraction

x(n) =

∞ X

sn+i 2−(i+1) ,

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y (n) =

∞ X

sn−i 2−i

(2)

i=1

The relation holds for positive as well as negative n, such that the action of B on the symbol sequences can be represented as shifting an offset in the bi-infinite sequence uniquely representing a trajectory, B : (· · · s(−2) s(−1) • s(0) s(1) s(2) · · ·) 7→ (· · · s(−2) s(−1) s(0) • s(1) s(2) · · ·)

(3)

After eight iterations the common property that all of the 2500 initial conditions followed in Fig. 1 have coordinates in the range 0 < x < 0.5 amounts to saying that the eighth digit in the binary representation of all the points in the lower left graph should be 0, i.e., s−8 = 0. Clearly this cannot be noticed by a meaningful observation, leading to the observed relaxation to a uniform density. Note that the uniform density can be viewed as representing a microcanonical distribution. It is approached since the map is strongly chaotic and area preserving (i.e., the Liouville theorem applies, cf. standard textbooks on statistical physics like Landau and Lifshitz [94], Reichl [136]). Dorfman [30] argued in his book that the prevalence of local equilibrium in non-equilibrium systems may be attributed to this rapid decay of initial atypical distributions.

1.3

Dissipation and phase-space contraction

The chaotic hypothesis was introduced in the context of transport in systems with periodic boundary conditions where transport is driven by external fields, and a stationary state is enforced by a deterministic reversible thermostat. In their numerical work Evans and Morriss [38], Hoover [75]) and collaborators noted that on a microscopic level a thermostat must not be viewed as a thermodynamic drag force, but should rather dynamically consume the energy released into the system by the external field. In particular, when due to some fluctuation the particles move against the field such the system cools down a thermostat should also be capable of releasing heat into the system rather than removing it. The removal or release of heat appears as a phase-space contraction or expansion, respectively, in the equations of motion. Such systems are called dissipative systems in the theory of dynamical systems. A surprising finding of the numerical studies was that in spite of dissipation a system can then still have a time-reversible dynamics that acts bijectively on the full phase space. At fist sight these properties seem incompatible. However, by a straightforward generalization of the baker map one can illustrate how they can be matched. To this end we consider (
x (1 − ξ) y for 0 ≤ x < ξ ξ,
Bξ : (x, y) 7→ (4) x−ξ for ξ ≤ x < 1 1−ξ , 1 + ξ(y − 1) For all values of 0 < ξ < 1 this map acts one-to-one on the unit square. For ξ = 1/2 it coincides with the conventional baker map (1), but for all other values it leads to nontrivial local contraction and expansion rates. Again it can be written in the form of Eq. (3) by saying that s(n) takes the value 0 or 1 when x(n) lies in the range 0 < x(n) < ξ or

10

1 Introduction

1

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Figure 2: Action of the asymmetric baker map for ξ = 1/3. The region ξ < x < 1 is uniformly shaded and marked by the letter B to illustrate the compression of vertical and expansion of horizontal distance due to the action of the map. The symbols in the region 0 < x < ξ indicate the position of 1667 randomly chosen initial conditions that are followed over the iterations (i.e., the density of initial conditions coincides with the one chosen in Fig. 1). The 1st, 2nd, 5th, and 8th iteration, and the recovered initial condition after eightfold application of the inverse map are shown.

ξ < x(n) < 1, respectively. The symbols 0 and 1 appear then with the probability l = ξ and r = 1 − ξ, respectively, i.e., not all symbol sequences appear with the same a priori probability any longer. This is reflected in the observation (Fig. 2) that an initial density no longer approaches the microcanonical distribution, but evolves towards a fractal attractor.4 We interpret the map as bringing forth a random-walk-like transport process by considering the symbol 0 (1) as a step to the right (left). Since the map is one-to-one on the unit square the attractor is supported on the full square, allowing it to comply with the requirement of time reversibility that both forward and backward trajectories are part of the attractor. On the other hand, one can show that due to the non-trivial contraction and expansion rates the time reverse of a typical trajectory only carries an exponentially small weight such that it will never be observed in practice. The importance of these difference in the dynamical weights or trajectories for the understanding of irreversible thermodynamic behavior was pointed out by Hoover, Kum, and Posch [78], Dettmann, Morriss, and Rondoni [27]. It will be one of the main points to address in the final parts of the review. Based on this work of Evans and Morriss, Hoover, and their collaborators, and on related work by Chernov, Eyink, Lebowitz, and Sinai [18, 19] and Ruelle [144], Gallavotti and Cohen [51, 52, 21] relate the dissipated heat with the rate of phase-space contraction. Then they discuss the properties of the entropy production connected to this dissipation, and its fluctuations. Green-Kubo relations and Onsager relations could straightforwardly be derived in this setting Gallavotti and Cohen [51], Gallavotti [47], Cohen [21], Gallavotti [48]. Closely related results were obtained by Searles and Evans [39, 40, 146] who even 4

The figure can of course only be an indication for the appearance of fractal structures . A more careful analysis underpinning the statement was given by Tasaki et al. [156].

1.4 A Gedankenexperiment: Particle transport through a wire

11

thermostat: T reservoir T, ρL

A

reservoir T, ρR

particle

a a a ...

L

a

A

x

Figure 3: Graphical illustration of the transport process modeled by the Drude gas. A system of spatial extension L is attached to reservoirs inducing particle and heat currents due to the differences in the densities ρ. The temperatures T along the wire is kept uniform by heat exchange with a thermostat.

succeed in their approach to give bounds on the validity of the linear response regime. In spite of these successes the description of transport properties based on the empirical identification of the phase-space contraction and entropy production rates meets severe difficulties in providing a satisfactory description of the thermodynamic entropy. These problems are so severe that Gallavotti [50] suggests to regard the problem of a definition of entropy in non-equilibrium states as entirely open: One of the key notions in equilibrium statistical physics is that of entropy ; its extension to non-equilibrium is surprisingly difficult, assuming that it really can be extended. [. . . ] We shall take the attitude that in a stationary state only the entropy creation rate is defined: the system entropy decreases indefinitely, but at a constant rate. Vollmer, T´el, and M´ aty´as [172] argued that the problems with the definition of entropy in Gallavotti’s exposition might originate from (i) the focus on the Gibbs entropy to characterize non-equilibrium states, and (ii) the fact that the approach (similar to many arguments based on Boltzmann’s approach to relate the entropy on to the number of microscopic realizations of a macrostate) only makes statements about a global entropy balance while the central results of non-equilibrium thermodynamics apply to a local balance equation. This will be elaborated in much more detail later on. First, we describe a gedankenexperiment, however, to further illuminate the difficulties to be tackled in the following.

1.4

A Gedankenexperiment: Particle transport through a wire

We consider particle transport through a quasi-one-dimensional wire of length L that is attached at its outermost ends to particle reservoirs of different particle density ρL and ρR , respectively. The density gradient sets up a particle flow along the direction x of the wire. Irrespective of the details of the dynamics this system relaxes to a steady state supporting a constant current j. We will discuss now the entropy production associated with this state. A sketch of this model is given in Fig. 3.

12 1.4.1

1 Introduction The Thermodynamic Entropy Balance

The thermodynamic entropy balance ∂t s = −∂x j (s) + σ (irr) .

(5)

relates the time derivative of the entropy density s to the sum of the divergence of an entropy current j (s) and the rate of entropy production σ (irr) per unit volume. The balance is based on the assumption of local equilibrium, i.e., the requirement that any small volume of the system locally fulfills the Gibbs relation ds =

1 µ du − dρ T T

where s, u and ρ are the entropy per unit volume, the internal energy per unit volume and the particle density, respectively. We look now into that relation for the conceptually simplest case (more general situations will be discussed later on) where the temperature and the internal energy are constant throughout the system. Since the number of particles is a conserved quantity, the time derivative ∂t ρ = −∂x j is related to the divergence of the particle current j. Consequently, the time derivative of the entropy density can be written in the form ∂t s = ∂x

µj j ∂x µ − T T

such that the entropy current as can be identified as j (s) = µj/T , and the entropy production as σ (irr) = j ∂x µ/T . The latter quantity characterizes the dissipative heating of the system. For instance, for particles of charge e in Ohm’s setting, where the current is driven by an external electric field E = −∂x φe , one has µ ≈ eφe such that the entropy production reduces to the well-known expression σ (irr) T = jE for Ohm’s heat released in electrical conduction. In the linear response regime one can rewrite this expression using the conductivity σel ≡ j/E and Einstein’s relation for the diffusion coefficient ρD = σel kB T to obtain σ (irr) =

j2 jE = kB . T ρD

(6)

The latter expression even holds for systems where the particle transport is not induced by an external field but by density or even temperature gradients. A general derivation is given in Appendix A. In this introduction we only make the point that (i) the entropy production is always positive since the density and the diffusion coefficient are positive, and it vanishes in equilibrium where j = 0; (ii) in non-equilibrium thermodynamics entropy production is found based on a local analysis. It localizes the entropy production in regions with non-vanishing currents. 1.4.2

Steady-state entropy production according to Boltzmann

Boltzmann relates the entropy of a macroscopic state to the number microscopic realizations of the state. For the considered flow we characterize the macroscopic state by prescribing

1.4 A Gedankenexperiment: Particle transport through a wire

13

that there should be Nm ≡ aρm particles in any cell m = 1 · · · M with x ∈ [(m − 1)a, ma], and N0 ≡ Aρ0 and NM +1 ≡ AρM +1 particles in the left and right reservoir, respectively. In this setting a stationary non-equilibrium state amounts to a state where N1 , · · · , NM do not change in time, while the changes of N0 and NM +1 are insignificant. From the point of view of the system including the reservoirs this is thermodynamic steady states are thus rather transient states that appear stationary on macroscopic time scales. Eventually they relax to equilibrium. The separation of time scales applies when the reservoirs are much larger P than the cells, A  a, such that the number M in the wire constitutes m=1 Nm of particles P +1 only a negligible fraction of the total number of particles Ntot ≡ M m=0 Nm . The Boltzmann entropy characterizes the dynamical disorder in a system. Up to a factor kB (the Boltzmann constant introduced by Planck) it amounts to the logarithm of the number of microscopic realizations of a state. Focusing only on the spatial degrees of freedom for the present system it takes the form S (B) = −kB ln

Ntot ! . N0 ! · · · NM +1 !

(7)

The entropy evolves in time due to the transfer of τ j particles per unit time τ from one reservoir to the other. The Boltzmann entropy characterizes the dynamical disorder in a system. Consequently, there is an associated change of the Boltzmann entropy 0

0 S (B) − S (B) kB N00 ! · · · NM +1 ! = ln τ τ N ! · · · NM +1 ! 0  kB (N0 − τ j)! (NM +1 + τ j)! ≈ ln τ N0 ! NM +1 ! NM +1 ≈ −j ln , N0

where the prime denotes quantities evaluated at time t + τ . In the linear response regime, where there is only a small density difference between the left and right reservoir, one can write for the logarithmic term M X ρM +1 −D ∂x ρ jL NM +1 ln = ln ≈ −a ≈L . N0 ρ0 ρD ρD i=0

Here, the particle current was introduced as j = −D∂x ρ, and the length of the system is identified as L ≡ a(M + 1). Altogether, one thus finds 0

j2 S (B) − S (B) (irr) = σtot ≈ L , τ ρD (irr)

(8)

where σtot stands for the rate of entropy production of the entire system. It fully agrees with the thermodynamic results Eqs. (5) and (6). After all, the integral of ∂x j (s) over a R closed system vanishes, and for the integrated entropy production one finds dx σ (irr) = j 2 L/(ρD). This agreement is remarkable for a number of reasons. (i) An entirely different perspective was taken in the derivations. The Boltzmann picture

14

1 Introduction

takes a global perspective, where the irreversible entropy production is entirely due to the relaxation of a density difference of the reservoirs. In contrast, thermodynamics only considers a local portion of the fluid, and does not make any assumptions on the reservoirs at all. (ii) The local thermodynamic balance localizes the entropy production in regions with nonvanishing currents, while (if at all) one would relate it to the equilibration of the reservoirs in the Boltzmann picture. This suggests that there is a certain degree of ambiguity in the perception of entropy production that does not influence, however, actual predictions of observables. (iii) The duality between the local thermodynamic and the global approach can apparently be analyzed already in terms of fairly simple models.

1.5

Objectives and Outline

The purpose of the present study is to revisit the results on the connection between microscopic chaos and thermodynamic properties from a unified point of view, and to strengthen the link between this approach and the formulation of equilibrium and non-equilibrium thermodynamics based on the Makovian postulate. I hope to contribute by that to setting up a common language for the various independent approaches in the field, and to make the results easier accessible for researchers with a background in statistical physics and/or the theory of classical dynamical systems. To this end the emphasis of the exposition is on the development of the conceptual results. In the first part of the review (Sects. 2–5) the motion of an assembly of non-interacting particles scattering from a fixed array of circular scatterers is treated. This model, the ordered Lorentz gas, was suggested by Lorentz [109] as a simple deterministic realization of the Drude model. It serves today as an elementary paradigm for transport in a many particle system. The relation between the chaotic dynamics of the particles and a description of the related macroscopic transport properties is described relying on graphical methods developed in the 80s to characterize deterministic dynamical systems. An important message of this part is the use of maps for the description of the dynamics and its reduction to a macroscopic description. In the second part (Sects. 6–9) a class of piecewise linear maps (multibaker maps) are introduced which preserve salient features of the maps studied before, but have a dynamics that can fully be treated analytically. For this model the relation between characteristics of the deterministic, reversible motion and of the thermodynamic transport is worked out by explicit calculations. A fully consistent description of particle and heat transport is given in this setting, that also covers the local entropy balance. This allows us to identify requirements on the dynamical system to achieve a consistent thermodynamic description of its transport, and an interpretation of the entropy production and entropy flows in molecular dynamics simulations involving Gaussian thermostats. Sects. 10–11 revisit the mentioned problems concerning local and global entropy production in the context of developments in the last decade. In particular they broaden the scope of the presentation to cover relaxation and fluctuation phenomena.

1.5 Objectives and Outline

15

In the concluding section (Sect. 12) the present status of the field is summarized, and some open problems are pointed out. The choice of references is biased to publications of historical interest to understand the background of the present discussion, and recent work on the relation between a deterministic chaotic dynamics and transport properties. Other reviews are available for early numerical work (Evans and Morriss [38]), and on a more general exposition of the relevant theory of dynamical systems (Gaspard [58]), while details on hard ball systems and ergodic theory can be found in a recent collection edited by Sz´asz [152].

British interest in heredity, spearheaded in its quantitative development by Sir Francis Galton, and the rapid development at the turn of the century of the physical theories of Brownian motion and statistical mechanics provided scientific sources of new problems. Encyclopædia Britannica [14]

2

The Galton Board: Coarse Graining and Markovian dynamics

Galton’s rule in genetics states that the development of features in a population of a certain species (approximately) takes a Gaussian distribution around a mean. In order to demonstrate the emergence of variation by external perturbations and the arising of a normal distribution, Sir Galton let balls fall down an inclined board with a triangular arrangement of scatterers and collected them at the lower end in bins of equal size (Fig. 4). The distribution of the number of balls in the bins was supposed to mimic the frequency of occurrence of the features. Based on a simple model we will now show how the deterministic dynamics can be reduced to a Markov chain describing the time evolution on the coarse-grained level, and how the normal distribution arises in this context. The model is chosen to introduce this relation in the simplest possible setting — a more realistic description of the Galton-board dynamics follows in Sects. 4 and 5.

2.1

Encoding and the probability distribution

We denote the coordinates of the inclined plane as q = (q1 , q2 ), and label the scatterers by the indices m1 = 0 · · · M1 + 1, m2 = 0 · · · M2 . In alternating rows m2 there are only scatterers at odd (even) values of m1 . The bins are located in row M2 +1. Each bin extends over the distance between scatterers in horizontal direction, i.e., its width comprises two columns of width a. The leftmost bin carries the label m1 = 0, and the rightmost one the label m1 = M1 + 1. For the arrangement of Fig. 4 considered in this section M1 + 1 = 2M2 . There is an external field acting in the negative q2 direction such that balls released in column m2 = 0 drop down the board towards larger m2 . The balls are viewed as point particles, and one considers the situation, where particles always fall down after a collision in order to hit either the last scatterer’s nearest neighbors to the lower left or lower right in the following collision. The probability r and l to respectively go right or left after the collision will be assumed not to depend on the previous history of the particle. Consequently, the probability to find a particle scattering from a scatterer m1 in row m2 in its nth collision can be written as δn,m2 P (n) m1 . The Kronecker-delta constituting the first factor accounts for the deterministic falling down

2.2 A one-dimensional map mimicking the Galton-board dynamics

17

m2=0

E

1 2 3 ...

√3a

M2

2a

q2

M2+1

q1 m1=

0

2

4

...

M1+1

Figure 4: Sketch of the experimental setup of the Galton broad, and the result of the experiment for unbiased scatterers. The width of the bins is taken to agree with the distance 2a between scatterers in a row, and E denotes the strength of an external field in the negative q2 direction (for particles subjected to gravitation it depends on the slope of the board). The indices m1 and m2 label the rows and columns of scatterers, respectively.

of the particles, which entails that scatters in row m2 can only be hit in the collision n = m2 . The second factor defines the distribution of test particles in horizontal direction. Fig. 4 shows the most probable outcome of the experiment for the case of unbiased collisions. Galton’s point was that for large n (i.e., M1 > M2 = n  1) the distribution approaches a normal distribution. The requirement of uncorrelated collisions adopted in the interpretation of this experiment, amounts to the Markovian postulate. For the sake of the present experiment, we characterize the time evolution of the particles by observing the sequence of scattering events (instantaneous measurements), and it is assumed that the probability of subsequent scattering events only depends on the outcome of the present observation and not on the detailed history of a given trajectory (Markovian postulate).

2.2

A one-dimensional map mimicking the Galton-board dynamics

The Galton-board experiment refers to the probability of the outcome of a large number of independent experiments. Following Gibbs [67] it is described by the time evolution of a statistical ensemble constituted by a very large number of identical systems (here: identical balls), all subject to exactly the same evolution laws (falling through the same board), but differing in their initial conditions.

18

2 The Galton Board

To gain further insight into the time evolution of such a density, we consider the map (cf. Fig. 5a)  x+a for x< 0        x − a(m − 1) + a(m − 1 − ∆) for a(m − 1) 0 is fulfilled (de Groot and Mazur [25], Prigogine [130]).

A.2

Identifying transport coefficients

It is worth expressing the kinetic coefficients Lij by means of directly measurable quantities. The total electro-chemical potential can be split as µ = µc + eφe , where µc is the chemical part, e is the charge of the particles, and φe is the electric potential. Since E = −∇φe and jel = ej is the electric current, we find that L11 is proportional to the electric conductivity σel > 0: L11 =

σel T . e2

(142a)

In the absence of a particle current (i.e., for j = 0), T j (s) provides the heat current, such that in view of (140c) L11 L22 − L12 L21 = λ, L11

(142b)

112

A Non-Equilibrium Thermodynamics

where λ > 0 is the heat conductivity. At zero particle current and constant chemical potential, a temperature gradient induces an electric field, which is conventionally written as α∇T , where α is called the thermoelectric power (or the Seebeck coefficient). Consequently, from (140a) one finds L12 =

ασel . Te

(142c)

Finally, in a system without temperature gradients, the entropy current due to the presence of an electric current ej amounts to Πej/T , where Π is the Peltier coefficient. Hence, Eq. (140c) implies L21 =

Πσel . e

(142d)

Using the phenomenological coefficients (142) we write σel (∇µ + eα∇T ), e2 ∇T eΠ = −λ + j. T T

j=− j (s)

(143a) (143b)

Note that the Onsager relation L12 = L21 makes the Peltier and Seebeck coefficients connected as Π = T α.

(144)

Substituting Eqs. (142) into (141) one finds σ

A.3

(irr)

  e2 j 2 ∂x T 2 = +λ . σel T T

(145)

Relating transport and diffusion coefficients

It is worth replacing the chemical potential in the expressions for the currents and entropy production by the density ρ and temperature T . We write ∇µc =

De2 ∇ρ − s∇T, σel

where the diffusion coefficient is defined as   σel ∂µc D= 2 , e ∂ρ T

(146)

(147a)

and

s≡−

∂µc ∂T ρ

is a quantity of dimension entropy per particle. Introducing the drift velocity

(147b)

A.4 Time evolution of the “multibaker gas”

v=

113

σel E . eρ

(148)

one can rewrite the particle current density (143a) as j = vρ − D∇ρ − k

ρD ∇T T

(149)

where k

ρD σel ≡ 2 (eα − s). T e

(150)

For the models considered in this review k vanishes identically such that in view of Eq. (147b) ∂µc eα = − , (151) ∂T ρ which implies in view of (144) ∂µc eΠ = −T . ∂T ρ

(152)

Moreover, the particle current takes then the form j = vρ − D∇ρ

(153)

Note that this form of the current does not at all exclude thermoelectric cross effects. After all, according to Eq. (146) in a system with a fixed chemical potential µc a temperature gradient induces a density gradient that in turn can give rise to a particle current.

A.4

Time evolution of the “multibaker gas”

By taking the limit E, e → 0 at constant eΠ, eα, and σel /e2 , the thermoelectric problem is formally mapped onto the problem of thermal diffusion in a binary mixture. In that case ρ stands for the concentration of one of the diffusing materials, and the quantity kD is the thermal diffusion coefficient (cf. Landau and Lifshitz [95]). Based on this analogy, we consider kD in Eq. (150) as the thermal diffusion coefficient of charged particles in the thermoelectric problem. It vanishes since the particle motion is the only source of heat conduction in multibaker maps. Since multibaker maps are dealing with an assembly of non-interacting particles the ideal gas law applies as equation of state of the local equilibrium. Consequently (Reichl [136]), the entropy density and the chemical potential take the respective forms, ρ T −c s = −ρkB ln ρ∗   s µc = T (1 + c)kB − ρ

(154a) (154b)

114

A Non-Equilibrium Thermodynamics

The constant c appearing in these equations is the dimensionless specific heat of the ideal gas. As an immediate consequence of Eqs. (147a) and (154b) the Einstein relation holds. After all,   ∂µc 2 ρDe ≡ ρσel = σel kB T. (155) ∂ρ T Similarly, one uses Eq. (152) to find for the Peltier coefficient eΠ s = kB − . T ρ

(156)

These equations also uniquely determine the evolution equation for the temperature. For this derivation, one first combines Eq. (156) with the definition Eq. (154a) and (136a) to obtain the explicit expression ∂t s =

eΠ ∂t T ∇j + ρkB c T T

for the time derivative of the entropy. A comparison with Eqs. (143a) and (145) inserted into Eq. (136c) yields then for the bulk of a thermodynamic system (i.e., for Φ(th) = 0): ∂t T =

T vj ∇(λ∇T ) ∇T + −j . c ρD kB c ρ ρ

(157)

The first term describes the effect of the dissipative heating vj/(ρD) on the temperature of the system, the second accounts for heat diffusion, and the latter for the advection of heat by the particle current j.

B

List of Symbols

As a rule the discrete spatial position is indicated by a subscript (typically m), and temporal evolution by superscripts in brackets [typically (n)]. When only the first iteration is required the indices for the time are suppressed, and variables at the discrete times n and n + 1 are denoted by unprimed and primed symbols, respectively. α a c, C D  e e E γν j j (e) j (s) κ kB l L ≡ a(M + 1) m M n N φe Φ(th) Π ψt (Γ) (cg) ψm P q r % ρ ς ς% σ (irr)

Seebeck coefficient width of cells of spatial partitioning dimensionless specific heat diffusion coefficient parameter characterizing phase-space compression in multibaker maps total energy per particle electric charge of particles external force decay rates of νs eigenmode of densities particle current energy current entropy current escape rate Boltzmann constant width of left column in multibaker maps spatial extension of the system index labeling volumes to take local thermodynamic averages m typically ranges between 0 and M + 1, where m = 0 and m = M + 1 are used to implement boundary conditions index labeling time steps number of particles in the system electric potential entropy flux into a thermostat Peltier coefficient normalized conditional phase-space density at position Γ density ψt (Γ) averaged over Γ in cell m probabilities source strength of kinetic energy width of right column in multibaker maps microscopic phase-space density coarse-grained particle density contraction rate of phase-space volume growth rate of the relative local phase-space density rate of entropy production

90, 112 17 67, 62 21 55 62 12 43 25 22, 53 62 76, 110 26 13 52 21 17, 33, 51 17, 33, 51 18 22 12, 111 76, 110 78, 112 84 85 16 64 52 53 53 52 70,92ff 12,77

116 (irr)

σtot

σel s s Sm S (B) S (G) de Sm di Sm τ T v ξ ζ

29

B List of Symbols integrated rate of entropy production for a spatially extended system conductivity entropy per unit volume29 width of center column in multibaker maps29 average entropy in region m Boltzmann entropy Gibbs entropy change of Sm due to entropy flow through the boundary of region m internal change of Sm mean free time of particles; time unit for maps temperature particle drift due to external force contracting eigenvalue of a map dynamical friction coefficient

13 12, 111 12 52 67 13 68 73 73 21, 53 62 21 9 48

There should be no confusion between the different meaning of the symbol s in the context of the parameterization of multibaker maps, and the continuum description of transport. (In the former setting it denotes the width of one of the columns, and in the latter the entropy density.) The ambiguity was preserved for consistency with the literature.

C

Translation of German citations

p. 1

Morgenstern [118] Palmstr¨ om denkt die Alpen sich als Kubus.. Und besteigt sie so mit seinem Tubus. Palmstr¨ om envisions the Alps as a cube.. and surmounts them thus with his tube.

p. 6

Boltzmann [9] Bring vor, was wahr ist; schreib’ so, daß klar ist und verficht’s, bis es dir gar ist Bring forward what is true; write it so that it’s clear and defend it to your last breath

p. 50

Hopf [81] Man kann diese Transformation, deren wiederholte Ausf¨ uhrung an die Herstellung von Bl¨atterteig erinnert, vermittelst dyadischer Br¨ uche [. . . ] darstellen. This transformation, whose repeated application reminds one of the preparation of fillo dough, can be represented by dyadic fractions.

p. 92

Boltzmann on Maxwell’s work [129] Gef¨ugig speien die Formeln nun Resultat auf Resultat aus! Compliantly the equations spit out now result after result!

p. 101

German proverb

Abwarten und Tee trinken . . . literally: figurative for:

wait and drink tea . . . . . . wait and see!

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