Hamiltonian dynamics,nanosystems, and nonequilibrium statistical mechanics

An overview is given of recent advances in nonequilibrium statistical mechanics on the basis of the theory of Hamiltonian dynamical systems and in the perspective provided by the nanosciences. It is shown how the properties of relaxation toward a state of equilibrium can be derived from Liouville's equation for Hamiltonian dynamical systems. The relaxation rates can be conceived in terms of the so-called Pollicott–Ruelle resonances. In spatially extended systems, the transport coefficients can also be obtained from the Pollicott–Ruelle resonances. The Liouvillian eigenstates associated with these resonances are in general singular and present fractal properties. The singular character of the nonequilibrium states is shown to be at the origin of the positive entropy production of nonequilibrium thermodynamics. Furthermore, large-deviation dynamical relationships are obtained, which relate the transport properties to the characteristic quantities of the microscopic dynamics such as the Lyapunov exponents, the Kolmogorov–Sinai entropy per unit time, and the fractal dimensions. We show that these large-deviation dynamical relationships belong to the same family of formulas as the fluctuation theorem, as well as a new formula relating the entropy production to the difference between an entropy per unit time of Kolmogorov–Sinai type and a time-reversed entropy per unit time. The connections to the nonequilibrium work theorem and the transient fluctuation theorem are also discussed. Applications to nanosystems are described.     

Thermal response in driven diffusive systems

Evaluating the linear response of a driven system to a change in environment temperature(s) is essential for understanding thermal properties of nonequilibrium systems. The system is kept in weak contact with possibly different fast relaxing mechanical, chemical or thermal equilibrium reservoirs. Modifying one of the temperatures creates both entropy fluxes and changes in dynamical activity. That is not unlike mechanical response of nonequilibrium systems but the extra difficulty for perturbation theory via path-integration is that for a Langevin dynamics temperature also affects the noise amplitude and not only the drift part. Using a discrete-time mesh adapted to the numerical integration one avoids that ultraviolet problem and we arrive at a fluctuation expression for its thermal susceptibility. The algorithm appears stable under taking even finer resolution.     

Gibbs–Jaynes Entropy Versus Relative Entropy

The maximum entropy formalism developed by Jaynes determines the relevant ensemble in nonequilibrium statistical mechanics by maximising the entropy functional subject to the constraints imposed by the available information. We present an alternative derivation of the relevant ensemble based on the Kullback–Leibler divergence from equilibrium. If the equilibrium ensemble is already known, then calculation of the relevant ensemble is considerably simplified. The constraints must be chosen with care in order to avoid contradictions between the two alternative derivations. The relative entropy functional measures how much a distribution departs from equilibrium. Therefore, it provides a distinct approach to the calculation of statistical ensembles that might be applicable to situations in which the formalism presented by Jaynes performs poorly (such as non-ergodic dynamical systems).     

Entropy Production in Continuous Phase Space Systems

We propose an alternative method to compute the environmental entropy production of a classical underdamped nonequilibrium system, not necessarily in detailed balance, in a continuous phase space. It is based on the idea that the Hamiltonian orbits of the corresponding isolated system can be regarded as microstates and that entropy is generated in the environment whenever the system moves from one microstate to another. This approach has the advantage that it is not necessary to distinguish between even and odd-parity variables. We show that the method leads to a different expression for the differential entropy production along an infinitesimal stochastic path. However, when integrating over all possible paths the local entropy production turns out to be the same as in previous studies. This demonstrates that the differential entropy production in continuous phase space systems is not uniquely defined.     

Positivity of Entropy Production

We discuss the positivity of the mean entropy production for stochastic systems driven from equilibrium, as it was defined in refs. 7 and 8. Non-zero entropy production is closely linked with violation of the detailed balance condition. This connection is rigorously obtained for spinflip dynamics. We remark that the positivity of entropy production depends on the choice of time-reversal transformation, hence on the choice of the dynamical variables in the system of interest.     

Equilibrium and nonequilibrium properties of systems with long-range interactions

We briefly review some equilibrium and nonequilibrium properties of systems with long-range interactions. Such systems, which are characterized by a potential that weakly decays at large distances, have striking properties at equilibrium, like negative specific heat in the microcanonical ensemble, temperature jumps at first order phase transitions, broken ergodicity. Here, we mainly restrict our analysis to mean-field models, where particles globally interact with the same strength. We show that relaxation to equilibrium proceeds through quasi-stationary states whose duration increases with system size. We propose a theoretical explanation, based on Lynden-Bell’s entropy, of this intriguing relaxation process. This allows to address problems related to nonequilibrium using an extension of standard equilibrium statistical mechanics. We discuss in some detail the example of the dynamics of the free electron laser, where the existence and features of quasi-stationary states is likely to be tested experimentally in the future. We conclude with some perspectives to study open problems and to find applications of these ideas to dipolar media.     

Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences

There are only a very few known relations in statistical dynamics that are valid for systems driven arbitrarily far-from-equilibrium. One of these is the fluctuation theorem, which places conditions on the entropy production probability distribution of nonequilibrium systems. Another recently discovered far from equilibrium expression relates nonequilibrium measurements of the work done on a system to equilibrium free energy differences. In this paper, we derive a generalized version of the fluctuation theorem for stochastic, microscopically reversible dynamics. Invoking this generalized theorem provides a succinct proof of the nonequilibrium work relation.     

Decrease in entropy as a result of measuring the time of escape from a potential well

The decrease in entropy when passing from an equilibrium thermodynamic system to a slightly nonequilibrium system is investigated. A quasi-equilibrium Boltzmann distribution is used to prove the conservation of free energy during this passage. Results are obtained for a Brownian particle in a potential well with a low escape probability. The escape is interpreted as a measurement. It is shown that because of the measurement itself, the distribution function is narrower than that for a system undisturbed by measurement, i.e., an equilibrium system. In this case, the entropy difference between the equilibrium and measurement-disturbed systems is equal to the amount of information entered into the system.     

Entropy production, fractals, and relaxation to equilibrium

The theory of entropy production in nonequilibrium, Hamiltonian systems, previously described for steady states using partitions of phase space, is here extended to time dependent systems relaxing to equilibrium. We illustrate the main ideas by using a simple multibaker model, with some nonequilibrium initial state, and we study its progress toward equilibrium. The central results are (i) the entropy production is governed by an underlying, exponentially decaying fractal structure in phase space, (ii) the rate of entropy production is largely independent of the scale of resolution used in the partitions, and (iii) the rate of entropy production is in agreement with the predictions of nonequilibrium thermodynamics.     

Geometrical Excess Entropy Production in Nonequilibrium Quantum Systems

For open systems described by the quantum Markovian master equation, we study a possible extension of the Clausius equality to quasistatic operations between nonequilibrium steady states (NESSs). We investigate the excess heat divided by temperature (i.e., excess entropy production) which is transferred into the system during the operations. We derive a geometrical expression for the excess entropy production, which is analogous to the Berry phase in unitary evolution. Our result implies that in general one cannot define a scalar potential whose difference coincides with the excess entropy production in a thermodynamic process, and that a vector potential plays a crucial role in the thermodynamics for NESSs. In the weakly nonequilibrium regime, we show that the geometrical expression reduces to the extended Clausius equality derived by Saito and Tasaki (J. Stat. Phys. 145:1275, 2011). As an example, we investigate a spinless electron system in quantum dots. We find that one can define a scalar potential when the parameters of only one of the reservoirs are modified in a non-interacting system, but this is no longer the case for an interacting system.     

Nonequilibrium Thermodynamics of Polymeric Liquids via Atomistic Simulation

The challenge of calculating nonequilibrium entropy in polymeric liquids undergoing flow was addressed from the perspective of extending equilibrium thermodynamics to include internal variables that quantify the internal microstructure of chain-like macromolecules and then applying these principles to nonequilibrium conditions under the presumption of an evolution of quasie equilibrium states in which the requisite internal variables relax on different time scales. The nonequilibrium entropy can be determined at various levels of coarse-graining of the polymer chains by statistical expressions involving nonequilibrium distribution functions that depend on the type of flow and the flow strength. Using nonequilibrium molecular dynamics simulations of a linear, monodisperse, entangled C₁₀₀₀H₂₀₀₂ polyethylene melt, nonequilibrium entropy was calculated directly from the nonequilibrium distribution functions, as well as from their second moments, and also using the radial distribution function at various levels of coarse-graining of the constituent macromolecular chains. Surprisingly, all these different methods of calculating the nonequilibrium entropy provide consistent values under both planar Couette and planar elongational flows. Combining the nonequilibrium entropy with the internal energy allows determination of the Helmholtz free energy, which is used as a generating function of flow dynamics in nonequilibrium thermodynamic theory.     

Diffusion,effusion, and chaotic scattering: An exactly solvable Liouvillian dynamics

We study diffusion and chaotic scattering in a chain of baker maps coupled together which forms an area-preserving mapping of an infinitely extended strip onto itself. This exactly solvable mapping sustains chaotic behaviors and diffusion processes. The relationship between the diffusion coefficient, the Lyapunov exponent, and the entropy per unit time is derived. The long-lived classical resonances of the Liouville evolution operator are proved to converge toward the eigenvalues of the phenomenological diffusion equation. In this sense, there is a quasi-isomorphism between the resonance spectrum of the Liouville evolution and the eigenvalue spectrum of the phenomenological diffusion equation. Furthermore, we show that a fractal repeller is associated to each non-equilibrium state in the isolated and finite multibaker chain. The nonequilibrium states are all unstable with respect to the equilibrium, validating a weak form of the second principle of thermodynamics for the present dynamical system. Consequences of nonequilibrium fractals on classical measurements are discussed. We then describe the open multibaker chain as a scattering system. Fractal properties of chaotic scattering are here shown to be related to diffusion in the chain.     

Entropy production and time asymmetry in nonequilibrium fluctuations

The time-reversal symmetry of nonequilibrium fluctuations is experimentally investigated in two out-of-equilibrium systems: namely, a Brownian particle in a trap moving at constant speed and an electric circuit with an imposed mean current. The dynamical randomness of their nonequilibrium fluctuations is characterized in terms of the standard and time-reversed entropies per unit time of dynamical systems theory. We present experimental results showing that their difference equals the thermodynamic entropy production in units of Boltzmann's constant.     

Spatial correlations in nonequilibrium systems: The effect of diffusion

We show how the ideas of the fluctuation-dissipation theory can be used to calculate spatial correlations in interacting systems away from equilibrium. The only spatially dependent dissipative process considered is diffusion, and spatial correlations are generated by the nonlocal spatial dependence of the chemical potential. The results are the lowest order in a hierarchy of generalized hydrodynamic type calculations applicable to nonequilibrium systems. We derive equations for the density correlation functions at stationary state supported by diffusive fluxes and show that they have the local equilibrium form. The static correlation function is obtained from the fluctuation-dissipation theorem, which we show to be equivalent to the Ornstein-Zernike integral equation. At equilibrium we demonstrate that the dynamical structure factor obtained by these methods includes the expected wave-vector dependent diffusion constant. Finally we derive a necessary and sufficient condition for local equilibrium to obtain at a stationary state and show by explicit calculation that chemical processes can give rise to significant nonequilibrium correlations.     

Comments on the Entropy of Nonequilibrium Steady States

We discuss the entropy of nonequilibrium steady states. We analyze the so-called spontaneous production of entropy in certain reversible deterministic nonequilibrium system, and its link with the collapse of such systems towards an attractor that is of lower dimension than the dimension of phase space. This means that in the steady state limit, the Gibbs entropy diverges to negative infinity. We argue that if the Gibbs entropy is expanded in a series involving 1, 2,... body terms, the divergence of the Gibbs entropy is manifest only in terms involving integrals whose dimension is higher than, approximately, the Kaplan–Yorke dimension of the steady state attractor. All the low order terms are finite and sum in the weak field limit to the local equilibrium entropy of linear irreversible thermodynamics.     

The Steady State Fluctuation Relation for the Dissipation Function

We give a proof of transient fluctuation relations for the entropy production (dissipation function) in nonequilibrium systems, which is valid for most time reversible dynamics. We then consider the conditions under which a transient fluctuation relation yields a steady state fluctuation relation for driven nonequilibrium systems whose transients relax, producing a unique nonequilibrium steady state. Although the necessary and sufficient conditions for the production of a unique nonequilibrium steady state are unknown, if such a steady state exists, the generation of the steady state fluctuation relation from the transient relation is shown to be very general. It is essentially a consequence of time reversibility and of a form of decay of correlations in the dissipation, which is needed also for, e.g., the existence of transport coefficients. Because of this generality the resulting steady state fluctuation relation has the same degree of robustness as do equilibrium thermodynamic equalities. The steady state fluctuation relation for the dissipation stands in contrast with the one for the phase space compression factor, whose convergence is problematic, for systems close to equilibrium. We examine some model dynamics that have been considered previously, and show how they are described in the context of this work.     

Extremal properties of distribution function and statistical operator in equilibrium and nonequilibrium thermodynamics

It is shown that the distribution function and the statistical operator, in the case that the considered system is close to the equilibrium state, can be received by the method relying upon minimizing the information gain, which is connected with the transition of the system from a nonequilibrium state to the equilibrium state. For the systems far from equilibrium the nonequilibrium distribution function or the nonequilibrium statistical operator can be derived using a variational principle based on Jaynes' maximum entropy formalism including memory effects.     

Generalized clausius inequality for nonequilibrium quantum processes

We show that the nonequilibrium entropy production for a driven quantum system is larger than the Bures length, the geometric distance between its actual state and the corresponding equilibrium state. This universal lower bound generalizes the Clausius inequality to arbitrary nonequilibrium processes beyond linear response. We further derive a fundamental upper bound for the quantum entropy production rate and discuss its connection to the Bremermann-Bekenstein bound.     

Time Evolution in Macroscopic Systems. II. The Entropy

The concept of entropy in nonequilibrium macroscopic systems is investigated in the light of an extended equation of motion for the density matrix obtained in a previous study. It is found that a time-dependent information entropy can be defined unambiguously, but it is the time derivative or entropy production that governs ongoing processes in these systems. The differences in physical interpretation and thermodynamic role of entropy in equilibrium and nonequilibrium systems is emphasized and the observable aspects of entropy production are noted. A basis for nonequilibrium thermodynamics is also outlined.     

Quantum dynamical entropies for classical stochastic systems

We compare two proposals for the dynamical entropy of quantum deterministic systems (CNT and AFL) by studying their extensions to classical stochastic systems. We show that the natural measurement procedure leads to a simple explicit expression for the stochastic dynamical entropy with a clear information-theoretical interpretation. Finally, we compare our construction with other recent proposals.