Modeling Thermostating, Entropy Currents, and Cross Effects by Dynamical Systems

A generalized multibaker map with periodic boundary conditions is shown to model boundary-driven transport, when the driving is applied by a perturbation of the dynamics localized in a macroscopically small region. In this case there are sustained density gradients in the steady state. A non-uniform stationary temperature profile can be maintained by incorporating a heat source into the dynamics, which deviates from the one of a bulk system only in a (macroscopically small) localized region such that a heat (or entropy) flux can enter an attached thermostat only in that region. For these settings the relation between the average phase-space contraction, the entropy flux to the thermostat and irreversible entropy production is clarified for stationary and non-stationary states. In addition, thermoelectric cross effects are described by a multibaker chain consisting of two parts with different transport properties, modeling a junction between two metals.     

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.     

Stationary nonequilibrium states in boundary-driven Hamiltonian systems: Shear flow

We investigate stationary nonequilibrium states of systems of particles moving according to Hamiltonian dynamics with specified potentials. The systems are driven away from equilibrium by Maxwell-demon reflection rules at the walls. These deterministic rules conserve energy but not phase space volume, and the resulting global dynamics may or may not be time reversible (or even invertible). Using rules designed to simulate moving walls, we can obtain a stationary shear flow. Assuming that for macroscopic systems this flow satisfies the Navier-Stokes equations, we compare the hydrodynamic entropy production with the average rate of phase-space volume compression. We find that they are equalwhen the velocity distribution of particles incident on the walls is a local Maxwellian. An argument for a general equality of this kind, based on the assumption of local thermodynamic equilibrium, is given. Molecular dynamic simulations of hard disks in a channel produce a steady shear flow with the predicted behavior.     

Thermodynamics based on the principle of least abbreviated action: Entropy production in a network of coupled oscillators

We present some novel thermodynamic ideas based on the Maupertuis principle. By considering Hamiltonians written in terms of appropriate action-angle variables we show that thermal states can be characterized by the action variables and by their evolution in time when the system is nonintegrable. We propose dynamical definitions for the equilibrium temperature and entropy as well as an expression for the nonequilibrium entropy valid for isolated systems with many degrees of freedom. This entropy is shown to increase in the relaxation to equilibrium of macroscopic systems with short-range interactions, which constitutes a dynamical justification of the Second Law of Thermodynamics. Several examples are worked out to show that this formalism yields the right microcanonical (equilibrium) quantities. The relevance of this approach to nonequilibrium situations is illustrated with an application to a network of coupled oscillators (Kuramoto model). We provide an expression for the entropy production in this system finding that its positive value is directly related to dissipation at the steady state in attaining order through synchronization.     

Sufficient and necessary condition for zero quantum entropy rates under any coupling to the environment

We find the necessary and sufficient conditions for the entropy rate of the system to be zero under any system-environment Hamiltonian interaction. We call the class of system-environment states that satisfy this condition lazy states. They are a generalization of classically correlated states defined by quantum discord, but based on projective measurements of any rank. The concept of lazy states permits the construction of a protocol for detecting global quantum correlations using only local dynamical information. We show how quantum correlations to the environment provide bounds to the entropy rate, and how to estimate dissipation rates for general non-Markovian open quantum systems.     

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.     

Dephasing and the steady state in quantum many-particle systems

We discuss relaxation in bosonic and fermionic many-particle systems. For integrable systems, time evolution can cause a dephasing effect, leading for finite subsystems to steady states. We explicitly derive those steady subsystem states and devise sufficient prerequisites for the dephasing to occur. We also find simple scenarios, in which dephasing is ineffective and discuss the dependence on dimensionality and criticality. It follows further that, after a quench of system parameters, entanglement entropy will become extensive. This provides a way of creating strong entanglement in a controlled fashion.     

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.     

Dynamical entropy of space translations of CAR and CCR algebras with respect to quasi-free states

We compute the dynamical entropy in the sense of Connes, Narnhofer and Thirring of space translations of the CAR and CCR algebras in -dimensional continuous spaces with respect to invariant quasi-free states. It turns out that the dynamical entropies are equal to the corresponding mean entropies of the systems under consideration. Computing the mean entropies explicitly we derive the entropy formulas for the systems.Research supported in part by the Basic Science Research Program, Ministry of Education, 1992Research supported in part by GARC in 1991–1992     

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.     

Time-Reversal and Entropy

There is a relation between the irreversibility of thermodynamic processes as expressed by the breaking of time-reversal symmetry, and the entropy production in such processes. We explain on an elementary mathematical level the relations between entropy production, phase-space contraction and time-reversal starting from a deterministic dynamics. Both closed and open systems, in the transient and in the steady regime, are considered. The main result identifies under general conditions the statistical mechanical entropy production as the source term of time-reversal breaking in the path space measure for the evolution of reduced variables. This provides a general algorithm for computing the entropy production and to understand in a unified way a number of useful (in)equalities. We also discuss the Markov approximation. Important are a number of old theoretical ideas for connecting the microscopic dynamics with thermodynamic behavior.     

A decomposed equation for local entropy and entropy production in volume-preserving coarse-grained systems

In this study an equation for the local entropy is derived based on the formulation of a master equation and is applied to volume-preserving maps. The equation consists of the following terms: unsteady, convection, diffusion, probability-weighted phase space volume expansion rate, nonnegative entropy production, and residuals. The decomposition makes it possible to evaluate entropy production in terms of microscopic dynamics and is expected to be applicable to many coarse-grained systems on the phase space. When it is applied to two volume-preserving multibaker chain systems it is confirmed that the summation of the nonnegative entropy production on each site numerically coincides with the entropy production introduced by Gilbert et al. [T. Gilbert, J.R. Dorfman, P. Gaspard, Entropy production, fractals, and relaxation to equilibrium, Phys. Rev. Lett. 85 (2000) 1606-1609] and the phenomenological expression both in nonequilibrium steady and unsteady states. The coincidence is brought about by the fact that the residual terms vanish in the thermodynamic limit when they are integrated on each site. It follows that the entropy production is dominated by the nonnegative entropy production term and becomes positive in nonequilibrium states.     

Extended Clausius Relation and Entropy for Nonequilibrium Steady States in Heat Conducting Quantum Systems

Recently, in their attempt to construct steady state thermodynamics (SST), Komatsu, Nakagawa, Sasa, and Tasaki found an extension of the Clausius relation to nonequilibrium steady states in classical stochastic processes. Here we derive a quantum mechanical version of the extended Clausius relation. We consider a small system of interest attached to large systems which play the role of heat baths. By only using the genuine quantum dynamics, we realize a heat conducting nonequilibrium steady state in the small system. We study the response of the steady state when the parameters of the system are changed abruptly, and show that the extended Clausius relation, in which “heat” is replaced by the “excess heat”, is valid when the temperature difference is small. Moreover we show that the entropy that appears in the relation is similar to von Neumann entropy but has an extra symmetrization with respect to time-reversal. We believe that the present work opens a new possibility in the study of nonequilibrium phenomena in quantum systems, and also confirms the robustness of the approach by Komatsu et al.     

Lyapunov Instability for a Hard-Disk Fluid in Equilibrium and Nonequilibrium Thermostated by Deterministic Scattering

We compute the full Lyapunov spectra for a hard-disk fluid under temperature gradient and under shear. The Lyapunov exponents are calculated using a recently developed formalism for systems with elastic hard collisions. The system is thermalized by deterministic and time-reversible scattering at the boundary, whereas the bulk dynamics remains Hamiltonian. This thermostating mechanism allows for energy fluctuations around a mean value which is reflected by only two vanishing Lyapunov exponents in equilibrium and nonequilibrium. In nonequilibrium steady states the phase-space volume is contracted on average, leading to a negative sum of the Lyapunov exponents. Since the system is driven inhomogeneously we do not expect the conjugate pairing rule to hold, which is indeed shown to be the case. Finally, the Kaplan–Yorke dimension and the Kolmogorov–Sinai entropy are calculated from the Lyapunov spectra.     

A Simple Discrete Model of Brownian Motors: Time-periodic Markov Chains

In this paper, we consider periodically inhomogeneous Markov chains, which can be regarded as a simple version of physical model—Brownian motors. We introduce for them the concepts of periodical reversibility, detailed balance, entropy production rate and circulation distribution. We prove the equivalence of the following statements: The time-periodic Markov chain is periodically reversible; It is in detailed balance; Kolmogorov's cycle condition is satisfied; Its entropy production rate vanishes; Every circuit and its reversed circuit have the same circulation weight. Hence, in our model of Markov chains, the directed transport phenomenon of Brownian motors, i.e. the existence of net circulation, can occur only in nonequilibrium and irreversible systems. Moreover, we verify the large deviation property and the Gallavotti-Cohen fluctuation theorem of sample entropy production rates of the Markov chain.     

Dynamics of Entropy Production Rate in Two Coupled Bosonic Modes Interacting with a Thermal Reservoir

The Markovian time evolution of the entropy production rate is studied as a measure of irreversibility generated in a bipartite quantum system consisting of two coupled bosonic modes immersed in a common thermal environment. The dynamics of the system is described in the framework of the formalism of the theory of open quantum systems based on completely positive quantum dynamical semigroups, for initial two-mode squeezed thermal states, squeezed vacuum states, thermal states and coherent states. We show that the rate of the entropy production of the initial state and nonequilibrium stationary state, and the time evolution of the rate of entropy production, strongly depend on the parameters of the initial Gaussian state (squeezing parameter and average thermal photon numbers), frequencies of modes, parameters characterising the thermal environment (temperature and dissipation coefficient), and the strength of coupling between the two modes. We also provide a comparison of the behaviour of entropy production rate and Rényi-2 mutual information present in the considered system.     

Discrete-step evaporation of an atomic beam

We present a theoretical analysis of the evaporative cooling of a magnetically guided atomic beam by means of discrete radio-frequency antennas. First we derive the changes in flux and temperature, as well as in collision rate and phase-space density, for a single evaporation step. Next we show how the occurrence of collisions during the propagation between two successive antennas can be probed. Finally, we discuss the optimization of the evaporation ramp with several antennas to reach quantum degeneracy. We estimate the number of antennas required to increase the phase-space density by several orders of magnitude. We find that at least 30 antennas are needed to gain a factor 108 in phase-space density.     

A model for the self-organization of microtubules driven by molecular motors

We propose a two-dimensional model for the organization of stabilized microtubules driven by molecular motors in an unconfined geometry. In this model two kinds of dynamics are competing. The first one is purely diffusive, with an interaction between the rotational degrees of freedom, while the second one is a local drive, dependent on microtubule polarity. As a result, there is a configuration dependent driving field. Applying a molecular field approximation, we are able to derive continuum equations. A study on the solutions of these equations shows non-equilibrium inhomogeneous steady states in various regions of the parameter space. The presence and stability of such self-organized states are investigated in terms of entropy production. Numerical simulations confirm our analytic results. Received 4 August 1999 and Received in final form 24 November 1999     

Monotonic return to steady nonequilibrium

We propose and analyze a new candidate Lyapunov function for relaxation towards general nonequilibrium steady states. The proposed functional is obtained from the large time asymptotics of time-symmetric fluctuations. For driven Markov jump or diffusion processes it measures an excess in dynamical activity rates. We present numerical evidence and we report on a rigorous argument for its monotonic time dependence close to the steady nonequilibrium or in general after a long enough time. This is in contrast with the behavior of approximate Lyapunov functions based on entropy production that when driven far from equilibrium often keep exhibiting temporal oscillations even close to stationarity.