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.     

Quantum Macrostatistical Picture of Nonequilibrium Steady States

We apply our quantum macrostatistical treatment of irreversible processes to prove that, in nonequilibrium steady states, (a) the hydrodynamical observables execute a generalised Onsager–Machlup process and (b) the spatial correlations of these observables are generically of long range. The key assumptions behind these results are a nonequilibrium version of Onsager regression hypothesis, together with certain hypotheses of chaoticity and local equilibrium for hydrodynamical fluctuations.     

Einstein Relation for Nonequilibrium Steady States

The Einstein relation, relating the steady state fluctuation properties to the linear response to a perturbation, is considered for steady states of stochastic models with a finite state space. We show how an Einstein relation always holds if the steady state satisfies detailed balance. More generally, we consider nonequilibrium steady states where detailed balance does not hold and show how a generalisation of the Einstein relation may be derived in certain cases. In particular, for the asymmetric simple exclusion process and a driven diffusive dimer model, the external perturbation creates and annihilates particles thus breaking the particle conservation of the unperturbed model.     

Steady state statistics of driven diffusions

We consider overdamped diffusion processes driven out of thermal equilibrium and we analyze their dynamical steady fluctuations. We discuss the thermodynamic interpretation of the joint fluctuations of occupation times and currents; they incorporate respectively the time-symmetric and the time-antisymmetric sector of the fluctuations. We highlight the canonical structure of the joint fluctuations. The novel concept of traffic complements the entropy production for the study of the occupation statistics. We explain how the occupation and current fluctuations get mutually coupled out of equilibrium. Their decoupling close-to-equilibrium explains the validity of entropy production principles.     

Onsager-Machlup Theory for Nonequilibrium Steady States and Fluctuation Theorems

A generalization of the Onsager-Machlup theory from equilibrium to nonequilibrium steady states and its connection with recent fluctuation theorems are discussed for a dragged particle restricted by a harmonic potential in a heat reservoir. Using a functional integral approach, the probability functional for a path is expressed in terms of a Lagrangian function from which an entropy production rate and dissipation functions are introduced, and nonequilibrium thermodynamic relations like the energy conservation law and the second law of thermodynamics are derived. Using this Lagrangian function we establish two nonequilibrium detailed balance relations, which not only lead to a fluctuation theorem for work but also to one related to energy loss by friction. In addition, we carried out the functional integral for heat explicitly, leading to the extended fluctuation theorem for heat. We also present a simple argument for this extended fluctuation theorem in the long time limit. PACS numbers: 05.70.Ln, 05.40.-a, 05.10.Gg.     

Theory on Morphological Instability in Driven Systems

Motivated by recent findings from simulation of a driven lattice gas under shifted periodic boundary conditions, we study within the context of a continuum model the interfacial stability of driven diffusive systems. In this model, an external driving field maintains the system away from equilibrium. Well below criticality, steady-state solutions of the associated bulk kinetic equation are obtained. Our results successfully account for the novel features found in simulation. In particular, the solution describing a pair of interfaces tilted with respect to the driving field under periodic boundary conditions shows a tilt-dependent bulk density (and internal energy), and boundary layers near one of the interfaces. Focusing on the interface dynamics, one finds that such an interface exhibits a characteristic Mullins-Sekerka instability. This is argued to be responsible for the onset of the single- to multistrip transformation observed in simulation.     

Fluctuation-Induced First-Order Transition in a Nonequilibrium Steady State

We present the first example of a phase transition in a nonequilibrium steady state that can be argued analytically to be first order. The system of interest is a two-species reaction-diffusion problem whose control parameter is the total density . Mean-field theory predicts a second-order transition between two stationary states at a critical density = _c. We develop a phenomenological picture that instead predicts a first-order transition below the upper critical dimension d _c=4. This picture is confirmed by hysteresis found in numerical simulations, and by the study of a renormalization-group improved equation of state. The latter approach is inspired by the Coleman–Weinberg mechanism in QED.     

Nonequilibrium thermodynamics and the transport phenomena in magnetically confined plasmas

The neoclassical theory of transport in magnetically confined plasmas is reviewed. The emphasis is laid on a set of relationships existing among the banana transport coefficients. The surface-averaged entropy production in such plasmas is evaluated. It is shown that neoclassical effects emerge from the entropy production due to parallel transport processes. The Pfirsch-Schlüter effect can be clearly interpreted as due to spatial fluctuations of parallel fluxes on a magnetic surface: the corresponding entropy production is the measure of these fluctuations. The banana fluxes can be formulated in a quasithermodynamic form in which the average entropy production is a bilinear form in the parallel fluxes and the conjugate generalized stresses. A formulation as a quadratic form in the thermodynamic forces is also possible, but leads to anomalies, which are discussed in some detail.     

Nonequilibrium Steady States of Some Simple 1-D Mechanical Chains

We study nonequilibrium steady states of some 1-D mechanical models with N moving particles on a line segment connected to unequal heat baths. For a system in which particles move freely, exchanging energy as they collide with one another, we prove that the mean energy along the chain is constant and equal to \(\frac{1}{2} \sqrt{T_{L}T_{R}}\) where T _L and T _R are the temperatures of the two baths. We then consider systems in which particles are trapped, i.e., each confined to its designated interval in the phase space, but these intervals overlap to permit interaction of neighbors. For these systems, we show numerically that the system has well defined local temperatures and obeys Fourier’s Law (with energy-dependent conductivity) provided we vary the masses randomly to enable the repartitioning of energy. Dynamical systems issues that arise in this study are discussed though their resolution is beyond reach.     

Computation of Current Cumulants for Small Nonequilibrium Systems

We analyze a systematic algorithm for the exact computation of the current cumulants in stochastic nonequilibrium systems, recently discussed in the framework of full counting statistics for mesoscopic systems. This method is based on identifying the current cumulants from a Rayleigh-Schrödinger perturbation expansion for the generating function. Here it is derived from a simple path-distribution identity and extended to the joint statistics of multiple currents. For a possible thermodynamical interpretation, we compare this approach to a generalized Onsager-Machlup formalism. We present calculations for a boundary driven Kawasaki dynamics on a one-dimensional chain, both for attractive and repulsive particle interactions.     

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.     

Correlations in Nonequilibrium Steady States of Random Halves Models

We present the results of a detailed study of energy correlations at steady state for a 1-D model of coupled energy and matter transport. Our aim is to discover—via theoretical arguments, conjectures, and numerical simulations—how spatial covariances scale with system size, their relations to local thermodynamic quantities, and the randomizing effects of heat baths. Among our findings are that short-range covariances respond quadratically to local temperature gradients, and long-range covariances decay linearly with macroscopic distance. These findings are consistent with exact results for the simple exclusion and KMP models. This research was supported by an NSF Postdoctoral Fellowship. This research was supported by a grant from the NSF.     

Nonequilibrium Green Functions Approach to Strongly Correlated Fermions in Lattice Systems

Quantum dynamics in strongly correlated systems are of high current interest in many fields including dense plasmas, nuclear matter and condensed matter and ultracold atoms. An important model case are fermions in lattice systems that is well suited to analyze, in detail, a variety of electronic and magnetic properties of strongly correlated solids. Such systems have recently been reproduced with fermionic atoms in optical lattices which allow for a very accurate experimental analysis of the dynamics and of transport processes such as diffusion. The theoretical analysis of such systems far from equilibrium is very challenging since quantum and spin effects as well as correlations have to be treated non‐perturbatively. The only accurate method that has been successful so far are density matrix renormalization group (DMRG) simulations. However, these simulations are presently limited to one‐dimensional (1D) systems and short times. Extension of quantum dynamics simulations to two and three dimensions is commonly viewed as one of the major challenges in this field. Recently we have reported a breakthrough in this area [N. Schlünzen et al., Phys. Rev. B (2016)] where we were able to simulate the expansion dynamics of strongly correlated fermions in a Hubbard lattice following a quench of the confinement potential in 1D, 2D and 3D. The results not only exhibited excellent agreement with the experimental data but, in addition, revealed new features of the short‐time dynamics where correlations and entanglement are being build up. The method used in this work are nonequilibrium Green functions (NEGF) which are found to be very powerful in the treatment of fermionic lattice systems filling the gap presently left open by DMRG in 2D and 3D. In this paper we present a detailed introduction in the NEGF approach and its application to inhomogeneous Hubbard clusters. In detail we discuss the proper strong coupling approximation which is given by T ‐matrix selfenergies that sum up two‐particle scattering processes to infinite order. The efficient numerical implemen‐tation of the method is discussed in detail as it has allowed us to achieve dramatic performance gains. This has been the basis for the treatment of more than 100 particles over large time intervals. The numerical results presented in this paper concentrate on the diffusion in 1D to 3D lattices. We find that the expansion dynamics consist of three different phases that are linked with the build‐up of correlations. In the long time limit, a universal scaling with the particle number is revealed. By extrapolating the expansion velocities to the macroscopic limit, the obtained results show excellent agreement with recent experiments on ultracold fermions in optical lattices. Moreover we present results for the site‐resolved behavior of correlations and entanglement that can be directly compared with experiments using the recently developed atomic microscope technique. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonequilibrium steady state in open quantum systems: Influence action,stochastic equation and power balance

The existence and uniqueness of a steady state for nonequilibrium systems (NESS) is a fundamental subject and a main theme of research in statistical mechanics for decades. For Gaussian systems, such as a chain of classical harmonic oscillators connected at each end to a heat bath, and for classical anharmonic oscillators under specified conditions, definitive answers exist in the form of proven theorems. Answering this question for quantum many-body systems poses a challenge for the present. In this work we address this issue by deriving the stochastic equations for the reduced system with self-consistent backaction from the two baths, calculating the energy flow from one bath to the chain to the other bath, and exhibiting a power balance relation in the total (chain + baths) system which testifies to the existence of a NESS in this system at late times. Its insensitivity to the initial conditions of the chain corroborates to its uniqueness. The functional method we adopt here entails the use of the influence functional, the coarse-grained and stochastic effective actions, from which one can derive the stochastic equations and calculate the average values of physical variables in open quantum systems. This involves both taking the expectation values of quantum operators of the system and the distributional averages of stochastic variables stemming from the coarse-grained environment. This method though formal in appearance is compact and complete. It can also easily accommodate perturbative techniques and diagrammatic methods from field theory. Taken all together it provides a solid platform for carrying out systematic investigations into the nonequilibrium dynamics of open quantum systems and quantum thermodynamics.     

A Remark on the Strict Positivity of the Entropy Production

We establish an algebraic criterion which ensures the strict positivity of the entropy production in quantum models consisting of a small system coupled to thermal reservoirs at different temperatures. Mathematics Subject Classifications (2000). 46L05, 81Q10, 82C10, 82C70.     

介观化学反应体系中非平衡相变最可几路径的熵产生

针对一个双稳的介观化学反应体系计算了在非平衡相变时最可几路径的熵产生. 利用概率产生函数和程函近似,将化学主方程转化为经典的哈密顿-雅可比方程并通过相空间的零能轨线找到双稳态之间转变的最可几路径. 通过计算前向和逆向最可几路径的熵产生,发现在共存点系统熵变和介质熵变都为零,而在非共存点系统熵变和介质熵变皆不为零.     

Entropy Production in Nonlinear,Thermally Driven Hamiltonian Systems

We consider a finite chain of nonlinear oscillators coupled at its ends to two infinite heat baths which are at different temperatures. Using our earlier results about the existence of a stationary state, we show rigorously that for arbitrary temperature differences and arbitrary couplings, such a system has a unique stationary state. (This extends our earlier results for small temperature differences.) In all these cases, any initial state will converge (at an unknown rate) to the stationary state. We show that this stationary state continually produces entropy. The rate of entropy production is strictly negative when the temperatures are unequal and is proportional to the mean energy flux through the system     

Canonical Analysis of Condensation in Factorised Steady States

We study the phenomenon of real space condensation in the steady state of a class of mass transport models where the steady state factorises. The grand canonical ensemble may be used to derive the criterion for the occurrence of a condensation transition but does not shed light on the nature of the condensate. Here, within the canonical ensemble, we analyse the condensation transition and the structure of the condensate, determining the precise shape and the size of the condensate in the condensed phase. We find two distinct condensate regimes: one where the condensate is gaussian distributed and the particle number fluctuations scale normally as L ^1/2 where L is the system size, and a second regime where the particle number fluctuations become anomalously large and the condensate peak is non-gaussian. Our results are asymptotically exact and can also be interpreted within the framework of sums of random variables. We further analyse two additional cases: one where the condensation transition is somewhat different from the usual second order phase transition and one where there is no true condensation transition but instead a pseudocondensate appears at superextensive densities. PACS numbers: 05.40.-a, 02.50.Ey, 64.60.-i.