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.     

Inertial Effects in Nonequilibrium Work Fluctuations by a Path Integral Approach

Inertial effects in fluctuations of the work to sustain a system in a nonequilibrium steady state are discussed for a dragged massive Brownian particle model using a path integral approach. We calculate the work distribution function in the laboratory and comoving frames and prove the asymptotic fluctuation theorem for these works for any initial condition. Important and observable differences between the work fluctuations in the two frames appear for finite times and are discussed concretely for a nonequilibrium steady state initial condition. We also show that for finite times a time oscillatory behavior appears in the work distribution function for masses larger than a nonzero critical value.     

Onsager-Machlup Theory and Work Fluctuation Theorem for a Harmonically Driven Brownian Particle

We extend Tooru-Cohen analysis for nonequilibrium steady state (NSS) of a Brownian particle to nonequilibrium oscillatory state (NOS) of Brownian particle by considering time dependent external drive protocol. We consider an unbounded charged Brownian particle in the presence of oscillating electric field and prove work fluctuation theorem, which is valid for any initial distribution and at all times. For harmonically bounded and constantly dragged Brownian particle considered by Tooru and Cohen, work fluctuation theorem is valid for any initial condition (also NSS), but only in large time limit. We use Onsager-Machlup Lagrangian with a constraint to obtain frequency dependent work distribution function, and describe entropy production rate and properties of dissipation functions for the present system using Onsager-Machlup functional.     

Nonequilibrium Thermodynamics of the First and Second Kind: Averages and Fluctuations

We elaborate and compare two approaches to nonequilibrium thermodynamics, the two-generator bracket formulation of time-evolution equations for averages and the macroscopic fluctuation theory, for a purely dissipative isothermal driven diffusive system under steady state conditions. The fluctuation dissipation relations of both approaches play an important role for a detailed comparison. The nonequilibrium Helmholtz free energies introduced in these two approaches differ as a result of boundary conditions. A Fokker-Planck equation derived by projection operator techniques properly reproduces long range fluctuations in nonequilibrium steady states and offers the most promising possibility to describe the physically relevant fluctuations around macroscopic averages for time-dependent nonequilibrium systems.     

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.     

Steady State Thermodynamics

The present paper reports our attempt to search for a new universal framework in nonequilibrium physics. We propose a thermodynamic formalism that is expected to apply to a large class of nonequilibrium steady states including a heat conducting fluid, a sheared fluid, and an electrically conducting fluid. We call our theory steady state thermodynamics (SST) after Oono and Paniconi's original proposal. The construction of SST is based on a careful examination of how the basic notions in thermodynamics should be modified in nonequilibrium steady states. We define all thermodynamic quantities through operational procedures which can be (in principle) realized experimentally. Based on SST thus constructed, we make some nontrivial predictions, including an extension of Einstein's formula on density fluctuation, an extension of the minimum work principle, the existence of a new osmotic pressure of a purely nonequilibrium origin, and a shift of coexistence temperature. All these predictions may be checked experimentally to test SST for its quantitative validity.     

Time-symmetric fluctuations in nonequilibrium systems

For nonequilibrium steady states, we identify observables whose fluctuations satisfy a general symmetry and for which a new reciprocity relation can be shown. Unlike the situation in recently discussed fluctuation theorems, these observables are time-reversal symmetric. That is essential for exploiting the fluctuation symmetry beyond linear response theory. In addition to time reversal, a crucial role is played by the reversal of the driving fields that further resolves the space-time action. In particular, the time-symmetric part in the space-time action determines the second order effects of the nonequilibrium driving.     

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.     

Dissipation and large thermodynamic fluctuations

The results of recent work of Kipnis, Olla, and Varadhan on the dynamic large deviations from a hydrodynamic limit for some interacting particle models are formally extended to a general hydrodynamic situation, including non-equilibrium steady states, as a fluctuation-dissipation hypothesis. The basic conjecture is that the exponent of decay in the probability of a large thermodynamic fluctuation is given by the dissipation of the force required to produce the fluctuation. It is shown that this hypothesis leads to a nonlinear version of Onsager-Machlup fluctuation theory that had previously been proposed by Graham. A direct consequence of the theory is a dynamic variational principle for the most probable thermodynamic history subject to imposed constraints (Onsager's principle of least dissipation). Following Graham, the theory leads also to a generalized potential, analogous to an equilibrium free energy, for the nonequilibrium steady state and an associated static variational principle. Finally, a formulation of nonlinear fluctuating hydrodynamics is proposed in which the noise enters multiplicatively so as to reproduce the conjectured large-deviations theory on a formal analogy with the results of Freidlin and Wentzell for finite-dimensional systems.     

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.     

Nonequilibrium charge susceptibility and dynamical conductance: identification of scattering processes in quantum transport

We calculate the nonequilibrium charge transport properties of nanoscale junctions in the steady state and extend the concept of charge susceptibility to the nonequilibrium conditions. We show that the nonequilibrium charge susceptibility is related to the nonlinear dynamical conductance. In spectroscopic terms, both contain the same features versus applied bias when charge fluctuation occurs in the corresponding electronic resonances. However, we show that, while the conductance exhibits features at biases corresponding to inelastic scattering with no charge fluctuations, the nonequilibrium charge susceptibility does not. We suggest that measuring both the nonequilibrium conductance and charge susceptibility in the same experiment will permit us to differentiate between different scattering processes in quantum transport.     

Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics

This paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics. We adopt a new point of view which has emerged progressively in recent years, and which takes seriously into account the chaotic character of the microscopic time evolution. The emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of Boltzmann. The nonequilibrium steady states, in presence of a Gaussian thermostat, are described by SRB measures. In terms of these one can prove the Gallavotti–Cohen fluctuation theorem. One can also prove a general linear response formula and study its consequences, which are not restricted to near-equilibrium situations. At equilibrium one recovers in particular the Onsager reciprocity relations. Under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of Dettmann and Morriss. The results just mentioned hold so far only for classical systems; they do not involve large size, i.e., they hold without a thermodynamic limit.     

Nonequilibrium thermodynamics of nonisothermal activation: Escape over mesoscopic barriers: I. The stochastic process

It is argued that in noisy “macroscopic” systems (e.g. Josephson junctions) the effective potential for the pertinent degree of freedom (e.g. the magnetic flux) generally depends on the specimen's - local - temperature and, hence, that it should be given a thermodynamical (rather than the usual mechanical) significance. A Brownian motion theory of the ensuing nonequilibrium thermodynamics will be presented, which generalizes Kramers' model for thermal activation to include nonisothermal effects. Inter alia it is shown how in the steady state the thermodynamic fluctuation potential (the “availability”) arises as the natural generalization of either the Gibbs or the Helmholtz free energy.     

Representation of Nonequilibrium Steady States in Large Mechanical Systems

Recently a novel concise representation of the probability distribution of heat conducting nonequilibrium steady states was derived. The representation is valid to the second order in the “degree of nonequilibrium”, and has a very suggestive form where the effective Hamiltonian is determined by the excess entropy production. Here we extend the representation to a wide class of nonequilibrium steady states realized in classical mechanical systems where baths (reservoirs) are also defined in terms of deterministic mechanics. The present extension covers such nonequilibrium steady states with a heat conduction, with particle flow (maintained either by external field or by particle reservoirs), and under an oscillating external field. We also simplify the derivation and discuss the corresponding representation to the full order.     

Crooks Fluctuation Theorem for Single Polymer Dynamics in Time-Dependent Flows: Understanding Viscoelastic Hysteresis

Nonequilibrium work relations have fundamentally advanced our understanding of molecular processes. In recent years, fluctuation theorems have been extensively applied to understand transitions between equilibrium steady-states, commonly described by simple control parameters such as molecular extension of a protein or polymer chain stretched by an external force in a quiescent fluid. Despite recent progress, far less is understood regarding the application of fluctuation theorems to processes involving nonequilibrium steady-states such as those described by polymer stretching dynamics in nonequilibrium fluid flows. In this work, we apply the Crooks fluctuation theorem to understand the nonequilibrium thermodynamics of dilute polymer solutions in flow. We directly determine the nonequilibrium free energy for single polymer molecules in flow using a combination of single molecule experiments and Brownian dynamics simulations. We further develop a time-dependent extensional flow protocol that allows for probing viscoelastic hysteresis over a wide range of flow strengths. Using this framework, we define quantities that uniquely characterize the coil-stretch transition for polymer chains in flow. Overall, generalized fluctuation theorems provide a powerful framework to understand polymer dynamics under far-from-equilibrium conditions.     

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.     

Nonequilibrium Thermodynamics in Biochemical Systems and Its Application

Living systems are open systems, where the laws of nonequilibrium thermodynamics play the important role. Therefore, studying living systems from a nonequilibrium thermodynamic aspect is interesting and useful. In this review, we briefly introduce the history and current development of nonequilibrium thermodynamics, especially that in biochemical systems. We first introduce historically how people realized the importance to study biological systems in the thermodynamic point of view. We then introduce the development of stochastic thermodynamics, especially three landmarks: Jarzynski equality, Crooks’ fluctuation theorem and thermodynamic uncertainty relation. We also summarize the current theoretical framework for stochastic thermodynamics in biochemical reaction networks, especially the thermodynamic concepts and instruments at nonequilibrium steady state. Finally, we show two applications and research paradigms for thermodynamic study in biological systems.     

Fluctuation Theorem for Currents and Schnakenberg Network Theory

A fluctuation theorem is proved for the macroscopic currents of a system in a nonequilibrium steady state, by using Schnakenberg network theory. The theorem can be applied, in particular, in reaction systems where the affinities or thermodynamic forces are defined globally in terms of the cycles of the graph associated with the stochastic process describing the time evolution.     

On the Second Fluctuation–Dissipation Theorem for Nonequilibrium Baths

Baths produce friction and random forcing on particles suspended in them. The relation between noise and friction in (generalized) Langevin equations is usually referred to as the second fluctuation–dissipation theorem. We show what is the proper nonequilibrium extension, to be applied when the environment is itself active and driven. In particular we determine the effective Langevin dynamics of a probe from integrating out a steady nonequilibrium environment. The friction kernel picks up a frenetic contribution, i.e., involving the environment’s dynamical activity, responsible for the breaking of the standard Einstein relation.