Nonequilibrium steady state thermodynamics and fluctuations for stochastic systems

We use the work done on and the heat removed from a system to maintain it in a nonequilibrium steady state for a thermodynamic-like description of such a system as well as of its fluctuations. Based on a generalized Onsager-Machlup theory for nonequilibrium steady states we indicate two ambiguities, not present in an equilibrium state, in defining such work and heat: one due to a non-uniqueness of time-reversal procedures and another due to multiple possibilities to separate heat into work and an energy difference in nonequilibrium steady states. As a consequence, for such systems, the work and heat satisfy multiple versions of the first and second laws of thermodynamics as well as of their fluctuation theorems. Unique laws and relations appear only to be obtainable for concretely defined systems, using physical arguments to choose the relevant physical quantities. This is illustrated on a number of systems, including a Brownian particle in an electric field, a driven torsion pendulum, electric circuits and an energy transfer driven by a temperature difference.Comment: 39 pages, 3 figur

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, Nakagwa, 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 Komtatsu et al.Comment: 19 pages, 2 figure

A quantum version of free energy - irreversible work relations

We give a quantum version of the Jarzynski relation between the distribution of work done over a certain time-interval on a system and the difference of equilibrium free energies. The main new ingredient is the identification of work depending on the quantum history of the system and the proper definition of various quantum ensembles over which the averages should be made. We also discuss a number of different regimes that have been considered by other authors and which are unified in the present set-up. In all cases, and quantum or classical, it is a general relation between heat and time-reversal that makes the Jarzynski relation so universally valid

Glassy behavior of a homopolymer from molecular dynamics simulations

We study at- and out-of-equilibrium dynamics of a single homopolymer chain at low temperature using molecular dynamics simulations. The main quantities of interest are the average root mean square displacement of the monomers below the theta point, and the structure factor, as a function of time. The observation of these quantities show a close resemblance to those measured in structural glasses and suggest that the polymer chain in its low temperature phase is in a glassy phase, with its dynamics dominated by traps. In equilibrium, at low temperature, we observe the trapping of the monomers and a slowing down of the overall motion of the polymer as well as non-exponential relaxation of the structure factor. In out-of-equilibrium, at low temperatures, we compute the two-time quantities and observe breaking of ergodicity in a range of waiting times, with the onset of aging.Comment: 11 pages, 4 figure

Quantum Jarzynski-Sagawa-Ueda relations

We consider a (small) quantum mechanical system which is operated by an external agent, who changes the Hamiltonian of the system according to a fixed scenario. In particular we assume that the agent (who may be called a demon) performs measurement followed by feedback, i.e., it makes a measurement of the system and changes the protocol according to the outcome. We extend to this setting the generalized Jarzynski relations, recently derived by Sagawa and Ueda for classical systems with feedback. One of the two relations by Sagawa and Ueda is derived here in error-free quantum processes, while the other is derived only when the measurement process involves classical errors. The first relation leads to a second law which takes into account the efficiency of the feedback.Comment: 11 pages. a major revision in v.2. Minor revision in v.3. The present version will appear in J. Stat. Phy

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.Comment: 27 pages, 3 figure

Onsager-Machlup theory and work fluctuation theorem for a harmonically driven Brownian particle

We extend Tooru-Cohen analysis for nonequilirium 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 an 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.Comment: 6 pages, 1 figur

Model of a fluid at small and large length scales and the hydrophobic effect

We present a statistical field theory to describe large length scale effects induced by solutes in a cold and otherwise placid liquid. The theory divides space into a cubic grid of cells. The side length of each cell is of the order of the bulk correlation length of the bulk liquid. Large length scale states of the cells are specified with an Ising variable. Finer length scale effects are described with a Gaussian field, with mean and variance affected by both the large length scale field and by the constraints imposed by solutes. In the absence of solutes and corresponding constraints, integration over the Gaussian field yields an effective lattice gas Hamiltonian for the large length scale field. In the presence of solutes, the integration adds additional terms to this Hamiltonian. We identify these terms analytically. They can provoke large length scale effects, such as the formation of interfaces and depletion layers. We apply our theory to compute the reversible work to form a bubble in liquid water, as a function of the bubble radius. Comparison with molecular simulation results for the same function indicates that the theory is reasonably accurate. Importantly, simulating the large length scale field involves binary arithmetic only. It thus provides a computationally convenient scheme to incorporate explicit solvent dynamics and structure in simulation studies of large molecular assemblies

Quantum Fluctuation Relations for the Lindblad Master Equation

An open quantum system interacting with its environment can be modeled under suitable assumptions as a Markov process, described by a Lindblad master equation. In this work, we derive a general set of fluctuation relations for systems governed by a Lindblad equation. These identities provide quantum versions of Jarzynski-Hatano-Sasa and Crooks relations. In the linear response regime, these fluctuation relations yield a fluctuation-dissipation theorem (FDT) valid for a stationary state arbitrarily far from equilibrium. For a closed system, this FDT reduces to the celebrated Callen-Welton-Kubo formula

On the validity of entropy production principles for linear electrical circuits

We discuss the validity of close-to-equilibrium entropy production principles in the context of linear electrical circuits. Both the minimum and the maximum entropy production principle are understood within dynamical fluctuation theory. The starting point are Langevin equations obtained by combining Kirchoff's laws with a Johnson-Nyquist noise at each dissipative element in the circuit. The main observation is that the fluctuation functional for time averages, that can be read off from the path-space action, is in first order around equilibrium given by an entropy production rate. That allows to understand beyond the schemes of irreversible thermodynamics (1) the validity of the least dissipation, the minimum entropy production, and the maximum entropy production principles close to equilibrium; (2) the role of the observables' parity under time-reversal and, in particular, the origin of Landauer's counterexample (1975) from the fact that the fluctuating observable there is odd under time-reversal; (3) the critical remark of Jaynes (1980) concerning the apparent inappropriateness of entropy production principles in temperature-inhomogeneous circuits.Comment: 19 pages, 1 fi