Nonequilibrium work relations: foundations and applications

When a macroscopic system in contact with a heat reservoir is driven away from equilibrium, the second law of thermodynamics places a strict bound on the amount of work performed on the system. With a microscopic system the situation is more subtle, as thermal fluctuations give rise to a statistical distribution of work values. In recent years it has been realized that such distributions encode surprisingly more information than one might expect from traditional thermodynamic arguments. I will discuss a number of exact results that relate equilibrium properties of the system, in particular free energy differences, to the fluctuations in the work performed during such a nonequilibrium process. I will describe the theoretical foundations of these relations, connections with irreversibility and the second law of thermodynamics, and potential experimental and computational applications.     

Fokker-Planck equation approach to fluctuations about nonequilibrium steady states

We examine the properties of steady states in systems which interact at the boundary with a nonequilibrium environment. The examination is based on a nonlinear Fokker-Planck equation, the structure of which is determined by the fact that it also governs the time evolution of the equilibrium fluctuations of the system. The nonlinearities in the Fokker-Planck equation may have two origins: thermodynamic nonlinearities which arise if the thermodynamic potential is not a bilinear function of the state variables, and nonlinear mode coupling which arises if the transport coefficients depend on the state. While these nonlinearities have only a small effect on the equilibrium fluctuations of a system away from critical points, they are shown to be important for the determination of fluctuations about nonequilibrium steady states. In particular the state dependence of the transport coefficients may lead to deviations from local equilibrium and to a breakdown of detail balance. An explicit formula for the time correlations of fluctuations about the nonequilibrium steady state is obtained. The formula leads to long-range correlations in fluids in the presence of a temperature gradient. The result is compared with earlier approaches to the same problem. Finally, we study the linear response to external forces and obtain a generalization of the fluctuation-dissipation formula relating the response functions with the nonequilibrium correlation functions.     

Thermodynamics of Currents in Nonequilibrium Diffusive Systems: Theory and Simulation

Understanding the physics of nonequilibrium systems remains as one of the major challenges of modern theoretical physics. We believe nowadays that this problem can be cracked in part by investigating the macroscopic fluctuations of the currents characterizing nonequilibrium behavior, their statistics, associated structures and microscopic origin. This fundamental line of research has been severely hampered by the overwhelming complexity of this problem. However, during the last years two new powerful and general methods have appeared to investigate fluctuating behavior that are changing radically our understanding of nonequilibrium physics: a powerful macroscopic fluctuation theory (MFT) and a set of advanced computational techniques to measure rare events. In this work we study the statistics of current fluctuations in nonequilibrium diffusive systems, using macroscopic fluctuation theory as theoretical framework, and advanced Monte Carlo simulations of several stochastic lattice gases as a laboratory to test the emerging picture. Our quest will bring us from (1) the confirmation of an additivity conjecture in one and two dimensions, which considerably simplifies the MFT complex variational problem to compute the thermodynamics of currents, to (2) the discovery of novel isometric fluctuation relations, which opens an unexplored route toward a deeper understanding of nonequilibrium physics by bringing symmetry principles to the realm of fluctuations, and to (3) the observation of coherent structures in fluctuations, which appear via dynamic phase transitions involving a spontaneous symmetry breaking event at the fluctuating level. The clear-cut observation, measurement and characterization of these unexpected phenomena, well described by MFT, strongly support this theoretical scheme as the natural theory to understand the thermodynamics of currents in nonequilibrium diffusive media, opening new avenues of research in nonequilibrium physics.     

Thermodynamics of particle systems in the presence of external macroscopic fields

We study the grand partition function of a system of identical particles interacting via a superstable potential in the presence of an external field depending on a scale factor. We discuss the case when the scale factor increases to infinity (macroscopic limit for the external potential) and we prove rigorously a link between the so obtained pressure and the usual one (barometric formula).Research partially supported by the Consiglio Nazionale delle Ricerche.     

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.     

Thermodynamic model for feedback control of systems with uncertain macroscopic states

Thermodynamics of feedback control processes, including the minimum work consumption of measurement, work extraction, and erasure processes of thermodynamic small systems have been investigated by researchers. We take systems with uncertain macroscopic states as the study object and study the feedback control processes of nonequilibrium macroscopic systems considering both the information entropy of microscopic states and macroscopic states. First we consider a system set that consists of systems with several macroscopic states and discuss the relations among the average information entropy of the system set, the thermodynamic entropy of the systems and the information entropy of macroscopic states of the systems. Then, we derive the expression of the average maximum net work obtained through feedback control, which relates to the free energy of the systems and the minimum work consumption of the measurement and erasure processes.     

Force-driven micro-rheology

Within a microscopic formalism for the nonequilibrium response of colloidal suspensions driven by an external force, we study the active micro-rheology of a glass-forming colloidal suspensions. In this technique, a probe particle is subject to an external force, and its nonequilibrium dynamics is monitored. Strong external forcing delocalizes the particle from its nearest-neighbor cage, resulting in a pronounced force-thinning behavior of the single-particle friction. We discuss the dynamics in the vicinity of this delocalization transition, and how long-range transport is induced for a particle that is localized in the quiescent case.     

Statistical physics and physiology: monofractal and multifractal approaches.

Even under healthy, basal conditions, physiologic systems show erratic fluctuations resembling those found in dynamical systems driven away from a single equilibrium state. Do such "nonequilibrium" fluctuations simply reflect the fact that physiologic systems are being constantly perturbed by external and intrinsic noise? Or, do these fluctuations actually, contain useful, "hidden" information about the underlying nonequilibrium control mechanisms? We report some recent attempts to understand the dynamics of complex physiologic fluctuations by adapting and extending concepts and methods developed very recently in statistical physics. Specifically, we focus on interbeat interval variability as an important quantity to help elucidate possibly non-homeostatic physiologic variability because (i) the heart rate is under direct neuroautonomic control, (ii) interbeat interval variability is readily measured by noninvasive means, and (iii) analysis of these heart rate dynamics may provide important practical diagnostic and prognostic information not obtainable with current approaches. The analytic tools we discuss may be used on a wider range of physiologic signals. We first review recent progress using two analysis methods--detrended fluctuation analysis and wavelets--sufficient for quantifying monofractual structures. We then describe recent work that quantifies multifractal features of interbeat interval series, and the discovery that the multifractal structure of healthy subjects is different than that of diseased subjects.     

Exact free energy functional for a driven diffusive open stationary nonequilibrium system

We obtain the exact probability exp[-LF([rho(x)])] of finding a macroscopic density profile rho(x) in the stationary nonequilibrium state of an open driven diffusive system, when the size of the system L-->infinity. F, which plays the role of a nonequilibrium free energy, has a very different structure from that found in the purely diffusive case. As there, F is nonlocal, but the shocks and dynamic phase transitions of the driven system are reflected in nonconvexity of F, in discontinuities in its second derivatives, and in non-Gaussian fluctuations in the steady state.     

The eigenvalue problem for linear macroscopic equations

The normal mode analysis of systems of linear macroscopic equations in irreversible thermodynamics is extended in several ways. When the characteristics equation has multiple roots, there may appear normal solutions that do not decay purely exponentially, but a closed form for the Green function and the autocorrelation function can still be given. Furthermore, nonexponential decay is associated only with accidental, not with systematic degeneracy. We also discuss the case of external parameters that break microscopic time reversibility. In this case the orthonormality relations between the normal mode vectors are replaced by biorthonormality relations between the normal modes of the system studied and those of the system with reversed external parameters. Finally we discuss systems in which the second order energy is only positive semi-definite.     

Large Deviations for the Boundary Driven Symmetric Simple Exclusion Process

The large deviation properties of equilibrium (reversible) lattice gases are mathematically reasonably well understood. Much less is known in nonequilibrium, namely for nonreversible systems. In this paper we consider a simple example of a nonequilibrium situation, the symmetric simple exclusion process in which we let the system exchange particles with the boundaries at two different rates. We prove a dynamical large deviation principle for the empirical density which describes the probability of fluctuations from the solutions of the hydrodynamic equation. The so-called quasi potential, which measures the cost of a fluctuation from the stationary state, is then defined by a variational problem for the dynamical large deviation rate function. By characterizing the optimal path, we prove that the quasi potential can also be obtained from a static variational problem introduced by Derrida, Lebowitz, and Speer.     

Energy fluctuations in a biharmonically driven nonlinear system

We study the fluctuations of work done and dissipated heat of a Brownian particle in a symmetric double well system. The system is driven by two periodic input signals that rock the potential simultaneously. Confinement in one preferred well can be achieved by modulating the relative phase between the drives. We show that in the presence of pumping the stochastic resonance signal is enhanced when analysed in terms of the average work done on the system per cycle. This is in contrast with the case when pumping is achieved by applying an external static bias, which degrades resonance. We analyse the nature of work and heat fluctuations and show that the steady state fluctuation theorem holds in this system.     

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.     

Fluctuations in a fluid under a stationary heat flux. I. General theory

We present the basic formulas for a unified treatment of the correlation functions of the hydrodynamic variables in a fluid between two horizontal plates which is exposed to a stationary heat flux in the presence of a gravity field (Rayleigh-Bénard system). Our analysis is based on fluctuating hydrodynamics. In this paper (I) we show that in the nonequilibrium stationary state the hydrodynamic fluctuations evolve on slow and fast time scales that are widely separated. A time scale perturbation theory is used to diagonalize the hydrodynamic operator partially. This enables us to derive the eigenvalue equations for the nonequilibrium hydrodynamic modes. Therein we take into account the variation of the macroscopic quantities with position. The correlation functions are formally expressed in terms of the nonequilibrium modes. In paper II the slow hydrodynamic modes (viscous and viscoheat modes) will be determined explicitly for ideal heat-conducting plates with stick boundary conditions and used to compute the slow part of the correlation functions; in paper III the fast hydrodynamic modes (sound modes) will be explicitly determined for stick boundary conditions and used to compute the fast part of the correlation functions. In these papers we will also compute the shape and intensity of the lines measured in light scattering experiments.     

The theory of shot noise in the space-charge-limited diffusive conduction regime

As is well known, fluctuations from a stable stationary nonequilibrium state are described by the linearized inhomogeneous Boltzmann-Langevin equation. The stationary state itself can be described by the nonlinear Boltzmann equation. The ways of its linearization sometimes seem to be not unique. We argue that there is actually a unique way to obtain a linear equation for the fluctuations. As an example, we consider an analytical theory of nonequilibrium shot noise in a diffusive conductor under the space-charge-limited regime. Our approach is compared to that in [11]. We find some difference between the present theory and the approach in [11] and discuss a possible origin of the difference. We believe that it is related to the fundamentals of the theory of fluctuation phenomena in a nonequilibrium electron gas.     

From superoperator formalism to nonequilibrium Thermo Field Dynamics

Emphasizing that the specification of the representation space or the quasiparticle picture is essential in nonequilibrium quantum field system, we have constructed the unique unperturbed representation of the interaction picture in the superoperator formalism. To achieve it, we put the three basic requirements (the existence of the quasiparticle picture at each instant of time, the macroscopic causality and the relaxation to equilibrium). From the resultant representation follows the formulation of nonequilibrium Thermo Field Dynamics (TFD). The two parameters, the number distribution and excitation energy, characterizing the representation, are to be determined by the renormalization condition. While we point out that the diagonalization condition by Chu and Umezawa is inconsistent with the equilibrium theory, we propose a new renormalization condition as a generalization of the on-shell renormalization on the self-energy which derives the quantum transport equation and determines the renormalized excitation energy.     

Dissipative dynamics of a quantum two-state system in presence of nonequilibrium quantum noise

We analyze the real-time dynamics of a quantum two-state system in the presence ofnonequilibrium quantum fluctuations. The latter are generated by a coupling of thetwo-state system to a single electronic level of a quantum dot which carries anonequilibrium tunneling current. We restrict to the sequential tunneling regime andcalculate the dynamics of the two-state system, of the dot population, and of thenonequilibrium charge current on the basis of a diagrammatic perturbative method valid fora weak tunneling coupling. We find a nontrivial dependence of the relaxation and dephasingrates of the two-state system due to the nonequilibrium fluctuations which is directlylinked to the structure of the unperturbed central system. In addition, aHeisenberg-Langevin-equation of motion allows us to calculate the correlation function ofthe nonequilibrium fluctuations. By this, we obtain a generalized nonequilibriumfluctuation relation which includes the equilibrium fluctuation-dissipation theorem in thelimit of zero transport voltage. A straightforward extension to the case with atime-periodic ac voltage is shown.