On thermodynamics near a steady state

The statistical deduction of a stability criterion for steady states (a generalization of the Glansdorff-Prigogine criterion) given in a previous paper is extended in some respects. Dynamical coefficients for the motion of small deviations from a steady state are expressed by correlation functions. The stability criterion is applied to some models of chemical reactions and to a model for second break-down in semiconductors.     

Essentially nonequilibrium steady state of atoms in an optical field

The kinetics of atoms in the monochromatic field of a standing wave has been considered taking into account the recoil effects that appear under the interaction of the atoms with photons of the field. It has been shown that the steady-state solution of the kinetic equation for the atomic density matrix can have considerable asymmetry in the phase space. There is also asymmetry in the fluxes of atoms in the ground and excited states, predominantly near the nodes of the optical field.     

Exact nonequilibrium steady state of a strongly driven open XXZ chain

An exact and explicit ladder-tensor-network ansatz is presented for the nonequilibrium steady state of an anisotropic Heisenberg XXZ spin-1/2 chain which is driven far from equilibrium with a pair of Lindblad operators acting on the edges of the chain only. We show that the steady-state density operator of a finite system of size n is-apart from a normalization constant-a polynomial of degree 2n - 2 in the coupling constant. Efficient computation of physical observables is facilitated in terms of a transfer operator reminiscent of a classical Markov process. In the isotropic case we find cosine spin profiles, 1/n(2) scaling of the spin current, and long-range correlations in the steady state. This is a fully nonperturbative extension of a recent result [Phys. Rev. Lett. 106, 217206 (2011)].     

Total entropy production fluctuation theorems in a nonequilibrium time-periodic steady state

We investigate the total entropy production of a Brownian particle in a driven bistable system. This system exhibits the phenomenon of stochastic resonance. We show that in the time-periodic steady state, the probability density function for the total entropy production satisfies Seifert’s integral and detailed fluctuation theorems over finite time trajectories.     

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.     

Light scattering from a fluid in a nonequilibrium steady state: Plane Couette flow

We calculate the Rayleigh-Brillouin light scattering spectrum from a newtonian fluid undergoing a plane Coutte flow using the Landau-Lifshitz method of fluctuating hydrodynamics. Our results for both the Rayleigh and the Brillouin lines differ significantly from all the previous results.     

Fluctuations about hydrodynamic nonequilibrium steady states

Using an extension of the Langevin method, we calculate the fluctuations of a fluid about a stationary state held away from global thermal equilibrium by a temperature gradient or shear flow. In the former case, the Brillouin peaks in the light scattering spectrum acquire an asymmetry that is qualitatively similat to earlier results but different in detail.     

Fractal dimension of steady nonequilibrium flows

The Kaplan-Yorke information dimension of phase-space attractors for two kinds of steady nonequilibrium many-body flows is evaluated. In both cases a set of Newtonian particles is considered which interacts with boundary particles. Time-averaged boundary temperatures are imposed by Nose-Hoover thermostat forces. For both kinds of nonequilibrium systems, it is demonstrated numerically that external isothermal boundaries can drive the otherwise purely Newtonian flow onto a multifractal attractor with a phase-space information dimension significantly less than that of the corresponding equilibrium flow. Thus the Gibbs' entropy of such nonequilibrium flows can diverge.     

Momentum transfer in nonequilibrium steady states

When a Brownian object interacts with noninteracting gas particles under nonequilibrium conditions, energy dissipation associated with Brownian motion causes an additional force on the object as a "momentum transfer deficit." This principle is demonstrated first by a new nonequilibrium steady state model and then applied to several known models such as an adiabatic piston for which a simple explanation has been lacking.     

Nondifferentiable potentials for nonequilibrium steady states

In the weak noise limit of Fokker-Planck models, the stationary probability density can be expressed in the form P∼exp(-1/ϵϕ(χ)), where ϵ measures the strength of the noise. We show firstly that, although in general ϕ is not differentiable, it is regular enough to allow its use as a Lyapunov function for the underlying deterministic dynamical system. This justifies the interpretation of ϕ as a generalized thermodynamic potential for systems away from equilibrium.Secondly, we show that the nondifferentiability of ϕ does not presuppose any kind of irregular behavior in the underlying deterministic system, nor the presence of several stationary points. We construct a simple model with one single attractor and trivial deterministic dynamics, which has a nondifferentiable quasipotential ϕ. We relate the nondifferentiability to the existence of caustics and shock lines in the associated variational problem.     

Open XXZ spin chain: nonequilibrium steady state and a strict bound on ballistic transport

An explicit matrix product ansatz is presented, in the first two orders in the (weak) coupling parameter, for the nonequilibrium steady state of the homogeneous, nearest neighbor Heisenberg XXZ spin 1/2 chain driven by Lindblad operators which act only at the edges of the chain. The first order of the density operator becomes, in the thermodynamic limit, an exact pseudolocal conservation law and yields-via the Mazur inequality-a rigorous lower bound on the high-temperature spin Drude weight. Such a Mazur bound is a nonvanishing fractal function of the anisotropy parameter Δ for |Δ|<1.     

Rules for transition rates in nonequilibrium steady states

Just as transition rates in a canonical ensemble must respect the principle of detailed balance, constraints exist on transition rates in driven steady states. I derive those constraints, by maximum information-entropy inference, and apply them to the steady states of driven diffusion and a sheared lattice fluid. The resulting ensemble can potentially explain nonequilibrium phase behavior and, for steady shear, gives rise to stress-mediated long-range interactions.     

Big entropy fluctuations in the nonequilibrium steady state: A simple model with the gauss heat bath

Large entropy fluctuations in a nonequilibrium steady state of classical mechanics are studied in extensive numerical experiments on a simple two-freedom model with the so-called Gauss time-reversible thermostat. The local fluctuations (on a set of fixed trajectory segments) from the average heat entropy absorbed in the thermostat are found to be non-Gaussian. The fluctuations can be approximately described by a two-Gaussian distribution with a crossover independent of the segment length and the number of trajectories (“particles”). The distribution itself does depend on both, approaching the single standard Gaussian distribution as any of those parameters increases. The global time-dependent fluctuations are qualitatively different in that they have a strict upper bound much less than the average entropy production. Thus, unlike the equilibrium steady state, the recovery of the initial low entropy becomes impossible after a sufficiently long time, even in the largest fluctuations. However, preliminary numerical experiments and the theoretical estimates in the special case of the critical dynamics with superdiffusion suggest the existence of infinitely many Poincaré recurrences to the initial state and beyond. This is a new interesting phenomenon to be further studied together with some other open questions. The relation of this particular example of a nonequilibrium steady state to the long-standing persistent controversy over statistical “irreversibility”, or the notorious “time arrow”, is also discussed. In conclusion, the unsolved problem of the origin of the causality “principle” is considered.     

Finite correlation time effects in nonequilibrium phase transitions: I. Dynamic equation and steady state properties

We determine an approximate renormalized equation of evolution for an arbitrary nonlinear single-degree-of-freedom system externally driven by Gaussian parametric fluctuations of finite correlation time. The renormalization scheme used here gives a second order equation with a time-and-state-dependent “diffusion coefficient”. We are able to calculate the diffusion coefficient in closed form. The steady-state distribution can easily be obtained from the evolution equation. We are thus able to determine the parameter dependence of the steady-state distribution and, in particular, the influence of a correlation time of the fluctuations, which does not vanish, on the steady-state distribution.     

Factorised steady states and condensation transitions in nonequilibrium systems

Systems driven out of equilibrium can often exhibit behaviour not seen in systems in thermal equilibrium —for example phase transitions in one-dimensional systems. In this talk I will review a simple model of a nonequilibrium system known as the ‘zero-range process’ and its recent developments. The nonequilibrium stationary state of this model factorises and this property allows a detailed analysis of several ‘condensation’ transitions wherein a finite fraction of the constituent particles condenses onto a single lattice site. I will then consider a more general class of mass transport models, encompassing continuous mass variables and discrete time updating, and present a necessary and sufficient condition for the steady state to factorise. The property of factorisation again allows an analysis of the condensation transitions which may occur.     

Expression for the stationary distribution in nonequilibrium steady states

We study the nonequilibrium steady state realized in a general stochastic system attached to multiple heat baths. Starting from the detailed fluctuation theorem, we derive concise and suggestive expressions for the corresponding stationary distribution which are correct up to the second order in thermodynamic forces. The probability of a microstate eta is proportional to exp[Phi(eta)] where Phi(eta)=-[under summation operator]kbeta_{k}E_{k}(eta) is the excess entropy change. Here, E_{k}(eta) is the difference between two kinds of conditioned path ensemble averages of excess heat transfer from the kth heat bath whose inverse temperature is beta_{k}. This result can be easily extended to steady states maintained with other sources, e.g., particle current driven by an external force. Our expression may be verified experimentally in nonequilibrium states realized, for example, in mesoscopic systems.     

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.