We describe a framework to significantly reduce the computational effort to evaluate large deviation functions of time integrated observables within nonequilibrium steady states. We do this by incorporating an auxiliary dynamics into trajectory based Monte Carlo calculations, through a transformation of the system's propagator using an approximate guiding function. This procedure importance samples the trajectories that most contribute to the large deviation function, mitigating the exponentially complexity of such calculations. We illustrate the method by studying driven diffusions and interacting lattice models in one and two dimensions. Our work offers an avenue to calculate large deviation functions for high dimensional systems driven far from equilibrium.
We present a strategy to construct guiding distribution functions (GDFs) based on variance minimization. Auxiliary dynamics via GDFs mitigates the exponential growth of variance as a function of bias in Monte Carlo estimators of large deviation functions. The variance minimization technique exploits the exact properties of eigenstates of the tilted operator that defines the biased dynamics in the nonequilibrium system. We demonstrate our techniques in two classes of problems. In the continuum, we show that GDFs can be optimized to study interacting driven diffusive systems where the efficiency…
We study the relaxation of a single colloidal sphere which is periodically driven between two nonequilibrium steady states. Experimentally, this is achieved by driving the particle along a toroidal trap imposed by scanned optical tweezers. We find that the relaxation time after which the probability distributions have been relaxed is identical to that obtained by a steady state measurement. In quantitative agreement with theoretical calculations the relaxation time strongly increases when driving the system further away from thermal equilibrium.
Large deviation functions contain information on the stability and response of systems driven into nonequilibrium steady states, and in such a way are similar to free energies for systems at equilibrium. As with equilibrium free energies, evaluating large deviation functions numerically for all but the simplest systems is difficult, because by construction they depend on exponentially rare events. In this first paper of a series, we evaluate different trajectory-based sampling methods capable of computing large deviation functions of time integrated observables within nonequilibrium steady sta…
We investigate nonequilibrium chemical reaction systems from the view point of steady state thermodynamics proposed by Oono and Paniconi [Prog. Theor. Phys. Suppl. 130, 29 (1998)]. The concentrations of some compounds are operated by an external system, so that a transition from a steady state to other steady state takes place. We show that an analogue of Clausius inequality holds macroscopically for the operation processes. This implies that the second law of thermodynamics can be generalized, including nonequilibrium steady states.
We study numerically phase separation in a binary fluid subject to an applied shear flow in two dimensions, with full hydrodynamics. To do so, we introduce a mixed finite-differencing/spectral simulation technique, with a transformation to render trivial the implementation of Lees-Edwards sheared periodic boundary conditions. For systems with inertia, we reproduce the nonequilibrium steady states reported in a recent lattice Boltzmann study. The domain coarsening that would occur in zero shear is arrested by the applied shear flow, which restores a finite domain size set by the inverse shear r…
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 behaviour and, for steady shear, gives rise to stress-mediated long-range interactions.
Fluctuation theorem is derived for a quantum current system around a nonequilibrium steady state. It is demonstrated that the fluctuation theorem can be a part of the generalized Green-Kubo formula or a nonlinear response theory of an external field or a change of the chemical potential difference.
Correlations in fluids in nonequilibrium steady states are long ranged. Hence, finite-size effects have important consequences in the nonequilibrium thermodynamics of fluids. One consequence is that nonequilibrium temperature fluctuations induce nonequilibrium Casimir-like pressures proportional to the square of the temperature gradient. Hence, fluctuations cause a breakdown of the concept of local thermal equilibrium. Furthermore, transport coefficients become dependent on boundary conditions and on gravity. Thus nonequilibrium fluctuations affect some traditional concepts in nonequilibrium t…
Transitions between nonequilibrium steady states obey a generalized Clausius inequality, which becomes an equality in the quasistatic limit. For slow but finite transitions, we show that the behavior of the system is described by a response matrix whose elements are given by a far-from-equilibrium Green-Kubo formula, involving the decay of correlations evaluated in the nonequilibrium steady state. This result leads to a fluctuation-dissipation relation between the mean and variance of the nonadiabatic entropy production, $Δs_{\rm na}$. Furthermore, our results extend -- to nonequilibrium stead…
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…
Nonequilibrium steady states in an open system connecting two reservoirs of platelike colloidal particles are investigated by means of a recently proposed phenomenological dynamic density functional theory [M. Bier and R. van Roij, Phys. Rev. E 76, 021405 (2007)]. The platelike colloidal particles are approximated within the Zwanzig model of restricted orientations, which exhibits an isotropic-nematic bulk phase transition. Inhomogeneities of the local chemical potential generate a diffusion current which relaxes to a nonvanishing value if the two reservoirs coupled to the system sustain diffe…
Jarzynski equality [Phys. Rev. E {\bf 56}, 5018 (1997)] is found to be valid with slight modefication for the transitions between nonequilibrium stationary states, as well as the one between equilibrium states. Also numerical results confirm its validity. Its relevance for nonequilibrium thermodynamics of the operational formalism is discussed.
We develop a controlled high-temperature expansion for nonequilibrium steady states of the driven lattice gas. We represent the steady state as $P(η)\propto e^{-H(η)-Ψ(η)}$, and evaluate the lowest order contribution to the nonequilibrium effective interaction $Ψ(η)$. We see that, in dimensions $d\ge2$, all models with nonsingular transition rates have the same $Ψ(η)$, which consists of a three-body effective interaction decaying like $1/r^{2d+1}$. The models with the Metropolis rule show exceptional behavior.
The steady state shock formation in processes like nonconserving asymmetric simple exclusion processes in varied situations is shown to be a nonequilibrium critical phenomenon. The diverging length scales and the quantitative description of the transition including the phase boundary are obtained from a few general properties of the dynamics without relying on specific details.
We derive universal properties of nonlinear response functions of nonequilibrium steady states. In particular, sum rules and asymptotic behaviors are derived. Their consequences are illustrated for nonlinear optical materials and nonlinear electrical conductors.
We propose a general classification of nonequilibrium steady states in terms of their stationary probability distribution and the associated probability currents. The stationary probabilities can be represented graph-theoretically as directed labelled trees; closing a single loop in such a graph leads to a representation of probability currents. This classification allows us to identify all choices of transition rates, based on a master equation, which generate the same nonequilibrium steady state. We explore the implications of this freedom, e.g., for entropy production.
We study a new balance condition multibalance to obtain the nonequilibrium steady states of a class of nonequilibrium lattice models on a ring where a particle hops from a particular site to its nearest and next nearest neighbours. For the well-known zero range process (ZRP) with asymmetric hop rates, with this balance condition, we obtain the conditions on hop rates that lead to a factorized steady state (FSS). We show that this balance condition gives the cluster-factorized steady state (CFSS) for finite range process (FRP) and other models. We also discuss the application of multibalance co…
In a nonequilibrium steady state, the violation of the fluctuation-dissipation theorem (FDT) is connected to breaking detailed balance. For the velocity correlations of a driven colloidal particle we calculate an explicit expression of the FDT violation. The equilibrium form of the FDT can be restored by measuring the velocity with respect to the local mean velocity.
We give the explicit structure of the functional governing the dynamical density and current fluctuations for a mesoscopic system in a nonequilibrium steady state. Its canonical form determines a generalised Onsager-Machlup theory. We assume that the system is described as a Markov jump process satisfying a local detailed balance condition such as typical for stochastic lattice gases and for chemical networks. We identify the entropy current and the traffic between the mesoscopic states as extra terms in the fluctuation functional with respect to the equilibrium dynamics. The density and curre…
We present a fully microscopics-based calculation of the Casimir effect in a nonequilibrium system, namely an energy flux driven quantum XX chain. The force between the walls (transverse-field impurities) is calculated in a nonequilibrium steady state which is prepared by letting the system evolve from an initial state with the two halves of the chain prepared at equilibrium at different temperatures. The steady state emerging in the large-time limit is homogeneous but carries an energy flux. The Casimir force in this nonequilibrium state is calculated analytically in the limit when the transv…
An expression for the energetic efficiency of a molecular motor is presented in terms of an effective temperature, which was defined based on the ratio of the correlation function to the susceptibility of its velocity. We also present a numerical result regarding this temperature for a thermal ratchet model. Furthermore, our expression of efficiency using known values for kinesin can adequately explain the experimental results. The framework presented here has a closed form phenomenologically, and is independent of any detailed models.
Bridging equilibrium and nonequilibrium statistical physics attracts sustained interest. Hallmarks of nonequilibrium systems include a breakdown of detailed balance, and an absence of a priori potential function corresponding to the Boltzmann-Gibbs distribution, without which classical equilibrium thermodynamical quantities could not be defined. Here, we construct dynamically the potential function through decomposing the system into a dissipative part and a conservative part, and develop a nonequilibrium theory by defining thermodynamical quantities based on the potential function. Concepts f…
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 a…
We derive the representation of the nonequilibrium steady-state distribution function which is expressed in terms of the excess free energy production. This representation resembles the one derived recently by Komatsu and Nakagawa [Phys. Rev. Lett. 100 (2008), 030601] resting on the use of microscopic time-reversal symmetry, but our representation applies also to sheared granular systems in which such a symmetry is broken.
We investigate a dissipatively driven XYZ spin-1/2 chain in the Zeno limit of strong dissipation, described by Lindblad master equation. The nonequilibrium steady state is expressed in terms of a matrix product ansatz using novel site-dependent Lax operators. The components of Lax operators satisfy a simple set of linear recurrence equations that generalize the defining algebraic relations of the quantum group $U_q(sl_2)$. We reveal connection between the nonequilibrium steady state of the nonunitary dynamics and the respective integrable model with edge magnetic fields, described by coherent …
When periodically driven by an external magnetic field, a spin system can enter a phase of steady entrained oscillations with nonequilibrium probability distribution function. We consider an arbitrary magnetic field switching its direction with frequency comparable with the spin-flip rate and show that the resulting nonequilibrium probability distribution can be related to the system equilibrium distribution in the presence of a constant magnetic field of the same magnitude. We derive convenient approximate expressions for this exact relation and discuss their implications.
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 monotonous 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 whe…
We give a decomposition expression for dissipative heat using the instantaneous diffusion coefficient in a nonequilibrium steady state. The dissipative heat can be expressed using three diffusion coefficients: instantaneous, equilibrium, and drift. An experimental application of the decomposition expression permits us to evaluate the heat dissipation rate from single-trajectory data only. We also numerically demonstrate this method.
A driven stochastic system in a constant temperature heat bath relaxes into a steady state which is characterized by the steady state probability distribution. We investigate the relationship between the driving force and the steady state probability distribution. We adopt the force decomposition method in which the force is decomposed as the sum of a gradient of a steady state potential and the remaining part. The decomposition method allows one to find a set of force fields each of which is compatible to a given steady state. Such a knowledge provides a useful insight on stochastic systems e…
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.
In this paper we revisit the notion of the "minus logarithm of stationary probability" as a generalized potential in nonequilibrium systems and attempt to illustrate its central role in an axiomatic approach to stochastic nonequilibrium thermodynamics of complex systems. It is demonstrated that this quantity arises naturally through both monotonicity results of Markov processes and as the rate function when a stochastic process approaches a deterministic limit. We then undertake a more detailed mathematical analysis of the consequences of this quantity, culminating in a necessary and sufficien…
Nonequlibrium phase transition of an open Takayama-Lin Liu-Maki chain coupled with two reservoirs is investigated. We will show that solitons connecting two uniform phases are possible, and the amplitude of solitons obeys the same self-consistent equation as that of uniform phases.
For systems in nonequilibrium steady states, a novel modulated Gaussian probability distribution is derived to incorporate a new phenomenon of biased current fluctuations, discovered by recent laboratory experiments and confirmed by molecular dynamics simulations. Our results consistently extend Onsager-Machlup fluctuation theory for systems in thermal equilibrium. Connections with the principles of Statistical Mechanics due to Boltzmann and Gibbs are discussed. At last, the modulated Gaussian distribution is of potential interest for other statistical disciplines, which make use of the Large …
Generic inhomogeneous steady states in an asymmetric exclusion process on a ring with a pair of point bottlenecks are studied. We show that, due to an underlying universal feature, measurements of coarse-grained steady-state densities in this model resolve the bottleneck structures only partially. Unexpectedly, it displays localization-delocalization transitions and confinement of delocalized domain walls, controlled by the interplay between particle number conservation and bottleneck competition for moderate particle densities.
Consistent thermodynamics reconstructed from transitions between nonequilibrium steady-states
Raymond & Beverly Sackler School of Chemistry, Tel Aviv University（特拉维夫大学雷蒙德和贝弗利萨克勒化学学院）；Princeton Center for Theoretical Science, Princeton University（普林斯顿大学普林斯顿理论科学中心）；Departamento de Física, Universidade Federal de Viçosa（维索萨联邦大学物理系）；Department of Physics, Humboldt-Universität zu Berlin（柏林洪堡大学物理系）；Bernstein Center for Computational Neuroscience（伯恩斯坦计算神经科学中心）；University of Strasbourg and CNRS, CESQ and ISIS, UMR 7006（斯特拉斯堡大学及法国国家科学研究中心CESQ和ISIS联合实验室）；Raymond & Beverly Sackler School of Physics and Astronomy, Tel Aviv University（特拉维夫大学雷蒙德和贝弗利萨克勒物理与天文学学院）
This paper derives a density matrix of the steady-state statistical mechanics compatible with the steady-state thermodynamics proposed by Oono and Paniconi [Prog. Theor. Phys. Suppl. {\bf 130}, 29 (1998)]. To this end, we adopt three plausible basic assumptions for uniform steady states: (i) equivalence between any two subsystems of the total, (ii) statistical independence between any two subsystems, and (iii) additivity of energy. With a suitable definition of energy, it is then shown that uniform steady states driven by mechanical forces may be described by the Gibbs distribution.
We simulate by lattice Boltzmann the steady shearing of a binary fluid mixture undergoing phase separation with full hydrodynamics in two dimensions. Contrary to some theoretical scenarios, a dynamical steady state is attained with finite domain lengths $L_{x,y}$ in the directions ($x,y)$ of velocity and velocity gradient. Apparent scaling exponents are estimated as $L_{x}\sim\dotγ^{-2/3}$ and $L_{y}\sim\dotγ^{-3/4}$. We discuss the relative roles of diffusivity and hydrodynamics in attaining steady state.
We address the question of whether transport coefficients obtained from a unitary closed system setting, i.e., the standard equilibrium Green-Kubo formula, are the same as the ones obtained from a weakly driven nonequilibrium steady-state calculation. We first derive a nonequilibrium Kubo-like expression for the steady-state diffusion constant expressed as a time-integral of either a current or a conserved density nonequilibrium correlation function. This expression has certain advantages over the equilibrium Green-Kubo formula, but is not clear if it gives the same value of the diffusion cons…
Excess work is a non-diverging part of the work during transition between nonequilibrium steady states (NESSs). It is a central quantity in the steady state thermodynamics (SST), which is a candidate for nonequilibrium thermodynamics theory. We derive an expression of excess work during quasistatic transitions between NESSs by using the macroscopic linear response relation of NESS. This expression is a line integral of a vector potential in the space of control parameters. We show a relationship between the vector potential and the response function of NESS, and thus obtain a relationship betw…
We study the nonequilibrium steady state (NESS) in a quantum system in contact with two heat baths at different temperatures. We use a time-independent perturbative expansion with respect to the coupling with the two heat baths to obtain the density matrix for the NESS. In particular, we show an explicit representation of the density matrix for the reflection symmetric and weakly nonequilibrium case. We also calculate the expectation value of the energy current and show that the Kubo formula holds in this case.
Thermodynamic quantities of the hard-sphere system in the steady state with a small heat flux are calculated within the continuous media approach. Analytical expressions for pressure, internal energy, and entropy are found in the approximation of the fourth order in temperature gradients. It is shown that the gradient contributions to the internal energy depend on the volume, while the entropy satisfies the second law of thermodynamics for nonequilibrium processes. The calculations are performed for dimensions 3D, 2D, and 1D.
We show that current fluctuations in a stochastic pump can be robustly mapped to fluctuations in a corresponding time-independent nonequilibrium steady state. We thus refine a recently proposed mapping so that it ensures equivalence of not only the averages, but also optimal representation of fluctuations in currents and density. Our mapping leads to a natural decomposition of the entropy production in stochastic pumps similar to the "housekeeping" heat. As a consequence of the decomposition of entropy production, the current fluctuations in weakly perturbed stochastic pumps are shown to satis…
We study nonequilibrium steady states (NESSs) in quantum spin-1/2 chains in contact with two heat baths at different temperatures. We consider the weak-coupling limit both for spin-spin coupling in the system and for system-bath coupling. This setting allows us to treat NESSs with a nonzero temperature gradient analytically. We develop a perturbation theory for this weak-coupling situation and show a simple condition for the existence of nonzero temperature gradient. This condition is independent of the integrability of the system.
We consider how the Lee-Yang description of phase transitions in terms of partition function zeros applies to nonequilibrium systems. Here one does not have a partition function, instead we consider the zeros of a steady-state normalization factor in the complex plane of the transition rates. We obtain the exact distribution of zeros in the thermodynamic limit for a specific model, the boundary-driven asymmetric simple exclusion process. We show that the distributions of zeros at the first and second order nonequilibrium phase transitions of this model follow the patterns known in the Lee-Yang…
The second law of ordinary thermodynamics and the second law of steady state thermodynamics, as proposed by Oono and Paniconi, are investigated from the microscopic point of view for the open quantum system. Based on the H-theorem of Lindblad, we show that both second laws are consistent with the quantum dynamics generated by the completely positive map. In addition, microscopic expressions of entropy production and ``housekeeping heat'' are obtained for some classes of dynamical 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 ar…
In equilibrium, the fluctuation-dissipation theorem (FDT) expresses the response of an observable to a small perturbation by a correlation function of this variable with another one that is conjugate to the perturbation with respect to \emph{energy}. For a nonequilibrium steady state (NESS), the corresponding FDT is shown to involve in the correlation function a variable that is conjugate with respect to \emph{entropy}. By splitting up entropy production into one of the system and one of the medium, it is shown that for systems with a genuine equilibrium state the FDT of the NESS differs from …
The understanding of the emergence of equilibrium statistical mechanics has progressed significantly thanks to developments from typicality, canonical and dynamical, and from the eigenstate thermalization hypothesis. Here we focus on a nonequilibrium scenario in which two nonintegrable systems prepared in different states are locally and non-extensively coupled to each other. Using both perturbative analysis and numerical exact simulations of up to 28 spin systems, we demonstrate the typical emergence of nonequilibrium (quasi-)steady current for weak coupling between the subsystems. We also id…
We examine stochastic processes that are used to model nonequilibrium processes (e.g, pulling RNA or dragging colloids) and so deliberately violate detailed balance. We argue that by combining an information-theoretic measure of irreversibility with nonequilibrium work theorems, the thermal physics implied by abstract dynamics can be determined. This measure is bounded above by thermodynamic entropy production and so may quantify how well a stochastic dynamics models reality. We also use our findings to critique various modeling approaches and notions arising in steady-state thermodynamics.
We calculate a nonequilibrium steady state of a quantum XX chain in the presence of dephasing and driving due to baths at chain ends. The obtained state is exact in the limit of weak driving while the expressions for one- and two-point correlations are exact for an arbitrary driving strength. In the steady state the magnetization profile and the spin current display diffusive behavior. Spin-spin correlation function on the other hand has long-range correlations which though decay to zero in either the thermodynamical limit or for equilibrium driving. At zero dephasing a nonequilibrium phase tr…
We explore driven lattice gases for the existence of an intensive thermodynamic variable which could determine "equilibration" between two nonequilibrium steady-state systems kept in weak contact. In simulations, we find that these systems satisfy surprisingly simple thermodynamic laws, such as the zero-th law and the fluctuation-response relation between the particle-number fluctuation and the corresponding susceptibility remarkably well. However at higher densities, small but observable deviations from these laws occur due to nontrivial contact dynamics and the presence of long-range spatial…
Nonequilibrium steady state of isothermal biochemical cycle kinetics has been extensively studied, but much less investigated under non-isothermal conditions. However, once the heat exchange between subsystems is rather slow, the isothermal assumption of the whole system meets great challenge, which is indeed the case inside many kinds of living organisms. Here we generalize the nonequilibrium steady-state theory of isothermal biochemical cycle kinetics, in the master-equation models, to the situation in which the temperatures of subsystems can be far from uniform. We first obtain a new thermo…
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 an…
We present a framework to determine nonequilibrium steady states in strongly correlated electron systems in the presence of dissipation. This is demonstrated for a correlated electron (Falicov-Kimball) model attached to a heat bath and irradiated by an intense pump light, for which an exact solution is obtained with the Floquet method combined with the nonequilibrium dynamical mean-field theory. On top of a Drude-like peak indicative of photometallization as observed in recent pump-probe experiments, new nonequilibrium phenomena are predicted to emerge, where the optical conductivity exhibits …
We provide a general macrostatistical formulation of nonequilibrium steady states of reservoir driven quantum systems. This formulation is centred on the large scale properties of the locally conserved hydrodynamical observables, and our basic assumptions comprise (a) a chaoticity hypothesis for the nonconserved currents carried by these observables, (b) an extension of Onsager's regression hypothesis to the fluctuations about nonequilibrium states, and (c) a certain mesoscopic local equilibrium hypothesis. On this basis we obtain a picture wherein the fluctuations of the hydrodynamical observ…
