Nonequilibrium Stationary State Research Articles

Entropic analysis of bistability and the general evolution criterion.

We present a detailed study of the entropy production, the entropy exchange and the entropy balance for the Schlögl model of chemical bi-stability for both the clamped and volumetric open-flow versions. The general evolution criterion (GEC) is validated for the transitions from the unstable to the stable non-equilibrium stationary states. The GEC is the sole theorem governing the temporal behavior of the entropy production in non-equilibrium thermodynamics, and we find no evidence for supporting a "principle" of maximum entropy production. We use stoichiometric network analysis (SNA) to calculate the distribution of the entropy production and the exchange entropy over the elementary flux modes of the clamped and open-flow models, and aim to reveal the underlying mechanisms of dissipation and entropy exchange.

Convergence of the likelihood ratio method for linear response of non-equilibrium stationary states

We consider numerical schemes for computing the linear response of steady-state averages with respect to a perturbation of the drift part of the stochastic differential equation. The schemes are based on the Girsanov change-of-measure theory in order to reweight trajectories with factors derived from a linearization of the Girsanov weights. The resulting estimator is the product of a time average and a martingale correlated to this time average. We investigate both its discretization and finite-time approximation errors. The designed numerical schemes are shown to be of a bounded variance with respect to the integration time which is desirable feature for long time simulations. We also show how the discretization error can be improved to second-order accuracy in the time step by modifying the weight process in an appropriate way.

Multiscale kinetic theory for heterogeneous granular and gas-solid flows

It has been recognized that the particle phase stress model derived from classical kinetic theory is valid only when sufficient scale resolution is offered to explicitly resolve the heterogeneous structures during numerical simulations, however, industrial applications prefer to use coarse computational grids where the heterogeneous structures are not explicitly resolved but implicitly modeled. Unfortunately, in this case a kinetic theory for heterogeneous granular and gas–solid flows is not available yet. To this end, an attempt was made to tracking this challenge: The single particle velocity distribution function at the nonequilibrium stationary state with heterogeneous structures was firstly derived by combining the idea of doubly stochastic Poisson processes or superstatistics with the concept of compromise in competition in the EMMS (Energy Minimization Multi-Scale) theory, the standard Chapman-Enskog method was then used to develop the constitutive relations of heterogeneous continuum theory. It was found that (i) seven state variables are needed to quantify the heterogeneous structures as compared to three for homogeneous systems; (ii) the constitutive relations not only include the contribution from microscale particle–particle interactions but also those due to the interactions between mesoscale structures; and (iii) the resultant constitutive relations are much more complex than those of homogeneous systems due to the simultaneous consideration of microscale and mesoscale contributions and the appearance of cross-coupling effects, but they correctly contain the constitutive relations of homogeneous systems as a limiting case. Finally, the theory was coupled with an EMMS drag model to offer a preliminary validation and to provide a unified EMMS-based constitutive relations for heterogeneous gas–solid flows.

Exact solution of the Floquet-PXP cellular automaton.

We study the dynamics of a bulk deterministic Floquet model, the Rule 201 synchronous one-dimensional reversible cellular automaton (RCA201). The system corresponds to a deterministic, reversible, and discrete version of the PXP model, whereby a site flips only if both its nearest neighbors are unexcited. We show that the RCA201 (Floquet-PXP) model exhibits ballistic propagation of interacting quasiparticles-or solitons-corresponding to the domain walls between nontrivial threefold vacuum states. Starting from the quasiparticle picture, we find the exact matrix product state form of the nonequilibrium stationary state for a range of boundary conditions, including both periodic and stochastic. We discuss further implications of the integrability of the model.

Convergence to stationary non-equilibrium states for Klein–Gordon equations

We consider Klein–Gordon equations in , , with constant or variable coefficients and study the Cauchy problem with random initial data. We investigate the distribution of a random solution at moments of time . We prove the convergence of correlation functions of the measure to a limit as . The explicit formulae for the limiting correlation functions and the energy current density (in mean) are obtained in terms of the initial covariance. Furthermore, we prove the weak convergence of to a limiting measure as . We apply these results to the case when the initial random function has the Gibbs distribution with different temperatures in some infinite “parts” of the space. In this case, we find states in which the limiting energy current density does not vanish. Thus, for the model being studied, we construct a new class of stationary non-equilibrium states.

Nonequilibrium phase transformations in alloys under severe plastic deformation

Experimental results and theoretical concepts of anomalous phase transformations in alloys under severe plastic deformation (SPD) are reviewed. The unconventional phase and structural state of alloys that emerges as a result of SPD determines the unique combination of physical and chemical properties that is of interest for various applications in technology. The driving forces and possible mechanisms that implement the anomalous transformations as a function of SPD intensity, alloy composition, and temperature are discussed. We distinguish among these mechanisms two fundamental and qualitatively different ones that are due to: (i) direct ‘mixing’ of atoms in slip bands or (ii) accelerated diffusion on defects under locally alternating thermodynamic conditions and subsequent ‘freezing’ of the nonequilibrium state that is reached. Summarizing the experimental data and theoretical concepts regarding the change in microscopic transformation mechanisms as a function of temperature and/or intensity of treatment, we suggest a diagram of nonequilibrium stationary states attainable during the development of phase and structural instability under SPD. The proposed approach enables the prediction of the structural state of alloys and compounds by controlling thermodynamic and kinetic parameters of the system.

Dissipative generation of pure steady states and a gambler's ruin problem

We consider an open quantum system, with dissipation applied only to a part of its degrees of freedom, evolving via a quantum Markov dynamics. We demonstrate that, in the Zeno regime of large dissipation, the relaxation of the quantum system towards a pure quantum state is linked to the evolution of a classical Markov process towards a single absorbing state. The rates of the associated classical Markov process are determined by the original quantum dynamics. Extension of this correspondence to absorbing states with internal structure allows us to establish a general criterion for having a Zeno-limit nonequilibrium stationary state of arbitrary finite rank. An application of this criterion is illustrated in the case of an open XXZ spin-1/2 chain dissipatively coupled at its edges to baths with fixed and different polarizations. For this system, we find exact nonequilibrium steady-state solutions of ranks 1 and 2.

Random acceleration process under stochastic resetting

We consider the motion of a randomly accelerated particle in one dimension under stochastic resetting mechanism. Denoting the position and velocity by x and v respectively, we consider two different resetting protocols—(i) complete resetting: here both x and v reset to their initial values x 0 and v 0 at a constant rate r, (ii) partial resetting: here only x resets to x 0 while v evolves without interruption. For complete resetting, we find that the particle attains stationary state in both x and v. We compute the non-equilibrium joint stationary state of x and v from which we obtain the stationary state distribution for position by integrating over v. We also study the late time relaxation of the position distribution function. On the other hand, for partial resetting, the joint distribution is always in the transient state. At large t, the position distribution possesses a scaling behaviour which we rigorously derive. Next, we study the first passage time properties with an absorbing wall at the origin. For complete resetting, we find that the mean first passage time (MFPT) is rendered finite by the resetting mechanism. We explicitly derive the expressions for the MFPT and the survival probability at large t. However, in stark contrast, for partial resetting, we find that resetting does not render finite MFPT. This is because even though x is brought to x 0, the large fluctuation in v (typically of the order ) can take the particle substantially far from the origin. All our analytic results are corroborated by the numerical simulations.

A Note on the Nonequilibrium Stationary State in Continuous-Activity Thermostatted Models

A Note on the Nonequilibrium Stationary State in Continuous-Activity Thermostatted Models

Quasiequilibrium self-gravitating systems

Because of the long-range nature of the gravitational interaction, self-gravitating systems never reach thermal equilibrium in the thermodynamic limit but remain trapped in nonequilibrium stationary states, or quasiequilibrium states. Here, we deal with quasiequilibrium self-gravitating systems by representing them as a collection of smaller subsystems that remain infinitely close to equilibrium. These subsystems represent regions from where thermalization spreads later over the whole system. Such a methodological attitude allows representing their statistical properties as a superposition of statistics, i.e., superstatistics. It has the advantage of producing Tsallis distributions, widely used in fitting observational data, as a special case of a more general family of distributions, while relying only on conventional statistical mechanics. Focusing on the three universality classes of superstatistics, namely, ${\ensuremath{\chi}}^{2}$, inverse-${\ensuremath{\chi}}^{2}$, and log-normal superstatistics, we discuss the velocity distributions arising in this picture and confront them with independent numerical simulations. Then, we study the consequences on typical phenomena arising in self-gravitating systems. We discuss the Jeans instability in the classical regime, for a static and an expanding universe, and extend our results to the quantum regime by applying the Wigner-Moyal procedure. Our results reveal that quasiequilibrium systems remain stable for larger perturbations, as compared to equilibrium systems, meaning that a larger mass is needed to initiate the gravitational collapse. This is particularly relevant for Bok globules because their mass is of the same order as their Jeans mass; hence, a small deviation from equilibrium may lead to a different prediction for their stability. We also discuss the Chandrasekhar dynamical friction in a quasiequilibrium medium and analyze the consequences on the decay of globular orbits. Our results suggest that the superstatistical picture may offer a partial solution to the problem of the large timescales shown by numerical $N$-body simulations and semianalytical models.

Bounding energy growth in frictionless stochastic oscillators.

This paper presents analytical and numerical results on the energetics of nonharmonic, undamped, single-well, stochastic oscillators driven by additive Gaussian white noises. The absence of damping and the action of noise are responsible for the lack of stationary states in such systems. We explore the properties of average kinetic, potential, and total energies along with the generalized equipartition relations. It is demonstrated that in frictionless dynamics, nonequilibrium stationary states can be produced by stochastic resetting. For an appropriate resetting protocol, the average energies become bounded. If the resetting protocol is not characterized by a finite variance of renewal intervals, stochastic resetting can only slow down the growth of the average energies but it does not bound them. Under special conditions regarding the frequency of resets, the ratios of the average energies follow the generalized equipartition relations.

Ising model with stochastic resetting

We study the stationary properties of the Ising model that, while evolving towards its equilibrium state at temperature $T$ according to the Glauber dynamics, is stochastically reset to its fixed initial configuration with magnetisation $m_0$ at a constant rate $r$. Resetting breaks detailed balance and drives the system to a non-equilibrium stationary state where the magnetisation acquires a nontrivial distribution, leading to a rich phase diagram in the $(T,r)$ plane. We establish these results exactly in one-dimension and present scaling arguments supported by numerical simulations in two-dimensions. We show that resetting gives rise to a novel "pseudo-ferro" phase in the $(T,r)$ plane for $r > r^*(T)$ and $T>T_c$ where $r^*(T)$ is a crossover line separating the pseudo-ferro phase from a paramagnetic phase. This pseudo-ferro phase is characterised by a non-zero typical magnetisation and a vanishing gap near $m=0$ of the magnetisation distribution.

Generating a nonequilibrium stationary state from a ground-state condensate through an almost adiabatic cycle

It is shown that the ground state of weakly interacting Bose particles in a quasi one-dimensional box trap can be converted into an excited stationary state by an adiabatic cyclic operation that involves a quench of interaction strength: A sharp impurity potential is applied and its strength is varied during the cycle, which induces a nonequilibrium stationary state exhibiting the inversion of population. This process is robust in the sense that the resultant stationary state is almost independent of the details of the cycle, such as the position of the impurity as long as the cycle is far enough from critical regions. The case of the failure of the population inversion due to the strong interparticle interactions is also examined.

Uncoupled Majorana fermions in open quantum systems: on the efficient simulation of non-equilibrium stationary states of quadratic Fermi models

A decomposition of the non-equilibrium stationary state of a quadratic Fermi system influenced by linear baths is obtained and used to establish a simulation protocol in terms of tensor states. The scheme is then applied to examine the occurrence of uncoupled Majorana fermions in Kitaev chains subject to baths on the ends. The resulting phase diagram is compared against the topological characterization of the equilibrium chain and the protocol efficiency is studied with respect to this model.

On Stationary Nonequilibrium Measures for Wave Equations

In the paper, the Cauchy problem for wave equations with constant and variable coefficients is considered. We assume that the initial data are a random function with finite mean energy density and study the convergence of distributions of the solutions to a limiting Gaussian measure for large times. We derive the formulas for the limiting energy current density (in mean) and find a new class of stationary nonequilibrium states for the studied model.

Constructing auxiliary dynamics for nonequilibrium stationary states by variance minimization

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 the interacting driven diffusive systems where the efficiency is systematically improved by incorporating higher correlations into the GDF. On the lattice, we use a correlator product state ansatz to study the 1D weakly asymmetric simple exclusion process. We show that with modest resources, we can capture the features of the susceptibility in large systems that mark the phase transition from uniform transport to a traveling wave state. Our work extends the repertoire of tools available to study nonequilibrium properties in realistic systems.

Approach to equilibrium and nonequilibrium stationary distributions of interacting many-particle systems that are coupled to different heat baths.

A Hamiltonian-based model of many harmonically interacting massive particles that are subject to linear friction and coupled to heat baths at different temperatures is used to study the dynamic approach to equilibrium and nonequilibrium stationary states. An equilibrium system is here defined as a system whose stationary distribution equals the Boltzmann distribution, the relation of this definition to the conditions of detailed balance and vanishing probability current is discussed both for underdamped as well as for overdamped systems. Based on the exactly calculated dynamic approach to the stationary distribution, the functional that governs this approach, which is called the free entropy S_{free}(t), is constructed. For the stationary distribution S_{free}(t) becomes maximal and its time derivative, the free entropy production S[over ̇]_{free}(t), is minimal and vanishes. Thus, S_{free}(t) characterizes equilibrium as well as nonequilibrium stationary distributions by their extremal and stability properties. For an equilibrium system, i.e., if all heat baths have the same temperature, the free entropy equals the negative free energy divided by temperature and thus corresponds to the Massieu function which was previously introduced in an alternative formulation of statistical mechanics. Using a systematic perturbative scheme for calculating velocity and position correlations in the overdamped massless limit, explicit results for few particles are presented: For two particles localization in position and momentum space is demonstrated in the nonequilibrium stationary state, indicative of a tendency to phase separate. For three elastically interacting particles heat flows from a particle coupled to a cold reservoir to a particle coupled to a warm reservoir if the third reservoir is sufficiently hot. This does not constitute a violation of the second law of thermodynamics, but rather demonstrates that a particle in such a nonequilibrium system is not characterized by an effective temperature which equals the temperature of the heat bath it is coupled to. Active particle models can be described in the same general framework, which thereby allows us to characterize their entropy production not only in the stationary state but also in the approach to the stationary nonequilibrium state. Finally, the connection to nonequilibrium thermodynamics formulations that include the reservoir entropy production is discussed.

Mathematical Analysis of a Thermostatted Equation with a Discrete Real Activity Variable

This paper deals with the mathematical analysis of a thermostatted kinetic theory equation. Specifically, the assumption on the domain of the activity variable is relaxed allowing for the discrete activity to attain real values. The existence and uniqueness of the solution of the related Cauchy problem and of the related non-equilibrium stationary state are established, generalizing the existing results.

Spontaneous mirror symmetry breaking: an entropy production survey of the racemate instability and the emergence of stable scalemic stationary states

We study the emergence of both stable and unstable non-equilibrium stationary states (NESS), as well as spontaneous mirror symmetry breaking (SMSB) provoked by the destabilization of the racemic thermodynamic branch, for an enantioselective autocatalytic reaction network in an open flow system, and for a continuous range n of autocatalytic orders. The system possesses a range of double bi-stability and also tri-stability depending on the autocatalytic order. We carry out entropy production and entropy flow calculations, from simulations of ordinary differential equations, stoichiometric network analysis (SNA), and consider a stability analysis of the NESS. The simulations provide a correct description of the relationship between energy state functions, the isothermal dissipated heat, entropy production and entropy flow exchange with the surroundings, and the correct solution of the balance of the entropy currents at the NESS. The validity of the General Evolution Criterion (GEC) is in full agreement with all the dynamic simulations.

Classical and quantum stochastic thermodynamics

The stochastic thermodynamics provides a framework for the description of systems that are out of thermodynamic equilibrium. It is based on the assumption that the elementary constituents are acted by random forces that generate a stochastic dynamics, which is here represented by a Fokker-Planck-Kramers equation. We emphasize the role of the irreversible probability current, the vanishing of which characterizes the thermodynamic equilibrium and yields a special relation between fluctuation and dissipation. The connection to thermodynamics is obtained by the definition of the energy function and the entropy as well as the rate at which entropy is generated. The extension to quantum systems is provided by a quantum evolution equation which is a canonical quantization of the Fokker-Planck-Kramers equation. An example of an irreversible systems is presented which shows a nonequilibrium stationary state with an unceasing production of entropy. A relationship between the fluxes and the path integral is also presented.