Sampling the collective, dynamical fluctuations that lead to nonequilibrium pattern formation requires probing rare regions of trajectory space. Recent approaches to this problem based on importance sampling, cloning, and spectral approximations, have yielded significant insight into nonequilibrium systems, but tend to scale poorly with the size of the system, especially near dynamical phase transitions. Here we propose a machine learning algorithm that samples rare trajectories and estimates the associated large deviation functions using a many-body control force by leveraging the flexible function representation provided by deep neural networks, importance sampling in trajectory space, and stochastic optimal control theory. We show that this approach scales to hundreds of interacting particles and remains robust at dynamical phase transitions.
We investigate the properties of the nonequilibrium steady state for the stochastic system driven by a nonlinear drift force and influenced by noises which are not identically and independently distributed. The nonequilibrium steady state (NESS) current results from a residual part of the drift force which is not cancelled by the diffusive action of noises. From our previous study for the linear drift force the NESS current was found to circulate on the equiprobability surface with the maximum at a stable fixed point of the drift force. For the nonlinear drift force, we use the perturbation th…
In a recent paper [P. Strasberg and M. Esposito, Phys. Rev. E {\bf 101}, 050101(R) (2020)] an attempt is presented to formulate the nonequilibrium thermodynamics of an open system in terms of the Hamiltonian of mean force. The purpose of the present comment is to clarify severe restrictions of this approach and also to stress that recently noted ambiguities [P. Talkner and P. Hänggi, Phys. Rev. E {\bf 94}, 022143 (2016)] of fluctuating thermodynamic potentials cannot be removed in the suggested way.
The force on a probe induced by a nonequilibrium medium is in general nongradient. We detail the mechanism of that feature via nonequilibrium response theory. The emergence of nongradient forces is due to a systematic "twist" of the excess frenesy with respect to the entropy flux, in response to changes in the coupling or in the position of the probe in the nonequilibrium medium.
We study nonequilibrium fluctuation theorems in the presence of a time-reversal symmetry-breaking field and nonconservative forces, in a stochastic as well as a deterministic set up. We consider a system and a heat bath, called the combined system, and show that the fluctuation theorems are valid even when the heat bath goes out of equilibrium during driving. The only requirement for the validity is that, when the driving is switched off, the combined system relaxes to a state having a uniform probability measure on a constant energy surface, consistent with microcanonical ensemble of an isola…
We show that the extension of Casimir-like forces to fluctuating fluids driven out of equilibrium can exhibit two interrelated phenomena forbidden at equilibrium: self-forces can be induced on single asymmetric objects and the action--reaction principle between two objects can be violated. These effects originate in asymmetric restrictions imposed by the objects' boundaries on the fluid's fluctuations. They are not ruled out by the second law of thermodynamics since the fluid is in a nonequilibrium state. Considering a simple reaction--diffusion model for the fluid, we explicitly calculate the…
We refute the criticism expressed in a comment by P. Talkner and P. Hänggi [Phys. Rev. E 102, 066101 (2020)] on our paper Phys. Rev. E 101, 050101(R) (2020). We first make clear that our paper is free of any technical mistakes. We then defend the statements of our manuscript which were claimed to be taken out of context. Finally, we give clear arguments showing that the basic concepts we rely on are meaningful and have a deep rational.
We study the statistical force of a nonequilibrium environment on a quasi-static probe. In the linear regime the isothermal work on the probe equals the excess work for the medium to relax to its new steady condition with displaced probe. Also the relative importance of reaction paths can be measured via statistical forces, and from second order onwards the force on the probe reveals information about nonequilibrium changes in the reactivity of the medium. We also show that statistical forces for nonequilibrium media are generally nonadditive, in contrast with the equilibrium situation. Both t…
Nonequilibrium thermodynamics (NET) investigates processes in systems out of global equilibrium. On a mesoscopic level, it provides a statistical dynamic description of various complex phenomena such as chemical reactions, ion transport, diffusion, thermochemical, thermomechanical and mechanochemical fluxes. In the present review, we introduce a mesoscopic stochastic formulation of NET by analyzing entropy production in several simple examples. The fundamental role of nonequilibrium steady-state cycle kinetics is emphasized. The statistical mechanics of Onsager's reciprocal relations in this c…
A macroscopically uniform model of a two-dimensional electron system is proposed to study nonequilibrium properties of electrical conduction. By molecular dynamics simulation, the steady state distribution function $P_y$ of electron velocity in a direction perpendicular to an external driving force is calculated. An explicit form of $P_y$ is determined within the accuracy of the numerical simulation, which fits the numerical data well even in the regime where a local equilibrium description is not valid. Although the entire structure of $P_y$ is different from that of a local equilibrium distr…
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 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…
We present the reduced dynamics of a bead in a Rouse chain which is submerged in a bath containing a driving agent that renders it out-of-equilibrium. We first review the generalized Langevin equation of the middle bead in an equilibrated bath. Thereafter, we introduce two driving forces. Firstly, we add a constant force that is applied to the first bead of the chain. We investigate how the generalized Langevin equation changes due to this perturbation for which the system evolves towards a new equilibrium state after some time. Secondly, we consider the case of stochastic active forces which …
A classical particle system coupled with a thermostat driven by an external constant force reaches its steady state when the ensemble-averaged drift velocity does not vary with time. The statistical mechanics of such a system is derived merely based on the equal probability and ergodicity principles, free from any conclusions drawn on equilibrium statistical mechanics or local equilibrium hypothesis. The momentum space distribution is determined by a random walk argument, and the position space distribution is determined by employing the equal probability and ergodicity principles. The express…
We analyze within mean-field theory as well as numerically a KPZ equation that describes nonequilibrium wetting. Both complete and critical wettitng transitions were found and characterized in detail. For one-dimensional substrates the critical wetting temperature is depressed by fluctuations. In addition, we have investigated a region in the space of parameters (temperature and chemical potential) where the wet and nonwet phases coexist. Finite-size scaling analysis of the interfacial detaching times indicates that the finite coexistence region survives in the thermodynamic limit. Within this…
We study a nonequilibrium Langevin many-body system containing two 'test' particles and many 'background' particles. The test particles are spatially confined by a harmonic potential, and the background particles are driven by an external driving force. Employing numerical simulations of the model, we formulate an effective description of the two test particles in a nonequilibrium steady state. In particular, we investigate several different definitions of the effective force acting between the test particles. We find that the law of action and reaction does not hold for the total mechanical f…
Dense assemblies of self-propelled particles undergo a nonequilibrium form of glassy dynamics. Physical intuition suggests that increasing departure from equilibrium due to active forces fluidifies a glassy system. We falsify this belief by devising a model of self-propelled particles where increasing departure from equilibrium can both enhance or depress glassy dynamics, depending on the chosen state point. We analyze a number of static and dynamic observables and suggest that the location of the nonequilibrium glass transition is primarily controlled by the evolution of two-point static dens…
A thermodynamic system is driven into a nonequilibrium condition when a time-dependent force or a nonconservative force represented by a protocol $λ(t)$ is applied. Such a system is time irreversible in the sense that the motion under a protocol $λ(0\leq t\leq τ)$ and the time-reversed motion under $λ(τ-t)$ are not equally probable. We investigate a stochastic echo phenomenon in which a system is controlled by an optimal protocol to display the time-reversed motion. The optimal protocol is derived for a Langevin system. For the time-dependent driving case, simple harmonic oscillator systems ar…
The time-dependent work probability distribution function $P(W)$ is investigated analytically for a diffusing particle trapped by an anisotropic harmonic potential and driven by a nonconservative drift force in two dimensions. We find that the exponential tail shape of $P(W)$ characterizing rare-event probabilities undergoes a sequence of dynamic transitions in time. These remarkable locking-unlocking type transitions result from an intricate interplay between a rotational mode induced by the nonconservative force and an anisotropic decaying mode due to the conservative attractive force. We ex…
For a colloidal particle driven by a constant force across a periodic potential, we investigate the distribution of entropy production both experimentally and theoretically. For short trajectories, the fluctuation theorem holds experimentally. The mean entropy production rate shows two regimes as a function of the applied force. Theoretically, both mean and variance of the pronounced non-Gaussian distribution can be obtained from a differential equation in good agreement with the experimental data.
As a first approximation, the forces acting on optically trapped particles are commonly assumed to be conservative. The influence of the nonconservative force has been shown to be negligible in overdamped liquid environments. However, its impact in the underdamped regime remains unexplored. Here, we experimentally study the combined effects of gradient and scattering forces on the nonlinear dynamics of trapped particles at various air pressures, leading to significant low-frequency noise excess along the optical axis and the emergence of toroidal Brownian vortices. Our results supported, by nu…
The standard formulation of tunneling transport rests on an open-boundary modeling. There, conserving approximations to nonequilibrium Green function or quantum-statistical mechanics provide consistent but computational costly approaches; alternatively, use of density-dependent ballistic-transport calculations [e.g., Phys. Rev. B 52, 5335 (1995)], here denoted `DBT', provide computationally efficient (approximate) atomistic characterizations of the electron behavior but has until now lacked a formal justification. This paper presents an exact, variational nonequilibrium thermodynamic theory fo…
We study the nonequilibrium steady state realized in a general stochastic system attached to multiple heat baths and/or driven by an external force. 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 $η$ is proportional to $\exp[Φ(η)]$ where $Φ(η)=-\sum_kβ_k\mathcal{E}_k(η)$ is the excess entropy change. Here $\mathcal{E}_k(η)$ is the difference between two kinds of conditioned path ensemble averages of excess h…
We study the induced interaction between multiple probes locally interacting with driven colloids and trapped in a toroidal geometry. The effective binary forces between the probes break the action-reaction principle and their range decreases with the driving. We demonstrate how in the stationary nonlinear nonequilibrium regime these interactions induce stability of a crystal-like pattern, where the probes are equidistant, when the probe-colloid interaction is either completely attractive or completely repulsive.
We discuss the physical meaning and significance of statistical forces on quasi-static probes in first order around detailed balance for driven media. Exploiting the quasi-static energetics and the structure of (McLennan) steady nonequilibrium ensembles, we find that the statistical force obtains a nonequilibrium correction deriving from the excess work of driving forces on the medium in its relaxation after probe displacement. This reformulates, within a more general context, the recent result by N. Nakagawa (Phys. Rev. E 90, 022108 (2014)) on thermodynamic aspects of weakly nonequilibrium ad…
The concept of work is studied in quantum thermostatistics of a system surrounded by an environment and driven by an external force. It is found that there emerges the gauge theoretical structure in a nonequilibrium process, the field of which is referred to as the work gauge field. The thermodynamic work as the flux of the work gauge field is considered for a cyclic process in the space of the external-force parameters. As an example, the system of a spin-1/2 interacting with an external magnetic field is analyzed. This geometric effect may be observed, for example, in an NMR experiment and c…
Long-range thermal fluctuations appear in fluids in nonequilibrium states leading to fluctuation-induced Casimir-like forces. Two distinct mechanisms have been identified for the origin of the long-range nonequilibrium fluctuations in fluids subjected to a temperature or concentration gradient. One is a coupling between the heat or mass-diffusion mode with a viscous mode in fluids subjected to a temperature or concentration gradient. Another one is the spatial inhomogeneity of thermal noise in the presence of a gradient. We show that in fluids fluctuation-induced forces arising from mode coupl…
The nonequilibrium thermodynamics of an open (classical or quantum) system in strong contact with a single heat bath can be conveniently described in terms of the Hamiltonian of mean force. However, the conventional formulation is limited by the necessity to measure differences in equilibrium properties of the system-bath composite. We make use of the freedom involved in defining thermodynamic quantities, which leaves the thermodynamics unchanged, to show that the Hamiltonian of mean force can be inferred from measurements on the system alone, up to that irrelevant freedom. In doing so, we ref…
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…
We investigate a novel type of Langevin model that describes the nonequilibrium dynamics of a classical particle interacting with a spatially extended environment. In this model, a particle, which interacts with the environment through the nonlinear interaction Hamiltonian, is driven by a constant external force, and subsequently, it reaches a nontrivial nonequilibrium steady state. We derive an effective Langevin equation for the particle in the nonequilibrium steady states. Using this equation, we calculate the effective temperature defined as the ratio of the correlation function of the vel…
We calculate the nonequilibrium mean-field 'temperature' of a Brownian system in contact with a heat bath. We consider two different cases: an equilibrium bath in the presence of strong external forces and a nonequilibrium bath. By proving the existence of a generalized fluctuation-dissipation relation this mean-field 'temperature' can be used to describe a nonequilibrium system as is if it were in thermal equilibrium with a thermal bath at the mean-field 'temperature' mentioned above. We apply our results to chemical reactions in the presence of external forces showing how chemical equilibriu…
We consider the motion of an overdamped particle in a force field in presence of an external, adiabatic noise, without the restriction that the noise process is Gaussian or the stochastic process is Markovian. We examine the condition for attainment of steady state for this nonequilibrium open system and calculate the adiabatic noise-induced rate of escape of the particle over a barrier.
We study long-range interacting systems driven by external stochastic forces that act collectively on all the particles constituting the system. Such a scenario is frequently encountered in the context of plasmas, self-gravitating systems, two-dimensional turbulence, and also in a broad class of other systems. Under the effect of stochastic driving, the system reaches a stationary state where external forces balance dissipation on average. These states have the invariant probability that does not respect detailed balance, and are characterized by non-vanishing currents of conserved quantities.…
Using an analytically tractable lattice model for reaction-diffusion processes of hard-core particles we demonstrate that under nonequilibrium conditions phase coexistence may arise even if the system is effectively one-dimensional as e.g. in the channel system of some zeolites or in artificial optical lattices. In our model involving two species of particles a steady-state particle current is maintained by a density gradient between the channel boundaries and by the influence of an external driving force. This leads to the development of a fluctuating but always microscopically sharp interfac…
We present a fundamental classification of forces relevant in nonequilibrium structure formation under collective flow in Brownian many-body systems. The internal one-body force field is systematically split into contributions relevant for the spatial structure and for the coupled motion. We demonstrate that both contributions can be obtained straightforwardly in computer simulations, and present a power functional theory that describes all types of forces quantitatively. Our conclusions and methods are relevant for flow in inertial systems, such as molecular liquids and granular media.
We introduce a variational algorithm to estimate the likelihood of a rare event within a nonequilibrium molecular dynamics simulation through the evaluation of an optimal control force. Optimization of a control force within a chosen basis is made possible by explicit forms for the gradients of a cost function in terms of the susceptibility of driven trajectories to changes in variational parameters. We consider probabilities of time-integrated dynamical observables as characterized by their large deviation functions, and find that in many cases the variational estimate is quantitatively accur…
The question of deriving general force/flux relationships that apply out of the linear response regime is a central topic of theories for nonequilibrium statistical mechanics. This work applies an information theory perspective to compute approximate force/flux relations and compares the result with traditional alternatives. If it can be said that there is a consensus on the form of response theories in driven, nonequilibrium transient dynamics, then that consensus is consistent with maximizing the entropy of a distribution over transition space. This agreement requires the problem of force/fl…
We investigate the nature of the effective dynamics and statistical forces obtained after integrating out nonequilibrium degrees of freedom. To be explicit, we consider the Rouse model for the conformational dynamics of an ideal polymer chain subject to steady driving. We compute the effective dynamics for one of the many monomers by integrating out the rest of the chain. The result is a generalized Langevin dynamics for which we give the memory and noise kernels and the effective force, and we discuss the inherited nonequilibrium aspects.
Nonequilibrium thermodynamics of a general second-order stochastic system is investigated. We prove that at steady state, under inversion of velocities, the condition of time-reversibility over the phase space is equivalent to the antisymmetry of spatial flux and the symmetry of velocity flux. Then we show that the condition of time-reversibility alone could not always guarantee the Maxwell-Boltzmann distribution. Comparing the two conditions together, we found that the frictional force naturally emerges as the unique odd term of the total force at thermodynamic equilibrium, and is followed by…
The study of stochastic systems has received considerable interest over the years. Their dynamics can describe many equilibrium and nonequilibrium fluctuating systems. At the same time, nonequilibrium constraints interact with the time evolution in various ways. Here we review the dynamics of stochastic systems from the viewpoint of nonequilibrium thermodynamics. We explore the effect of external thermodynamic forces on the possible dynamical regimes and show that the time evolution can become intrinsically different under nonequilibrium conditions. For example, nonequilibrium systems with rea…
In net-neutral systems correlations between charge fluctuations generate strong attractive thermal Casimir forces and engineering these forces to optimize nanodevice performance is an important challenge. We show how the normal and lateral thermal Casimir forces between two plates containing Brownian charges can be modulated by decorrelating the system through the application of an electric field, which generates a nonequilibrium steady state with a constant current in one or both plates, reducing the ensuing fluctuation-generated normal force while at the same time generating a lateral drag f…
In stationary nonequilibrium states a coupling between hydrodynamic modes causes thermal fluctuations to become long ranged inducing nonequilibrium Casimir forces or pressures. Here we consider nonequilibrium Casimir pressures induced in liquids by a velocity gradient. Specifically, we have obtained explicit expressions for the magnitude of the shear-induced pressure enhancement in a liquid layer between two horizontal plates that complete and correct results previously presented in the literature. In contrast to nonequiibrium Casimir pressures induced by a temperature gradient, kinetic theory…
In this paper we presented an overview on our works. More than ten years ago, we proposed a new fundamental equation of nonequilibrium statistical physics in place of the present Liouville equation. That is the stochastic velocity type's Langevin equation in 6N dimensional phase space or its equivalent Liouville diffusion equation. This equation is time-reversed asymmetrical. It shows that the form of motion of particles in statistical thermodynamic systems has the drift-diffusion duality, and the law of motion of statistical thermodynamics is expressed by a superposition of both the law of dy…
External biasing forces are often applied to enhance sampling in regions of phase space which would otherwise be rarely observed. While the typical goal of these experiments is to calculate the potential of mean force (PMF) along the biasing coordinate, here I present a method to construct PMFs in multiple dimensions and along arbitary alternative degrees of freedom. A protocol for multidimensional PMF reconstruction from nonequilibrium single-molecule pulling experiments is introduced and tested on a series of two dimensional potential surfaces with varying levels of correlation. Reconstructi…
We present a Monte Carlo study of a lattice gas driven out of equilibrium by a local hopping bias. Sites can be empty or occupied by one of two types of particles, which are distinguished by their response to the hopping bias. All particles interact via excluded volume and a nearest-neighbor attractive force. The main result is a phase diagram with three phases: a homogeneous phase, and two distinct ordered phases. Continuous boundaries separate the homogeneous phase from the ordered phases, and a first-order line separates the two ordered phases. The three lines merge in a nonequilibrium bicr…
The time-reversal symmetry of nonequilibrium fluctuations is experimentally investigated in two out-of-equilibrium systems namely, a Brownian particle in a trap moving at constant speed and an electric circuit with an imposed mean current. The dynamical randomness of their nonequilibrium fluctuations is characterized in terms of the standard and time-reversed entropies per unit time of dynamical systems theory. We present experimental results showing that their difference equals the thermodynamic entropy production in units of Boltzmann's constant.
We establish a general theory of feedback control on classical stochastic thermodynamic systems, and generalize nonequilibrium equalities such as the fluctuation theorem and the Jarzynski equality in the presence of feedback control with multiple measurements. Our results are generalizations of the previous relevant works to the situations with general measurements and multi-heat baths. The obtained equalities involve additional terms that characterize the information obtained by measurements or the efficacy of feedback control. A generalized Szilard engine and a feedback-controlled ratchet ar…
As a rough model for the collective motions of cells and organisms we develop here the statistical mechanics of swarms of self-propelled particles. Our approach is closely related to the recently developed theory of active Brownian motion and the theory of canonical-dissipative systems. Free motion and motion of a swarms confined in an external field is studied. Briefly the case of particles confined on a ring and interacting by repulsive forces is studied. In more detail we investigate self-confinement by Morse-type attracting forces. We begin with pairs N = 2; the attractors and distribution…
We demonstrate the power of 2D tensor networks for obtaining large deviation functions of dynamical observables in a classical nonequilibrium setting. Using these methods, we analyze the previously unstudied dynamical phase behavior of the fully 2D asymmetric simple exclusion process with biases in both the x and y directions. We identify a dynamical phase transition, from a jammed to a flowing phase, and characterize the phases and the transition, with an estimate of the critical point and exponents.
The separation of substances into different phases is ubiquitous in nature and important scientifically and technologically. This phenomenon may become drastically different if the species involved, whether molecules or supramolecular assemblies, interconvert. In the presence of an external force large enough to overcome energetic differences between the interconvertible species (forced interconversion), the two alternative species will be present in equal amounts, and the striking phenomenon of steady-state, restricted phase separation into mesoscales is observed. Such microphase separation i…
We investigate motility-induced phase separation of active Brownian particles, which are modeled as purely repulsive spheres that move due to a constant swim force with freely diffusing orientation. We develop on the basis of power functional concepts an analytical theory for nonequilibrium phase coexistence and interfacial structure. Theoretical predictions are validated against Brownian dynamics computer simulations. We show that the internal one-body force field has four nonequilibrium contributions: (i) isotropic drag and (ii) interfacial drag forces against the forward motion, (iii) a sup…
Chemical transformations, such as ion exchange, are commonly employed to modify nanocrystal compositions. Yet the mechanisms of these transformations, which often operate far from equilibrium and entail mixing diverse chemical species, remain poorly understood. Here, we explore an idealized model for ion exchange in which a chemical potential drives compositional defects to accumulate at a crystal's surface. These impurities subsequently diffuse inward. We find that the nature of interactions between sites in a compositionally impure crystal strongly impacts exchange trajectories. In particula…
Most computer simulations of molecular dynamics take place under equilibrium conditions--in a closed, isolated system, or perhaps one held at constant temperature or pressure. Sometimes, extra tensions, shears, or temperature gradients are introduced to those simulations to probe one type of nonequilibrium response to external forces. Catalysts and molecular motors, however, function based on the nonequilibrium dynamics induced by a chemical reaction's thermodynamic driving force. In this scenario, simulations require chemostats capable of preserving the chemical concentrations of the nonequil…
When a flexible polymer is sucked into a localized small hole, the chain can initially respond only locally and the sequential nonequilibrium processes follow in line with the propagation of the tensile force along the chain backbone. We analyze this dynamical process by taking the nonuniform stretching of the polymer into account both with and without hydrodynamics interactions. Earlier conjectures on the absorption time are criticized and new formulae are proposed together with time evolutions of relevant dynamical variables.
