Nonequilibrium Stationary State Research Articles

Scaling limits of random walks, harmonic profiles, and stationary nonequilibrium states in Lipschitz domains

Scaling limits of random walks, harmonic profiles, and stationary nonequilibrium states in Lipschitz domains

Transition path theory for diffusive search with stochastic resetting

Many chemical reactions can be formulated in terms of particle diffusion in a complex energy landscape. Transition path theory (TPT) is a theoretical framework for describing the direct (reaction) pathways from reactant to product states within this energy landscape, and calculating the effective reaction rate. It is now the standard method for analyzing rare events between long lived states. In this paper, we consider a completely different application of TPT, namely, a dual-aspect diffusive search process in which a particle alternates between collecting cargo from a source domain A and then delivering it to a target domain B. The rate of resource accumulation at the target, k AB , is determined by the statistics of direct (reactive or transport) paths from A to B. Rather than considering diffusion in a complex energy landscape, we focus on pure diffusion with stochastic resetting. Resetting introduces two non-trivial problems in the application of TPT. First, the process is not time-reversal invariant, which is reflected by the fact that there exists a unique non-equilibrium stationary state (NESS). Second, calculating k AB involves determining the total probability flux of direct transport paths across a dividing surface S between A and B. This requires taking into account discontinuous jumps across S due to resetting. We derive a general expression for k AB and show that it is independent of the choice of dividing surface. Finally, using the example of diffusion in a finite interval, we show that there exists an optimal resetting rate at which k AB is maximized. We explore how this feature depends on model parameters.

Temperature inversion in a gravitationally bound plasma: Case of the solar corona

The temperature of the solar atmosphere increases from thousands to millions of degrees moving from the lower layer (chromosphere) to the outermost one (corona), while the density drops accordingly. The mechanism behind this phenomenon, known as a temperature inversion, is still unknown. In this work, we model a coronal loop as a collisionless plasma confined in a semicircular tube that is subject to the Sun’s gravity and in thermal contact with a fully collisional chromosphere behaving as a thermostat at the loop’s feet. By using kinetic N-particle simulations and analytical calculations, we show that rapid, intermittent, and short-lived heating events in the chromosphere drive the coronal plasma towards a non-equilibrium stationary state. The latter is characterized by suprathermal tails in the particles’ velocity distribution functions, exhibiting temperature and density profiles strikingly similar to those observed in the atmosphere of the Sun. These results suggest that a million-Kelvin solar corona can be produced without the local deposition of heat in the upper layer of the atmosphere that is typically assumed by standard approaches. We find that suprathermal distribution functions in the corona are self-consistently produced instead of postulated a priori, in contrast to classical kinetic models based on a velocity filtration mechanism.

Extending the thermodynamic form factor bootstrap program: multiple particle-hole excitations, crossing symmetry, and reparameterization invariance

In this study, we further the thermodynamic bootstrap program which involves a set of recently developed ideas used to determine thermodynamic form factors of local operators in integrable quantum field theories. These form factors are essential building blocks for dynamic correlation functions at finite temperatures or non-equilibrium stationary states. In this work we extend this program in three ways. Firstly, we demonstrate that the conjectured annihilation pole axiom is valid in the low energy particle-hole excitations. Secondly, we introduce a crossing relation, which establishes a connection between form factors with different excitation content. Typically, the crossing relation is a consequence of Lorentz invariance, but due to the finite energy density of the considered states, Lorentz invariance is broken. Nonetheless a crossing relation involving excitations with both particles and holes can established using the finite volume representation of the thermodynamic form factors. Finally, we demonstrate that the thermodynamic form factors satisfy a reparameterization invariance, an invariance which encompasses crossing. Reparameterization invariance exploits the fact that the details of the representation of the thermodynamic state are unimportant. In the course of developing these results, we demonstrate the internal consistency of the thermodynamic form factor bootstrap program in a number of ways. Finally, we provide explicit computations of form factors of conserved charges and densities with crossed excitations and show our results can be used to infer information about thermodynamic form factors in the Lieb-Liniger model.

Ornstein–Uhlenbeck Process on Three-Dimensional Comb under Stochastic Resetting

The Ornstein–Uhlenbeck (O-U) process with resetting is considered as the anomalous transport taking place on a three-dimensional comb. The three-dimensional comb is a comb inside a comb structure, consisting of backbones and fingers in the following geometrical correspondence x–backbone →y–fingers–backbone →z–fingers. Realisation of the O-U process on the three-dimensional comb leads to anomalous (non-Markovian) diffusion. This specific anomalous transport in the presence of resets results in non-equilibrium stationary states. Explicit analytical expressions for the mean values and the mean squared displacements along all three directions of the comb are obtained and verified numerically. The marginal probability density functions for each direction are obtained numerically by Monte Carlo simulation of a random transport described by a system of coupled Langevin equations for the comb geometry.

Resetting induced multimodality.

Properties of stochastic systems are defined by the noise type and deterministic forces acting on the system. In out-of-equilibrium setups, e.g., for motions under action of Lévy noises, the existence of the stationary state is not only determined by the potential but also by the noise. Potential wells need to be steeper than parabolic in order to assure the existence of stationary states. The existence of stationary states, in sub-harmonic potential wells, can be restored by stochastic resetting, which is the protocol of starting over at random times. Herein, we demonstrate that the combined action of Lévy noise and Poissonian stochastic resetting can result in the phase transition between non-equilibrium stationary states of various multimodality in the overdamped system in super-harmonic potentials. Fine-tuned resetting rates can increase the modality of stationary states, while for high resetting rates, the multimodality is destroyed as the stochastic resetting limits the spread of particles.

Ornstein-Uhlenbeck process and generalizations: Particle dynamics under comb constraints and stochastic resetting.

The Ornstein-Uhlenbeck process is interpreted as Brownian motion in a harmonic potential. This Gaussian Markov process has a bounded variance and admits a stationary probability distribution, in contrast to the standard Brownian motion. It also tends to a drift towards its mean function, and such a process is called mean reverting. Two examples of the generalized Ornstein-Uhlenbeck process are considered. In the first one, we study the Ornstein-Uhlenbeck process on a comb model, as an example of the harmonically bounded random motion in the topologically constrained geometry. The main dynamical characteristics (as the first and the second moments) and the probability density function are studied in the framework of both the Langevin stochastic equationand the Fokker-Planck equation. The second example is devoted to the study of the effects of stochastic resetting on the Ornstein-Uhlenbeck process, including stochastic resetting in the comb geometry. Here the nonequilibrium stationary state is the main question in task, where the two divergent forces, namely, the resetting and the drift towards the mean, lead to compelling results in the cases of both the Ornstein-Uhlenbeck process with resetting and its generalization on the two-dimensional comb structure.

Extreme Statistics and Spacing Distribution in a Brownian Gas Correlated by Resetting.

We study a one-dimensional gas of N Brownian particles that diffuse independently, but are simultaneously reset to the origin at a constant rate r. The system approaches a nonequilibrium stationary state with long-range interactions induced by the simultaneous resetting. Despite the presence of strong correlations, we show that several observables can be computed exactly, which include the global average density, the distribution of the position of the kth rightmost particle, and the spacing distribution between two successive particles. Our analytical results are confirmed by numerical simulations. We also discuss a possible experimental realization of this resetting gas using optical traps.

Nonstationary but quasisteady states in self-organized criticality.

The notion of self-organized criticality (SOC) was conceived to interpret the spontaneous emergence of long-range correlations in nature. Since then many different models have been introduced to study SOC. All of them have a few common features: externally driven dynamical systems self-organize themselves to nonequilibrium stationary states exhibiting fluctuations of all length scales as the signatures of criticality. In contrast, we have studied here in the framework of the sandpile model a system that has mass inflow but no outflow. There is no boundary, and particles cannot escape from the system by any means. Therefore, there is no current balance, and consequently it is not expected that the system would arrive at a stationary state. In spite of that, it is observed that the bulk of the system self-organizes to a quasisteady state where the grain density is maintained at a nearly constant value. Power law distributed fluctuations of all lengths and time scales have been observed, which are the signatures of criticality. Our detailed computer simulation study gives the set of critical exponents whose values are very close to their counterparts in the original sandpile model. This study indicates that (i) a physical boundary and (ii) the stationary state, though sufficient, may not be the necessary criteria for achieving SOC.

Irreversible thermodynamics of surfaces and interfaces: Special reference to the strained thin solid films on the substrates: Theory and practice

The realization of nanoscale devices largely depends on our ability to control and manipulate interfacial interactions and, thus, understanding of the mechanisms of surface/interface instabilities. In this work, theoretically as well as technologically important and distinct two thermodynamic systems, which are exposed to (isobaric) and isolated from (isochoric) external body forces and surface tractions, are formulated by using irreversible thermodynamics in combination with the generalized variational method. The starting point for the present formulation closely follows up the Fowler and Guggenheim [Statistical Thermodynamics (University Press, Cambridge, 1952)] interpretation of the Planck inequality [Über Prinzip Vermehrung Entropie: Ann. Phys. Series 2(32), 462 (1887)] for isothermal reversible and irreversible (natural) infinitesimal changes in heterogeneous systems (multi-phase and multi-component). By combining this fundamental principle with the interlink between the dissipation function and global internal entropy production postulates, two distinct sets of governing equations for the surface drift-diffusion flux as well as the rate of evaporation/condensation and/or the growth/recrystallization of amorphous solid thin films are obtained for isochoric and isobaric systems. The role of Eshelby's energy-momentum tensor in the generalized potential for the interface displacement is found to differ (opposite in sign) for isochoric and isobaric systems. To demonstrate the importance of these sign conflicts, two sets of computer experiments are performed on isochoric and isobaric systems. They showed us that the elastic strain energy density contribution to the generalized driving force for surface drift-diffusion alone favoring flat and smooth surfaces in isobaric systems regardless of the sign of the uniaxial stress (healing), rather than causing the surface roughness and even catastrophic crack initiation as the case in internally strained isochoric systems. Computer simulations allowed us to track down the dynamical behavior of test modules by furnishing surface and strain energy variations, combined with the Global Helmholtz free change, which indicates the existence of two regimes: initial smooth surface undulations followed up by the rather chaotic crack formation and propagation stage at the middle of the thin film supported by the stiff substrate. In this study, we mainly focused on the development kinetics of “Stranski–Krastanow” island-type morphology, initiated by the nucleation route rather than the surface roughening scheme. The physicomathematical model, which is based on the irreversible thermodynamics treatment of surfaces and interfaces with singularities [T. O. Ogurtani, J. Chem. Phys. 124, 144706 (2006)], furnishes us to have autocontrol on otherwise free-motion of the triple junction contour line between the substrate and the droplet without presuming any equilibrium dihedral contact (wetting) angles at edges. We have also demonstrated the formation of the Stranski–Krastanow (SK)-type doublet islanding (quantum dots) as a stationary nonequilibrium state in an epitaxially strained thin flat droplet on a rigid substrate by introducing the wetting potential—invoked by the quantum confinement—into the scenario and carefully selecting the system parameters (size and shape) for the isochoric system represented by [Ge/Si (100)]. It has been also shown that on the contrary to common perceptions, the Stranski–Krastanow islands are in genuine stationary nonequilibrium states in the sense of Prigogine if one invokes proper free-moving boundary conditions at triple junctions deduced from the irreversible thermodynamics rather than ad hoc periodic or reflecting constrains at the edges.

Nonequilibrium dynamics of the Ising model on heterogeneous networks with an arbitrary distribution of threshold noise.

The Ising model on networks plays a fundamental role as a testing ground for understanding cooperative phenomena in complex systems. Here we solve the synchronous dynamics of the Ising model on random graphs with an arbitrary degree distribution in the high-connectivity limit. Depending on the distribution of the threshold noise that governs the microscopic dynamics, the model evolves to nonequilibrium stationary states. We obtain an exact dynamical equation for the distribution of local magnetizations, from which we find the critical line that separates the paramagnetic from the ferromagnetic phase. For random graphs with a negative binomial degree distribution, we demonstrate that the stationary critical behavior as well as the long-time critical dynamics of the first two moments of the local magnetizations depend on the distribution of the threshold noise. In particular, for an algebraic threshold noise, these critical properties are determined by the power-law tails of the distribution of thresholds. We further show that the relaxation time of the average magnetization inside each phase exhibits the standard mean-field critical scaling. The values of all critical exponents considered here are independent of the variance of the negative binomial degree distribution. Our work highlights the importance of certain details of the microscopic dynamics for the critical behavior of nonequilibrium spin systems.

Fluctuation-response relation of time-symmetric quantities around general nonequilibrium stationary state

A connection between the response and fluctuation in general nonequilibrium stationary states is investigated. We focus on time-symmetric quantities and find that the fluctuation of a kind of empirical measure can be expressed with the response of the empirical measure, current, and the time-symmetric current. This relation is proven by using the fictitious stalling decomposition: we decompose a single observed transition (edge in the state space) between two microscopic states into two transitions such that one of the transitions stalls in this stationary state. Through this trick, relations for stalling stationary states apply to general nonequilibrium stationary states, which leads to the desired relation.

Effects of correlation in an information ratchet with finite tape.

With the finite-tape autonomous information ratchet modeled by He etal. [Phys. Rev. E 105, 054131 (2022)2470-004510.1103/PhysRevE.105.054131], we recast the information processing second law, giving a tighter bound on the work extracted, in terms of the marginal bit-ratchet distribution defined from the joint tape-ratchet distribution. The marginal distribution is further utilized to probe and elucidate the conditions that lead to the presence of equilibrium and nonequilibrium stationary states in general, which are related to the effects of correlation. Applying our analysis to two designs of this information ratchet, where correlations within manifest differently, we uncover the mathematical condition for equilibrium stationary states for information ratchets that harness correlation, to identify them for engine operation during the transient phase.

Stationary states of activity-driven harmonic chains.

We study the stationary state of a chain of harmonic oscillators driven by two active reservoirs at the two ends. These reservoirs exert correlated stochastic forces on the boundary oscillators which eventually leads to a nonequilibrium stationary state of the system. We consider three most well-known dynamics for the active force, namely, the active Ornstein-Uhlenbeck process, run-and-tumble process, and active Brownian process, all of which have exponentially decaying two-point temporal correlations but very different higher-order fluctuations. We show that, irrespective of the specific dynamics of the drive, the stationary velocity fluctuations are Gaussian in nature with a kinetic temperature which remains uniform in the bulk. Moreover, we find the emergence of an "equipartition of energy" in the bulk of the system-the bulk kinetic temperature equals the bulk potential temperature in the thermodynamic limit. We also calculate the stationary distribution of the instantaneous energy current in the bulk which always shows a logarithmic divergence near the origin and asymmetric exponential tails. The signatures of specific active driving become visible in the behavior of the oscillators near the boundary. This is most prominent for the RTP- and ABP-driven chains where the boundary velocity distributions become non-Gaussian and the current distribution has a finite cutoff.

Persistent ATP-Concentration Gradients in a Hydrogel Sustained by Chemical Fuel Consumption.

We show the formation of macroscopic ATP-concentrations in an agarose gel and demonstrate that these gradients can be sustained in time at the expense of the consumption of a chemical fuel. The approach relies on the spatially controlled activation of ATP-producing and ATP-consuming reactions through the local injection of enzymes in the matrix. The reaction-diffusion system is maintained in a stationary non-equilibrium state as long as chemical fuel, phosphocreatine, is present. The reaction-diffusion system is coupled to a supramolecular system composed of monolayer protected gold nanoparticles and a fluorescent probe. As a result of this coupling, fluorescence signals emerge spontaneously in response to the ATP-concentration gradients. We show that the approach permits the rational formation of complex fluorescence patterns that change over time as a function of the evolution of the ATP-concentrations present in the system.

Emergent quantum correlations and collective behavior in noninteracting quantum systems subject to stochastic resetting

We investigate the dynamics of a non-interacting spin system, undergoing coherent Rabi oscillations, in the presence of stochastic resetting. We show that resetting generally induces long-range quantum and classical correlations both in the emergent dissipative dynamics and in the non-equilibrium stationary state. Moreover, for the case of conditional reset protocols -- where the system is reinitialized to a state dependent on the outcome of a preceding measurement -- we show that, in the thermodynamic limit, the spin system can feature collective behavior which results in a phenomenology reminiscent of that occurring in non-equilibrium phase transitions. The discussed reset protocols can be implemented on quantum simulators and quantum devices that permit fast measurement and readout of macroscopic observables, such as the magnetisation. Our approach does not require the control of coherent interactions and may therefore highlight a route towards a simple and robust creation of quantum correlations and collective non-equilibrium states, with potential applications in quantum enhanced metrology and sensing.

Stochastic dynamics with multiplicative dichotomic noise: Heterogeneous telegrapher’s equation, anomalous crossovers and resetting

We analyze diffusion processes with finite propagation speed in a non-homogeneous medium in terms of the heterogeneous telegrapher’s equation. In the diffusion limit of infinite-velocity propagation we recover the results for the heterogeneous diffusion process. The heterogeneous telegrapher’s process exhibits a rich variety of diffusion regimes including hyperdiffusion, ballistic motion, superdiffusion, normal diffusion and subdiffusion, and different crossover dynamics characteristic for complex systems in which anomalous diffusion is observed. The anomalous diffusion exponent in the short time limit is twice the exponent in the long time limit, in accordance to the crossover dynamics from ballistic diffusion to normal diffusion in the standard telegrapher’s process. We also analyze the finite-velocity heterogeneous diffusion process in presence of stochastic Poissonian resetting. We show that the system reaches a non-equilibrium stationary state. The transition to this non-equilibrium steady state is analyzed in terms of the large deviation function.

Metastable discrete time-crystal resonances in a dissipative central spin system

We consider the non-equilibrium behavior of a central spin system where the central spin is periodically reset to its ground state. The quantum mechanical evolution under this effectively dissipative dynamics is described by a discrete-time quantum map. Despite its simplicity this problem shows surprisingly complex dynamical features. In particular, we identify several metastable time-crystal resonances. Here the system does not relax rapidly to a stationary state but undergoes long-lived oscillations with a period that is an integer multiple of the reset period. At these resonances the evolution becomes restricted to a low-dimensional state space within which the system undergoes a periodic motion. Generalizing the theory of metastability in open quantum systems, we develop an effective description for the evolution within this long-lived metastable subspace and show that in the long-time limit a non-equilibrium stationary state is approached. Our study links to timely questions concerning emergent collective behavior in the 'prethermal' stage of a dissipative quantum many-body evolution and may establish an intriguing link to the phenomenon of quantum synchronization.

Effect of stochastic resetting on Brownian motion with stochastic diffusion coefficient

We study the dynamics of a Brownian motion with a diffusion coefficient which evolves stochastically. We first study this process in arbitrary dimensions and find the scaling form and the corresponding scaling function of the position distribution. We find that the tails of the distribution have exponential tails with a ballistic scaling. We then introduce the resetting dynamics where, at a constant rate, both the position and the diffusion coefficient are reset to zero. This eventually leads to a nonequilibrium stationary state, which we study in arbitrary dimensions. In stark contrast to ordinary Brownian motion under resetting, the stationary position distribution in one dimension has a logarithmic divergence at the origin. For higher dimensions, however, the divergence disappears and the distribution attains a dimension-dependent constant value at the origin, which we compute exactly. The distribution has a generic stretched exponential tail in all dimensions. We also study the approach to the stationary state and find that, as time increases, an inner core region around the origin attains the stationary state, while the outside region still has a transient distribution—this inner stationary region grows ∼t 2, i.e., with a constant acceleration, much faster than ordinary Brownian motion.

Non-equilibrium Stationary Properties of the Boundary Driven Zero-Range Process with Long Jumps

We consider the zero-range process with long jumps and in contact with infinitely many reservoirs in its non-equilibrium stationary state. We derive the hydrostatic limit and the Fick's law, which are a consequence of a relationship between the exclusion process and the zero-range process. We also obtain the large deviation principle for the empirical density, i.e. we compute the non-equilibrium free energy.