Nonlinear driven diffusive systems with dissipation: fluctuating hydrodynamics

Probing the large deviations for the beta random walk in random medium

We consider a discrete-time random walk on a one-dimensional lattice with space- and time-dependent random jump probabilities, known as the beta random walk. We are interested in the probability that, for a given realization of the jump probabilities (a sample), a walker starting at the origin at time t=0 is at position beyond ξsqrt[T/2] at time T. This probability fluctuates from sample to sample and we study the large-deviation rate function, which characterizes the tails of its distribution at large time T≫1. It is argued that, up to a simple rescaling, this rate function is identical to the one recently obtained exactly by two of the authors for the continuum version of the model. That continuum model also appears in the macroscopic fluctuation theory of a class of lattice gases, e.g., in the so-called KMP model of heat transfer. An extensive numerical simulation of the beta random walk, based on an importance sampling algorithm, is found in good agreement with the detailed analytical predictions. A first-order transition in the tilted measure, predicted to occur in the continuum model, is also observed in the numerics.

Spectral signatures of symmetry-breaking dynamical phase transitions

Large deviation theory provides the framework to study the probability of rare fluctuations of time-averaged observables, opening new avenues of research in nonequilibrium physics. Some of the most appealing results within this context are dynamical phase transitions (DPTs), which might occur at the level of trajectories in order to maximize the probability of sustaining a rare event. While macroscopic fluctuation theory has underpinned much recent progress on the understanding of symmetry-breaking DPTs in driven diffusive systems, their microscopic characterization is still challenging. In this work we shed light on the general spectral mechanism giving rise to continuous DPTs not only for driven diffusive systems, but for any jump process in which a discrete Z_{n} symmetry is broken. By means of a symmetry-aided spectral analysis of the Doob-transformed dynamics, we provide the conditions whereby symmetry-breaking DPTs might emerge and how the different dynamical phases arise from the specific structure of the degenerate eigenvectors. In particular, we show explicitly how all symmetry-breaking features are encoded in the subleading eigenvectors of the degenerate subspace. Moreover, by partitioning configuration space into equivalence classes according to a proper order parameter, we achieve a substantial dimensional reduction which allows for the quantitative characterization of the spectral fingerprints of DPTs. We illustrate our predictions in several paradigmatic many-body systems, including (1) the one-dimensional boundary-driven weakly asymmetric exclusion process (WASEP), which exhibits a particle-hole symmetry-breaking DPT for current fluctuations, (2) the three- and four-state Potts model for spin dynamics, which displays discrete rotational symmetry-breaking DPTs for energy fluctuations, and (3) the closed WASEP which presents a continuous symmetry-breaking DPT into a time-crystal phase characterized by a rotating condensate.

Crossover from the macroscopic fluctuation theory to the Kardar-Parisi-Zhang equation controls the large deviations beyond Einstein's diffusion

We study the crossover from the macroscopic fluctuation theory (MFT), which describes one-dimensional stochastic diffusive systems at late times, to the weak noise theory (WNT), which describes the Kardar-Parisi-Zhang (KPZ) equation at early times. We focus on the example of the diffusion in a time-dependent random field, observed in an atypical direction which induces an asymmetry. The crossover is described by a nonlinear system which interpolates between the derivative and the standard nonlinear Schrodinger equations in imaginary time. We solve this system using the inverse scattering method for mixed-time boundary conditions introduced by us to solve the WNT. We obtain the rate function which describes the large deviations of the sample-to-sample fluctuations of the cumulative distribution of the tracer position. It exhibits a crossover as the asymmetry is varied, recovering both MFT and KPZ limits. We sketch how it is consistent with extracting the asymptotics of a Fredholm determinant formula, recently derived for sticky Brownian motions. The crossover mechanism studied here should generalize to a larger class of models described by the MFT. Our results apply to study extremal diffusion beyond Einstein's theory.

Inverse Scattering Method Solves the Problem of Full Statistics of Nonstationary Heat Transfer in the Kipnis-Marchioro-Presutti Model

We determine the full statistics of nonstationary heat transfer in the Kipnis-Marchioro-Presutti lattice gas model at long times by uncovering and exploiting complete integrability of the underlying equations of the macroscopic fluctuation theory. These equations are closely related to the derivative nonlinear Schrödinger equation (DNLS), and we solve them by the Zakharov-Shabat inverse scattering method (ISM) adapted by D. J. Kaup and A. C. Newell, J. Math. Phys. 19, 798 (1978)JMAPAQ0022-248810.1063/1.523737 for the DNLS. We obtain explicit results for the exact large deviation function of the transferred heat for an initially localized heat pulse, where we uncover a nontrivial symmetry relation.

Building Continuous Time Crystals from Rare Events

Symmetry-breaking dynamical phase transitions (DPTs) abound in the fluctuations of nonequilibrium systems. Here, we show that the spectral features of a particular class of DPTs exhibit the fingerprints of the recently discovered time-crystal phase of matter. Using Doob's transform as a tool, we provide a mechanism to build classical time-crystal generators from the rare event statistics of some driven diffusive systems. An analysis of the Doob's smart field in terms of the order parameter of the transition then leads to the time-crystal lattice gas (TCLG), a model of driven fluid subject to an external packing field, which presents a clear-cut steady-state phase transition to a time-crystalline phase characterized by a matter density wave, which breaks continuous time-translation symmetry and displays rigidity and long-range spatiotemporal order, as required for a time crystal. A hydrodynamic analysis of the TCLG transition uncovers striking similarities, but also key differences, with the Kuramoto synchronization transition. Possible experimental realizations of the TCLG in colloidal fluids are also discussed.

Induced correlations and rupture of molecular chaos by anisotropic dissipative Janus hard disks

A system of smooth "frozen" Janus-type disks is studied. Such disks cannot rotate and are divided by their diameter into two sides of different inelasticities. Taking as a reference a system of colored elastic disks, we find differences in the behavior of the collisions once the anisotropy is included. A homogeneous state, akin to the homogeneous cooling state of granular gases, is seen to arise and the singular behavior of both the collisions and the precollisional correlations are highlighted.

Dynamical phase behavior of the single- and multi-lane asymmetric simple exclusion process via matrix product states

The open asymmetric simple exclusion process (ASEP) has emerged as a paradigmatic model of nonequilibrium behavior, in part due to its complex dynamical behavior and wide physical applicability as a model of driven diffusion. We compare the dynamical phase behavior of the one-dimensional (1D) ASEP and the multi-lane ASEP, a previously unstudied extension of the 1D model that may be thought of as a finite-width strip of the fully two-dimensional (2D) system. Our characterization employs large deviation theory (LDT), matrix product states (MPS), and the density matrix renormalization group (DMRG) algorithm, to compute the current cumulant generating function and its derivatives, which serve as dynamical order parameters. We use this measure to show that when particles cannot exit or enter the lattice vertically, the phase behavior of the multi-lane ASEP mimics that of its 1D counterpart, exhibiting the macroscopic and microscopic signatures of the maximal current, shock, and high-density-low-density coexistence phases. Conversely, when particles are allowed to freely enter and exit the lattice, no such transition is observed. This contrast emphasizes the complex interplay between latitudinal and longitudinal hopping rates and the effect of current biasing. Our results support the potential of tensor networks as a framework to understand classical nonequilibrium statistical mechanics.

Sampling rare events across dynamical phase transitions

Interacting particle systems with many degrees of freedom may undergo phase transitions to sustain atypical fluctuations of dynamical observables such as the current or the activity. In some cases, this leads to symmetry-broken space-time trajectories which enhance the probability of such events due to the emergence of ordered structures. Despite their conceptual and practical importance, these dynamical phase transitions (DPTs) at the trajectory level are difficult to characterize due to the low probability of their occurrence. However, during the last decade, advanced computational techniques have been developed to measure rare events in simulations of many-particle systems that allow the direct observation and characterization of these DPTs. Here we review the application of a particular rare-event simulation technique, based on cloning Monte Carlo methods, to characterize DPTs in paradigmatic stochastic lattice gases. In particular, we describe in detail some tricks and tips of the trade, paying special attention to the measurement of order parameters capturing the physics of the different DPTs, as well as to the finite-size effects (both in the system size and in the number of clones) that affect the measurements. Overall, we provide a consistent picture of the phenomenology associated with DPTs and their measurement.

Additivity and density fluctuations in Vicsek-like models of self-propelled particles

We study coarse-grained density fluctuations in the disordered phase of the paradigmatic Vicsek-like models of self-propelled particles with alignment interactions and random self-propulsion velocities. By numerically integrating a fluctuation-response relation-the direct consequence of an additivity property-we compute logarithm of the large-deviation probabilities of the coarse-grained subsystem density, while the system is in the disordered fluid phase with vanishing macroscopic velocity. The large-deviation probabilities, computed within additivity, agree remarkably well with that obtained from direct microscopic simulations of the models. Our results provide evidence of the existence of an equilibriumlike chemical potential, which governs the coarse-grained density fluctuations in the Vicsek-like models. Moreover, comparison of the particle-number fluctuations among several self-propelled particle systems suggests a common mechanism through which the number fluctuations arise in such systems.

Infinite family of universal profiles for heat current statistics in Fourier's law

Using tools from large deviation theory, we study fluctuations of the heat current in a model of d-dimensional incompressible fluid driven out of equilibrium by a temperature gradient. We find that the most probable temperature fields sustaining atypical values of the global current can be naturally classified in an infinite set of curves, allowing us to exhaustively analyze their topological properties and to define universal profiles onto which all optimal fields collapse. We also compute the statistics of empirical heat current, where we find remarkable logarithmic tails for large current fluctuations orthogonal to the thermal gradient. Finally, we determine explicitly a number of cumulants of the current distribution, finding interesting relations between them.

Hydrodynamics of granular particles on a line

We investigate a lattice model representing a granular gas in a thin channel. We deduce the hydrodynamic description for the model from the microscopic dynamics in the large-system limit, including the lowest finite-size corrections. The main prediction from hydrodynamics, when finite-size corrections are neglected, is the existence of a steady "uniform longitudinal flow" (ULF), with the granular temperature and the velocity gradient both uniform and directly related. Extensive numerical simulations of the system show that such a state can be observed in the bulk of a finite-size system by attaching two thermostats with the same temperature at its boundaries. The relation between the ULF state and the shocks appearing in the late stage of a cooling gas of inelastic hard rods is discussed.

Lattice Model to Derive the Fluctuating Hydrodynamics of Active Particles with Inertia

We derive the hydrodynamic equations with fluctuating currents for the density, momentum, and energy fields for an active system in the dilute limit. In our model, nonoverdamped self-propelled particles (such as grains or birds) move on a lattice, interacting by means of aligning dissipative forces and excluded volume repulsion. Our macroscopic equations, in a specific case, reproduce a transition line from a disordered phase to a swarming phase and a linear dispersion law accounting for underdamped wave propagation. Numerical simulations up to a packing fraction ∼10% are in fair agreement with the theory, including the macroscopic noise amplitudes. At a higher packing fraction, a dense-diluted coexistence emerges. We underline the analogies with the granular kinetic theories, elucidating the relation between the active swarming phase and granular shear instability.

Einstein relation and hydrodynamics of nonequilibrium mass transport processes

We derive hydrodynamics of paradigmatic conserved-mass transport processes on a ring. The systems, governed by chipping, diffusion, and coalescence of masses, eventually reach a nonequilibrium steady state, having nontrivial correlations, with steady-state measures in most cases not known. In these processes, we analytically calculate two transport coefficients, bulk-diffusion coefficient and conductivity. Remarkably, the two transport coefficients obey an equilibrium-like Einstein relation even when the microscopic dynamics violates detailed balance and systems are far from equilibrium. Moreover, we show, using a macroscopic fluctuation theory, that the probability of large deviation in density, obtained from the above hydrodynamics, is in complete agreement with the same derived earlier by Das et al. [Phys. Rev. E 93, 062135 (2016)2470-004510.1103/PhysRevE.93.062135] using an additivity property.

Global stability and H theorem in lattice models with nonconservative interactions

In kinetic theory, a system is usually described by its one-particle distribution function f(r,v,t), such that f(r,v,t)drdv is the fraction of particles with positions and velocities in the intervals (r,r+dr) and (v,v+dv), respectively. Therein, global stability and the possible existence of an associated Lyapunov function or H theorem are open problems when nonconservative interactions are present, as in granular fluids. Here, we address this issue in the framework of a lattice model for granularlike velocity fields. For a quite general driving mechanism, including both boundary and bulk driving, we show that the steady state reached by the system in the long-time limit is globally stable. This is done by proving analytically that a certain H functional is nonincreasing in the long-time limit. Moreover, for a quite general energy injection mechanism, we are able to demonstrate that the proposed H functional is nonincreasing for all times. Also, we put forward a proof that clearly illustrates why the "classical" Boltzmann functional H_{B}[f]=∫drdvf(r,v,t)lnf(r,v,t) is inadequate for systems with nonconservative interactions. This is done not only for the simplified kinetic description that holds in the lattice models analyzed here but also for a general kinetic equation, like Boltzmann's or Enskog's.

Structure of the optimal path to a fluctuation

Macroscopic fluctuations have become an essential tool to understand physics far from equilibrium due to the link between their statistics and nonequilibrium ensembles. The optimal path leading to a fluctuation encodes key information on this problem, shedding light on, e.g., the physics behind the enhanced probability of rare events out of equilibrium, the possibility of dynamic phase transitions, and new symmetries. This makes the understanding of the properties of these optimal paths a central issue. Here we derive a fundamental relation which strongly constrains the architecture of these optimal paths for general d-dimensional nonequilibrium diffusive systems, and implies a nontrivial structure for the dominant current vector fields. Interestingly, this general relation (which encompasses and explains previous results) makes manifest the spatiotemporal nonlocality of the current statistics and the associated optimal trajectories.

Driven inelastic Maxwell gas in one dimension

A lattice version of the driven inelastic Maxwell gas is studied in one dimension with periodic boundary conditions. Each site i of the lattice is assigned with a scalar "velocity," v_{i}. Nearest neighbors on the lattice interact, with a rate τ_{c}^{-1}, according to an inelastic collision rule. External driving, occurring with a rate τ_{w}^{-1}, sustains a steady state in the system. A set of closed coupled equations for the evolution of the variance and the two-point correlation is found. Steady-state values of the variance, as well as spatial correlation functions, are calculated. It is shown exactly that the correlation function decays exponentially with distance, and the correlation length for a large system is determined. Furthermore, the spatiotemporal correlation C(x,t)=〈v_{i}(0)v_{i+x}(t)〉 can also be obtained. We find that there is an interior region -x^{*}<x<x^{*}, where C(x,t) has a time-dependent form, whereas in the exterior region |x|>x^{*}, the correlation function remains the same as the initial form. C(x,t) exhibits second-order discontinuity at the transition points x=±x^{*}, and these transition points move away from the x=0 with a constant speed.

Weak additivity principle for current statistics in d dimensions

The additivity principle (AP) allows one to compute the current distribution in many one-dimensional nonequilibrium systems. Here we extend this conjecture to general d-dimensional driven diffusive systems, and validate its predictions against both numerical simulations of rare events and microscopic exact calculations of three paradigmatic models of diffusive transport in d=2. Crucially, the existence of a structured current vector field at the fluctuating level, coupled to the local mobility, turns out to be essential to understand current statistics in d>1. We prove that, when compared to the straightforward extension of the AP to high d, the so-called weak AP always yields a better minimizer of the macroscopic fluctuation theory action for current statistics.

Statistics of the dissipated energy in driven diffusive systems

Understanding the physics of non-equilibrium systems remains one of the major open questions in statistical physics. This problem can be partially handled by investigating macroscopic fluctuations of key magnitudes that characterise the non-equilibrium behaviour of the system of interest; their statistics, associated structures and microscopic origin. During the last years, some new general and powerful methods have appeared to delve into fluctuating behaviour that have drastically changed the way to address this problem in the realm of diffusive systems: macroscopic fluctuation theory (MFT) and a set of advanced computational techniques that make it possible to measure the probability of rare events. Notwithstanding, a satisfactory theory is still lacking in a particular case of intrinsically non-equilibrium systems, namely those in which energy is not conserved but dissipated continuously in the bulk of the system (e.g. granular media). In this work, we put forward the dissipated energy as a relevant quantity in this case and analyse in a pedagogical way its fluctuations, by making use of a suitable generalisation of macroscopic fluctuation theory to driven dissipative media.

An itinerant oscillator model with cage inertia for mesorheological granular experiments

Recent experiments with a rotating probe immersed in weakly fluidized granular materials show a complex behavior on a wide range of time scales. Viscous-like relaxation at high frequency is accompanied by an almost harmonic dynamical trapping at intermediate times, with possibly anomalous long time behavior in the form of super-diffusion. Inspired by the itinerant oscillator model for diffusion in molecular liquids, and other models with coupled thermostats acting at different time scales, here we discuss a new model able to account for fast viscous relaxation, dynamical trapping, and super-diffusion at long times. The main difference with respect to liquids is a non-negligible cage inertia for the surrounding (granular) fluid, which allows it to sustain a slow but persistent motion for long times. The computed velocity power density spectra and mean-squared displacement qualitatively reproduce the experimental findings. We also discuss the linear response to external perturbations and the tail of the distribution of persistency time, which is associated with superdiffusion, and whose cut-off time is determined by cage inertia.

Probing local equilibrium in nonequilibrium fluids

We use extensive computer simulations to probe local thermodynamic equilibrium (LTE) in a quintessential model fluid, the two-dimensional hard-disks system. We show that macroscopic LTE is a property much stronger than previously anticipated, even in the presence of important finite-size effects, revealing a remarkable bulk-boundary decoupling phenomenon in fluids out of equilibrium. This allows us to measure the fluid's equation of state in simulations far from equilibrium, with an excellent accuracy comparable to the best equilibrium simulations. Subtle corrections to LTE are found in the fluctuations of the total energy which strongly point to the nonlocality of the nonequilibrium potential governing the fluid's macroscopic behavior out of equilibrium.

Scaling laws and bulk-boundary decoupling in heat flow

When driven out of equilibrium by a temperature gradient, fluids respond by developing a nontrivial, inhomogeneous structure according to the governing macroscopic laws. Here we show that such structure obeys strikingly simple scaling laws arbitrarily far from equilibrium, provided that both macroscopic local equilibrium and Fourier's law hold. Extensive simulations of hard disk fluids confirm the scaling laws even under strong temperature gradients, implying that Fourier's law remains valid in this highly nonlinear regime, with putative corrections absorbed into a nonlinear conductivity functional. In addition, our results show that the scaling laws are robust in the presence of strong finite-size effects, hinting at a subtle bulk-boundary decoupling mechanism which enforces the macroscopic laws on the bulk of the finite-sized fluid. This allows one to measure the marginal anomaly of the heat conductivity predicted for hard disks.

Memory effect in uniformly heated granular gases

We evidence a Kovacs-like memory effect in a uniformly driven granular gas. A system of inelastic hard particles, in the low density limit, can reach a nonequilibrium steady state when properly forced. By following a certain protocol for the drive time dependence, we prepare the gas in a state where the granular temperature coincides with its long time value. The temperature subsequently does not remain constant but exhibits a nonmonotonic evolution with either a maximum or a minimum, depending on the dissipation and on the protocol. We present a theoretical analysis of this memory effect at Boltzmann-Fokker-Planck equation level and show that when dissipation exceeds a threshold, the response can be called anomalous. We find excellent agreement between the analytical predictions and direct Monte Carlo simulations.

Kovacs-like memory effect in driven granular gases

While memory effects have been reported for dense enough disordered systems such as glasses, we show here by a combination of analytical and simulation techniques that they are also intrinsic to the dynamics of dilute granular gases. By means of a certain driving protocol, we prepare the gas in a state where the granular temperature T coincides with its long time limit. However, T does not subsequently remain constant but exhibits a nonmonotonic evolution before reaching its nonequilibrium steady value. The corresponding so-called Kovacs hump displays a normal behavior for weak dissipation (as observed in molecular systems) but is reversed under strong dissipation, where it, thus, becomes anomalous.

Typical and rare fluctuations in nonlinear driven diffusive systems with dissipation

We consider fluctuations of the dissipated energy in nonlinear driven diffusive systems subject to bulk dissipation and boundary driving. With this aim, we extend the recently introduced macroscopic fluctuation theory to nonlinear driven dissipative media, starting from the fluctuating hydrodynamic equations describing the system mesoscopic evolution. Interestingly, the action associated with a path in mesoscopic phase space, from which large-deviation functions for macroscopic observables can be derived, has the same simple form as in nondissipative systems. This is a consequence of the quasielasticity of microscopic dynamics, required in order to have a nontrivial competition between diffusion and dissipation at the mesoscale. Euler-Lagrange equations for the optimal density and current fields that sustain an arbitrary dissipation fluctuation are also derived. A perturbative solution thereof shows that the probability distribution of small fluctuations is always Gaussian, as expected from the central limit theorem. On the other hand, strong separation from the Gaussian behavior is observed for large fluctuations, with a distribution which shows no negative branch, thus violating the Gallavotti-Cohen fluctuation theorem, as expected from the irreversibility of the dynamics. The dissipation large-deviation function exhibits simple and general scaling forms for weakly and strongly dissipative systems, with large fluctuations favored in the former case but heavily suppressed in the latter. We apply our results to a general class of diffusive lattice models for which dissipation, nonlinear diffusion, and driving are the key ingredients. The theoretical predictions are compared to extensive numerical simulations of the microscopic models, and excellent agreement is found. Interestingly, the large-deviation function is in some cases nonconvex beyond some dissipation. These results show that a suitable generalization of macroscopic fluctuation theory is capable of describing in detail the fluctuating behavior of nonlinear driven dissipative media.