Exactly Solvable Models for Many-Body Systems Far from Equilibrium

Spectral boundary of the asymmetric simple exclusion process: Free fermions, Bethe ansatz, and random matrix theory

In nonequilibrium statistical mechanics, the asymmetric simple exclusion process (ASEP) serves as a paradigmatic example. We investigate the spectral characteristics of the ASEP, focusing on the spectral boundary of its generator matrix. We examine finite ASEP chains of length L, under periodic boundary conditions (PBCs) and open boundary conditions (OBCs). Notably, the spectral boundary exhibits L spikes for PBCs and L+1 spikes for OBCs. Treating the ASEP generator as an interacting non-Hermitian fermionic model, we extend the model to have tunable interaction. In the noninteracting case, the analytically computed many-body spectrum shows a spectral boundary with prominent spikes. For PBCs, we use the coordinate Bethe ansatz to interpolate between the noninteracting case to the ASEP limit and show that these spikes stem from clustering of Bethe roots. The robustness of the spikes in the spectral boundary is demonstrated by linking the ASEP generator to random matrices with trace correlations or, equivalently, random graphs with distinct cycle structures, both displaying similar spiked spectral boundaries.

Efficient simulations of epidemic models with tensor networks: Application to the one-dimensional susceptible-infected-susceptible model

The contact process is an emblematic model of a nonequilibrium system, containing a phase transition between inactive and active dynamical regimes. In the epidemiological context, the model is known as the susceptible-infected-susceptible model, and it is widely used to describe contagious spreading. In this work, we demonstrate how accurate and efficient representations of the full probability distribution over all configurations of the contact process on a one-dimensional chain can be obtained by means of matrix product states (MPSs). We modify and adapt MPS methods from many-body quantum systems to study the classical distributions of the driven contact process at late times. We give accurate and efficient results for the distribution of large gaps, and illustrate the advantage of our methods over Monte Carlo simulations. Furthermore, we study the large deviation statistics of the dynamical activity, defined as the total number of configuration changes along a trajectory, and investigate quantum-inspired entropic measures, based on the second Rényi entropy.

Random sparse generators of Markovian evolution and their spectral properties

The evolution of a complex multistate system is often interpreted as a continuous-time Markovian process. To model the relaxation dynamics of such systems, we introduce an ensemble of random sparse matrices which can be used as generators of Markovian evolution. The sparsity is controlled by a parameter φ, which is the number of nonzero elements per row and column in the generator matrix. Thus, a member of the ensemble is characterized by the Laplacian of a directed regular graph with D vertices (number of system states) and 2φD edges with randomly distributed weights. We study the effects of sparsity on the spectrum of the generator. Sparsity is shown to close the large spectral gap that is characteristic of nonsparse random generators. We show that the first moment of the eigenvalue distribution scales as ∼φ, while its variance is ∼sqrt[φ]. By using extreme value theory, we demonstrate how the shape of the spectral edges is determined by the tails of the corresponding weight distributions and clarify the behavior of the spectral gap as a function of D. Finally, we analyze complex spacing ratio statistics of ultrasparse generators, φ=const, and find that starting already at φ⩾2, spectra of the generators exhibit universal properties typical of Ginibre's orthogonal ensemble.

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.

Model of Processive Catalysis with Site Clustering and Blocking and Its Application to Cellulose Hydrolysis

Interactions between particles moving on a linear track and their possible blocking by obstacles can lead to crowding, impeding the particles' transport kinetics. When the particles are enzymes processively catalyzing a reaction along a linear polymeric substrate, these crowding and blocking effects may substantially reduce the overall catalytic rate. Cellulose hydrolysis by exocellulases processively moving along cellulose chains assembled into insoluble cellulose particles is an example of such a catalytic transport process. The details of the kinetics of cellulose hydrolysis and the causes of the often observed reduction of hydrolysis rate over time are not yet fully understood. Crowding and blocking of enzyme particles are thought to be one of the important factors affecting the cellulose hydrolysis, but its exact role and mechanism are not clear. Here, we introduce a simple model based on an elementary transport process that incorporates the crowding and blocking effects in a straightforward way. This is achieved by making a distinction between binding and non-binding sites on the chain. The model reproduces a range of experimental results, mainly related to the early phase of cellulose hydrolysis. Our results indicate that the combined effects of clustering of binding sites together with the occupancy pattern of these sites by the enzyme molecules play a decisive role in the overall kinetics of cellulose hydrolysis. It is suggested that periodic desorption and rebinding of enzyme molecules could be a basis of a strategy to partially counter the clustering of and blocking by the binding sites and so enhance the rate of cellulose hydrolysis. The general nature of the model means that it could be applicable also to other transport processes that make a distinction between binding and non-binding sites, where crowding and blocking are expected to be relevant.

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.

Symmetry-induced fluctuation relations for dynamical observables irrespective of their behavior under time reversal

We extend previous work to describe a class of fluctuation relations (FRs) that emerge as a consequence of symmetries at the level of stochastic trajectories in Markov chains. We prove that given such a symmetry, and for a suitable dynamical observable, it is always possible to obtain a FR under a biased dynamics corresponding to the so-called generalized Doob transform. The general transformations of the dynamics that we consider go beyond time-reversal or spatial isometries, and an implication is the existence of FRs for observables irrespective of their behavior under time reversal, for example for time-symmetric observables rather than currents. We further show how to deduce in the long-time limit these FRs from the symmetry properties of the generator of the dynamics. We illustrate our results with four examples that highlight the novel features of our work.

Exponential decay of spatial correlation in driven diffusive system: A universal feature of macroscopic homogeneous state

Driven diffusive systems have been a paradigm for modelling many physical, chemical, and biological transport processes. In the systems, spatial correlation plays an important role in the emergence of a variety of nonequilibrium phenomena and exhibits rich features such as pronounced oscillations. However, the lack of analytical results of spatial correlation precludes us from fully understanding the effect of spatial correlation on the dynamics of the system. Here we offer precise analytical predictions of the spatial correlation in a typical driven diffusive system, namely facilitated asymmetric exclusion process. We find theoretically that the correlation between two sites decays exponentially as their distance increases, which is in good agreement with numerical simulations. Furthermore, we find the exponential decay is a universal property of macroscopic homogeneous state in a broad class of 1D driven diffusive systems. Our findings deepen the understanding of many nonequilibrium phenomena resulting from spatial correlation in driven diffusive systems.

Temporally correlated zero-range process with open boundaries: Steady state and fluctuations

We study an open-boundary version of the on-off zero-range process introduced in Hirschberg et al. [Phys. Rev. Lett. 103, 090602 (2009)]. This model includes temporal correlations which can promote the condensation of particles, a situation observed in real-world dynamics. We derive the exact solution for the steady state of the one-site system, as well as a mean-field approximation for larger one-dimensional lattices, and also explore the large deviation properties of the particle current. Analytical and numerical calculations show that, although the particle distribution is well described by an effective Markovian solution, the probability of rare currents differs from the memoryless case. In particular, we find evidence for a memory-induced dynamical phase transition.

Bulk induced phase transition in driven diffusive systems

This paper studies a weakly and asymmetrically coupled three-lane driven diffusive system. A non-monotonically changing density profile in the middle lane has been observed. When the extreme value of the density profile reaches ρ = 0.5, a bulk induced phase transition occurs which exhibits a shock and a continuously and smoothly decreasing density profile which crosses ρ = 0.5 upstream or downstream of the shock. The existence of double shocks has also been observed. A mean-field approach has been used to interpret the numerical results obtained by Monte Carlo simulations. The current minimization principle has excluded the occurrence of two or more bulk induced shocks in the general case of nonzero lane changing rates.

Exact relaxation dynamics in the totally asymmetric simple exclusion process

The relaxation dynamics of the one-dimensional totally asymmetric simple exclusion process on a ring is considered in the case of step initial condition. Analyzing the time evolution of the local particle densities and currents by the Bethe ansatz method, we examine their full relaxation dynamics. As a result, we observe peculiar behaviors, such as the emergence of a ripple in the density profile and the existence of the excessive particle currents. Moreover, by making a finite-size scaling analysis of the asymptotic amplitudes of the local densities and currents, we find the scaling exponents with respect to the total number of sites to be -3/2 and -1, respectively.

Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review

We briefly review hyperbolic and kinetic models for self-organized biological aggregations and traffic-like movement. We begin with the simplest models described by an advection-reaction equation in one spatial dimension. We then increase the complexity of models in steps. To this end, we begin investigating local hyperbolic systems of conservation laws with constant velocity. Next, we proceed to investigate local hyperbolic systems with density-dependent speed, systems that consider population dynamics (i.e., birth and death processes), and nonlocal hyperbolic systems. We conclude by discussing kinetic models in two spatial dimensions and their limiting hyperbolic models. This structural approach allows us to discuss the complexity of the biological problems investigated, and the necessity for deriving complex mathematical models that would explain the observed spatial and spatiotemporal group patterns.

Autocorrelations in the totally asymmetric simple exclusion process and Nagel-Schreckenberg model

We study via Monte Carlo simulation the dynamics of the Nagel-Schreckenberg model on a finite system of length L with open boundary conditions and parallel updates. We find numerically that in both the high and low density regimes the autocorrelation function of the system density behaves like 1-|t|/τ with a finite support [-τ,τ] . This is in contrast to the usual exponential decay typical of equilibrium systems. Furthermore, our results suggest that in fact τ=L/c , and in the special case of maximum velocity v{max}=1 (corresponding to the totally asymmetric simple exclusion process) we can identify the exact dependence of c on the input, output and hopping rates. We also emphasize that the parameter τ corresponds to the integrated autocorrelation time, which plays a fundamental role in quantifying the statistical errors in Monte Carlo simulations of these models.

Role of the particle's stepping cycle in an asymmetric exclusion process: a model of mRNA translation

Messenger RNA translation is often studied by means of statistical-mechanical models based on the asymmetric simple exclusion process (ASEP), which considers hopping particles (the ribosomes) on a lattice (the polynucleotide chain). In this work we extend this class of models and consider the two fundamental steps of the ribosome's biochemical cycle following a coarse-grained perspective. In order to achieve a better understanding of the underlying biological processes and compare the theoretical predictions with experimental results, we provide a description lying between the minimal ASEP-like models and the more detailed models, which are analytically hard to treat. We use a mean-field approach to study the dynamics of particles associated with an internal stepping cycle. In this framework it is possible to characterize analytically different phases of the system (high density, low density or maximal current phase). Crucially, we show that the transitions between these different phases occur at different parameter values than the equivalent transitions in a standard ASEP, indicating the importance of including the two fundamental steps of the ribosome's biochemical cycle into the model.

Large fluctuations of the macroscopic current in diffusive systems: a numerical test of the additivity principle

Most systems, when pushed out of equilibrium, respond by building up currents of locally conserved observables. Understanding how microscopic dynamics determines the averages and fluctuations of these currents is one of the main open problems in nonequilibrium statistical physics. The additivity principle is a theoretical proposal that allows to compute the current distribution in many one-dimensional nonequilibrium systems. Using simulations, we validate this conjecture in a simple and general model of energy transport, both in the presence of a temperature gradient and in canonical equilibrium. In particular, we show that the current distribution displays a Gaussian regime for small current fluctuations, as prescribed by the central limit theorem, and non-Gaussian (exponential) tails for large current deviations, obeying in all cases the Gallavotti-Cohen fluctuation theorem. In order to facilitate a given current fluctuation, the system adopts a well-defined temperature profile different from that of the steady state and in accordance with the additivity hypothesis predictions. System statistics during a large current fluctuation is independent of the sign of the current, which implies that the optimal profile (as well as higher-order profiles and spatial correlations) are invariant upon current inversion. We also demonstrate that finite-time joint fluctuations of the current and the profile are well described by the additivity functional. These results suggest the additivity hypothesis as a general and powerful tool to compute current distributions in many nonequilibrium systems.

Equivalence of a one-dimensional driven-diffusive system and an equilibrium two-dimensional walk model

It is known that a single product shock measure in some of the one-dimensional driven-diffusive systems with nearest-neighbor interactions might evolve in time quite similarly to a random walker moving on a one-dimensional lattice with reflecting boundaries. The nonequilibrium steady state of the system in this case can be written in terms of a linear superposition of such uncorrelated shocks. Equivalently, one can write the steady state of this system using a matrix-product approach with two-dimensional matrices. In this paper, we present an equilibrium two-dimensional one-transit walk model and find its partition function using a transfer matrix method. We will show that there is a direct connection between the partition functions of these two systems. We will explicitly show that in the steady state, the transfer matrix of the one-transit walk model is related to the matrix representation of the algebra of the driven-diffusive model through a similarity transformation. The physical quantities are also related through the same transformation.

Stochasticity and traffic jams in the transcription of ribosomal RNA: Intriguing role of termination and antitermination

In fast-growing bacteria, ribosomal RNA (rRNA) is required to be transcribed at very high rates to sustain the high cellular demand on ribosome synthesis. This results in dense traffic of RNA polymerases (RNAP). We developed a stochastic model, integrating results of single-molecule and quantitative in vivo studies of Escherichia coli, to evaluate the quantitative effect of pausing, termination, and antitermination (AT) on rRNA transcription. Our calculations reveal that in dense RNAP traffic, spontaneous pausing of RNAP can lead to severe "traffic jams," as manifested in the broad distribution of inter-RNAP distances and can be a major factor limiting transcription and hence growth. Our results suggest the suppression of these pauses by the ribosomal AT complex to be essential at fast growth. Moreover, unsuppressed pausing by even a few nonantiterminated RNAPs can already reduce transcription drastically under dense traffic. However, the termination factor Rho can remove the nonantiterminated RNAPs and restore fast transcription. The results thus suggest an intriguing role by Rho to enhance rather than attenuate rRNA transcription.

Molecular Spiders in One Dimension

Molecular spiders are synthetic bio-molecular systems which have "legs" made of short single-stranded segments of DNA. Spiders move on a surface covered with single-stranded DNA segments complementary to legs. Different mappings are established between various models of spiders and simple exclusion processes. For spiders with simple gait and varying number of legs we compute the diffusion coefficient; when the hopping is biased we also compute their velocity.

Amplification of molecular traffic control in catalytic grains with novel channel topology design

We investigate the conditions for reactivity enhancement of catalytic processes in porous solids by the use of molecular traffic control (MTC). With dynamic Monte-Carlo simulations and continuous-time master equation theory applied to the high concentration regime, we obtain a quantitative description of the MTC effect for a network of intersecting single-file channels in a wide range of grain parameters and for optimal external operating conditions. Implementing the concept of MTC in models with specially designed alternating bimodal channels, we find the efficiency ratio (compared with a topologically and structurally similar reference system without MTC) to be enhanced with increasing grain diameter, a property verified for the first time for a MTC system. Even for short intersection channels, MTC leads to a reactivity enhancement of up to approximately 65%. This suggests that MTC may significantly enhance the efficiency of a catalytic process for small as well as large porous particles with a suitably chosen binary channel topology.

Molecular motor traffic in a half-open tube

The traffic of molecular motors which interact through mutual exclusion is studied theoretically for half-open tube-like compartments. These half-open tubes mimic the shapes of axons. The mutual exclusion leads to traffic jams or density plateaus on the filaments. A phase transition is obtained when the motor velocity changes sign. We identify the relevant length scales and characterize the jamming behaviour using both analytical approximations and Monte Carlo simulations of lattice models.