We revisit the problem of constructing the stationary states of the multispecies asymmetric simple exclusion process on a one-dimensional periodic lattice. Central to our approach is a quantum oscillator weighted five vertex model which features a strange weight conservation distinct from the conventional one. Our results clarify the interrelations among several known results and refine their derivations. For instance, the stationary probability derived from the multiline queue construction by Martin (2020) and Corteel–Mandelshtam–Williams (2022) is identified with the partition function of a three-dimensional system. The matrix product operators by Prolhac–Evans–Mallick (2009) acquire a natural diagrammatic interpretation as corner transfer matrices (CTM). The origin of their recursive tensor structure, as questioned by Aggarwal–Nicoletti–Petrov (2023), is revealed through the CTM diagrams. Finally, the derivation of the Zamolodchikov–Faddeev algebra by Cantini–de Gier–Wheeler (2015) is made intrinsic by elucidating its precise connection to a solution to the Yang–Baxter equation originating from quantum group representations.

A century after Ising introduced the Ising measure to study equilibrium systems, its relevance has expanded well beyond equilibrium contexts, notably appearing in nonequilibrium frameworks such as the Katz-Lebowitz-Spohn (KLS) model. In this work, we investigate a class of generalized asymmetric simple exclusion processes (ASEPs) for which the Ising measure serves as the stationary state. We show that the average stationary current in these models can display current reversal and other unconventional behaviors, offering insights into transport phenomena in nonequilibrium systems. Moreover, although long-range interaction rates often give rise to long-range interactions in the potential function, our model provides a counterexample: Even with long-range interactions in the dynamics, the resulting potential remains short ranged. Finally, our framework encompasses several well-known models as special cases, including ASEPs, the KLS model, the facilitated exclusion process, the cooperative exclusion process, and the assisted exchange model.

Time crystals are many-body systems whose ground state spontaneously breaks time-translation symmetry and thus exhibits long-range spatiotemporal order and robust periodic motion. Using hydrodynamics, we have recently shown how an m-th-order external packing field coupled to density fluctuations in driven diffusive fluids can induce the spontaneous emergence of time-crystalline order in the form of m rotating condensates, which can be further controlled and modulated. Here, we analyze this phenomenon at the microscopic level in a paradigmatic model of particle diffusion under exclusion interactions, a generalization of the weakly asymmetric simple exclusion process with a configuration-dependent field called the time-crystal lattice gas. Using extensive Monte Carlo simulations, we characterize the nonequilibrium phase transition to these complex time-crystal phases for different values of m, including the order parameter, the susceptibility, and the Binder cumulant, from which we measure the critical exponents, which turn out to be within the Kuramoto universality class for oscillator synchronization. We also elucidate the condensate density profiles and velocities, confirming along the way a scaling property predicted for the higher-order condensate shapes in terms of first-order ones, discussing also some possibilities for this promising route to time crystals.

We discuss nonreversible Markov-chain MonteCarlo algorithms that, for particle systems, rigorously sample the positional Boltzmann distribution and that have faster than physical dynamics. These algorithms all feature a nonthermal velocity distribution. They are exemplified by the lifted totally asymmetric simple exclusion process (lifted TASEP), a one-dimensional lattice reduction of event-chain MonteCarlo. We analyze its dynamics in terms of a velocity trapping that arises from correlations between the local density and the particle velocities. This allows us to formulate a conjecture for its out-of-equilibrium mixing timescale, and to rationalize its equilibrium superdiffusive timescale. Both scales are faster than for the (unlifted) TASEP. They are further justified by our analysis of the lifted TASEP in terms of many-particle realizations of true self-avoiding random walks. We discuss velocity trapping beyond the case of one-dimensional lattice models and in more than one physical dimensions. Possible applications beyond physics are pointed out.

Abstract Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems, motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials). In this paper, we present a unified approach to constructing stationary measures for most of the known colored particle systems on the ring and the line, including (1) the Asymmetric Simple Exclusion Process (multi-species ASEP, or mASEP); (2) the $q$ -deformed Totally Asymmetric Zero Range Process (TAZRP) also known as the $q$ -Boson particle system; (3) the $q$ -deformed Pushing Totally Asymmetric Simple Exclusion Process ( $q$ -PushTASEP). Our method is based on integrable stochastic vertex models and the Yang–Baxter equation. We express the stationary measures as partition functions of new ‘queue vertex models’ on the cylinder. The stationarity property is a direct consequence of the Yang–Baxter equation. For the mASEP on the ring, a particular case of our vertex model is equivalent to the multiline queues of Martin ( Stationary distributions of the multi-type ASEP , Electron. J. Probab. 25 (2020), 1–41). For the colored $q$ -Boson process and the $q$ -PushTASEP on the ring, we recover and generalize known stationary measures constructed using multiline queues or other methods by Ayyer, Mandelshtam and Martin ( Modified Macdonald polynomials and the multispecies zero range process: II , Algebr. Comb. 6 (2022), 243–284; Modified Macdonald polynomials and the multispecies zero-range process: I , Algebr. Comb. 6 (2023), 243–284) and Bukh and Cox ( Periodic words, common subsequences and frogs , Ann. Appl. Probab. 32 (2022), 1295–1332). Our proofs of stationarity use the Yang–Baxter equation and bypass the Matrix Product Ansatz (used for the mASEP by Prolhac, Evans and Mallick ( The matrix product solution of the multispecies partially asymmetric exclusion process , J. Phys. A. 42 (2009), 165004)). On the line and in a quadrant, we use the Yang–Baxter equation to establish a general colored Burke’s theorem, which implies that suitable specializations of our queue vertex models produce stationary measures for particle systems on the line. We also compute the colored particle currents in stationarity.

Abstract We study the behavior of colloidal active particles interacting via steric repulsion in various quasi-one-dimensional geometries. We mainly focus on active particles with a high Péclet number. We discuss three phenomena closely tied to these systems: motility-induced phase separation (or dynamical freezing), the ratchet effect (which takes place if the geometry has broken spatial symmetry), and the enhanced diffusion coefficient. We study these particles using numerical simulations, employing an asymmetric simple exclusion process (ASEP)-like model. In addition to direct numerical simulations, we study the model by mean-field approximation and by solving the coarse-grained hydrodynamic equations.

We describe limit fluctuations of the height function for the open TASEP on the coexistence line under the stationary measure. It is known that the height function satisfies a law of large numbers as the number of sites n goes to infinity which at the coexistence line is exotic in the sense that the first-order limit is random. Here, we study the functional central limit theorem: we show that with a random centering and normalized by n, the second-order limit of the height functions is a (random) mixture of two independent Brownian motions.

We study the directed transport of bosons along a one dimensional lattice in a dissipative setting, where the hopping is only facilitated by coupling to a Markovian reservoir. By combining numerical simulations with a field-theoretic analysis, we investigate the current fluctuations for this process and determine its asymptotic behavior. These findings demonstrate that dissipative bosonic transport belongs to the Kardar-Parisi-Zhang universality class and therefore, in spite of the drastic difference in the underlying particle statistics, it features the same coarse-grained behavior as the corresponding asymmetric simple exclusion process for fermions. However, crucial differences between the two processes emerge when focusing on the full counting statistics of current fluctuations. By mapping both models to the physics of fluctuating interfaces, we find that dissipative transport of bosons and fermions can be understood as surface growth and erosion processes, respectively. Within this unified description, both the similarities and discrepancies between the full counting statistics of the transport are reconciled. Beyond purely theoretical interest, these findings are relevant for experiments with cold atoms or long-lived quasiparticles in nanophotonic lattices, where such transport scenarios can be realized.

We consider a family of totally asymmetric simple exclusion processes (TASEPs), consisting of particles on a lattice that require binding by a "token" in various physical configurations to advance over the lattice. Using a combination of theory and simulations, we address the following questions: (i) How does token binding kinetics affect the current-density relation on the lattice? (ii) How does this current-density relation depend on the scarcity of tokens? (iii) How do tokens propagate the effects of the locally imposed disorder (such as a slow site) over the entire lattice? (iv) How does a shared pool of tokens couple concurrent TASEPs running on multiple lattices? and (v) How do our results translate to TASEPs with open boundaries that exchange particles with the reservoir? Since real particle motion (including in biological systems that inspired the standard TASEP model, e.g., protein synthesis or movement of molecular motors) is often catalyzed, regulated, actuated, or otherwise mediated, the token-driven TASEP dynamics analyzed in this paper should allow for a better understanding of real systems and enable a closer match between TASEP theory and experimental observations.

Abstract We address the dynamics of interacting particles on a disordered lattice formed by a random comb (RC). The dynamics comprises that of the asymmetric simple exclusion process, whereby motion to nearest-neighour sites that are empty is more likely in the direction of a bias than in the opposite direction. The RC comprises a backbone lattice from each site of which emanates a branch with a random number of sites. The backbone and the branches run in the direction of the bias. The number of branch sites or alternatively the branch lengths are sampled independently from a common distribution, specifically, an exponential distribution. The system relaxes at long times into a nonequilibrium stationary state. We analyze the stationary-state density of sites across the RC, and also explore the transport properties, in particular, the stationary-state drift velocity of particles along the backbone. We show that in the stationary state, the density is uniform along the backbone and nonuniform along the branches, decreasing monotonically from the free-end of a branch to its intersection with the backbone. On the other hand, the drift velocity as a function of the bias strength has a non-monotonic dependence, first increasing and then decreasing with increase of bias. However, remarkably, as the particle density increases, the dependence becomes no more non-monotonic. We understand this effect as a consequence of an interplay between biased hopping and hard-core exclusion, whereby sites towards the free end of the branches remain occupied for long times and become effectively non-participatory in the dynamics of the system. This results in an effective reduction of the branch lengths and a motion of the particles that takes place primarily along the backbone.

An important mechanism enabling fast ion diffusion in solid electrolytes is considered to be the significant lowering of the activation barrier when two or more ion particles hop simultaneously between sites, i.e., concerted migration, compared to single-ion hopping. In this study we incorporate a mechanism of simultaneous particle hopping into the asymmetric simple exclusion process, which is an archetypal model for many-particle transportation phenomena, and investigate its impact on particle transport properties. In our model, reflecting ion dynamics, the hopping rates are controlled by the activation energy, inverse temperature, and strength of an external driving field. We first construct an exact solution that describes the steady state of the proposed model. By using this solution, we find that concerted migration substantially increases the particle current and induces a shift of the peak in the fundamental diagram, i.e., the density-current relationship. We also show the presence of a critical temperature that maximizes the current. Additionally, we discuss the implications within parameter regions corresponding to actual materials. Published by the American Physical Society 2025

Abstract We examine the behavior of a single impurity particle embedded within a totally asymmetric simple exclusion process (TASEP). By analyzing the impurity’s dynamics, characterized by two arbitrary hopping parameters α and β, we investigate both its macroscopic impact on the system and its individual trajectory, providing new insights into the interaction between the impurity and the TASEP environment. We classify the induced hydrodynamic limit shapes based on the initial densities to the left and right of the impurity, along with the values of the parameters α, β. We develop a new method that enables the analysis of the impurity’s behavior within an arbitrary density field, thereby generalizing the traditional coupling technique used for second-class particles. With this tool, we extend to the impurity case under certain parameter conditions, Ferrari and Kipnis’s results on the distribution of the asymptotic speed of a second-class particle within a rarefaction fan.

We propose and study a conceptual one-dimensional model to explore how the combined interplay between fixed resources with unlimited carrying capacity and particle exchanges between different parts of an extended system can affect the stationary densities in a current-carrying channel connecting different parts of the system. Our model is composed of a totally asymmetric simple exclusion process (TASEP) connecting two particle reservoirs without any internal dynamics, but which can directly exchange particles between each other, ensuring nonvanishing currents in the TASEP lane in the steady states. The total particle number in the system that defines the "resources" available, although held conserved by the model dynamics, can take any value giving unrestricted carrying capacity. We show how the resulting phase diagrams of the model are controlled by relevant parameters, together with the total available resources. These control parameters can be tuned to make the density on the TASEP lane globally uniform or piecewise continuous with localized domain walls, and can also control populations of the two reservoirs. In general, the phase diagrams are quite different from a TASEP with open boundaries. However, in the limit of a large amount of resources, the phase diagrams in the plane of the control parameters become topologically identical to that for an open TASEP along with delocalization of domain walls.

Abstract The asymmetric simple exclusion process (ASEP) is a paradigmatic driven-diffusive system that describes the asymmetric diffusion of particles with hardcore interactions in a lattice. Although the ASEP is known as an exactly solvable model where physical quantities can be evaluated without approximations, most exact results are limited to one-dimensional systems. Recently, the exact steady states in the multi-dimensional ASEP have been proposed (Ishiguro and Sato 2024 Phys. Rev. Res. 6 033030). The research focused on the situations where the number of particles is conserved. In this paper, we consider the two-dimensional ASEP with attachment and detachment of particles (ASEP-LK), where particle number conservation is violated. By applying the results from (Ishiguro and Sato 2024 Phys. Rev. Res. 6 033030), we construct the exact steady states of the ASEP-LK and reveal their properties through the exact computation of physical quantities.

We exploit the equivalence between the partition function of an adsorbing Dyck walk model and the Asymmetric Simple Exclusion Process (ASEP) normalization to obtain the thermodynamic limit of the locus of the ASEP normalization zeros from a conformal map. We discuss the equivalence between this approach and using an electrostatic analogy to determine the locus, both in the case of the ASEP and the random allocation model.

We address how the interplay between the finite availability and carrying capacity of particles at different parts of a spatially extended system can control the steady-state currents and density profiles in the one-dimensional current-carrying lanes connecting the different parts of the system. To study this, we set up a minimal model consisting of two particle reservoirs of the same finite carrying capacity connected by two equally sized antiparallel totally asymmetric simple exclusion processes(TASEPs). We focus on the steady-state currents and particle density profiles in the two TASEP lanes. The ensuing phases and the phase diagrams, which can be remarkably complex, are parametrized by the model parameters defining particle exchange between the TASEP lanes and the reservoirs and the filling fraction of the particles that determine the total resources available. These parameters may be tuned to make the densities of the two TASEP lanes globally uniform or piece-wise continuous in the form of a combination of a single localized domain wall and a spatially constant density or a pair of delocalized domain walls. Our model reveals that the two reservoirs can be preferentially populated or depopulated in the steady states.

Motivated by the significant influence of the defects in the dynamics of the natural or man-made transportation systems, we propose an open, dynamically disordered, totally asymmetric simple exclusion process featuring bulk particle attachment and detachment. The site-wise dynamic defects might randomly emerge or vanish at any lattice location, and their presence slows down the motion of the particles. Using a mean-field approach, we obtain an analytical expression for both particle and defect density and validate them using Monte Carlo simulation. The study investigates the steady-state characteristics of the system, including phase transitions, analysis of boundary layers, and phase diagrams. Our approach streamlines the defect dynamics by integrating two parameters into one called the obstruction factor, which helps in determining an effective binding constant. The impact of the obstruction factor on the phase diagram is explored across various combinations of binding constants and detachment rates. A critical value of the obstruction factor is obtained, about which an infinitesimal change results in a substantial qualitative change in the structure of the phase diagrams. Further, the effect of the detachment rate is studied, and critical values along which the system observes a quantitative transition of the stationary phases are obtained as a function of the obstruction factor. Overall, the system shows stationary phases ranging from three to seven depending upon the value of the obstruction factor, the binding constant, and the detachment rate. Moreover, we scrutinized the impact of the obstruction factor on the shock dynamics and found no finite-size effect.

Bethe ansatz approach for the steady state of the asymmetric simple exclusion process with open boundaries

Abstract We formulate and establish symmetries of certain integrable half space models, analogous to recent results on symmetries for models in a full space. Our starting point is the colored stochastic six vertex model in a half space, from which we obtain results on the asymmetric simple exclusion process, as well as for the beta polymer through a fusion procedure which may be of independent interest. As an application, we establish a distributional identity between the absorption time in a type B analogue of the oriented swap process and last passage times in a half space, establishing the Baik–Ben Arous–Péché phase transition for the absorption time. The proof uses Hecke algebras and integrability of the six vertex model through the Yang–Baxter and reflection equations.

As the simplest model of transport of interacting particles in a disordered medium, we consider the asymmetric simple exclusion process (ASEP) in which particles with hard-core interactions perform biased random walks, on the supercritical percolation cluster. In this process, the long time trajectory of a marked particle consists of steps on the backbone, punctuated by time spent in side branches. We study the probability distribution in the steady state of the waiting time T_{w} of a randomly chosen particle, in a side branch since its last step along the backbone. Exact numerical evaluation of this on a single side branch of length L=1 to 9 shows that for large fields, the probability distribution of logT_{w} has multiple well separated peaks. We extend this result to a regular comb, and to the ASEP on the percolation cluster. We show that in the steady state, the fractional number of particles that have been in the same side branch for a time interval greater than T_{w} varies as exp(-csqrt[logT_{w}]) for large T_{w}, where c depends only on the bias field. However, these long timescales are not reflected in the eigenvalue spectrum of the Markov evolution matrix. The system shows dynamical heterogeneity, with particles segregating into pockets of high and low mobilities.