We address the problem of exactly describing stochastic nonequilibrium systems that are widely used to model one-dimensional transport, in biology, traffic flow and others. We review the matrix product states ansatz to interacting multiparticle systems and its extension to a tridiagonal (generalized Onsager) algebra approach. The stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra defined by the dynamics of the process. The states involved in the matrix dements are determined by the boundary conditions. This reflects the intriguing feature of open systems that the bulk behaviour in the steady state strongly depends on the boundary rates. The importance of the boundary conditions manifests itself in the fact that the boundary operators are generators of a tridiagonal algebra whose irreducible modules are the Askey-Wilson polynomials. The matrices of the matrix product ansatz obey the tridiagonal algebraic relations for particular values of the structure, constants. Previously known representations, both infinite dimensional and finite dimensional ones, are recovered within the tridiagonal framework. The boundary Askey-Wilson and. tridiagonal symmetry is the deep algebraic property of driven diffusive systems allowing for the exact solvability in the steady state and the exact description of the stochastic dynamics.
We present three works on Interacting Particle Systems where some form of non-linearity arises. In the context of the description of Non-Equilibrium Steady States (NESS), we formulate a Matrix Product Ansatz in order to describe the NESS of a process defined on the lattice
ΛN={1,…,N}, evolving as the SSEP in
{2,…,N−1} and coupled with two-site reaction-diffusion processes at the boundaries, acting on the pairs of sites
{1,2} and
{N−1,N}. These pairs of sites can take four different states each, leading to 12 possible transitions between them. We derive a set of constraints where the underlying quadratic algebra is consistent, which are related with the correlation of the reservoirs, present the representation of the objects in the formulation, and provide examples of rates satisfying these constraints. In the second work, our focus is in generalizing the Porous Media Model (PMM), associated with the hydrodynamic equation
∂tρ=∂2uρm for
m∈N+, into a universal exclusion family parametrized by
m∈R, and, in particular, representing the transition from the slow diffusion regime (
m>1) to the fast diffusion regime (
m<1). We successfully treat the case
m∈(0,2), encompassing the phase transition
m=1, in this manner deriving the Porous Media Equation in the range
m∈(1,2), and the Fast Diffusion Equation for
m>0. Next, we generalize the PMM in another direction: constructing the diffusion coefficient
D(ρ)=ρn(1−ρ)k with
n,k∈N+. The generalized PMM that we introduce is a gradient model that inherits theoretical properties from PMM, as, for instance, the presence of mobile clusters and blocked configurations. The construction of the model is indirect and delicate, in order to maintain the gradient property. We then generalize this into a long-range dynamics inheriting the gradient property. The long-range generalization of the dynamics is simple and can be applied to any exclusion process.
It is known that exact travelling wave solutions exist for families of (n + 1)-states stochastic one-dimensional non-equilibrium lattice models with open boundaries provided that some constraints on the reaction rates are fulfilled. These solutions describe the diffusive motion of a product shock or a domain wall with the dynamics of a simple biased random walker. The steady state of these systems can be written in terms of linear superposition of such shocks or domain walls. These steady states can also be expressed in a matrix product form. We show that in this case the associated quadratic algebra of the system has always a twodimensional representation with a generic structure. A couple of examples for n = 1 and n = 2 cases are presented. On the macroscopic level nonlinear hydrodynamic equations, such as the Burgers equation or the Fisher equation, might exhibit shocks in some cases [1, 2]. On the microscopic level some stochastic non-equilibrium one-dimensional lattice models with open boundaries might also develop shocks provided that some constrained on the reaction rates are fulfilled [3, 4, 5]. In [6] exact travelling wave solutions have been obtained for three families of two-states drivendiffusive models in one-dimension with nearest-neighbors interactions. These three models are the Partially Asymmetric Simple Exclusion Process (PASEP) [7, 8, 9], the Branching-Coalescing Random Walk (BCRW) and the Asymmetric Kawasaki-Glauber Process (AKGP). By taking a Bernoulli shock distribution as initial distribution, the shock distribution in these models evolves in time into a linear combination of similar distributions with different shock positions. The time evolution of the shock position in these models is similar to the dynamics of a random walker moving on a one-dimensional lattice with reflecting boundaries [6]. This property of the travelling wave solution is not limited to the two-states systems with nearest-neighbor interactions. Shocks in threestates
It is known that exact traveling wave solutions exist for families of ( n +1)-states stochastic one-dimensional non-equilibrium lattice models with open boundaries provided that some constraints on the reaction rates are fulfilled. These solutions describe the diffusive motion of a product shock or a domain wall with the dynamics of a simple biased random walker. The steady state of these systems can be written in terms of linear superposition of such shocks or domain walls. These steady states can also be expressed in a matrix product form. We show that, in this case, the associated quadratic algebra of the system always has a two-dimensional representation with a generic structure. A couple of examples for the n = 1 and 2 cases are presented.
Abstract We review recent results on an exactly solvable model of nonequilibrium statistical mechanics, specifically the classical rule 54 reversible cellular automaton and some of its quantum extensions. We discuss the exact microscopic description of nonequilibrium dynamics as well as the equilibrium and nonequilibrium stationary states. This allows us to obtain a rigorous handle on the corresponding emergent hydrodynamic description, which is treated as well. Specifically, we focus on two different paradigms of rule 54 dynamics. Firstly, we consider a finite chain driven by stochastic boundaries, where we provide exact matrix product descriptions of the nonequilibrium steady state, most relevant decay modes, as well as the eigenvector of the tilted Markov chain yielding exact large deviations for a broad class of local and extensive observables. Secondly, we treat the explicit dynamics of macro-states on an infinite lattice and discuss exact closed form results for dynamical structure factor, multi-time-correlation functions and inhomogeneous quenches. Remarkably, these results prove that the model, despite its simplicity, behaves like a regular fluid with coexistence of ballistic (sound) and diffusive (heat) transport. Finally, we briefly discuss quantum interpretation of rule 54 dynamics and explicit results on dynamical spreading of local operators and operator entanglement.
Abstract We construct matrix product steady states for a class of interacting particle systems where particles do not obey hardcore exclusion, meaning each site can occupy any number of particles subjected to the global conservation of the total number of particles in the system. To represent the arbitrary occupancy of the sites, the matrix product ansatz here requires an infinite set of matrices which in turn leads to an algebra involving an infinite number of matrix equations. We show that these matrix equations, in fact, can be reduced to a single functional relation when the matrices are parametric functions of the representative occupation number. We demonstrate this matrix formulation in a class of stochastic particle hopping processes on a one dimensional periodic lattice where hop rates depend on the occupation numbers of the departure site and its neighbors within a finite range; this includes some well known stochastic processes like, the totally or partially asymmetric zero range process, misanthrope process and finite range process.