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Reaction-Diffusion Processes of Hard-Core Particles

Gunter Markus Schutz

Published Jan 1, 1994• 38 citations
OpenAlex

Abstract

We study a 12-parameter stochastic process involving particles with two-site interaction and hardcore repulsion on a d-dimensional lattice. In this model, which includes the asymmetric exclusion process, contact processes and other processes, the stochastic variables are particle occupation numbers taking values n⃗x = 0, 1. We show that on a 10-parameter submanifold the k-point equaltime correlation functions 〈n⃗x1 · · · n⃗xk 〉 satisfy linear differential-difference equations involving no higher correlators. In particular, the average density 〈n⃗x 〉 satisfies an integrable diffusion-type equation. These properties are explained in terms of dual processes and various duality relations are derived. By defining the time evolution of the stochastic process in terms of a quantum Hamiltonian H, the model becomes equivalent to a lattice model in thermal equilibrium in d + 1 dimensions. We show that the spectrum of H is identical to the spectrum of the quantum Hamiltonian of a d-dimensional, anisotropic spin-1/2 Heisenberg model. In one dimension our results hint at some new algebraic structure behind the integrability of the system. Keywords: Reaction-diffusion processes, correlation functions, integrable models An interesting class of non-equilibrium problems with a rich dynamical behaviour are

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We study a 12-parameter stochastic process involving particles with two-site interaction and hardcore repulsion on a d-dimensional lattice. In this model, which includes the asymmetric exclusion process, contact processes and other processes, the stochastic variables are particle occupation numbers taking values n⃗x = 0, 1. We show that on a 10-parameter submanifold the k-point equaltime correlation functions 〈n⃗x1 · · · n⃗xk 〉 satisfy linear differential-difference equations involving no higher correlators. In particular, the average density 〈n⃗x 〉 satisfies an integrable diffusion-type equation. These properties are explained in terms of dual processes and various duality relations are derived. By defining the time evolution of the stochastic process in terms of a quantum Hamiltonian H, the model becomes equivalent to a lattice model in thermal equilibrium in d + 1 dimensions. We show that the spectrum of H is identical to the spectrum of the quantum Hamiltonian of a d-dimensional, anisotropic spin-1/2 Heisenberg model. In one dimension our results hint at some new algebraic structure behind the integrability of the system. Keywords: Reaction-diffusion processes, correlation functions, integrable models An interesting class of non-equilibrium problems with a rich dynamical behaviour are
We show that the stochastic dynamics of a large class of one-dimensional interacting particle systems may be presented by integrable quantum spin Hamiltonians. Generalizing earlier work [1, 2] we present an alternative description of these processes in terms of a time-dependent operator algebra with quadratic relations. These relations generate the Bethe ansatz equations for the spectrum and turn the calculation of time-dependent expectation values into the problem of either finding representations of this algebra or of solving functional equations for the initial values of the operators. We use both strategies for the study of two specific models: (i) We construct a two-dimensional time-dependent representation of the algebra for the symmetric exclusion process with open boundary conditions. In this way we obtain new results on the dynamics of this system and on the eigenvectors and eigenvalues of the corresponding quantum spin chain, which is the isotropic Heisenberg ferromagnet with non-diagonal, symmetry-breaking boundary fields. (ii) We consider the non-equilibrium spin relaxation of Ising spins with zero-temperature Glauber dynamics and an additional coupling to an infinite-temperature heat bath
We show that the stochastic dynamics of a large class of one-dimensional interacting particle systems may be presented by integrable quantum spin Hamiltonians. Using the Bethe ansatz and similarity transformations this yields new exact results. In a complementary approach we generalize previous work and present a new description of these and other processes and the related quantum chains in terms of an operator algebra with quadratic relations. The full solution of the master equation of the process is thus turned into the problem of finding representations of this algebra. We find a two-dimensional time-dependent representation of the algebra for the symmetric exclusion process with open boundary conditions. We obtain new results on the dynamics of this system and on the eigenvectors and eigenvalues of the corresponding quantum spin chain, which is the isotropic Heisenberg ferromagnet with non-diagonal boundary fields.
We consider the one-dimensional partially asymmetric exclusion model with open boundaries. The model describes a system of hard-core particles that hop stochastically in both directions with different rates. At both boundaries particles are injected and extracted. By means of the method of Derrida et al the stationary probability measure can be expressed as a matrix-product state involving two matrices forming a Fock-like representation of a general quadratic algebra. We obtain the representations of this algebra, which were unknown in the mathematical literature and use the two-dimensional one to derive exact expressions for the density profile and correlation functions. Using the correspondence between the stochastic model and a quantum spin chain, we obtain exact correlation functions for a spin- Heisenberg XXZ chain with non-diagonal boundary terms. Generalizations to other reaction - diffusion models are discussed.
We study reaction-diffusion processes with multi-species of particles on a hypercubic lattice. We define the particle density operators for multi-species case and express the time evolution equation of correlation functions in quantum Hamiltonian formalism. We consider the conditions that the time evolution of correlation functions does not generate higher-order correlation functions. These conditions are called the closed conditions. Moreover we generalize the dual processes which are known in the symmetric exclusion processes and find that the Hamiltonian of the dual processes fixes the time evolution of correlation functions of processes with multi-species of particles.
We consider a process in which there are two types of particles, A and B, on an infinite one-dimensional lattice. The particles hop to their adjacent sites, like the totally asymmetric exclusion process (ASEP), and have also the following interactions: A+B-->B+B and B+A-->B+B, which all occur with equal rate. We study this process by imposing four boundary conditions on the ASEP master equation. It is shown that this model is integrable, in the sense that its N-particle S matrix is factorized into a product of two-particle S matrices and, more importantly, the two-particle S matrix satisfies the quantum Yang-Baxter equation. Using the coordinate Bethe-ansatz, the N-particle wave functions and the two-particle conditional probabilities are found exactly. Further, by imposing four reasonable physical conditions on two-species diffusion-reaction processes (where the most important ones are the equality of the reaction rates and the conservation of the number of particles in each reaction), we show that among the 4096 types of interactions which have these properties and can be modeled by a master equation and an appropriate set of boundary conditions, there are only 28 independent interactions which are integrable. We find all these interactions and also their corresponding wave functions. Some of these may be new solutions of the quantum Yang-Baxter equation.
We consider systems of particles hopping stochastically on d-dimensional lattices with space-dependent probabilities. We map the master equation onto an evolution equation in a Fock space where the dynamics are given by a quantum Hamiltonian (continuous time) or a transfer matrix (discrete time). Using non-Abelian symmetries of these operators we derive duality relations, expressing the time evolution of a given initial configuration in terms of correlation functions of simpler dual processes. Particularly simple results are obtained for the time evolution of the density profile. As a special case we show that for any SU(2) symmetric system the two-point and three-point density correlation functions in the N-particle steady state can be computed from the probability distribution of a single particle moving in the same environment. We apply our results to various models, among them partial exclusion, a simple diffusion-reaction system, and the two-dimensional six-vertex model with space-dependent vertex weights. For a random distribution of the vertex weights one obtains a version of the random-barrier model describing diffusion of particles in disordered media. We derive exact expressions for the averaged two-point density correlation functions in the presence of weak, correlated disorder.