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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Non-Abelian symmetries of stochastic processes: Derivation of correlation functions for random-vertex models and disordered-interacting-particle systems

Gunter Markus Schutz, Sven Sandow

Published Apr 1, 1994• 148 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.49.2726

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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.
We consider systems of particles hopping stochastically on ddd-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 resp. (discrete time). We show that under certain conditions the time-dependent two-point density correlation function in the NNN-particle steady state can be computed from the probability distribution of a single particle moving in the same environment. Focussing on exclusion models where each lattice site can be occupied by at most one particle we discuss as an example for such a stochastic process a generalized Heisenberg antiferromagnet where the strength of the spin-spin coupling is space-dependent. In discrete time one obtains for one-dimensional systems the diagonal-to-diagonal transfer matrix of 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.
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
Non-universal dynamics is shown to occur in a one-dimensional non-equilibrium system of hard-core particles. The stochastic processes included are pair creation and annihilation (with rates and ) and symmetric hopping rates which alternate from one bond to the next ( , ). A dynamical scaling relation between the relaxation time and the correlation length in the steady state is derived in a simple way for the case . We find that the dynamical exponent takes the non-universal value . For the special condition , where the stochastic system is in principle soluble by reduction to a free fermion system, the model is mapped to the Glauber dynamics of an Ising chain with alternating ferromagnetic bonds of values and , in contact with a quantum thermal bath. The full time dependence of the space-dependent magnetization and of the equal time spin-spin correlation function are studied by writing the master equation for this system in the quantum Hamiltonian formalism. In particular, we obtain the dispersion relations and rigorously confirm the results obtained for the correlation length and for the dynamical exponent.
We construct a time evolution operator for a dynamical quantum models with a multi-component random Gaussian field characterized by a long correlation time. We analyze three paradigmatic cases: non-Abelian SU(2)SU(2)SU(2) symmetry class for the three-component Gaussian noise and Abelian symmetry classes U(1)U(1)U(1) and Z2Z_2Z2​ for the two- and one-component noises correspondingly. Starting with a limiting case of the infinite correlation time (Keldysh model) we shown that statistics of the two-component random field with a non-zero mean is described by the Rice distribution. We derived and solved the system of coupled linear differential equations for the exact diagonal and off-diagonal single-particle Green's functions. We showed how the finite time correlation in Keldysh models are accounted by a continuous fractions method.
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 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.