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arXiv:cond-mat/9305040 · Unknown eBooks

Exclusive Many-Particle Diffusion in Disordered Media and Correlation Functions for Random Vertex Models

Gunter Markus Schutz, Sven Sandow

Published May 29, 1993• 0 citations
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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 resp. (discrete time). We show that under certain conditions the time-dependent two-point density correlation function in the $N$-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.

DOI
10.48550/arxiv.cond-mat/9305040

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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 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 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 study analytically a model where particles with hard-core repulsion diffuse on a finite one-dimensional lattice with space-dependent, asymmetric hopping rates. The dynamics are given by the U q [ SU (2)]-symmetric Hamiltonian of a generalized anisotropic Heisenberg ferromagnet. Exploiting this symmetry, we derive exact expressions for various correlation functions. In the weakly driven system correlation length ξ s and correlation time ξ t are related by ξ t ∼ ξ s 2 indicating a dynamical exponent z = 2 as for symmetric diffusion. But also in strongly driven systems with finite density we find correlation functions with a dynamical exponent z = 2. This is not expected from the finite-size scaling analysis of the model.
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We discuss a reaction–diffusion model in one dimension subjected to an external driving force. Each lattice site may be occupied by at most one particle. The particles hop with rates (1±η)/2 to the right or left nearest neighbor site if it is vacant, and annihilate with rate one if it is occupied. The representation of an uncorrelated random initial state in terms of free fermions allows the calculation of multiple time-dependent higher order correlation functions of the local density. We outline the procedure using a field theoretic approach. We also compute the long time behavior of the density profile if the initial density profile is a step function. The fermion representation of the step function state also allows the calculation of multiple time-dependent correlation functions. Experimental applications of these results are pointed out.
We study a spin-1/2-particle moving on a one-dimensional lattice subject to disorder induced by a random, space-dependent quantum coin. The discrete time evolution is given by a family of random unitary quantum walk operators, where the shift operation is assumed to be deterministic. Each coin is an independent identically distributed random variable with values in the group of two-dimensional unitary matrices. We derive sufficient conditions on the probability distribution of the coins such that the system exhibits dynamical localization. Put differently, the tunneling probability between two lattice sites decays rapidly for almost all choices of random coins and after arbitrary many time steps with increasing distance. Our findings imply that this effect takes place if the coin is chosen at random from the Haar measure, or some measure continuous with respect to it, but also for a class of discrete probability measures which support consists of two coins, one of them being the Hadamard coin.