Gunter M. Schütz

Phase transitions in an exactly soluble one-dimensional exclusion process

We consider an exclusion process with particles injected with rate α at the origin and removed with rate β at the right boundary of a one-dimensional chain of sites. The particles are allowed to hop onto unoccupied sites, to the right only. For the special case of α =β = 1 the model was solved previously by Derridaet al. Here we extend the solution to general α, β. The phase diagram obtained from our exact solution differs from the one predicted by the mean-field approximation.

Generalized Bethe ansatz solution of a one-dimensional asymmetric exclusion process on a ring with blockage

We present a model for a one-dimensional anisotropic exclusion process describing particles moving deterministically on a ring of lengthL with a single defect, across which they move with probability 0 ⩽p ⩽ 1. This model is equivalent to a two-dimensional, six-vertex model in an extreme anisotropic limit with a defect line interpolating between open and periodic boundary conditions. We solve this model with a Bethe ansatz generalized to this kind of boundary condition. We discuss in detail the steady state and derive exact expressions for the currentj, the density profilen(x), and the two-point density correlation function. In the thermodynamic limitL → ∞ the phase diagram shows three phases, a low-density phase, a coexistence phase, and a high-density phase related to the low-density phase by a particle-hole symmetry. In the low-density phase the density profile decays exponentially with the distance from the boundary to its bulk value on a length scale ξ. On the phase transition line ξ diverges and the currentj approaches its critical valuejc = p as a power law,jc − j ∞ ξ−1/2. In the coexistence phase the widthδ of the interface between the high-density region and the low-density region is proportional toL1/2 if the densityρf 1/2 andδ=0 independent ofL ifρ = 1/2. The (connected) two-point correlation function turns out to be of a scaling form with a space-dependent amplitude n(x1, x2) =A(x2)AKe−r/ξ withr = x2 −x1 and a critical exponent κ = 0.

Zero-range process with open boundaries

We calculate the exact stationary distribution of the one-dimensional zero-range process with open boundaries for arbitrary bulk and boundary hopping rates. When such a distribution exists, the steady state has no correlations between sites and is uniquely characterized by a space-dependent fugacity which is a function of the boundary rates and the hopping asymmetry. For strong boundary drive the system has no stationary distribution. In systems which on a ring geometry allow for a condensation transition, a condensate develops at one or both boundary sites. On all other sites the particle distribution approaches a product measure with the finite critical density ρc. In systems which do not support condensation on a ring, strong boundary drive leads to a condensate at the boundary. However, in this case the local particle density in the interior exhibits a complex algebraic growth in time. We calculate the bulk and boundary growth exponents as a function of the system parameters.

Time-dependent correlation functions in a one-dimensional asymmetric exclusion process

We study a one-dimensional anisotropic exclusion process describing particles injected at the origin, moving to the right on a chain of

Pairwise balance and invariant measures for generalized exclusion processes

L

REACTION-DIFFUSION PROCESSES OF HARD-CORE PARTICLES

sites and being removed at the (right) boundary. We construct the steady state and compute the density profile, exact expressions for all equal-time n-point density correlation functions and the time-dependent two-point function in the steady state as functions of the injection and absorption rates. We determine the phase diagram of the model and compare our results with predictions from dynamical scaling and discuss some conjectures for other exclusion models.

Equivalences between stochastic systems

We characterize the steady state of a driven diffusive lattice gas in which each site holds several particles, and the dynamics is activated and asymmetric. Using a quantum Hamiltonian formalism, we show that for arbitrary transition rates the model has product invariant measure. In the steady state, a pairwise balance condition is shown to hold. Configurations and leading respectively into and out of a given configuration are matched in pairs so that the flux of transitions from to is equal to the flux from to . Pairwise balance is more general than the condition of detailed balance and holds in the non-equilibrium steady state of a number of stochastic models.

Boundary-induced phase transitions in equilibrium and non-equilibrium systems

We study a 12-parameter stochastic process involving particles with two-site interaction and hard-core repulsion on ad-dimensional lattice. In this model, which includes the asymmetric exclusion process, contact processes, and other processes, the stochastic variables are particle occupation numbers taking valuesnx=0,1. We show that on a ten-parameter submanifold thek-point equal-time correlation functions 〈nx1...nxk〉 satisfy linear differential-difference equations involving no higher correlators. In particular, the average density 〈nx〉 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 HamiltonianH, the model becomes equivalent to a lattice model in thermal equilibrium ind+1 dimensions. We show that the spectrum ofH is identical to the spectrum of the quantum Hamiltonian of ad-dimensional anisotropic, spin-1/2 Heisenberg model. In one dimension our results hint at some new algebraic structure behind the integrability of the system.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 n~x = 0, 1. We show that on a 10-parameter submanifold the k-point equaltime correlation functions 〈n~x1 · · ·n~xk 〉 satisfy linear differential-difference equations involving no higher correlators. In particular, the average density 〈n~x 〉 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.

Global Phase Diagram of a One-Dimensional Driven Lattice Gas

Time-dependent correlation functions of (unstable) particles undergoing biased or unbiased diffusion, coagulation and annihilation are studied. This is achieved by similarity transformations between different stochastic models and between stochastic and soluble nonstochastic models. For special cases we obtain exact results which are in good agreement with experiments on one-dimensional annihilation-coagulation processes.

THE HEISENBERG CHAIN AS A DYNAMICAL MODEL FOR PROTEIN SYNTHESIS : SOME THEORETICAL AND EXPERIMENTAL RESULTS

Boundary conditions may change the phase diagram of non-equilibrium statistical systems like the one-dimensional asymmetric simple exclusion process with and without particle number conservation. Using the quantum Hamiltonian approach, the model is mapped onto an XXZ quantum chain and solved using the Bethe ansatz. This system is related to a two-dimensional vertex model in thermal equilibrium. The phase transition caused by a point-like boundary defect in the dynamics of the one-dimensional exclusion model is in the same universality class as a continuous (bulk) phase transition of the two-dimensional vertex model caused by a line defect at its boundary.