Diffusion-limited annihilation in inhomogeneous environments

We study diffusion-limited (on-site) pair annihilation $A+A\to 0$ and (on-site) fusion $A+A\to A$ which we show to be equivalent for arbitrary space-dependent diffusion and reaction rates. For one-dimensional lattices with nearest neighbour hopping we find that in the limit of infinite reaction rate the time-dependent $n$-point density correlations for many-particle initial states are determined by the correlation functions of a dual diffusion-limited annihilation process with at most $2n$ particles initially. By reformulating general properties of annihilating random walks in one dimension in terms of fermionic anticommutation relations we derive an exact representation for these correlation functions in terms of conditional probabilities for a single particle performing a random walk with dual hopping rates. This allows for the exact and explicit calculation of a wide range of universal and non-universal types of behaviour for the decay of the density and density correlations.Comment: 27 pages, Latex, to appear in Z. Phys.

Dynamic Matrix Ansatz for Integrable Reaction-Diffusion Processes

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 \cite{Stin95a,Stin95b} 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 with Kawasaki dynamics. We solve the functional equations arising from the algebraic description and show non-perturbatively on the level of all finite-order correlation functions that the coupling to the infinite-temperature heat bath does not change the late-time behaviour of the zero-temperature process. The associated quantum chain is a non-hermitian anisotropic Heisenberg chain related to the seven-vertex model.Comment: Latex, 23 pages, to appear in European Physical Journal

An exactly solvable lattice model for inhomogeneous interface growth

We study the dynamics of an exactly solvable lattice model for inhomogeneous interface growth. The interface grows deterministically with constant velocity except along a defect line where the growth process is random. We obtain exact expressions for the average height and height fluctuations as functions of space and time for an initially flat interface. For a given defect strength there is a critical angle between the defect line and the growth direction above which a cusp in the interface develops. In the mapping to polymers in random media this is an example for the transverse Meissner effect. Fluctuations around the mean shape of the interface are Gaussian.Comment: 10 pages, late

Totally asymmetric exclusion process with long-range hopping

Generalization of the one-dimensional totally asymmetric exclusion process (TASEP) with open boundary conditions in which particles are allowed to jump $l$ sites ahead with the probability $p_l\sim 1/l^{\sigma+1}$ is studied by Monte Carlo simulations and the domain-wall approach. For $\sigma>1$ the standard TASEP phase diagram is recovered, but the density profiles near the transition lines display new features when $1<\sigma<2$. At the first-order transition line, the domain-wall is localized and phase separation is observed. In the maximum-current phase the profile has an algebraic decay with a $\sigma$-dependent exponent. Within the $\sigma \leq 1$ regime, where the transitions are found to be absent, analytical results in the continuum mean-field approximation are derived in the limit $\sigma=-1$.Comment: 10 pages, 9 figure

Infinite reflections of shock fronts in driven diffusive systems with two species

Interaction of a domain wall with boundaries of a system is studied for a class of stochastic driven particle models. Reflection maps are introduced for the description of this process. We show that, generically, a domain wall reflects infinitely many times from the boundaries before a stationary state can be reached. This is in an evident contrast with one-species models where the stationary density is attained after just one reflection.Comment: 11 pages, 8 eps figs, to appearin JPhysA 01.200

Diffusion of a hydrocarbon mixture in a one-dimensional zeolite channel: an exclusion model approach

Zeolite channels can be used as effective hydrocarbon traps. Earlier experiments (Czaplewski {\sl et al.}, 2002) show that the presence of large aromatic molecules (toluene) block the diffusion of light hydrocarbon molecules (propane) inside the narrow pore of a zeolite sample. As a result, the desorption temperature of propane is significantly higher in the binary mixture than in the single component case. In order to obtain further insight into these results, we use a simple lattice gas model of diffusion of hard-core particles to describe the diffusive transport of two species of molecules in a one-dimensional zeolite channel. Our dynamical Monte Carlo simulations show that taking into account an Arrhenius dependence of the single molecule diffusion coefficient on temperature, one can explain many significant features of the temperature programmed desorption profile observed in experiments. However, on a closer comparison of the experimental curve and our simulation data, we find that it is not possible to reproduce the higher propane current than toluene current near the desorption peak seen in experiment. We argue that this is caused by a violation of strict single-file behavior.Comment: Accepted for publication in the special issue "Diffusion in Micropores" of the journal Microporous and Mesoporous Material

Exact shock measures and steady-state selection in a driven diffusive system with two conserved densities

We study driven 1d lattice gas models with two types of particles and nearest neighbor hopping. We find the most general case when there is a shock solution with a product measure which has a density-profile of a step function for both densities. The position of the shock performs a biased random walk. We calculate the microscopic hopping rates of the shock. We also construct the hydrodynamic limit of the model and solve the resulting hyperbolic system of conservation laws. In case of open boundaries the selected steady state is given in terms of the boundary densities.Comment: 12 pages, 4 figure