Exact correlations in the one-dimensional coagulation-diffusion process by the empty-interval method

Journal of Statistical Mechanics Theory and Experiment (Impact Factor: 2.06). 01/2010; 2010(04). DOI: 10.1088/1742-5468/2010/04/P04002
Source: arXiv

ABSTRACT The long-time dynamics of reaction-diffusion processes in low dimensions is
dominated by fluctuation effects. The one-dimensional coagulation-diffusion
process describes the kinetics of particles which freely hop between the sites
of a chain and where upon encounter of two particles, one of them disappears
with probability one. The empty-interval method has, since a long time, been a
convenient tool for the exact calculation of time-dependent particle densities
in this model. We generalise the empty-interval method by considering the
probability distributions of two simultaneous empty intervals at a given
distance. While the equations of motion of these probabilities reduce for the
coagulation-diffusion process to a simple diffusion equation in the continuum
limit, consistency with the single-interval distribution introduces several
non-trivial boundary conditions which are solved for the first time for
arbitrary initial configurations. In this way, exact space-time-dependent
correlation functions can be directly obtained and their dynamic scaling
behaviour is analysed for large classes of initial conditions.

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    ABSTRACT: One-dimensional models of particles subjected to a coagulation-diffusion process are important in understanding non-equilibrium dynamics, and fluctuation-dissipation relation. We consider in this paper transport properties in finite and semi-infinite one-dimensional chains. A set of particles freely hop between nearest-neighbor sites, with the additional condition that, when two particles meet, they merge instantaneously into one particle. By imposing a localized source of particle-current at the origin and a non-symmetric hopping rate between the left and right directions (particle drift), we obtain exact expressions for the particle-density, current and coagulation rate in the continuum limit as function of time and position. These results are derived from the empty-interval-particle method, where the probability of finding an empty interval between two given sites is considered as the fundamental function for evaluating transport quantities. Closed equations are obtained for this quantity, depending on initial and boundary conditions, and a crossover behavior is studied between an algebraic and exponential decay of the particle density.
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