IWNET12
IWNET12
Spatial correlation of concentration uctuations in reaction diusion
problems by the Gillespie algorithm
Jorge Luis Hita1, José María Ortiz de Zárate1
1 Department of Applied Physics I, Universidad Complutense de Madrid
Abstract
During the last decades the spectrum of thermal uctuations in non-equilibrium thermodynamic systems has
been widely studied. It has been shown that static correlations have generically long range behaviour, while the
uctuations around equilibrium states are always short ranged (except in the neighbourhood of critical points).
These investigations may be helpful to elucidate whether or not there exist thermodynamic potentials out of
equilibrium, and could exclude the local potentials.
Among the non-equilibrium thermodynamic states that have been studied theoretically, there are examples of
binary reaction-diusion systems in which the inverse chemical reaction and the direct chemical reaction occur
through dierent paths, avoiding chemical equilibrium. It is hard to develop an experimental set up to verify
the theoretical conclusions in this case, because it is dicult to nd a two-component system whose kinetics
is as simple as the ones theoretically studied. A numerical simulation could be a better way to conrm the
theory for this kind of problems. More over, there are so far very few simulations that focus on non-equilibrium
uctuations. This is why a simulation of equilibrium and non-equilibrium reaction-diusion problem is of current
interest.
The Gillespie algorithm is a Montecarlo method that simulates eciently a system of chemical reactions through
the chemical master equation. Although it was developed in 1976, the interest in this algorithm has grown
during the last years, particularly because of its proven utility in Biophysical problems. The Gillespie algorithm
was originally conceived to describe chemical kinetics, but it can be applied to diusion processes as well, by
assuming that the diusion of one particle from one domain of the system to other can be handled as a chemical
reaction with a certain probability.
We have performed extensive simulations of one dimensional reaction diusion problems using the Gillespie
algorithm. We studied the system with an equilibrium chemical reaction (a simple association-dissociation),
and as an example of a non-equilibrium reaction we studied the WOH model. We have conrmed numerically
that for the equilibrium chemical reaction, spatial correlations are short ranged, while for the non-equilibrium
chemical reaction spatial correlations are long raged, as predicted by the theory.
E-mail: jmortizz@fis.ucm.es

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