Fluctuating fronts as correlated extreme value problems: An example of Gaussian statistics.
In this paper, we view fluctuating fronts made of particles on a one-dimensional lattice as an extreme value problem. The idea is to denote the configuration for a single front realization at time t by the set of co-ordinates [k(i) (t)] identical with [ k(1) (t) , k(2) (t) ,..., k(N (t) ) (t) ] of the constituent particles, where N (t) is the total number of particles in that realization at time t . When [k(i) (t)] are arranged in the ascending order of magnitudes, the instantaneous front position can be denoted by the location of the rightmost particle, i.e., by the extremal value k(f) (t) =max [ k(1) (t) , k(2) (t) ,..., k(N (t) ) (t) ] . Due to interparticle interactions, [k(i) (t)] at two different times for a single front realization are naturally not independent of each other, and thus the probability distribution P (k(f) ) (t) (based on an ensemble of such front realizations) describes extreme value statistics for a set of correlated random variables. In view of the fact that exact results for correlated extreme value statistics are rather rare, here we show that for a Fermionic front model in a reaction-diffusion system, P (k(f) ) (t) is Gaussian. In a Bosonic front model, however, we observe small deviations from the Gaussian.
Panja
Debabrata
D
Institute for Theoretical Physics, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands.