Fluctuating Fronts as Correlated Extreme Value Problems: An Example of Gaussian Statistics

Details Fluctuating Fronts as Correlated Extreme Value Problems: An Example of Gaussian Statistics Fluctuating Fronts as Correlated Extreme Value Problems: An Example of Gaussian Statistics

2004-01-23

arxiv.org/abs/cond-mat/0401471v3

Phys. Rev. E 70, 036101 (2004)

Physics

Condensed Matter

Statistical Mechanics

6 pages, 3 figures, miniscule changes on the previous version, to appear in Phys. Rev. E

Scientific paper

10.1103/PhysRevE.70.036101

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 $\{k_i(t)\}\equiv[k_1(t),k_2(t),...,k_{N(t)}(t)]$ 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 $k_f(t)=\text{max}[k_1(t),k_2(t),...,k_{N(t)}(t)]$. 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.

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