Development and regression of a large fluctuation

Abstract

We study the evolution leading to (or regressing from) a large fluctuation in a Statistical Mechanical system. We introduce and study analytically a simple model of many identically and independently distributed microscopic variables $n_m$ ($m=1,M$) evolving by means of a master equation. We show that the process producing a non-typical fluctuation with a value of $N=\sum_{m=1}^Mn_m$ well above the average $\langle N\rangle$ is slow. Such process is characterized by the power-law growth of the largest possible observable value of $N$ at a given time $t$. We find similar features also for the reverse process of the regression from a rare state with $N\gg \langle N\rangle$ to a typical one with $N \simeq \langle N\rangle$.