A Study of Persistence in Different Non Equilibrium Systems

Abstract

Considerable interest has been generated recently in understanding the statistics of first newlinepassage events in spatially extended non-equilibrium systems. A persistence probability newlineP(t), is defined as the probability that the position of the step edge at a point along a newlinefluctuating step does not return to its initial value (at time t = 0) over time t is found in newlinethese studies to decay in time as a power-law, P(t) and#8764; t and#8722;and#952; , for large t, where and#952; is the so- newlinecalled persistence exponent. Similar power-law behavior of the persistence probability newlinehas also been found in experiments for other physical processes. The persistence proba- newlinebility has been obtained both analytically and numerically for a large class of stochastic newlineprocesses, Markovian as well as non-Markovian. For single particle systems such as the newlineBrownian motion, which is also Markovian in nature, the persistence probability is easy newlineto calculate since the stationary correlator of such a process decays exponentially at all newlinetimes. For many body systems where the field and#966; has a space dependence, the calculation newlineof the zero crossing probability becomes complicated. newlineIn the first part ( Chapter 2 and Chapter 3) of the thesis, we investigate the persis- newlinetence probability p(t) of the position of a Brownian particle with shape asymmetry in newlinetwo dimensions. We explicitly consider two cases diffusion of a free particle and that newlineof a harmonically trapped particle. The latter is particularly relevant in experiments newlinethat use trapping and tracking techniques to measure the displacements. We provide newlineanalytical expressions of p(t) for both the scenarios and show that in the absence of newlineshape asymmetry, the results reduce to the case of an isotropic particle. The analytical newlineexpressions of p(t) are further validated against a numerical simulation of the underly- newlineing overdamped dynamics. We also illustrate that p(t) can be a measure to determine newlinethe shape asymmetry of a colloid and the translational and rotational diffusivities can newlinebe estimated from the measured persistence probabilit