Persistence of a Brownian particle in a Time Dependent Potential

TL;DR

This paper studies how the likelihood of a Brownian particle remaining in a certain state evolves over time when confined by a potential that weakens and eventually disappears, analyzing different decay behaviors.

Contribution

It provides analytical and numerical insights into the persistence probability of Brownian particles in time-dependent harmonic potentials with exponential and algebraic decay.

Findings

Persistence at short and long times is independent of decay parameters for exponential relaxation.

Long-time dynamics depend on the decay exponent in algebraic decay.

Constructed the persistence probability using the two-time correlation function.

Abstract

We investigate the persistence probability of a Brownian particle in a harmonic potential, which decays to zero at long times -- leading to an unbounded motion of the Brownian particle. We consider two functional forms for the decay of the confinement, an exponential and an algebraic decay. Analytical calculations and numerical simulations show, that for the case of the exponential relaxation, the dynamics of Brownian particle at short and long times are independent of the parameters of the relaxation. On the contrary, for the algebraic decay of the confinement, the dynamics at long times is determined by the exponent of the decay. Finally, using the two-time correlation function for the position of the Brownian particle, we construct the persistence probability for the Brownian walker in such a scenario.