The OU$^2$ process: Characterising dissipative confinement in noisy traps

TL;DR

This paper introduces the OU$^2$ process, an extension of the Ornstein-Uhlenbeck model where the confining potential's stiffness fluctuates stochastically, revealing new statistical and thermodynamic behaviors relevant to out-of-equilibrium systems.

Contribution

The paper proposes the OU$^2$ process, a novel stochastic model with fluctuating potential stiffness, and analyzes its properties, extending the classical OU process to more realistic, noisy trapping scenarios.

Findings

Probability density exhibits power-law tails, unlike Gaussian decay.

Trapping, extreme value, and first passage statistics differ qualitatively from standard OU.

Entropy production shows distinct behavior due to stochastic potential fluctuations.

Abstract

The Ornstein-Uhlenbeck (OU) process describes the dynamics of Brownian particles in a confining harmonic potential, thereby constituting the paradigmatic model of overdamped, mean-reverting Langevin dynamics. Despite its widespread applicability, this model falls short when describing physical systems where the confining potential is itself subjected to stochastic fluctuations. However, such stochastic fluctuations generically emerge in numerous situations, including in the context of colloidal manipulation by optical tweezers, leading to inherently out-of-equilibrium trapped dynamics. To explore the consequences of stochasticity at this level, we introduce a natural extension of the OU process, in which the stiffness of the harmonic potential is itself subjected to OU-like fluctuations. We call this model the OU$^2$ process. We examine its statistical, dynamic, and thermodynamic properties through a combination of analytical and numerical methods. Importantly, we show that the probability density for the particle position presents power-law tails, in contrast to the Gaussian decay of the standard OU process. In turn, this causes the trapping behavior, extreme value statistics, first passage statistics, and entropy production of the OU$^2$ process to differ qualitatively from their standard OU counterpart. Due to the wide applicability of the standard OU process and of the proposed OU$^2$ generalisation, our study sheds light on the peculiar properties of stochastic dynamics in random potentials and lays the foundation for the refined analysis of the dynamics and thermodynamics of numerous experimental systems.