Survival of interacting Brownian particles in crowded 1D environment

TL;DR

This paper provides an exact analytical solution for the diffusive behavior and escape dynamics of interacting Brownian particles in a crowded 1D environment with an absorbing boundary, revealing how interactions influence long-term dynamics.

Contribution

It introduces a novel exact solution method for interacting particles in 1D with absorbing boundaries, highlighting the impact of interactions on escape times and dynamical exponents.

Findings

Interactions induce entropic repulsive forces affecting particle escape.

Interaction alters long-time asymptotic dynamical exponents.

Exact probabilistic descriptions of particle survival and escape times.

Abstract

We investigate a diffusive motion of a system of interacting Brownian particles in quasi-one-dimensional micropores. In particular, we consider a semi-infinite 1D geometry with a partially absorbing boundary and the hard-core inter-particle interaction. Due to the absorbing boundary the number of particles in the pore gradually decreases. We present the exact analytical solution of the problem. Our procedure merely requires the knowledge of the corresponding single-particle problem. First, we calculate the simultaneous probability density of having still a definite number $N-k$ of surviving particles at definite coordinates. Focusing on an arbitrary tagged particle, we derive the exact probability density of its coordinate. Secondly, we present a complete probabilistic description of the emerging escape process. The survival probabilities for the individual particles are calculated, the first and the second moments of the exit times are discussed. Generally speaking, although the original inter-particle interaction possesses a point-like character, it induces entropic repulsive forces which, e.g., push the leftmost (rightmost) particle towards (opposite) the absorbing boundary thereby accelerating (decelerating) its escape. More importantly, as compared to the reference problem for the non-interacting particles, the interaction changes the dynamical exponents which characterize the long-time asymptotic dynamics. Interesting new insights emerge after we interpret our model in terms of a) diffusion of a single particle in a $N$-dimensional space, and b) order statistics defined on a system of $N$ independent, identically distributed random variables.