Single file diffusion meets Feynman path integral

The path-integral representation of the Smoluchowski equation is exploited to explore the stochastic dynamics of a tagged Brownian particle within an interacting system where hydrodynamic effects are neglected. In particular, this formalism is applied to a particle system confined to a one-dimensional infinite line aiming to investigate the single-file diffusion phenomenon in this scenario. In particular, the path-integral method is contrasted against the standard many-particle Langevin equation for a system of interacting Brownian particles in a harmonic chain model, exhibiting excellent agreement; in this case of study a formula defined on the whole time-scale for the mean-square displacement, in the thermodynamic limit, is found for the tracer particle in terms of Bessel functions, recovering also the single-file regime. Additionally, a Brownian particle system with paramagnetic interactions is considered near crystallization where the total interaction potential is roughly a harmonic potential. Taking advantage of the path-integral formalism a simple perturbation treatment is carried out to investigate the single file diffusion behaviour when temperature is increased from the crystal phase, where we also validate our findings using Brownian dynamics simulation.

Automated summary

The study examines the stochastic dynamics of a tagged Brownian particle in an interacting system using the path-integral representation and contrasts it with the standard equations. It investigates the single-file diffusion phenomenon and the mean-square displacement of the tracer particle in terms of Bessel functions. Additionally, it explores the behavior of a Brownian particle system with paramagnetic interactions near crystallization using a perturbation treatment and validates the findings through Brownian dynamics simulation.