Velocity auto correlation function of a confined Brownian particle

TL;DR

This paper provides exact calculations of the velocity autocorrelation function for a Brownian particle under a linear velocity-dependent force, exploring its behavior in various confining and driving conditions, revealing exponential decay, oscillations, and effects of external forces.

Contribution

It presents exact analytical results for the velocity autocorrelation function of a Brownian particle with a velocity-dependent force in different scenarios, including confinement and external driving.

Findings

Velocity autocorrelation decays exponentially and saturates over time.

In harmonic confinement, the autocorrelation shows damped oscillations.

External sinusoidal driving increases velocity correlations in the transient regime.

Abstract

Motivated by the simple models of molecular motor obeying a linear force-velocity relation, we have studied the stochastic dynamics of a Brownian particle in the presence of a linear velocity dependent force, $f_s(1-\frac{v}{v_0})$ where $f_{s}$ is a constant. The position and velocity auto correlation functions in different situations of the dynamics are calculated exactly. We observed that the velocity auto correlation function shows an exponentially decaying behaviour with time and saturates to a constant value in the time asymptotic limit, for a fixed $f_s$. It attains saturation faster with increase in the $f_{s}$ value. When the particle is confined in a harmonic well, the spectral density exhibits a symmetric behaviour and the corresponding velocity auto correlation function shows a damped oscillatory behaviour before decaying to zero in the long time limit. With viscous coefficient, a non-systematic variation of the velocity auto correlation function is observed. Further, in the presence of a sinusoidal driving force, the correlation in velocities increases with increase in the amplitude of driving in the transient regime. For the particle confined in a harmonic well, the correlation corresponding to the shift relative to the average position is basically the thermal contribution to the total position correlation. Moreover, in the athermal regime, the total correlation is entirely due to the velocity dependent force.