Irregular gyration of a two-dimensional random-acceleration process in a confining potential

TL;DR

This paper investigates the complex stochastic dynamics of a 2D particle in a confining potential with unequal noise amplitudes, revealing irregular rotational behavior, non-zero average torque, and distinctive probability distributions of angular momentum and velocity.

Contribution

It introduces a detailed analysis of the irregular gyration and statistical properties of a coupled 2D random-acceleration process with unequal noise amplitudes in a confining potential.

Findings

Non-zero average torque causes irregular oscillations in angular momentum and velocity.

The probability density function of angular momentum has exponential tails and converges to a uniform distribution over time.

The velocity's pdf has heavy power-law tails, with the mean velocity being the only finite moment.

Abstract

We study the stochastic dynamics of a two-dimensional particle assuming that the components of its position are two coupled random-acceleration processes evolving in a confining parabolic potential and are the subjects of independent Gaussian white noises with different amplitudes (temperatures). We determine the standard characteristic properties, i.e., the moments of position's components and their velocities, mixed moments and two-time correlations, as well as the position-velocity probability density function (pdf). We show that if the amplitudes of the noises are not equal, then the particle experiences a non-zero (on average) torque, such that the angular momentum L and the angular velocity W have non-zero mean values. Both are (irregularly) oscillating with time t, such that the characteristics of a rotational motion are changing their signs. We also evaluate the pdf-s of L and W and show that the former has exponential tails for any fixed t, and hence, all moments. In addition, in the large-time limit this pdf converges to a uniform distribution with a diverging variance. The pdf of W possesses heavy power-law tails such that the mean W is the only existing moment. This pdf converges to a limiting form which, surprisingly, is completely independent of the amplitudes of noises.