A Paradox in the Langevin Equation with Long-Time Noise Correlations

TL;DR

This paper investigates the effects of long-time correlated noise on the Langevin equation, revealing a paradoxical diminishing velocity variance and discussing implications for diffusion processes.

Contribution

It introduces a novel analysis of the Langevin equation driven by power-law correlated noise, highlighting non-stationary behavior and comparing stochastic and deterministic noise models.

Findings

Velocity variance diminishes over time under long-time correlated noise.

Algebraic distributions can cause non-stationary effects in the system.

Implications for diffusion processes are discussed.

Abstract

We solve the generalized Langevin equation driven by a stochastic force with power-law autocorrelation function. A stationary Markov process has been applied as a model of the noise. However, the resulting velocity variance does not stabilizes but diminishes with time. It is shown that algebraic distributions can induce such non-stationary affects. Results are compared to those obtained with a deterministic random force. Consequences for the diffusion process are also discussed.