A matrix form solution of the multi-dimensional generalized Langevin equation in the quadratic potential

TL;DR

This paper derives an exact matrix analytical solution for the multi-dimensional generalized Langevin equation with quadratic potentials, enhancing understanding of stochastic dynamics in harmonic systems.

Contribution

It introduces a novel matrix form solution for the multi-dimensional generalized Langevin equation with quadratic potentials, using inverse Laplace transform techniques.

Findings

Provides explicit probability distribution expressions

Demonstrates the solution with specific examples

Clarifies the limitations of the approach

Abstract

In this research paper, we present an exact matrix form analytical solution of the multi-dimensional generalized Langevin equation with quadratic potentials. Our investigation provides detailed expressions for the two-dimensional probability distribution and extends the understanding of the dynamics governed by harmonic potentials. By utilizing the inverse Laplace transformation, we offer a precise method to solve these equations, corroborated by specific examples. This study contributes to the fundamental understanding of stochastic processes in multi-dimensional systems with harmonic potentials and clarifies the limitations of our approach. While the findings are specific to quadratic potentials, they provide a robust framework for exploring related phenomena within this context.