Fractional stochastic differential equations satisfying fluctuation-dissipation theorem

TL;DR

This paper introduces a fractional stochastic differential equation model aligned with the fluctuation-dissipation theorem, establishing existence, ergodicity, and convergence properties relevant for systems with memory effects and subdiffusion.

Contribution

It proposes a novel FSDE model consistent with the fluctuation-dissipation theorem and proves key properties like existence and algebraic convergence to Gibbs measure.

Findings

Established existence of strong solutions for the FSDE model.

Proved algebraic convergence to Gibbs measure under linear forcing.

Validated the physical consistency of the model with fluctuation-dissipation theorem.

Abstract

We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the `fluctuation-dissipation theorem', the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the `fluctuation-dissipation theorem' is satisfied, and this verifies that satisfying `fluctuation-dissipation theorem' indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors.