Mathematical foundation of nonequilibrium fluctuation-dissipation theorems for inhomogeneous diffusion processes with unbounded coefficients

TL;DR

This paper rigorously proves two types of nonequilibrium fluctuation-dissipation theorems for inhomogeneous diffusion processes with unbounded coefficients, expanding their applicability and clarifying conditions for use.

Contribution

It provides the first rigorous mathematical proofs of these FDTs for complex diffusion processes with unbounded coefficients, using advanced PDE and semigroup theories.

Findings

Proved two types of nonequilibrium FDTs for inhomogeneous diffusions.

Established the uniqueness of conjugate observables.

Clarified conditions and applicability ranges for the FDTs.

Abstract

Nonequilibrium fluctuation-dissipation theorems (FDTs) are one of the most important advances in stochastic thermodynamics over the past two decades. Here we provide rigorous mathematical proofs of two types of nonequilibrium FDTs for inhomogeneous diffusion processes with unbounded drift and diffusion coefficients by using the Schauder estimates for partial differential equations of parabolic type and the theory of weakly continuous semigroups. The FDTs proved in this paper apply to any forms of inhomogeneous and nonlinear external perturbations. Furthermore, we prove the uniqueness of the conjugate observables and clarify the precise mathematical conditions and ranges of applicability for the two types of FDTs. Examples are also given to illustrate the main results of this paper.