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

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  • Chen, Xian
  • Jia, Chen

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.

Suggested Citation

  • Chen, Xian & Jia, Chen, 2020. "Mathematical foundation of nonequilibrium fluctuation–dissipation theorems for inhomogeneous diffusion processes with unbounded coefficients," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 171-202.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:1:p:171-202
    DOI: 10.1016/j.spa.2019.02.005
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    References listed on IDEAS

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    1. Van den Broeck, C. & Esposito, M., 2015. "Ensemble and trajectory thermodynamics: A brief introduction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 418(C), pages 6-16.
    2. Chen Jia & Minping Qian, 2016. "Nonequilibrium Enhances Adaptation Efficiency of Stochastic Biochemical Systems," PLOS ONE, Public Library of Science, vol. 11(5), pages 1-19, May.
    3. Ge, Hao & Jia, Chen & Jiang, Da-Quan, 2017. "Cycle symmetry, limit theorems, and fluctuation theorems for diffusion processes on the circle," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1897-1925.
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