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Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem

Publication ,  Journal Article
Li, L; Liu, JG; Lu, J
Published in: Journal of Statistical Physics
October 1, 2017

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.

Duke Scholars

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Published In

Journal of Statistical Physics

DOI

EISSN

1572-9613

ISSN

0022-4715

Publication Date

October 1, 2017

Volume

169

Issue

2

Start / End Page

316 / 339

Related Subject Headings

  • Fluids & Plasmas
  • 51 Physical sciences
  • 49 Mathematical sciences
  • 02 Physical Sciences
  • 01 Mathematical Sciences
 

Citation

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Li, L., Liu, J. G., & Lu, J. (2017). Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem. Journal of Statistical Physics, 169(2), 316–339. https://doi.org/10.1007/s10955-017-1866-z
Li, L., J. G. Liu, and J. Lu. “Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem.” Journal of Statistical Physics 169, no. 2 (October 1, 2017): 316–39. https://doi.org/10.1007/s10955-017-1866-z.
Li L, Liu JG, Lu J. Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem. Journal of Statistical Physics. 2017 Oct 1;169(2):316–39.
Li, L., et al. “Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem.” Journal of Statistical Physics, vol. 169, no. 2, Oct. 2017, pp. 316–39. Scopus, doi:10.1007/s10955-017-1866-z.
Li L, Liu JG, Lu J. Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem. Journal of Statistical Physics. 2017 Oct 1;169(2):316–339.
Journal cover image

Published In

Journal of Statistical Physics

DOI

EISSN

1572-9613

ISSN

0022-4715

Publication Date

October 1, 2017

Volume

169

Issue

2

Start / End Page

316 / 339

Related Subject Headings

  • Fluids & Plasmas
  • 51 Physical sciences
  • 49 Mathematical sciences
  • 02 Physical Sciences
  • 01 Mathematical Sciences