Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem
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© 2017, Springer Science+Business Media, LLC. 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.
Published Version (Please cite this version)10.1007/s10955-017-1866-z
Publication InfoLi, Lei; Liu, JG; & Lu, Jianfeng (2017). Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem. Journal of Statistical Physics, 169(2). pp. 316-339. 10.1007/s10955-017-1866-z. Retrieved from http://hdl.handle.net/10161/15666.
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William W. Elliott Assistant Research Professor
This author no longer has a Scholars@Duke profile, so the information shown here reflects their Duke status at the time this item was deposited.
Associate Professor of Mathematics
Jianfeng Lu is an applied mathematician interested in mathematical analysis and algorithm development for problems from computational physics, theoretical chemistry, materials science and other related fields.More specifically, his current research focuses include:Electronic structure and many body problems; quantum molecular dynamics; multiscale modeling and analysis; rare events and sampling techniques.
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