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2014 | OriginalPaper | Chapter

9. Linear Response Theory for Diffusion Processes

Author : Grigorios A. Pavliotis

Published in: Stochastic Processes and Applications

Publisher: Springer New York

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Abstract

In this chapter, we study the effect of a weak external forcing on a system at equilibrium. The forcing moves the system away from equilibrium, and we are interested in understanding the response of the system to this forcing. We study this problem for ergodic diffusion processes using perturbation theory. In particular, we develop linear response theory. The analysis of weakly perturbed systems leads to fundamental results such as the fluctuation–dissipation theorem and to the Green–Kubo formula, which enables us to calculate transport coefficients.

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previous chapter Derivation of the Langevin Equation

The natural choice is t ₀ = 0. Sometimes, it is convenient to take \(t_{0} = -\infty\).

Note that to be consistent with the notation that we have used previously in the book, in (9.1a) we use \(\mathcal{L}^{{\ast}}\) instead of \(\mathcal{L}\), since the operator that appears in the Liouville or the Fokker–Planck equation is the adjoint of the generator.

Note that this is the Green’s function for the damped harmonic oscillator.

In fact, all we need is \(f,g \in D(\mathcal{L})\) and \(fg \in D(\mathcal{L})\).

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Title: Linear Response Theory for Diffusion Processes
Author: Grigorios A. Pavliotis
Publisher: Springer New York
Book: Stochastic Processes and Applications
Print ISBN: 978-1-4939-1322-0

Electronic ISBN: 978-1-4939-1323-7

Copyright Year: 2014
DOI: https://doi.org/10.1007/978-1-4939-1323-7_9

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