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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Notes
- 1.
The natural choice is t 0 = 0. Sometimes, it is convenient to take \(t_{0} = -\infty\).
- 2.
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.
- 3.
Note that this is the Green’s function for the damped harmonic oscillator.
- 4.
In fact, all we need is \(f,g \in D(\mathcal{L})\) and \(fg \in D(\mathcal{L})\).
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Pavliotis, G.A. (2014). Linear Response Theory for Diffusion Processes. In: Stochastic Processes and Applications. Texts in Applied Mathematics, vol 60. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1323-7_9
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