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By Eric Bertin

Introduction.- Equilibrium Systems.- Nonequlibrium Systems.- References

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A Gaussian tail C(u) ∼ e−αu (α > 0) would for instance be suitable here. 190) where we have also approximated the original integral by its extension over the whole real axis, which is justified by fast decay of C(u). 192) 1 − e−2γ t . 193) we eventually find t Var[v(t)] = e −2γ t 0 which is readily integrated into Var[v(t)] = 2γ Hence the variance starts from a zero value at t = 0 (the value v0 at t = 0 is nonrandom), and progressively grows until reaching the asymptotic limit /(2γ ). As Note that from this definition, is of the order of C(0)τcol .

3 Fokker–Planck Equation The Fokker–Planck equation basically describes the evolution of the probability distribution p(x, t) of a variable x obeying a Langevin equation. It can be derived in several ways, one of the simplest being to start from the above biased random walk problem, and to derive the continuous limit of the master equation, following the same lines as for the derivation of the diffusion equation—see Sect. 4. Starting from the biased random walk model of Sect. 207) W (n |n) = ν2 (1 − aqn ) if n = n − 1 ⎩ 0 otherwise.

173) where v is the velocity of the probe particle. Since the collisional force is strongly fluctuating, the basic idea is to decompose Fcoll into an average force which depends on the velocity, and a purely fluctuating (or noise) part: Fcoll = Fcoll + ξ(t). 174) Here also, the ensemble average . . is computed as an average over a large number of samples of the process, so that Fcoll a priori depends on time. By definition, the noise ξ(t) has zero mean, ξ(t) = 0. To proceed further, it is necessary to choose a specific model for both Fcoll and ξ(t).

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