By Eric Bertin
Introduction.- Equilibrium Systems.- Nonequlibrium Systems.- References
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Extra info for A Concise Introduction to the Statistical Physics of Complex Systems
R. E. Fisher, Ann. Phys. F. W. Anderson, J. Phys. F 5, 965 (1975) B. Derrida, Phys. Rev. Lett. 45, 79 (1980) B. Derrida, Phys. Rev. B 24, 2613 (1981) Chapter 2 Non-stationary Dynamics and Stochastic Formalism In the first part of this book, we have considered the stationary properties of physical systems composed of a large number of particles, using as fundamental statistical object the joint distribution of all the degrees of freedom of the system (for instance positions and velocities, or spin variables).
For instance, if the walk starts with probability 1 from x0 = 0, then all subsequent averages xt = 0. Let us now compute the variance of the process, defined as Var(xt ) = xt2 − xt 2 . 158) We assume for simplicity that xt = 0, so that Var(xt ) = xt2 (the generalization to xt = 0 is however straightforward). 159) = 1. Computing the ensemble average of Eq. 160) using the fact that xt depends only on t with t < t, so that xt and t are independent 2 = xt2 + 1, so that xt2 = random variables. 1 Markovian Stochastic Processes and Master Equation 39 x02 + t.
175) where the function C(u) converges rapidly to zero when u → ∞. Altogether, Eq. 173 can be rewritten as: dv = −γ v + ξ(t). 176) Such an equation is called a linear Langevin equation. For simplicity, we take as initial condition a fixed value v(0) = v0 . We first note, computing the ensemble average of Eq. 177) that the ensemble-averaged velocity v(t) obeys the same equation as the nonaveraged velocity, except that noise is now absent. , Fcoll = −γ v − γ3 v 3 . The solution of Eq. 177 is a decaying exponential: v(t) = v0 e−γ t .