By Jeff Markell

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X (1 - x) + x (1 - x) 17(t}. 35) The steady states of the system are found now, similarly to the previous example, by deriving the Fokker-Planck equation obeyed by the probability density of x(t}, solving it in the stationary case and calculating the extrema of the resulting steady probability density. In the particular case of 0: = 1/2 and >. 36) For values of the noise intensity below a certain critical value, D < Dc = 2, Xml is a maximum of the steady probability density, whereas for D > Dc, Xml becomes a minimum and Xm2 become maxima.

68) T O , ~ T-+oo A2(t')dt' - A 2 which has been renormalized so that C(O) = 1 and lim C(t) = t-+oo o. 69) The linear relaxation time TL of observable A(t) is defined as the characteristic time scale of C(t). Its value can be estimated as the time integral of the correlation function from 0 to 00, or as the largest relaxation time Ti (which measures the time scale of the decay of the time correlation function at long times). On the other hand, an initial relaxation time T] can be established to measure the decay of C (t) at short times.

90) 36 1. Introduction where the gaussian spatiotemporal noise", (x, t) is no longer white, but is assumed to be colored in space and/or time: x-x't-t') (",(x,t)",(x',t'))=2DC ( -"\-'-7- . 91) This stochastic process is characterized by three parameters: its intensity D, its correlation length"\, and its correlation time 7. \ and 7 measure the "distance" of the system from equilibrium. 1. The coupling constant r is related to the state of the environment surrounding the system. 92) where eis an additive internal noise and TJ is a new multiplicative external noise term, which can be assumed for simplicity to be white with intensity D.