GENERALIZED LANGEVIN EQUATION WITH CHAOTIC FORCE

The generalized Langevin equation with chaotic force is investigated: x(t) = - integral(0)(t) dt'phi(t,t')x(t') + f (t) where phi(t,t') = << f (t) f (t') >>/<< x(2) >>. force f(t) is defined by f(t) (y(n+1) - << y >>)/root tau for n tau < t less than or equal to (n + 1)tau (n = 0, 1, 2,...), where y(n+1) is a chaotic sequence: y(n+1) = F (y(n)). The time evolution of x(t), which is generated by the chaotic force, is discussed. The approach of the distribution function of x to a stationary distribution is studied. It is shown that the distribution function satisfies the Fokker-Planck type equation with the memory effect in the small tau limit. The relation between the invariant density of F (y) and the stationary distribution of x is discussed.