Statistics of finite-time Lyapunov exponents in a random time-dependent potential

The sensitivity of trajectories over finite-time intervals t to perturbations of the initial conditions can be associated with a finite-time Lyapunov exponent lambda, obtained from the elements M-ij of the stability matrix M. For globally chaotic dynamics, lambda tends to a unique value (the usual Lyapunov exponent lambda(infinity)) for almost all trajectories as t is sent to infinity, but for finite t it depends on the initial conditions of the trajectory and can be considered as a statistical quantity. We compute for a particle moving in a randomly time-dependent, one-dimensional potential how the distribution function P(lambda;t) approaches the limiting distribution P(lambda;infinity) = delta(lambda-lambda(infinity)). Our method also applies to the tail of the distribution, which determines the growth rates of moments of M-ij. The results are also applicable to the problem of wave-function localization in a disordered one-dimensional potential.