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 $M_{\mathrm{ij}}$ of the stability matrix M. For globally chaotic dynamics, $\lambda$ tends to a unique value (the usual Lyapunov exponent ${\lambda }_{\infty })$ 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 $P(\lambda ;\infty )=\delta \left(\lambda -{\lambda }_{\infty }\right).\mathrm{}$ Our method also applies to the tail of the distribution, which determines the growth rates of moments of $M_{\mathrm{ij}}.$ The results are also applicable to the problem of wave-function localization in a disordered one-dimensional potential.