Relaxation to the Invariant Density for Kicked Rotor

Abstract

The relaxation rates to the invariant density in the chaotic phase space component of the kicked rotor (standard map) are calculated analytically for a large stochasticity parameter, $K$. These rates are the logarithms of the poles of the matrix elements of the resolvent, $ \hat{R}(z)=(z-\hat{U})^{-1} $, of the classical evolution operator $\hat{U} $. The resolvent poles are located inside the unit circle. For hyperbolic systems this is a rigorous result, but little is known about mixed systems such as the kicked rotor. In this work, the leading relaxation rates of the kicked rotor are calculated in presence of noise, to the leading order in $1/\sqrt{K}$. Then the limit of vanishing noise is taken and the relaxation rates are found to be finite, corresponding to poles lying inside the unit circle. It is found that the slow relaxation rates, in essence, correspond to diffusion modes in the momentum direction. Faster relaxation modes intermix the motion in the momentum and the angle space. The slowest relaxation rate of distributions in the angle space is calculated analytically by studying the dynamics of inhomogeneities projected down to this space. The analytical results are verified by numerical simulations.