Finite-Planck's-Constant Scaling at Stochastic Transitions of Dynamical Systems

Quantum effects of the transition to chaos of an (almost) classical dynamical system are studied by renormalization-group methods. Planck's constant is a relevant variable which determines the time $t_h\propto {\hslash }^{-\frac{1}{\gamma }}$ at which the system crosses over to quantal behavior. The exponent $\gamma$ is 6.039 for the limit of period-doubling bifurcations and is 3.04 for the disappearance of the final Kolmogorov-Arnold-Moser trajectory in the standard map. For the standard map the scaling of the energy diffusion coefficient is also calculated.