Transport and scaling analysis in the relativistic standard map

We investigate some statistical and transport properties of the relativistic standard map. Through the Hamiltonian of a wave packet under an electric potential, we are able to obtain a relativistic version of the standard map, where there are two control parameters that rule the dynamics: K, which is the classical intensity parameter, and β, which controls the relativity. The phase space is mixed and exhibits confined local chaos for β near unity, approaching integrability. As β is diminished (entering the semi-classical regime), diffusion in the action variable begins to occur. However, the phase space loses its axial symmetry and an invariant curve appears to limit the diffusion as β gets smaller. We investigate the diffusion in the action variable as a function of the number of iterations, showing that the root mean square action grows initially and bends towards a saturation regime for long times. Scaling properties were established for this behavior as a function of β, and a perfect collapse of the curves was obtained, indicating scaling invariance. Additionally, we investigated the transport properties concerning the survival probability of initial conditions. The decay rates of the survival probability are mainly exponential, followed by power-law tails. As we vary the value of β, the escape rates become slower and also obey a scaling law in their decay.