A proposed common decay law for the quantum Loschmidt echo (fidelity) M(t) in the Chirikov Standard Map when the classical limit sits in mixed-type phase space, at the edge of chaos and in the chaotic sea. The law M(t) approx exp(-c0(t) sigma^nu(t) t^alpha(sigma,t)) treats the exponent alpha as the characterizing indicator of decay when c0 and nu are fixed, and distinguishes three decay stages (initial, transitive, followed). A statistical semi-classical method tracks the time-evolving probability P(s) of the classical action s and uses a Levy-distribution approximation to explain the logarithmic alpha-versus-perturbation relation in the chaotic sea, locate a critical perturbation against strong-chaos reference decay, and derive a time-scale rule tau proportional to sigma^(-gamma) with gamma generally not equal to 1 or 2.
A numerical and semi-classical study of how the quantum Loschmidt echo (fidelity) M(t) decays in the Chirikov Standard Map when the classical limit lies in mixed-type phase space, specifically at the…