Statistical properties of 1D parametrically kicked Hamilton systems

Abstract: We study the 1D Hamilton systems and their statistical behaviour, assuming the initial microcanonical distribution and describing its change under a parametric kick, which by definition means a discontinuous jump of a control parameter of the system. Following a previous work by Papamikos and Robnik (J. Phys. A: Math. Theor. 44 (2012) 315102) we specifically analyze the change of the adiabatic invariant (the action) of the system under a parametric kick: A conjecture has been put forward that the change of the action at the mean energy always increases, which means, for the given statistical ensemble, that the entropy in the mean increases. By means of a detailed analysis of a great number of case studies we show that the conjecture largely is satisfied, except if either the potential is not smooth enough, or if the energy is too close to a stationary point of the potential (separatrix in the phase space). Very fast changes in a time dependent system quite generally can be well described by such a picture and by the approximation of a parametric kick, if the change of the parameter is sufficiently fast and takes place on the time scale of less than one oscillation period.