Statistical Properties of One-dimensional Time-dependent Hamilton Oscillators: From the Parametrically Adiabatic Driving to the Kicked Systems

Abstract

Time-dependence of a Hamilton system models its interaction with the environment and such systems are recently of great interest in many di˙erent contexts. We review the recent studies of the parametrically driven one-dimensional Hamilton oscillators, their time evolution and the statistical properties of the energy, starting from a microcanonical ensemble. In the case of adiabatic driving the energy remains sharply distributed (Dirac delta function), and its value follows the adiabatic law. If the driving is not adiabatic, the energy becomes distributed, as its value depends on the initial condition. For the linear oscillator this distribution is rigorously derived to be the arcsine distribution for any parametric driving law and the value of the adiabatic invariant (or action) at the mean energy always increases. The mean energy and the variance can be calculated for a few exactly solvable cases, and in the general case we apply the rigorous WKB method developed by Robnik and Romanovski (2000). In the nonlinear oscillators this universality is lost. The adiabatic invariant (action) at the mean energy can decrease for nonadiabatic but slow changes, while in another extreme case of fast parametric variation, in particular for a kick (jump), almost always increases, and so does the Gibbs entropy. This is shown for a number of exactly solvable cases, and a local analysis is o˙ered for the general oscillators. In the case of a monotonic driving of homogeneous power law potentials the nonlinear WKB method developed by Papamikos and Robnik (2012) is applied and shown to be highly accurate. The periodic kicking is also investigated. The relation to the statistical mechanics in the sense of Gibbs is explained.