Statistical properties of 1D time-dependent Hamiltonian systems: from the adiabatic limit to the parametrically kicked systems

Abstract

We consider 1D time-dependent Hamiltonian systems and their statistical properties, namely the time evolution of microcanonical distributions, whose properties are very closely related to the existence and preservation of the adiabatic invariants. We review the elements of the recent developments by Robnik and Romanovski (during 2006–2008) for the entirely general 1D time-dependent linear oscillator and try to generalize the results to the 1D nonlinear Hamilton oscillators, in particular the power-law potentials like e.g. the quartic oscillator. Furthermore, we consider the limit opposite to the adiabatic limit, namely parametrically kicked 1D Hamiltonian systems. Even for the linear oscillator, interesting properties are revealed: an initial kick disperses the microcanonical distribution to a spread one, but an anti-kick at an appropriate moment of time can annihilate it, kicking it back to the microcanonical distribution. The case of periodic parametric kicking is also interesting. Finally, we propose that in the parametric kicking of a general 1D Hamilton system, the average value of the adiabatic invariant always increases, which we prove for the power-law potentials. We find that the approximation of kicking is good for quite long times of the parameter variation, up to the order of not much less than one period of the oscillator. We also look at the behavior of the quartic oscillator for the case of the kick and anti-kick, and also the periodic kicking.

Statistical properties of one-dimensional parametrically kicked Hamilton systems

“…In this work we have analyzed the statistical properties of one degree of freedom parametrically kicked Hamiltonian systems, which is the extreme case of fast time dependence, being the opposite extreme to an adiabatic, infinitely slow, changing. As such, the parametric kick behaviour is a very good approximation for the behaviour of the systems under very fast changes in the system parameters, within a time scale of less than one period of oscillation, as has been demonstrated already by Papamikos and Robnik [1]. The most natural ensemble, and the most important one is the microcanonical ensemble, because if we have a large ensemble of identical systems with the same ("prepared") energy, and we do not have any further information about them, the uniform distribution with respect to the canonical angle ("the phases") is the most appropriate one.…”

“…It has been realized by Gibbs [30], Hertz [32] and Einstein [31], that the fundamental quantity of classical statistical mechanics is Ω(E), defined in (1). It is precisely the adiabatic invariant of the system, which is conserved under adiabatic infinitely slow changes, as proven by Paul Hertz [32] for ergodic systems.…”

“…This led us to demonstrate analytically by a rigorous calculation for the case of homogeneous power law potentials (of which the quartic oscillator and the harmonic oscillators are two special cases), that the adiabatic invariant at the average final energy indeed increases under a parametric kick. Therefore, we [1] have put forward a Conjecture (henceforth called PR Conjecture) that the adiabatic invariant for an initial microcanonical ensemble at the mean energy always increases under a parametric kick. The purpose of the present work is to investigate the validity and the conditions under which it is true.…”

“…One example is the kicked rotator (standard map) and many other time periodic systems [20,22]. More recent works included the periodic parametric kicking of the quartic oscillator [1] and the periodic linear driving (sawtooth driving law) of the quartic oscillator [23].…”

Entropy and the second law for driven, or quenched, thermally isolated systems

“…For the present crucial micro-canonical initial conditions, earlier work in this direction includes the observation that for quenched one-dimensional non-linear oscillators the Gibbs volume entropy can decrease [19,20]. For large systems, Sasa and Komatsu show first that for a small quench the change in volume entropy is not extensively negative.…”