Statistical properties of the energy in time-dependent homogeneous power law potentials

TL;DR

This paper analyzes the statistical behavior of 1D Hamiltonian systems with power law potentials under time-dependent scaling, showing how energy and variance grow while the action remains invariant, validated through detailed case studies.

Contribution

It applies a nonlinear WKB-like method to derive the asymptotic behavior of energy, variance, and action in time-dependent power law potentials, providing new analytical insights.

Findings

Mean energy and variance grow as powers of the scaling function a(t).

Action oscillates but remains constant at large times.

Theoretical predictions match detailed case studies.

Abstract

We study 1D Hamilton systems with homogeneous power law potential and their statistical behaviour, assuming the microcanonical distribution of the initial conditions and describing its change under monotonically increasing time-dependent function $a(t)$ (prefactor of the potential). Using the nonlinear WKB-like method by Papamikos and Robnik {\em J. Phys. A: Math. Theor. {\bf 44} (2012) 315102} and following a previous work by Papamikos G and Robnik M {\em J. Phys. A: Math. Theor. {\bf 45} (2011) 015206} we specifically analyze the mean energy, the variance and the adiabatic invariant (action) of the systems for large time $t\rightarrow\infty$ and we show that the mean energy and variance increase as powers of $a(t)$, while the action oscillates and finally remains constant. By means of a number of detailed case studies we show that the theoretical prediction is excellent which demonstrates the usefulness of the method in such applications.