The Liouville Dynamics of the q-Deformed 1-D Classical Harmonic Oscillator

TL;DR

This paper derives and solves the Liouville equation for the q-deformed 1-D classical harmonic oscillator, revealing dynamics similar to anharmonic oscillators and providing insights into the physical nature of q-deformation.

Contribution

It introduces a novel derivation of the Liouville equation for the q-deformed oscillator using two different Hamiltonian representations and compares the dynamics with classical anharmonic oscillators.

Findings

q-deformed oscillator exhibits whorl-shaped probability distributions

Liouville dynamics reveal details about q-deformation's physical nature

Results suggest similarities between q-deformed and anharmonic oscillators

Abstract

The Liouville equation for the q-deformed 1-D classical harmonic oscillator is derived for two definitions of q-deformation. This derivation is achieved by using two different representations for the q-deformed Hamiltonian of this oscillator corresponding to undeformed and deformed phase spaces. The resulting Liouville equation is solved by using the method of characteristics in order to obtain the classical probability distribution function for this system. The 2-D and 3-D behaviors of this function are then investigated using a computer visualization method. The results are compared with those for the classical anharmonic oscillator. This comparison reveals that there are some similarities between these two models, where the results for the q-deformed oscillator exhibit similar whorl shapes that evolve with time as for the anharmonic oscillator. It is concluded that studying the Liouville dynamics gives more details about the physical nature of q-deformation than using the equation of motion method. It is also concluded that this result could have reflections on the interpretation of the quantized version of this q-deformed oscillator.