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

The Liouville equation for the q-deformed 1-D classical harmonic oscillator is derived for twodefinitions of q-deformation. This derivation is achieved by using two different representations forthe q-deformed Hamiltonian of this oscillator corresponding to undeformed and deformed phasespaces. The resulting Liouville equation is solved by using the method of characteristics in order toobtain the classical probability distribution function for this system. The 2-D and 3-D behaviors ofthis function are then investigated using a computer visualization method. The results are comparedwith those for the classical anharmonic oscillator. This comparison reveals that there are somesimilarities between these two models, where the results for the q-deformed oscillator exhibitsimilar whorl shapes that evolve with time as for the anharmonic oscillator. It is concluded thatstudying the Liouville dynamics gives more details about the physical nature of q-deformation thanusing the equation of motion method. It is also concluded that this result could have reflections onthe interpretation of the quantized version of this q-deformed oscillator.