Transcription of Liouville’s Theorem - Inside Mines
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Liouville’s Theorem - Inside Mines
liouville s TheoremPHGN 505 ReportbyMarc DvorakDecember 2, 2011 Prof. Tim OhnoColorado School of Mines1 Introduction and Basic FormulationLiouville s Theorem describes the evolution of the distribution function in phase space for aHamiltonian system. It is a fundamental theory in classical mechanics and has a straight-forward generalization to quantum systems. The basic idea of liouville s Theorem can bepresented in a basic, geometric fashion. The mathematical derivation is also rather straight-forward. liouville s Theorem applies to all Hamiltonian systems with number of single particle case is important but rather boring. The Theorem becomes more im-portant in classical statistical mechanics as the particle number becomes very large. Inparticular, liouville s Theorem , along with its partners ergodicity, entropy, and Poincare srecurrence Theorem , describe the equilibration and fluctuations of a dynamical system.
Thus, Liouville’s theorem states that the phase space density of a certain element as it moves in phase space is xed, df=dt= 0. One can return to the geometric …
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