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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Liouville operator ： ウィキペディア英語版 Liouville's theorem (Hamiltonian) In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that the phase-space distribution function is constant along the trajectories of the system — that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. There are also related mathematical results in symplectic topology and ergodic theory. ==Liouville equations== The Liouville equation describes the time evolution of the phase space distribution function. Although the equation is usually referred to as the "Liouville equation", Josiah Willard Gibbs was the first to recognize the importance of this equation as the fundamental equation of statistical mechanics.〔J. W. Gibbs, "On the Fundamental Formula of Statistical Mechanics, with Applications to Astronomy and Thermodynamics." Proceedings of the American Association for the Advancement of Science, 33, 57-58 (1884). Reproduced in ''The Scientific Papers of J. Willard Gibbs, Vol II'' (1906), (pp. 16 ).〕 It is referred to as the Liouville equation because its derivation for non-canonical systems utilises an identity first derived by Liouville in 1838.〔(Liouville, Journ. de Math., 3, 349(1838) ).〕 Consider a Hamiltonian dynamical system with canonical coordinates $q\_i$ and conjugate momenta $p\_i$, where $i=1,\backslash dots,n$. Then the phase space distribution $\backslash rho(p,q)$ determines the probability $\backslash rho(p,q)\backslash ,d^nq\backslash ,d^n\; p$ that the system will be found in the infinitesimal phase space volume $d^nq\backslash ,d^n\; p$. The Liouville equation governs the evolution of $\backslash rho(p,q;t)$ in time $t$: :$\backslash frac=$ \frac +\sum_^n\left(\frac\dot_i +\frac\dot_i\right)=0. Time derivatives are denoted by dots, and are evaluated according to Hamilton's equations for the system. This equation demonstrates the conservation of density in phase space (which was Gibbs's name for the theorem). Liouville's theorem states that :''The distribution function is constant along any trajectory in phase space.'' A proof of Liouville's theorem uses the n-dimensional divergence theorem. This proof is based on the fact that the evolution of $\backslash rho$ obeys an n-dimensional version of the continuity equation: :$\backslash frac+\backslash sum\_^n\backslash left(\backslash frac+\backslash frac\backslash right)=0.$ That is, the tuplet $(\backslash rho,\; \backslash rho\backslash dot\_i,\backslash rho\backslash dot\_i)$ is a conserved current. Notice that the difference between this and Liouville's equation are the terms :$\backslash rho\backslash sum\_^n\backslash left($ \frac +\frac\right) =\rho\sum_^n\left( \frac -\frac\right)=0, where $H$ is the Hamiltonian, and Hamilton's equations have been used. That is, viewing the motion through phase space as a 'fluid flow' of system points, the theorem that the convective derivative of the density, $d\; \backslash rho/dt$, is zero follows from the equation of continuity by noting that the 'velocity field' $(\backslash dot\; p\; ,\; \backslash dot\; q)$ in phase space has zero divergence (which follows from Hamilton's relations). Another illustration is to consider the trajectory of a cloud of points through phase space. It is straightforward to show that as the cloud stretches in one coordinate – $p\_i$ say – it shrinks in the corresponding $q^i$ direction so that the product $\backslash Delta\; p\_i\; \backslash ,\; \backslash Delta\; q^i$ remains constant. Equivalently, the existence of a conserved current implies, via Noether's theorem, the existence of a symmetry. The symmetry is invariance under time translations, and the generator (or Noether charge) of the symmetry is the Hamiltonian. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Liouville's theorem (Hamiltonian)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.