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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 Langevin equation ： ウィキペディア英語版 Langevin equation In statistical physics, a Langevin equation (Paul Langevin, 1908) is a stochastic differential equation describing the time evolution of a subset of the degrees of freedom. These degrees of freedom typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation. == Brownian motion as a prototype == The original Langevin equation〔; reviewed by D. S. Lemons & A. Gythiel: ''Paul Langevin’s 1908 paper "On the Theory of Brownian Motion" ()'', Am. J. Phys. 65, 1079 (1997), DOI:10.1119/1.18725〕 describes Brownian motion, the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid, :$m\backslash frac\}\}+\backslash boldsymbol\backslash left(\; t\backslash right).$ The degree of freedom of interest here is the position ''x'' of the particle, ''m'' denotes the particle's mass. The force acting on the particle is written as a sum of a viscous force proportional to the particle's velocity (Stokes' law), and a ''noise term'' ''η(t)'' (the name given in physical contexts to terms in stochastic differential equations which are stochastic processes) representing the effect of the collisions with the molecules of the fluid. The force ''η(t)'' has a Gaussian probability distribution with correlation function :$\backslash left\backslash langle\; \backslash eta\_\backslash left(\; t\backslash right)\backslash eta\_\backslash left(\; t^\backslash right)\; \backslash right\backslash rangle\; =2\backslash lambda\; k\_T\backslash delta\; \_\backslash delta\; \backslash left(t-t^\backslash right)\; ,$ where ''kB'' is Boltzmann's constant and ''T'' is the temperature. The δ-function form of the correlations in time means that the force at a time ''t'' is assumed to be completely uncorrelated with it at any other time. This is an approximation; the actual random force has a nonzero correlation time corresponding to the collision time of the molecules. However, the Langevin equation is used to describe the motion of a "macroscopic" particle at a much longer time scale, and in this limit the δ-correlation and the Langevin equation become exact. Another prototypical feature of the Langevin equation is the occurrence of the damping coefficient λ in the correlation function of the random force, a fact also known as Einstein relation. 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「Langevin equation」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.