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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 Fokker-Planck equation ： ウィキペディア英語版 Fokker–Planck equation In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. It is named after Adriaan Fokker〔A. D. Fokker, ''Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld'', Ann. Phys. 348 (4. Folge 43), 810–820 (1914).〕 and Max Planck〔 M. Planck, Sitz.ber. Preuß. Akad. (1917).〕 and is also known as the Kolmogorov forward equation (diffusion), named after Andrey Kolmogorov, who first introduced it in a 1931 paper.〔Andrei Kolmogorov, "On Analytical Methods in the Theory of Probability", 448-451, (1931), (in German).〕 When applied to particle position distributions, it is better known as the Smoluchowski equation. The case with zero diffusion is known in statistical mechanics as Liouville equation. The first consistent microscopic derivation of the Fokker–Planck equation in the single scheme of classical and quantum mechanics was performed〔N. N. Bogolyubov (jr) and D. P. Sankovich (1994). "(N. N. Bogolyubov and statistical mechanics )". ''Russian Math. Surveys'' 49(5): 19—49.〕 by Nikolay Bogoliubov and Nikolay Krylov.〔N. N. Bogoliubov and N. M. Krylov (1939). ''Fokker–Planck equations generated in perturbation theory by a method based on the spectral properties of a perturbed Hamiltonian''. Zapiski Kafedry Fiziki Akademii Nauk Ukrainian SSR 4: 81–157 (in Ukrainian).〕 The Smoluchowski equation (after Marian Smoluchowski) is the Fokker–Planck equation for the probability density function of the particle positions of Brownian particles.〔Dhont, An Introduction to Dynamics of Colloids, p. 183〕 == One dimension == In one spatial dimension ''x'', for an Itō process driven by the standard Wiener process $W\_t$ and described by the stochastic differential equation (SDE) :$dX\_t\; =\; \backslash mu(X\_t,t)dt\; +\; \backslash sigma(X\_t,t)dW\_t$ with drift $\backslash mu(X\_t,t)$ and diffusion coefficient $D(X\_t,t)\; =\; \backslash sigma^2(X\_t,t)/2$, the Fokker–Planck equation for the probability density $p(x,t)$ of the random variable $X\_t$ is :$\backslash frac\; p(x,t)\; =\; -\backslash frac\backslash left(p(x,t)\backslash right)\; +\; \backslash frac\backslash left(p(x,t)\backslash right).$ In the following, use $\backslash sigma=\backslash sqrt$. Define the Infinitesimal Generator $\backslash mathcal$ (the following can be found in Ref.) : :$$ \mathcalp(X_t)=\lim_\frac1\left(\mathbb\big(\mathcalp(x) \right ) \mathbb_(x|x') \, dx = \int p(x)\, \partial_t \mathbb_(x|x')\, dx From here, the Kolmogorov Backward Equation can be deduced. If we instead use the adjoint operator of $\backslash mathcal$, $\backslash mathcal^$, defined such that :$$ \int \left(\mathcal p(x) \right ) \mathbb_(x|x') dx = \int p(x) \left(\mathcal^ \mathbb_(x|x') \right ) dx then we arrive to the Kolmogorov Forward Equation, or Fokker-Planck Equation which, simplifying the notation $p(x,t)=\backslash mathbb\_(x|x\text{'})$, in its differential form reads :$$ \mathcal^ p(x,t) = \partial_t p(x,t) Remains the issue of defining explicitly $\backslash mathcal$. This can be done taking the expectation from the integral form of the Itō's lemma, :$$ \mathbb(p(X_t))= p(X_0)+\mathbb(\int_0^t\left(\partial_t + \mu\partial_x + \frac\partial_x^2 \right) p(X_) dt' ) Notice that the part that depends on $dW\_t$ vanished because of the martingale property. Then, for a particle subject to an Itō equation, using :$$ \mathcal = \mu\partial_x + \frac\partial_x^2 it can be easily calculated, using integration by parts, that :$$ \mathcal^ = -\partial_x(\mu \cdot) + \frac12\partial_x^2(\sigma^2 \cdot) which bring us to the Fokker–Planck equation, :$$ \partial_t p(x,t) = -\partial_x \left(\mu(x,t)\cdot p(x,t)\right) + \partial_x^2\left( \frac \, p(x,t)\right). While the Fokker-Planck equation is used with problems where the initial distribution is known, if the problem is to know the distribution at previous times, the Feynman-Kac formula can be used, which is a consequence of the Kolmogorov backward equation. The stochastic process defined above in the Itō sense can be rewritten within the Stratonovich convention as a Stratonovich SDE: :$dX\_t\; =\; \backslash left(-\; \backslash frac\; \backslash fracD(X\_t,t)\backslash right)dt\; +\; \backslash sqrt\; \backslash circ\; dW\_t.$ It includes an added noise-induced drift term due to diffusion gradient effects if the noise is state-dependent. This convention is more often used in physical applications. Indeed, it is well known that any solution to the Stratonovich SDE is a solution to the Itō SDE. The zero drift equation with constant diffusion can be considered as a model of classical Brownian motion: $\backslash frac\; p(x,t)\; =D\_0\backslash frac\backslash left()$ This model has discrete spectrum of solutions if the condition of fixed boundaries is added for $\backslash$: $p(0,t)=p(L,t)=0$ $p(x,0)=p\_0(x)$ It has been shown that in this case an analytical spectrum of solutions allows deriving a local uncertainty relation for the coordinate-velocity phase volume: $\backslash Delta\; x\; \backslash Delta\; v\; \backslash geqslant\; D\_0$ Here $D\_0$ is a minimal value of a corresponding diffusion spectrum $$, while $\backslash Delta\; x$ and $\backslash Delta\; v$ represent the uncertainty of coordinate-velocity definition. 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「Fokker–Planck 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.