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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 Kardar–Parisi–Zhang equation ： ウィキペディア英語版 Kardar–Parisi–Zhang equation The Kardar–Parisi–Zhang (KPZ) equation〔M. Kardar, G. Parisi, and Y.-C. Zhang, ''Dynamic Scaling of Growing Interfaces'', Physical Review Letters, Vol. 56, 889–892 (1986). (APS )〕 (named after its creators Mehran Kardar, Giorgio Parisi, and Yi-Cheng Zhang) is a non-linear stochastic partial differential equation. It describes the temporal change of the height $h(\backslash vec\; x,t)$ at place $\backslash vec\; x$ and time $t$. It is formally : $\backslash frac\; =\; \backslash nu\; \backslash nabla^2\; h\; +\; \backslash frac\; \backslash left(\backslash nabla\; h\backslash right)^2\; +\; \backslash eta(\backslash vec\; x,t)\; \backslash ;\; ,$ where $\backslash eta(\backslash vec\; x,t)$ is white Gaussian noise with average $\backslash langle\; \backslash eta(\backslash vec\; x,t)\; \backslash rangle\; =\; 0$ and second moment : $\backslash langle\; \backslash eta(\backslash vec\; x,t)\; \backslash eta(\backslash vec\; x\text{'},t\text{'})\; \backslash rangle\; =\; 2D\backslash delta^d(\backslash vec\; x-\backslash vec\; x\text{'})\backslash delta(t-t\text{'}).$ $\backslash nu$, $\backslash lambda$, and $D$ are parameters of the model and $d$ is the dimension. In one spatial dimension the KPZ equation corresponds to a stochastic version of the well known Burgers' equation, in a field $u(x,t)$ say, via the substitution $u=-\backslash lambda\backslash ,\; \backslash partial\; h/\backslash partial\; x$. By use of renormalization group techniques it has been conjectured that the KPZ equation is the field theory of many surface growth models, such as the Eden model, ballistic deposition, and the SOS model. A rigorous proof has been given by Bertini and Giacomin〔L. Bertini and G. Giacomin, ''Stochastic Burgers and KPZ equations from particle systems'', Comm. Math. Phys., Vol. 183, 571–607 (1997) ().〕 in the case of the SOS model. Many models in the field of interacting particle systems, such as the totally asymmetric simple exclusion process, also lie in the KPZ universality class. This class is characterised by models which, in one spatial dimension (1 + 1 dimension) have a roughness exponent ''α'' = 1/2, growth exponent ''β'' = 1/3 and dynamic exponent ''z'' = 3/2. In order to check if a growth model is within the KPZ class, one can calculate the width of the surface, $W(L,t)$, defined as : $W(L,t)=\backslash left\backslash langle\backslash frac1L\backslash int\_0^L\; \backslash big(\; h(x,t)-\backslash bar(t)\backslash big)^2\; \backslash ,\; dx\backslash right\backslash rangle^,$ where $\backslash bar(t)$ is the mean surface height at time t and L is the size of the system. For models within the KPZ class, the main properties of the surface $h(x,t)$ can be characterized by the Family–Vicsek scaling relation〔F. Family and T. Vicsek, ''Scaling of the active zone in the Eden process on percolation networks and the ballistic deposition model'', J. Phys. A: Math. Gen., Vol. 18, L75–L81 (1985) ().〕 of the roughness, where : $$ W(L,t) \approx L^ f(t/L^z), with a scaling function $f(u)$ satisfying : $$ f(u) \propto \begin u^ & \ u\ll 1 \\ 1 & \ u\gg1\end Due to the nonlinearity in the equation and the presence of space-time white-noise, the mathematical study of the KPZ equation has proven to be quite challenging: indeed, even without the nonlinear term, the equation reduces to the stochastic heat equation, whose solution is not differentiable in the space variable but verifies a Hölder condition with exponent < 1/2. Thus, the nonlinear term $\backslash left(\backslash nabla\; h\backslash right)^2$ is ill-defined in a classical sense. A breakthrough in the mathematical study of the KPZ equation was achieved by Martin Hairer, whose work on the KPZ equation 〔M. Hairer: Solving the KPZ equation, Annals of Mathematics, 178 (2013), no. 2, pp. 559–664. ()〕 earned him the 2014 Fields Medal. Hairer and Quastel〔(M Hairer and J Quastel (2014) Weak universality of the KPZ equation )〕 have recently shown that equations of the type : $\backslash frac\; =\; \backslash nu\; \backslash nabla^2\; h\; +\; P\backslash left(\backslash nabla\; h\backslash right)\; +\; \backslash eta(\backslash vec\; x,t)\; \backslash ;\; ,$ where $P$ is any even polynomial, lie in the KPZ universality class. == Sources == 〔 * A.-L. Barabási and H.E. Stanley, ''Fractal concepts in surface growth'' (Cambridge University Press, 1995) * Lecture Notes by Jeremy Quastel http://math.arizona.edu/~mathphys/school_2012/IntroKPZ-Arizona.pdf * Lecture Notes by Ivan Corwin http://arxiv.org/abs/1106.1596 * M. Hairer: Solving the KPZ equation, Annals of Mathematics, 178 (2013), no. 2, pp. 559–664. http://annals.math.princeton.edu/2013/178-2/p04 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「Kardar–Parisi–Zhang 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.