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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 Kicked rotator ： ウィキペディア英語版 Kicked rotator The kicked rotator, also spelled as kicked rotor, is a prototype model for chaos and quantum chaos studies. It describes a particle that is constrained to move on a ring (equivalently: a rotating stick). The particle is kicked periodically by an homogeneous field (equivalently: the gravitation is switched on periodically in short pulses). The model is described by the Hamiltonian : $\backslash mathcal(p,x,t)=\; \backslash fracp^2\; +\; K\; \backslash cos(x)\; \backslash sum\_^\backslash infty\; \backslash delta(t-n)$ Where $\backslash textstyle\; \backslash delta$ is the Dirac delta function, $\backslash textstyle\; x$ is the angular position (for example, on a ring), taken modulo $\backslash textstyle\; 2\backslash pi$, $\backslash textstyle\; p$ is the momentum, and $\backslash textstyle\; K$ is the kicking strength. Its dynamics are described by the standard map : $p\_=p\_n+K\backslash sin(x\_n),\backslash ;\; \backslash ;\; x\_=x\_n+p\_$ With the caveat that $\backslash textstyle\; p$ is not periodic, as it is in the standard map. See more details and references on the standard map here, or better in the associated (Scholarpedia entry ). ==Main properties (classical)== In the classical analysis, if the kicks are strong enough, $K>K\_c\backslash approx0.971635\backslash dots$, the system is chaotic and has a positive Maximal Lyapunov exponent (MLE). The averaged diffusion of the momentum-squared is a useful parameter in characterizing the delocalization of nearby trajectories. The inductive result of the standard map yields the following equation for momentum〔Y. Zheng, D.H. Kobe, Chaos, Solitons and Fractals 28, 385 (2006)〕 : $p(n)\; =\; p(0)\; +\; K\backslash sum\_^\backslash sin(x(i))$ The diffusion can then be calculated by squaring the difference in momentum after the $^$ kick and the initial momentum, and then averaging, yielding : $\backslash left\; \backslash langle\; ^\; \backslash right\; \backslash rangle\; =\; \backslash left\; \backslash langle\; ^\; \backslash right\; \backslash rangle\; =\; K^2\backslash sum\_^\backslash left\; \backslash langle\; ^(x(i))\; \backslash right\; \backslash rangle\; +\; K^2\; \backslash sum\_^\backslash left\; \backslash langle\; \backslash sin(x(i))\backslash sin(x(j))\; \backslash right\; \backslash rangle$ In the chaotic domain, the momenta at different time points may be anywhere from entirely uncorrelated to highly correlated. If they are assumed uncorrelated due to the quasi-random behaviour, the sum involving the cross-terms $\backslash textstyle\; i\backslash neq\; j$ is neglected. In this limit, since the first term is a sum of $\backslash textstyle\; n$ terms all equalling $\backslash textstyle\; \backslash frac$, the momentum diffusion becomes $\backslash textstyle\; \backslash left\; \backslash langle\; ^\; \backslash right\; \backslash rangle\; =\; \backslash fracK^2n$. However, if the momenta at different time points are assumed highly correlated, the sum involving the cross-terms is not neglected, and so it contributes more terms equalling $\backslash textstyle\; \backslash frac$. Altogether, there are $\backslash textstyle\; n^2$ terms to sum, all of the form $\backslash textstyle\; \backslash sim\; \backslash left\; \backslash langle\; ^x\; \backslash right\; \backslash rangle\; =\; \backslash frac$. This gives an upper bound on the momentum diffusion of $\backslash textstyle\; \backslash left\; \backslash langle\; ^\; \backslash right\; \backslash rangle\; =\; \backslash fracK^2n^2$. Therefore, in the chaotic domain, the momentum diffusion is between : $\backslash fracK^2n\; \backslash leq\; \backslash left\; \backslash langle\; ^\; \backslash right\; \backslash rangle\; \backslash leq\; \backslash fracK^2n^2$ That is, the momentum diffusion in the chaotic domain has somewhere between a linear and a quadratic dependence on the number of kicks. An exact expression for $\backslash textstyle\; \backslash left\; \backslash langle\; ^\; \backslash right\; \backslash rangle$ can be obtained in principle by calculating the sums explicitly for an ensemble of trajectories. 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「Kicked rotator」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.