非中性等离子体和带电粒子束中的混沌和连续极限
Chaos and the continuum limit in nonneutral plasmas and charged particle beams
摘要：
本文研究了带电粒子通过未屏蔽的r^-2力相互作用的近无碰撞N体系统中的离散效应，考虑到体势允许规则和混沌轨道。无论是对合奏和个人的轨道，随着N的增加，有一个光滑的收敛到一个连续极限。离散效应可以用高斯白噪声很好地建模，驰豫时间t_R = const *（N/log L）t_D，其中L是库仑对数，t_D是动态时间尺度。离散效应加速发射度的增长，最初本地化的团块。然而，即使考虑到离散效应，人们也可以区分在连续极限中感觉到规则势的轨道，使得发射率随时间按幂律增长，以及发射率呈指数增长的混沌轨道。对于足够大的N，人们可以区分两种不同的混沌。短程微混沌是一个普遍的特征，它与电荷之间的紧密接触有关，产生大的正李雅普诺夫指数X_N，即使体势是可积的，X_N也不随N的增加而减小。或者，有可能是更大规模的宏混沌，其特征在于较小的李雅普诺夫指数X_S，这是目前只有当体势是混沌的。传统的李雅普诺夫指数计算方法探测X_N，得出了一个自相矛盾的结论，即看起来几乎规则的、具有尖锐的傅立叶谱峰的N体轨道是“非常混沌的”。然而，由典型的粒子间间距设定的微观混沌的“范围”随着N的增加而减小，因此，对于大N，这种微观混沌尽管非常强，但在宏观上基本上是无关紧要的。一个更仔细的数值分析允许一个估计X_N和X_S。
英文摘要：
This paper examines discreteness effects in nearly collisionless N-body systems of charged particles interacting via an unscreened r^-2 force, allowing for bulk potentials admitting both regular and chaotic orbits. Both for ensembles and individual orbits, as N increases there is a smooth convergence towards a continuum limit. Discreteness effects are well modeled by Gaussian white noise with relaxation time t_R = const * (N/log L)t_D, with L the Coulomb logarithm and t_D the dynamical time scale. Discreteness effects accelerate emittance growth for initially localised clumps. However, even allowing for discreteness effects one can distinguish between orbits which, in the continuum limit, feel a regular potential, so that emittance grows as a power law in time, and chaotic orbits, where emittance grows exponentially. For sufficiently large N, one can distinguish two different `kinds' of chaos. Short range microchaos, associated with close encounters between charges, is a generic feature, yielding large positive Lyapunov exponents X_N which do not decrease with increasing N even if the bulk potential is integrable. Alternatively, there is the possibility of larger scale macrochaos, characterised by smaller Lyapunov exponents X_S, which is present only if the bulk potential is chaotic. Conventional computations of Lyapunov exponents probe X_N, leading to the oxymoronic conclusion that N-body orbits which look nearly regular and have sharply peaked Fourier spectra are `very chaotic.' However, the `range' of the microchaos, set by the typical interparticle spacing, decreases as N increases, so that, for large N, this microchaos, albeit very strong, is largely irrelevant macroscopically. A more careful numerical analysis allows one to estimate both X_N and X_S.
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