Phase-space dynamics and ergodicity

J. Sethna
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Abstract

This chapter provides the mathematical justification for the theory of equilibrium statistical mechanics. A Hamiltonian system which is ergodic is shown to have time-average behavior equal to the average behavior in the energy shell. Liouville’s theorem is used to justify the use of phase-space volume in taking this average. Exercises explore the breakdown of ergodicity in planetary motion and in dissipative systems, the application of Liouville’s theorem by Crooks and Jarzynski to non-equilibrium statistical mechanics, and generalizations of statistical mechanics to chaotic systems and to two-dimensional turbulence and Jupiter’s great red spot.
相空间动力学和遍历性
本章提供平衡统计力学理论的数学证明。一个遍历的哈密顿系统的时间平均行为等于能量层的平均行为。刘维尔定理被用来证明在取这个平均值时使用相空间体积的合理性。练习探讨了行星运动和耗散系统中遍历性的分解,克鲁克斯和雅津斯基将刘维尔定理应用于非平衡统计力学,以及将统计力学推广到混沌系统、二维湍流和木星的大红斑。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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