经典微正则系综的Vlasov连续极限
The Vlasov continuum limit for the classical microcanonical ensemble
摘要：
对于具有弱正则对相互作用势的经典Hamilton N体系统，当N在固定的有界区域趋于无穷大时，能量E在N中按二次标度与e成正比，则Boltzmann遍历系综熵S（N，E）有渐近展开式S（N，E）= -NlogN + s（e）N + o（N）;这里，Nlog N项在起源上是组合的并且独立于重新标度的哈密顿算子，而s（e）是每个粒子的系统特定玻尔兹曼熵，即-s（e）是“能量”为e的理想气体在外加场和自生场共同作用下的玻尔兹曼H函数的最小值。本文还证明了n点边际系综测度的任一极限点都是H函数极小化单点函数的n重乘积的线性凸叠加。证明是直接的，在这个意义上，（a）研究的是E到S（E）的映射，而不是它的逆S到E（S）;（b）没有调用微正则测度狄拉克（E-H）的正则化;（c）没有通过正则系综绕道。无论微正则系综和正则系综是否等价，证明都成立。
英文摘要：
For classical Hamiltonian N-body systems with mildly regular pair interaction potential it is shown that when N tends to infinity in a fixed bounded domain, with energy E scaling quadratically in N proportional to e, then Boltzmann's ergodic ensemble entropy S(N,E) has the asymptotic expansion S(N,E) = - N log N + s(e) N + o(N); here, the N log N term is combinatorial in origin and independent of the rescaled Hamiltonian while s(e) is the system-specific Boltzmann entropy per particle, i.e. -s(e) is the minimum of Boltzmann's H-function for a perfect gas of "energy" e subjected to a combination of externally and self-generated fields. It is also shown that any limit point of the n-point marginal ensemble measures is a linear convex superposition of n-fold products of the H-function-minimizing one-point functions. The proofs are direct, in the sense that (a) the map E to S(E) is studied rather than its inverse S to E(S); (b) no regularization of the microcanonical measure Dirac(E-H) is invoked, and (c) no detour via the canonical ensemble. The proofs hold irrespective of whether microcanonical and canonical ensembles are equivalent or not.
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