无碰撞系统中的熵产生。II。任意相空间占有数
Entropy Production in Collisionless Systems. II. Arbitrary Phase-Space Occupation Numbers
摘要：
本文分析了两种确定自引力系统平衡态的热力学方法。一个是林登-贝尔熵最大化分析，它引入了剧烈的弛豫。由于我们没有使用斯特林近似，这是无效的小占领数，我们的系统有有限的质量，不像林登-贝尔的等温球。（代替Stirling，我们对$\ln{x!}$使用非常精确的光滑近似。）第二个分析扩展熵产生极值的自引力系统，也没有使用斯特林近似。除了Lynden-Bell（LB）统计家族的特征在于相空间中的不相容原理，并设计用于治疗无碰撞系统，我们也适用于麦克斯韦-玻尔兹曼（MB）的家庭，没有不相容原理，因此代表碰撞系统的两种方法。我们隐含地假设所有的相空间都是同样可达的。我们推导了这两个族的熵产生表达式，并给出了熵产生的极值条件。令人惊讶的是，我们的分析表明，极值熵产生率的结果在系统中具有最大熵，在LB和MB统计。换句话说，两种热力学方法导致相同的平衡结构。
英文摘要：
We present an analysis of two thermodynamic techniques for determining equilibria of self-gravitating systems. One is the Lynden-Bell entropy maximization analysis that introduced violent relaxation. Since we do not use the Stirling approximation which is invalid at small occupation numbers, our systems have finite mass, unlike Lynden-Bell's isothermal spheres. (Instead of Stirling, we utilize a very accurate smooth approximation for $\ln{x!}$.) The second analysis extends entropy production extremization to self-gravitating systems, also without the use of the Stirling approximation. In addition to the Lynden-Bell (LB) statistical family characterized by the exclusion principle in phase-space, and designed to treat collisionless systems, we also apply the two approaches to the Maxwell-Boltzmann (MB) families, which have no exclusion principle and hence represent collisional systems. We implicitly assume that all of the phase-space is equally accessible. We derive entropy production expressions for both families, and give the extremum conditions for entropy production. Surprisingly, our analysis indicates that extremizing entropy production rate results in systems that have maximum entropy, in both LB and MB statistics. In other words, both thermodynamic approaches lead to the same equilibrium structures.
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