Chavanis Pierre-Henri
Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, 31062 Toulouse, France.
Entropy (Basel). 2024 Sep 4;26(9):757. doi: 10.3390/e26090757.
After reviewing the peculiar thermodynamics and statistical mechanics of self-gravitating systems, we consider the case of a "binary star" consisting of two particles of size in gravitational interaction in a box of radius . The caloric curve of this system displays a region of negative specific heat in the microcanonical ensemble, which is replaced by a first-order phase transition in the canonical ensemble. The free energy viewed as a thermodynamic potential exhibits two local minima that correspond to two metastable states separated by an unstable maximum forming a barrier of potential. By introducing a Langevin equation to model the interaction of the particles with the thermal bath, we study the random transitions of the system between a "dilute" state, where the particles are well separated, and a "condensed" state, where the particles are bound together. We show that the evolution of the system is given by a Fokker-Planck equation in energy space and that the lifetime of a metastable state is given by the Kramers formula involving the barrier of free energy. This is a particular case of the theory developed in a previous paper (Chavanis, 2005) for Brownian particles in gravitational interaction associated with the canonical ensemble. In the case of a binary star (N=2), all the quantities can be calculated exactly analytically. We compare these results with those obtained in the mean field limit N→+∞.
在回顾了自引力系统独特的热力学和统计力学之后，我们考虑一个“双星”的情况，它由两个大小为 的粒子在半径为 的盒子中通过引力相互作用组成。该系统的热容量曲线在微正则系综中显示出一个负比热容区域，在正则系综中则被一阶相变所取代。作为热力学势的自由能呈现出两个局部最小值，它们对应于两个亚稳态，由一个不稳定的最大值分隔开，形成一个势垒。通过引入一个朗之万方程来模拟粒子与热库的相互作用，我们研究了系统在“稀薄”状态（粒子相距很远）和“凝聚”状态（粒子束缚在一起）之间的随机跃迁。我们表明，系统的演化由能量空间中的福克 - 普朗克方程给出，并且亚稳态的寿命由涉及自由能势垒的克莱默斯公式给出。这是先前一篇论文（沙瓦尼斯，2005年）中为与正则系综相关的引力相互作用的布朗粒子所发展理论的一个特殊情况。在双星（N = 2）的情况下，所有量都可以精确地解析计算。我们将这些结果与在平均场极限N→ +∞时获得的结果进行比较。
