引力成团现象的热力学：$N$ -球体上的自引力气体$\mathbb{S}^{3}\subset\mathbb{R}^{4}$
Thermodynamics of gravitational clustering phenomena: $N$-body self-gravitating gas on the sphere $\mathbb{S}^{3}\subset\mathbb{R}^{4}$
摘要：
这项工作致力于热力学的引力集群，一个集体的现象有很大的相关性，在$N$体宇宙学问题。我们研究了定义在球$\mathbb{S}^{3}\subset \mathbb{R}^{4}$上的全同非相对论粒子的经典自引力气体，考虑了与这个几何空间相对应的引力相互作用。在孤立哈密顿系统的微正则描述下，结合连续近似和最速下降法进行分析。根据所得方程的数值解，引力团簇可以与两个微正则相变相联系。具有连续特征的第一相变与该模型的$SO(4)$对称性的破坏有关。第二类是引力坍缩，它的连续性或不连续性主要取决于引力势短程发散的正则化。我们还导出了该模型系统的热力学极限、Gibbs-Duhem关系的天体物理对应物、表征其相变的序参量和状态方程。另一个有趣的行为是存在负热容的状态，当引力的影响在能量足够低的情况下占主导地位时，就会出现这种状态。最后，我们评论的相关性，这些结果在天体物理学和宇宙学的情况下的研究。特别值得注意的是，由于物质分布的局部不均匀性，状态方程的引力修正。尽管在宇宙膨胀的研究中，这个特征被系统地忽略了，但同样的一个特征能够模拟归因于暗能量的效应：负压。
英文摘要：
This work is devoted to the thermodynamics of gravitational clustering, a collective phenomenon with a great relevance in the $N$-body cosmological problem. We study a classical self-gravitating gas of identical non-relativistic particles defined on the sphere $\mathbb{S}^{3}\subset \mathbb{R}^{4}$ by considering gravitational interaction that corresponds to this geometric space. The analysis is performed within microcanonical description of an isolated Hamiltonian system by combining continuum approximation and steepest descend method. According to numerical solution of resulting equations, the gravitational clustering can be associated with two microcanonical phase transitions. A first phase transition with a continuous character is associated with breakdown of $SO(4)$ symmetry of this model. The second one is the gravitational collapse, whose continuous or discontinuous character crucially depends on the regularization of short-range divergence of gravitation potential. We also derive the thermodynamic limit of this model system, the astrophysical counterpart of Gibbs-Duhem relation, the order parameters that characterize its phase transitions and the equation of state. Other interesting behavior is the existence of states with negative heat capacities, which appear when the effects of gravitation turn dominant for energies sufficiently low. Finally, we comment the relevance of some of these results in the study of astrophysical and cosmological situations. Special interest deserves the gravitational modification of the equation of state due to the local inhomogeneities of matter distribution. Although this feature is systematically neglected in studies about Universe expansion, the same one is able to mimic an effect that is attributed to the dark energy: a negative pressure.
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