标题： 理想气体在弯曲时空中的研究 显示英文标题

标题： Study of ideal gases in curved spacetimes

摘要： 弯曲时空$M$对理想气体$N$粒子物理行为的影响是通过将系统的相空间视为余切丛$T^{*}M^{N}$的一个区域，并使用 Souriau 的李群{热力学}来定义相应的概率分布函数来分析的。 虽然相空间的构造尊重系统的可分离性，通过强制每个粒子满足所谓的质量壳约束，概率分布是通过将余切丛的自然辛结构与系统的哈密顿描述相结合来构建的。 这样，时空被引入统计中，其等距变换显得特别重要，因为分布由等距群在$T^{*}M^{N}$上作用的动量映射所参数化的李代数元素来参数化。 我们找到了 Gibbs 分布，在平坦时空的最简单情况下，它会简化为所谓的修正 Jüttner 分布，用于描述狭义相对论范围内的理想气体。 我们还利用一个 Killing 向量的范数定义了一个类似温度的函数，这使我们能够恢复所谓的 Tolman-Ehrenfest 效应。 作为一个具体例子，我们研究了 Schwarzschild 黑洞的外部区域，对于该区域，Schwarzschild 半径的幂级数展开使我们能够用闵可夫斯基时空的相应量来表示配分函数和 Gibbs 分布。 显示英文摘要

摘要： The influence of a curved spacetime $M$ on the physical behavior of an ideal gas of $N$ particles is analyzed by considering the phase space of the system as a region of the cotangent bundle $T^{*}M^{N}$ and using Souriau's Lie group {thermodynamics} to define the corresponding probability distribution function. While the construction of the phase space respects the separability of the system, by forcing each particle to satisfy the so-called mass-shell constraint, the probability distribution is constructed by mixing the natural symplectic structure of the cotangent bundle with a Hamiltonian description of the system. In this way, the spacetime is introduced into the statistics and its isometries turn out to be of special interest because the distributions are parameterized by the elements of the Lie algebra of the isometry group, through the momentum map of the action of the isometries in $T^{*}M^{N}$. We find the Gibbs distribution that, in the simplest case of a flat spacetime, reduces to the so-called modified J\"{u}ttner distribution, used to describe ideal gases in the regime of special relativity. We also define a temperature-like function using the norm of a Killing vector, which allows us to recover the so-called Tolman-Ehrenfest effect. As a particular example, we study the outer region of a Schwarzschild black hole, for which a power series expansion of the Schwarzschild radius allows us to represent the partition function and the Gibbs distribution in terms of the corresponding quantities of the Minkowski spacetime.