具有两种或更多种粒子的自引力气体的统计力学。
Statistical mechanics of the self-gravitating gas with two or more kinds of particles.
作者信息
de Vega H J, Siebert J A
机构信息
Laboratoire de Physique et Hautes Energies, Université Paris VI, Tour 16, 1er étage, 4 Place Jussieu, 75252 Paris Cedex 05, France.
出版信息
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Jul;66(1 Pt 2):016112. doi: 10.1103/PhysRevE.66.016112. Epub 2002 Jul 19.
We study the statistical mechanics of the self-gravitating gas at thermal equilibrium with two kinds of particles. We start from the partition function in the canonical ensemble, which we express as a functional integral over the densities of the two kinds of particles for a large number of particles. The system is shown to possess an infinite volume limit when (N(1),N(2),V)--> infinity, keeping N(1)/V(1/3) and N(2)/V(1/3) fixed. The saddle point approximation becomes here exact for (N1,N2,V)--> infinity. It provides a nonlinear differential equation for the densities of each kind of particle. For the spherically symmetric case, we compute the densities as functions of two dimensionless physical parameters: eta(1)=Gm(2)(1)N(1)/V(1/3)T and eta(2)=Gm(2)(2)N(2)/V(1/3)T (where G is Newton's constant, m(1) and m(2) the masses of the two kinds of particles, and T the temperature). According to the values of eta(1) and eta(2) the system can be either in a gaseous phase or in a highly condensed phase. The gaseous phase is stable for eta(1) and eta(2) between the origin and their collapse values. We have thus generalized the well-known isothermal sphere for two kinds of particles. The gas is inhomogeneous and the mass M(R) inside a sphere of radius R scales with R as M(R) proportional to R(d) suggesting a fractal structure. The value of d depends in general on eta(1) and eta(2) except on the critical line for the canonical ensemble in the (eta(1),eta(2)) plane where it takes the universal value d approximately 1.6 for all values of N(1)/N(2). The equation of state is computed. It is found to be locally a perfect gas equation of state. The thermodynamic functions (free energy, energy, entropy) are expressed and plotted as functions of eta(1) and eta(2). They exhibit a square root Riemann sheet with the branch points on the critical canonical line. The behavior of the energy and the specific heat at the critical line is computed. This treatment is further generalized to the self-gravitating gas with n types of particles.
我们研究了具有两种粒子的处于热平衡的自引力气体的统计力学。我们从正则系综中的配分函数出发，对于大量粒子，将其表示为两种粒子密度的泛函积分。当((N(1),N(2),V)\to\infty)，同时保持(N(1)/V^{1/3})和(N(2)/V^{1/3})固定时，该系统具有无限体积极限。对于((N1,N2,V)\to\infty)，鞍点近似在这里变得精确。它为每种粒子的密度提供了一个非线性微分方程。对于球对称情况，我们将密度计算为两个无量纲物理参数的函数：(\eta(1)=Gm(2)(1)N(1)/V^{1/3}T)和(\eta(2)=Gm(2)(2)N(2)/V^{1/3}T)（其中(G)是牛顿引力常数，(m(1))和(m(2))是两种粒子的质量，(T)是温度）。根据(\eta(1))和(\eta(2))的值，系统可以处于气相或高度凝聚相。对于(\eta(1))和(\eta(2))在原点与其坍缩值之间，气相是稳定的。因此，我们将著名的等温球推广到了两种粒子的情况。气体是不均匀的，半径为(R)的球内的质量(M(R))与(R)的关系为(M(R))与(R^d)成正比，这表明存在分形结构。一般来说，(d)的值取决于(\eta(1))和(\eta(2))，但在((\eta(1),\eta(2)))平面中对于正则系综的临界线上，对于所有(N(1)/N(2))的值，它都取通用值(d\approx1.6)。计算了状态方程。发现它在局部是理想气体状态方程。将热力学函数（自由能、能量、熵）表示并绘制为(\eta(1))和(\eta(2))的函数。它们呈现出一个平方根黎曼曲面，分支点在临界正则线上。计算了临界线上能量和比热的行为。这种处理方法进一步推广到了具有(n)种粒子的自引力气体。
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