三维晶格统计模型的一种封闭的近似解

引用四元数(或转动群的双值表示)推广处理二维晶格统计模型的无规行走法,为三维简立方晶格求得了统计配分函数的一个封闭的近似解析表达式。它在从零到无限大的温度区间内都成立,可以过渡到正确的二维结果,具有一个温度奇点、非对数型的比热奇异性和较好的高温展开系数。

A CLOSED-FORM APPROXIMATION FOR THE 3-DIMENSIONAL ISING MODEL

An analytical expression for the partition function of the 3-dimensional Ising Model is obtained (see formula (32)). It is approximate, but has several rational features: (1) It is valid on the whole temperature range from zero to infinity and has only one singular point at u_c = 0.2612 in comparision to 0.22 from the series expan-sion; (2) It leads to a nonlogarithmic singularity for the specific heat; (3) It can be reduced into the exact Onsager 2-dimensional solution for planar rectangle lattice in the anisotropic case (see formula (34)) ; (4) It has a high temperature expansion, the first two coefficients are correct and the higher ones diminish systematically but remain positive.