Monroe James L
Department of Physics, Penn State University--Beaver Campus, 100 University Dr., Monaca, Pennsylvania 15061, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Aug;68(2 Pt 2):027103. doi: 10.1103/PhysRevE.68.027103. Epub 2003 Aug 12.
The mean-field approximation and the Bethe approximation are two of the most often used approximations when one wants to obtain approximations of the phase diagrams and the critical temperature of lattice spin systems. Both can and have often been generalized to produce what are known as cluster mean-field and Bethe approximations. Generally, three characteristics are associated with these approximations. First, they give upper bounds to the critical temperature; second, considering larger clusters will result in better approximations; and third, the Bethe approximation is better than the corresponding mean-field approximation. We show what we believe to be a rather surprising result that, for one-dimensional Ising models with algebraically decaying interactions falling off slowly enough, the Bethe cluster approximations violate all three of these characteristics.
当人们想要获得晶格自旋系统的相图和临界温度的近似值时，平均场近似和贝叶斯近似是最常使用的两种近似方法。这两种方法都可以并且经常被推广以产生所谓的团簇平均场和贝叶斯近似。一般来说，这些近似方法有三个特点。第一，它们给出了临界温度的上限；第二，考虑更大的团簇会得到更好的近似；第三，贝叶斯近似比相应的平均场近似更好。我们展示了一个我们认为相当惊人的结果，即对于具有足够缓慢代数衰减相互作用的一维伊辛模型，贝叶斯团簇近似违反了所有这三个特点。
