高度非线性伊辛模型与社会隔离
Highly Nonlinear Ising Model and Social Segregation
摘要：
对统计物理学中常用的随机场伊辛模型的相互作用能进行了修正，即在通常的伊辛模型的能量上增加一个非线性项S^n，其中S是相邻自旋之和，n= 0，1，3，5，7，9，11.在谢林的城市隔离模型中，这种修正对应于住房价格取决于邻近地区。在不同温度、晶格尺寸、磁场、近邻数和不同时间间隔下的模拟结果表明，除了n=3违反普适性原理和对应态定律外，所有n的结果都是相似的。为了找到临界温度，对于大的n，我们不再从所有的自旋平行开始，而是从随机的构型开始，以促进自旋翻转。然而，在所有情况下，我们都有一个居里温度，只有在低于这个居里温度时才有相分离或长程偏析，并且它可以用一个简单的公式近似：当n=1时，Tc与1+m成正比，而当n >> 1时，Tc与m大致成正比。
英文摘要：
The usual interaction energy of the random field Ising model in statistical physics is modified by complementing the random field by added to the energy of the usual Ising model a nonlinear term S^n were S is the sum of the neighbor spins, and n=0,1,3,5,7,9,11. Within the Schelling model of urban segregation, this modification corresponds to housing prices depending on the immediate neighborhood. Simulations at different temperatures, lattice size, magnetic field, number of neighbors and different time intervals showed that results for all n are similar, expect for n=3 in violation of the universality principle and the law of corresponding states. In order to find the critical temperatures, for large n we no longer start with all spins parallel but instead with a random configuration, in order to facilitate spin flips. However, in all cases we have a Curie temperature with phase separation or long-range segregation only below this Curie temperature, and it is approximated by a simple formula: Tc is proportional to 1+m for n=1, while Tc is roughly proportional to m for n >> 1.
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