二维伊辛模型临界参数对晶格尺寸的依赖性
Dependence of critical parameters of 2D Ising model on lattice size
摘要：
对于二维伊辛模型，我们通过计算机模拟分析了自旋数对热力学特性的影响。我们比较了实验数据，使用Fisher-Kasteleyn算法与$N=l{\times}l$自旋和渐近Onsager解决方案（$N\to\infty$）的正方形晶格。我们推导出经验表达式的临界参数作为$N$的功能和推广的Onsager解决方案的情况下，有限尺寸的晶格。我们的自由能及其衍生物（内能，能量色散和热容）的解析表达式准确地描述了计算机模拟的结果。我们发现，当$N$增加时，临界点的热容随着$lnN$而增加。由于系统的尺寸有限，我们对临界温度的准确度进行了限制。此外，在有限维的情况下，我们得到的表达式描述的磁化强度和相关长度的温度依赖性。它们是在一个很好的定性协议与计算机模拟的结果，通过动态的大都会蒙特卡罗方法。
英文摘要：
For the 2D Ising model, we analyzed dependences of thermodynamic characteristics on number of spins by means of computer simulations. We compared experimental data obtained using the Fisher-Kasteleyn algorithm on a square lattice with $N=l{\times}l$ spins and the asymptotic Onsager solution ($N\to\infty$). We derived empirical expressions for critical parameters as functions of $N$ and generalized the Onsager solution on the case of a finite-size lattice. Our analytical expressions for the free energy and its derivatives (the internal energy, the energy dispersion and the heat capacity) describe accurately the results of computer simulations. We showed that when $N$ increased the heat capacity in the critical point increased as $lnN$. We specified restrictions on the accuracy of the critical temperature due to finite size of our system. Also in the finite-dimensional case, we obtained expressions describing temperature dependences of the magnetization and the correlation length. They are in a good qualitative agreement with the results of computer simulations by means of the dynamic Metropolis Monte Carlo method.
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