不同边界条件下$2d$伊辛模型的非确定性Hernial元元胞自动机比较研究
A comparative study of $2d$ Ising model at different boundary conditions using non-deterministic Hexagonal Cellular Automata
摘要：
用非确定性元胞自动机模拟了具有六角格子的$2d$伊辛模型的自旋系统。该方案的实施方法进行了概述，我们的结果显示出很好的近似精确的解析解。本文用$40\times40$六角点阵研究了$2d$伊辛模型中的相变，绝热的、周期性的、自反的、固定的$+1$和固定的$-1$，在没有外部施加磁场的情况下，以自旋的随机取向作为初始条件。计算了临界温度，在此温度以下自发磁化出现，以及其他物理量，如磁化强度，能量，比热，磁化率和熵与每个BCS。相变发生在$T^H_c$ = 1.5附近，这与Wannier和Houtappel的精确解析解结果很接近。我们通过计算接近$T>T^H_c$的零磁化线的点数，比较了五种不同类型BCS的每个单元的磁化行为。我们发现，周期性的，绝热的和自反的BCS给更接近的$T^H_c$的值比固定的$+1$和固定的$-1$ BCS与所有三个初始条件的晶格尺寸小于$50\times50$。然而，对于$50\times50$和$200\times200$之间的晶格尺寸，固定的$+1$ bc和固定的$-1$ bc给出了更接近于$T^H_c$的初始条件，其中所有自旋分别处于下配置和所有自旋分别处于上配置。
英文摘要：
The spin system of the $2d$ Ising model having a hexagonal-lattice is simulated using non-deterministic Cellular Automata. The method to implement this program is outlined and our results show a good approximation to the exact analytic solution. The phase transition in $2d$ Ising model is studied with a $40\times40$ hexagonal-lattice with five different boundary conditions (bcs) i.e., adiabatic, periodic, reflexive, fixed $+1$ and fixed $-1$ with random orientation of spins as initial conditions in the absence of an external applied magnetic field. The critical temperature below which the spontaneous magnetization appears as well as other physical quantities such as the magnetization, energy, specific heat, susceptibility and entropy with each of the bcs are calculated. The phase transition occurs around $T^H_c$ = 1.5 which approximates well with the result obtained from exact analytic solution by Wannier and Houtappel. We compare the behavior of magnetisation per cell for five different types of bcs by calculating the number of points close to the line of zero magnetization for $T>T^H_c$. We find that the periodic, adiabatic and reflexive bcs give closer approximation to the value of $T^H_c$ than fixed $+1$ and fixed $-1$ bcs with all three initial conditions for lattice size less than $50\times50$. However, for lattice size between $50\times50$ and $200\times200$, fixed $+1$ bc and fixed $-1$ bc give closer approximation to the $T^H_c$ with initial conditions in which all spins are in down configuration and all spins are in up configuration respectively.
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