不同边界条件下$2d$伊辛模型的元胞自动机比较研究
A comparative study of $2d$ Ising model at different boundary conditions using Cellular Automata
摘要：
利用元胞自动机，模拟了$2d$伊辛模型对应的自旋系统在各种边界条件下的运动.用具有五种不同bcs的$64\times64$正方形晶格研究了在没有磁场的情况下自发磁化的出现，周期性的、绝热的、自反的、固定的（$+1$或$-1$）BCS，具有三个初始条件（所有自旋向上、所有自旋向下和自旋的随机取向）。在$2d$伊辛模型的背景下，我们已经计算了磁化强度，能量，比热，磁化率和熵与每一个bcs和观察到的相变发生在$T_c$ = 2.269附近获得的Onsager。我们比较不同类型的BCS的磁化强度与温度的行为，通过计算在各种晶格尺寸的$T>T_c$后接近零磁化强度线的点的数量。我们观察到，周期性的，绝热的和自反的bcs给出更接近的值$T_c$比固定+1和固定-1 bcs与所有三个初始条件的晶格尺寸小于$70\times70$。然而，对于$70\times70$和$100\times100$之间的晶格尺寸，固定的+1 bc和固定的-1 bc分别给出了与初始条件下的全自旋向下构型和全自旋向上构型的$T_c$更接近的近似。
英文摘要：
Using Cellular Automata, we simulate spin systems corresponding to $2d$ Ising model with various kinds of boundary conditions (bcs). The appearance of spontaneous magnetization in the absence of magnetic field is studied with a $64\times64$ square lattice with five different bcs, i.e., periodic, adiabatic, reflexive, fixed ($+1$ or $-1$) bcs with three initial conditions (all spins up, all spins down and random orientation of spins). In the context of $2d$ Ising model, we have calculated the magnetisation, energy, specific heat, susceptibility and entropy with each of the bcs and observed that the phase transition occurs around $T_c$ = 2.269 as obtained by Onsager. We compare the behaviour of magnetisation vs temperature for different types of bcs by calculating the number of points close to the line of zero magnetisation after $T>T_c$ at various lattice sizes. We observe that the periodic, adiabatic and reflexive bcs give closer approximation to the value of $T_c$ than fixed +1 and fixed -1 bcs with all three initial conditions for lattice size less than $70\times70$. However, for lattice size between $70\times70$ and $100\times100$, fixed +1 bc and fixed -1 bc give closer approximation to the $T_c$ with initial conditions all spin down configuration and all spin up configuration respectively.
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