Numerical Studies of Phase Transitions in Fluid Membranes

(1)定义了一种具有表面张力、弯曲弹性和流动性的液膜模型。模型的哈密顿量被认为是基本粒子模型的Nambu弦的离散作用量。(2)对哈密顿量与文献(1)的哈密顿量相同的晶膜模型进行了MC模拟，发现晶体模型发生了预期的二级相变。由这一结果和(1)的结果发现，具有表面张力和弯曲刚性的膜中形状涨落的二级相变与膜的流动性无关。(3)我们研究了作为Polyakov刚体弦离散模型的流体膜的普通模型。MC分析发现，普通的流体膜模型中存在形状起伏的二级相变。(4)对普通的结晶膜模型进行MC模拟，研究了形状起伏的相变。通过比较晶体模型和文献(3)的计算结果，我们发现膜中的流动性增强了相变，并且在相变时流体模型的Hausdorff维大于晶体模型的Hausdorff维。(5)对普通晶体模型进行了朗之万法模拟和蒙特卡罗模拟。用朗之万法和蒙特卡罗法求出了相变的临界指数和相变时的Housdorff维数，两种方法得到的结果基本一致。从而证实了我们所用的两种技术对晶膜的模拟都是正确的。

(1) We defined a model of fluid membrane which has a surface tension, a bending elasticity and a fluidity. The Hamiltonian of the model is considered as a discrete action of Nambu string for a model of elementary particles. We found by MC that this model undergoes a second order phase transition.(2) We performed MC simulations for a model of crystalline membranes whose Hamiltonian is identical with that of (1), and found that the crystalline model undergoes a second order phase transition as expected. It is found from this result together with that of (1) that the second order phase transition of shape fluctuations in the membrane, which has a surface tension and a bending rigidity, is independent of the fluidity of membranes.(3) We studied the ordinary model of fluid membrane that is considered as a discrete model of Polyakov rigid string. We found by MC that there is a second order phase transition of shape fluctuations in the ordinary model of fluid membranes.(4) Performing MC simulations for the ordinary model of crystalline membranes, we studied the phase transitions of shape fluctuations. By comparing the results of crystalline model with those of (3), we found that the fluidity in membranes strengthen the phase transition and that the Hausdorff dimension of the fluid model is larger that that of the crystalline model at the phase transition.(5) Langevin simulations as well as the Monte Carlo were performed for the ordinary crystalline model. The critical exponent of the phase transition and the Housdorff dimension at the phase Transition were obtained by both Langevin and MC, and results of the two techniques were almost identical with each other. Thus it was confirmed that the both techniques we used are correct for simulations of crystalline membranes.