Numerical Studies of Phase Transitions in Fluid Membranes

(1)定义了具有表面张力、弯曲弹性和流动性的流体膜模型。模型的哈密顿量被认为是基本粒子模型中Nambu弦的离散作用。我们发现该模型经历了一个二阶相变。(2)我们对与(1)的哈密顿量相同的晶体膜模型进行MC模拟，发现晶体模型如预期的那样经历了二级相变。由该结果与式(1)的结果可以发现，具有表面张力和弯曲刚度的膜的形状波动的二阶相变与膜的流动性无关。(3)研究了流体膜的普通模型，将其视为Polyakov刚性弦的离散模型。通过MC计算发现，在普通流体膜模型中存在着形状波动的二阶相变。(4)对普通晶体膜模型进行MC模拟，研究了形状波动的相变。通过对比结晶模型与(3)的结果，我们发现膜内的流动性加强了相变，并且流体模型在相变时的Hausdorff维数大于结晶模型。(5)对普通晶体模型进行了朗格万模拟和蒙特卡罗模拟。用Langevin和MC两种方法分别得到了相变的临界指数和相变时的Housdorff维数，两种方法的结果基本一致。因此，这证实了我们使用的两种技术对晶体膜的模拟是正确的。

(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.