Numerical Studies of Phase Transitions in Fluid Membranes

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(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.