A Landau Model Of A Kind Of One-Dimentional Quasicrystal

Quasicrystal is a kind of peculiar material structure which has quasiperiodic order. And it is still an unsolved problem that how quasicrystal can be stable.In this paper, we firstly review the predecessor’s research on the geometry description and thermodynamic stability, and introduce the definition of Quasicrystal on Mathematics and on Physics. We focus on a kind of simple quasiperiodic function which only has4spectral points, according to the demand of discrete spectrum of Quasicrystal. Under the framework of Landau phenomenological theory, we model it’s stability using the ideas proposed by Lifshitz and Petrich in1997.Then we analyze the models using single-wave approximate method and numerical simulation method. The stability condition of our1D Quasicrystal and the phase transitions are given. Single-wave approximate method is to deal with the ideal case of the c, a parameter in our models, approaching to infinite. In the other case, for finite positive c, we employ the numerical simulation method. In order to improve the numerical efficiency, we use higher dimensional projection simulation method instead of the classical periodical approximation methods. This method has been constructed according to the fact that quasicrystal can be seen as a projected higher dimensional periodic crystal. It only requests a higher dimensional rectangular area to numerical calculation but the whole1D space.Finally we compare the results of the single-wave approximation method and the numerical simulation method. It is shown that single-wave approximate not only can be used to the extreme situation of parameters c tend to infinite, but also to assist the normal situation to get the solutions and phase diagrams.