Statistical Mechanics of Nonlinear Lattice Dynamic Models Exhibiting Phase Transitions

Abstract

To imitate real systems exhibiting distortive phase transitions, it becomes customary to consider Hamiltonians of the form [1]

$$H = \sum\limits_{\ell ,\alpha } {\frac{{P_{\ell \alpha }^2}}{{2M}} + \frac{A}{2}} \sum\limits_{\ell ,\alpha } {X_{\ell \alpha }^2 + \frac{B}{{4n}}{{\sum\limits_\ell {\left( {\sum\limits_\alpha {X_{\ell \alpha }^2} } \right)} }^2}} + \frac{{{B_1}}}{4}\sum\limits_{\ell ,\alpha } {X_{\ell \alpha }^4 + \sum\limits_{\ell ,m,\alpha } {{V_{\ell ,\ell + m,\alpha \alpha }}{X_{\ell \alpha }}{X_{\ell + m\alpha }}} } .$$

(1)

Lectures given at NATO Advanced Study Institute on Nonlinear Equations in Physics and Mathematics, Istanbul, August 1977.