Application of optimization method to the $x^4$ model in the Tsallis nonextensive statistics

Abstract

We study the effects of the environment described by the Tsallis nonextensive statistics on physical quantities using an optimization method in the case of small deviation from the Boltzmann-Gibbs statistics. The $x^4$ model is used and the density operator is restricted to be a gaussian form. The variational parameter is the frequency $\Omega$ of a particle in the optimization method. We obtain an approximate expression of free energy and of the expectation value of $\beta m \Omega^2 x^2 /2$, where $\beta$ is the inverse of the temperature and $m$ is the mass of a particle. Numerically, the optimized frequency is estimated and the expectation value of $\beta m \Omega^2 x^2 /2$ is calculated. The effects of the Tsallis nonextensive statistic for small deviation from the Boltzmann-Gibbs statistics are found: 1) the frequency modulation of a particle and 2) the variation of the expectation value of $\beta m \Omega^2 x^2 /2$ at high temperature.