  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.
