Stability of the Boltzmann equation with potential forces on torus

In this paper, we are concerned with the stability of solutions to the Cauchy problem of the Boltzmann equation with potential forces on torus. It is shown that the natural steady state with the symmetry of origin is asymptotically stable in the Sobolev space with exponential rate in time for any initially smooth, peri-odic, origin symmetric small perturbation, which preserves the same total mass, momentum and mechanical energy. For the non-symmetric steady state, it is also shown that it is stable in L1-norm for any initial data with the finite total mass