  We investigate in this work the effects of interaction on the fluctuation of
empirical measures. The systems with positive definite interaction potentials
tend to exhibit smaller fluctuation compared to the fluctuation in standard
Monte Carlo sampling while systems with negative definite potentials tend to
exhibit larger fluctuation. Moreover, if the temperature goes to zero, the
fluctuation for positive definite kernels in the long time tends to vanish to
zero, while the fluctuation for negative definite kernels in the long time
tends to blow up to infinity. This phenomenon may gain deeper understanding to
some physical systems like the Poisson-Boltzmann system, and may help to
understand the properties of some particle based variational inference sampling
methods.
