  The classical Poisson-Boltzmann equation (CPBE), which is a mean field theory
by averaging the ion fluctuation, has been widely used to study ion
distributions in charged fluids. In this study, we derive a modified
Poisson-Boltzmann equation with a random field from the field theory and
recover the ion fluctuation through a multiplicative noise added in the CPBE.
The Poisson-Boltzmann equation with a random field (RFPBE) captures the effect
of the ion fluctuation and gives different ion distributions in the charged
fluids compared to the CPBE. To solve the RFPBE, we propose a Monte Carlo
method based on the path integral representation. Numerical results show that
the effect of the ion fluctuation strengthens the ion diffusion into the domain
and intends to distribute the ions in the fluid uniformly. The final ion
distribution in the fluid is determined by the competition between the ion
fluctuation and the electrostatic forces exerted by the boundaries. The RFPBE
is general and feasible for high dimensional systems by taking the advantage of
the Monte Carlo method. We use the RFPBE to study a two dimensional system as
an example, in which the effect of ion fluctuation is clearly captured.
