text: Consider an overdamped Brownian particle under external force :where the  term is negligible (the meaning of "overdamped"). Thus, it is just . The Fokker–Planck equation for this particle is the Smoluchowski diffusion equation:
  Where  is the diffusion constant and . The importance of this equation is it allows for both the inclusion of the effect of temperature on the system of particles and a spatially dependent diffusion constant.
  Starting with the Langevin Equation of a Brownian particle in external field , where  is the friction term,  is a fluctuating force on the particle, and  is the amplitude of the fluctuation.
  At equilibrium the frictional force is much greater than the inertial force, . Therefore, the Langevin equation becomes,
  Which generates the following Fokker–Planck equation,
  Rearranging the Fokker–Planck equation,
  Where . Note, the diffusion coefficient may not necessarily be spatially independent if  or  are spatially dependent.
  Next, the total number of particles in any particular volume is given by,
  Therefore, the flux of particles can be determined by taking the time derivative of the number of particles in a given volume, plugging in the Fokker–Planck equation, and then applying Gauss's Theorem.
  In equilibrium, it is assumed that the flux goes to zero. Therefore, Boltzmann statistics can be applied for the probability of a particles location at equilibrium, where  is a conservative force and the probability of a particle being in a state  is given as .
  This relation is a realization of the fluctuation–dissipation theorem. Now applying  to  and using the Fluctuation-dissipation theorem,
  Rearranging,
  Therefore, the Fokker–Planck equation becomes the Smoluchowski equation,
  for an arbitrary force .