A new microscopic model for diffusion was described which captured diffusion-induced fluctuations at scales where the concept of concentration gave way to discrete particles. It was shown that, in the limit where the number of particles tended to infinity, the model was equivalent to the classical stochastic diffusion equation. The new model and the stochastic diffusion equation were tested against Langevin dynamics in numerical simulations, and it was shown that the present model successfully reproduced the correct ensemble statistics, while the classical model failed. Thus the new discrete model sat between the multivariate master equation (a mesoscopic model) and the classical diffusion stochastic diffusion equation (a field model); a microscopic particle model occupying an even smaller regime. Using this model, another and more intuitive derivation of the classical diffusion stochastic diffusion equation could be given. Simulations were performed which showed that, at very small particle densities, the new discrete model of diffusion closely approximated the statistical properties of a particle simulation, whereas the stochastic diffusion equation did not. This appeared to be the first time that a classical diffusion stochastic diffusion equation had been tested against a particle model. In addition to the new theoretical model for diffusion, important practical applications of the new discrete model of diffusion were suggested. Although the multivariate master equation was a lower-level description, it was not well-suited to being coupled to other grid-based methods. The multivariate master equation was often very slow, and each domain modelled by the multivariate master equation would generate its own time-step. Particle simulations could be more efficient for modelling diffusion for small particle densities when no reactions were involved, but did not integrate perfectly with finite-difference methods. When even simple reactions were involved, particle methods became less efficient than the new discrete model of diffusion.

Multinomial Diffusion Equation. A.Balter, A.M.Tartakovsky: Physical Review E, 2011, 83[6], 061143