Stochastic model of slow dynamics in supercooled liquids and dense colloidal suspensions

We propose a stochastic model equation for the density variable which is intended for slow dynamics occurring in supercooled liquids and dense colloidal suspensions near glass transitions. The equation takes the following form for the probability distribution functional P({ϱ},t) of the density variable ϱ(r): (∂∂t)P({ϱ},t) = -D0 ∫ dr(δδϱ(r))τϱ(r) · τ[δδϱ(r) + βδH{ϱ}δϱ(r)]P({ϱ},t), where D0 is the microscopic diffusion constant, β ≡ 1kBT, and H{ϱ} the density functional of Ramakrishnan-Yussouff type. We show that thermally activated so-called hopping processes enter in a rather essential manner here in contrast to the earlier works where such processes occurred only on extending the theory. Further use of the decoupling approximation for this model leads to the low frequency form of the well-known self-consistent mode coupling equation for the density auto-correlation function.