Time correlation functions of the Smoluchowski level of description of solutions and suspensions

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Abstract
Time-correlation functions at the Smoluchowski level of description, in which the variables are only the coordinates of the solute particles, apparently violate the stationarity and interchange symmetries that are well known characteristics of time correlation functions formulated in the full Hamiltonian level of description. It is shown that the evolution of a dynamical variable A that depends only on the solute coordinates may be described by the random Smoluchowski equation dA(t)/dt=S†A(t)+RA(t), where S† is the adjoint of the operator S that governs the evolution of the distribution function f (q, t) of solute configurations, i.e., df/dt=Sf. The ‘‘random force’’ RA is determined so as to resolve the anomalies mentioned above. The theory of the random force term itself leads to the formulation of a simulation procedure for Smoluchowski-level systems that reduces to the Ermak–McCammon algorithm in the case that A is the set of solute coordinates.