The meaning of the irreducible memory function in stochastic theories of dynamics with detailed balance

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We investigate the relationship between the memory functions that arise in stochastic theories of fluctuations at equilibrium and those appropriate for an underlying microscopic (deterministic) description. We consider the class of stochastic theories that are Markovian with transition rates that satisfy the detailed balance condition. This class includes, for example, Smoluchowski dynamics, kinetic lattice gas models, and kinetic Ising models. When a time autocorrelation function is calculated using stochastic and deterministic descriptions, and the projection operator method of Mori is used, first and second order memory functions arise in both descriptions. We find a close and simple relationship between the first order memory functions of the two descriptions but not for the second order memory functions. Instead, the second order memory function of the microscopic description is simply related to the so-called irreducible memory function of the stochastic description. The latter was introduced for Smoluchowski dynamics by Cichocki and Hess and generalized by Kawasaki. This explains the empirical findings that for stochastic dynamics the irreducible memory function, rather than the second order memory function, has a more fundamental physical interpretation and is more useful for constructing mode coupling theories.