Variational Derivation of the Steady State

Abstract

For a system undergoing Markoffian irreversible change of state, a concept of persistency (shown to arise naturally from a physical microscopic description of the system) is introduced; it is defined as the probability that a certain state persists during a short time interval without drifting away from the state. This concept is used in deriving a theorem which says that "the steady state of a Markoffian irreversible process corresponds to the maximum of the persistency." This theorem holds even outside the range of validity of Prigogine's principle of minimum entropy production, namely, even when the steady state is not close to the equilibrium state or even when the process is nonlinear. The proposed function is derived from the path probability for irreversible statistical dynamics previously reported by the author. The theorem is first applied to a system of two level atoms interacting simultaneously with a heat bath and a radiation field. Next, Kohler's variational derivation of the Boltzmann transport equation is shown to follow from the theorem.