Action Principle in Statistical Dynamics

We discuss a least-action principle characterizing ensemble-averages in statistical dynamics, based upon “effective actions” defined as in quantum field-theory. These generalize to all system variables the Onsager-Machlup actions of thermodynamic fluctuation theory. In the statistical steady-state, the variational principles discussed are related to the “thermodynamical formalism”for chaotic dynamical systems. Non-perturbative methods of field-theory can be applied to approximate the effective actions: instantons, 1/N-expansion, Hartree-Fock, Rayleigh-Ritz, etc. In particular, the Rayleigh-Ritz method is shown to be closely related to traditional moment-closure schemes. Some concrete applications of the variational methods are outlined, e.g., to free decay of homogeneous, isotropic Navier-Stokes turbulence at high Reynolds number. “Fluctuation-dissipation relations” are obtained for the strength of turbulence-generated eddy noise in terms of mean dissipation characteristics. The relation of the effective action to dissipation and transport characteristics was already noted by Onsager, who pointed out that the associated variational principles generalize the hydrodynamic least-dissipation principle of Rayleigh. We briefly discuss the application of such principles to pattern-selection far from equilibrium.