A path integral formalism for the closure of autonomous statistical systems

TL;DR

This paper extends a path integral formalism to close autonomous statistical systems, enabling the analysis of realistic dynamical systems like turbulence by evaluating informational discrepancies of paths.

Contribution

The formalism is generalized to handle non-Hamiltonian autonomous systems with non-vanishing divergence, broadening its applicability to realistic turbulence and complex dynamics.

Findings

Applicable to realistic turbulence modeling

Addresses divergence in autonomous systems

Provides a framework for practical implementation

Abstract

Recently a path integral formalism has been proposed by the author which gives the time evolution of moments of slow variables in a Hamiltonian statistical system. This closure relies on evaluating the informational discrepancy of a time sequence (path) of approximating densities from the Liouvillian evolution that an exact density must follow. The discrepancy is then used to weight all possible paths using a generalized Boltzmann principle. That formalism is extended here to deal with more general and realistic autonomous dynamical systems. There the divergence of the time derivative of dynamical variables need not vanish as it does in the Hamiltonian case and this property complicates the closure derivation. Many interesting and realistic applications are covered by this new formalism including those describing realistic turbulence and the relevant specifics of this situation are outlined. The practical issues associated with the implementation of the outlined formalism are also discussed.