Thursday, July 4, 2013

1307.1102 (Richard Kleeman)

A path integral formalism for non-equilibrium Hamiltonian statistical

Richard Kleeman
In a near equilibrium statistical system the Onsager-Machlup path integral has a long and useful history. The situation far from equilibrium has remained less clear. In this contribution a new general formulation for path integrals is proposed based on mixtures of an appropriate family of quasi-equilibrium probability densities. The path integral introduced here uses a generalized Boltzmann principle to associate path likelihoods with a multiple of the information loss of a particular path with respect to Liouvillean evolution. The loss at a particular time is given by a Lagrangian function of the thermodynamical variables and their time derivatives. An important implication of the present formulation is that future thermodynamical evolution depends not just on the instantaneous thermodynamic variables but also on the particular mixture of quasi-equilibrium distributions present. This behaviour has been previously seen in direct numerical simulations of turbulent dynamical systems and is also a fundamental property of Wiener path integrals. The Lagrangian derived is formally identical to that used in quantum mechanics to describe a particle moving non-relativistically in a particular vector and scalar electromagnetic field and also within a manifold with a metric tensor equal to the Fisher information matrix of the exponential family manifold. It is shown that a simple transformation due to Roncadelli enables the derived Lagrangian to be recast into standard Onsager-Machlup form. This transformation thus enables a derivation of the thermodynamical trajectory which is the most likely path. Also revealed is a decomposition of the thermodynamical trajectory into a reversible and irreversible piece which takes the form of the non-equilibrium thermodynamical equations recently proposed by \"Ottinger.
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