IWNET12
IWNET12
Derivation of the GENERIC form of nonequilibrium thermodynamics
from a statistical optimization principle
Bruce Turkington1
1 Department of Mathematics and Statistics, University of Massachusetts, Amherst, USA
Abstract
We show that the form of the governing equations for nonequilibrium thermodynamics called GENERIC
by Grmela and Öttinger can be derived from a natural variational principle. Our approach employs quasi-
equilibrium ensembles corresponding to a given set of macroscopic variables, as in the standard projection
operator methodology of statistical mechanics. The novelty of our derivation is to consider these ensembles as
trial probability densities in an optimization principle, and to characterize a macroscopic evolution as a path of
these trial pdfs that is a best t to the underlying microscopic dynamics. The lack-of-t of such a path of pdfs
is quantied by its residual with respect to the Liouville equation, and a time-integrated, ensemble-averaged,
squared norm of this residual denes the cost functional in our optimization principle. Optimal paths which
connect nonequilibrium initial macrostates to equilibrium determine the relaxation of the macroscopic variables.
The closed reduced equations governing the macroscopic variables are deduced by the techniques of the calculus
of variations, and they are found to have the GENERIC form; namely, they have a reversible term that is a
Hamiltonian vector eld and an irreversible term that is a gradient vector eld. The potential in the irreversible
term is the optimal value function associated with the cost functional, and near equilibrium it coincides with
(half) the entropy production.
Our theoretical framework can be viewed as a general statistical model reduction procedure that applies to any
Hamiltonian microdynamics and any set of resolved variables, and is valid far from equilibrium. Adjustable
parameters enter in this statistical closure as weights assigned by the cost functional to represent the inuence of
unresolved uctuations on the resolved evolution. The optimal value function is a thermodynamic potential that
establishes the relation between thermodynamic forces and uxes, and in this way our optimization framework
extends the classical structure of linear irreversible thermodynamics.
E-mail: turk@math.umass.edu

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