Steady shear flow thermodynamics based on a canonical distribution approach

A nonequilibrium steady-state thermodynamics to describe shear flow is developed using a canonical distribution approach. We construct a canonical distribution for shear flow based on the energy in the moving frame using the Lagrangian formalism of the classical mechanics. From this distribution, we derive the Evans-Hanley shear flow thermodynamics, which is characterized by the first law of thermodynamics dE=TdS-Qdgamma relating infinitesimal changes in energy E, entropy S, and shear rate gamma with kinetic temperature T. Our central result is that the coefficient Q is given by Helfand's moment for viscosity. This approach leads to thermodynamic stability conditions for shear flow, one of which is equivalent to the positivity of the correlation function for Q. We show the consistency of this approach with the Kawasaki distribution function for shear flow, from which a response formula for viscosity is derived in the form of a correlation function for the time-derivative of Q. We emphasize the role of the external work required to sustain the steady shear flow in this approach, and show theoretically that the ensemble average of its power W must be non-negative. A nonequilibrium entropy, increasing in time, is introduced, so that the amount of heat based on this entropy is equal to the average of W. Numerical results from nonequilibrium molecular-dynamics simulation of two-dimensional many-particle systems with soft-core interactions are presented which support our interpretation.