  A non-equilibrium steady state thermodynamics to describe shear flows 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 $dE = T dS - Q d\gamma$ 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 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 $\dot{W}$ must be
non-negative. A non-equilibrium entropy, increasing in time, is introduced, so
that the amount of heat based on this entropy is equal to the average of
$\dot{W}$. Numerical results from non-equilibrium molecular dynamics simulation
of two-dimensional many-particle systems with soft-core interactions are
presented which support our interpretation.
