Toward a thermodynamic theory of hydrodynamics: The Lorenz equations

Earlier work on the thermodynamics of nonlinear systems is extended to the Lorenz model in a first attempt to apply the theory to hydrodynamics. An excess work, Φ, related to the work necessary for displacement of the system from stationary states is defined. The excess work Φ is shown to have the following properties: (1) The differential of Φ is expressed in terms of thermodynamic functions: the energy for viscous flow and the entropy for thermal conduction when taken separately; (2) Φ is an extremum at all stationary states, a minimum (maximum) at stable (unstable) stationary states, and thus yields necessary and sufficient criteria for stability; (3) Φ describes the bifurcation from homogeneous to inhomogeneous stationary states; (4) Φ is a Lyapunov function with physical significance parallel to that of the Gibbs free energy change (maximum work) on relaxation to an equilibrium state; (5) Φ is the thermodynamic ‘‘driving force’’ (potential) toward stable stationary states; (6) Φ̇ is a component of the total dissipation during the relaxation toward a stable stationary state; (7) for constraints leading to equilibrium Φ reduces to the known thermodynamic function, which is the work of displacing the system from the equilibrium for those given constraints; and (8) Φ qualitatively explains the positive energy release in both the destruction and formation of a convective structure in a Rayleigh–Bénard experiment.