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 hydrodynam ics. An excess work, PHI, related to the work necessary for displaceme nt of the system from stationary states is defined. The excess work PH I is shown to have the following properties: (1) The differential of P HI is expressed in terms of thermodynamic functions: the energy for vi scous flow and the entropy for thermal conduction when taken separatel y; (2) PHI is an extremum at all stationary states, a minimum (maximum ) at stable (unstable) stationary states, and thus yields necessary an d sufficient criteria for stability; (3) PHI describes the bifurcation from homogeneous to inhomogeneous stationary states; (4) PHI is a Lya punov function with physical significance parallel to that of the Gibb s free energy change (maximum work) on relaxation to an equilibrium st ate; (5) PHI is the thermodynamic ''driving force'' (potential) toward stable stationary states; (6) PHI is a component of the total dissipa tion during the relaxation toward a stable stationary state; (7) for c onstraints leading to equilibrium PHI reduces to the known thermodynam ic function, which is the work of displacing the system from the equil ibrium for those given constraints; and (8) PHI qualitatively explains the positive energy release in both the destruction and formation of a convective structure in a Rayleigh-Benard experiment.