Lorenz Equations and an Interesting Experiment

We present a brief introduction to coupled transport processes described macroscopically by hydrodynamic equations, the Navier-Stokes equations [4]. These are difficult, highly non-linear coupled partial differential equations they are frequently approximated. One such approximation consists of the Lorenz equations [5,6], which are obtained from the Navier-Stokes equations by Fourier transform of the spatial variables in those equations, retention of first order Fourier modes and restriction to small deviations from a bifurcation of an homogeneous motionless stationary state (a conductive state) to an inhomogeneous convective state in Rayleigh-Benard convection (see the next paragraph). The Lorenz equations have been applied successfully in various fields ranging from meteorology to laser physics. 

What is the purpose of a thermodynamic and stochastic theory of hydrodynamics The thermodynamic potential (state) functions for irreversible processes approaching equilibrium are known, for example the Gibbs free energy change for a process at constant temperature and pressure. Changes in that energy yield the maximum work, other than pressure volume work, available from that process. Then, by analogy, the aims of a theory of thermodynamics for hydrodynamics are the establishment of evolution criteria 

We developed our thermodynamic theory for the Lorenz equations, obtained with approximations from the Navier-Stokes equations (we present almost no mathematics here that is given in detail in [10]). The Lorenz equations are 

One solution of the Lorenz equations is (X,Y,Z) = (0,0,0). When the control parameter r is less than unity, that is the Rayleigh number is less than its critical value Rc, then the zero solution is unique and stable, and it corresponds to the motionless conductive state of the fluid. At the bifurcation point r = 1 this solution becomes unstable, and a new solution becomes stable corresponding to convective modes. These solutions can be used to construct an excess work function, just as we did for single transport properties. 

Our theory based on the concept of exess work accounts for these experiments, at least qualitatively. According to our theory, when the system approaches a stable stationary state, either convective or conductive, there is a decrease in, the excess work, and a positive excess work is released, which 

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