The Ubiquitous Van der Waals Gas

Relative to the choice of a projective renormalization, a generalization of the Gauss map permits a Jacobian matrix, J, of rank N-1 to be defined for any given vector field of dimension N. The characteristic polynomial of such a three dimensional matrix is always a cubic. If three universal coordinate functions are defined in terms of the similarity invariants of the Jacobian matrix, then the characteristic polynomial equation, in terms of these universal variables, defines a universal surface which has the topological features of the Gibbs surface for a Van der Waals gas. The method permits the evaluation of those thermodynamic properties which correspond to the limiting spinodal line of single phase stability, the binodal line of mixed phase stability, and the critical point, for any dynamical process constructed from a dynamical system. The method establishes a theoretical foundation, in terms of projective geometry, for the chemical law of corresponding states. All differentiable three dimensional systems can be mapped to the Van der Waals gas.


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Last update 01/23/2009
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