Thermodynamics as a Finsler Space with Torsion

One of the most striking features of thermodynamics is the division into two sets of those variables that appear in thermodynamic constraints: there are intensities (like Pressure and Temperature, E and B) which appear in the thermodynamic functions as variables homogeneous of degree 0, and there are extensive quantities (like Volume, Energy and Entropy, D and H) which appear as variables homogeneous of degree 1. Classically, there is no way to distinguish (geometrically) these two different species. However, it has long been recognized in the calculus of variations of a singly parameterized function, L(x,V;t), with dx-Vdt=0 as a constraint, that homogeneous functions play a critical role. The variational integral is independent of the choice of parameterization, t, if and only if the function L(x,V,t) is homogeneous of degree one in the variables, V. This means that V, or anything proportional to V, such as U=b(x,V) V can be used in the constraints, dx-Vdt = 0 = dx-Uds, where the new parameterization function satisfies ds - bdt = 0. Then the variational procedure on the variational integrand L(x,V,t)dt or on the variational integrand L(x,U,s)ds will give the same minimal result.

When a vector field, V, is considered to be the same vector field, independent from any factor of scaling (or renormalization), then Chern calls the vector field, V, a projectivized vector field. A projective geometry is a geometry that is independent from scales, and can be modeled as the set of all rays through some point of perspective, named the origin. The mathematical vehicle of parametric independent variational calculus distinguishes functions which are homogeneous of degree 1. Such is the arena of thermodynamics, and therefor projective geometries share some of the properties of an (equilibrium) thermodynamic system.

But there is more to the story, for in 1919 Finsler developed a thesis in which he studied a geometry which, if unconstrained, was not only non-euclidean, but also non-Riemannian. Special constraints would reduce Finsler geometries to the more common Riemannian geometries. The new feature permitted in Finsler spaces (a feature that is missing in Riemannian geometries because of quadratic, or quartic constraints) is the property of Torsion. Chern has shown how to interpret Finsler spaces in terms of the parametric independent calculus of variations, where the dual field, V, is projectivized.

Cartan recognized in 1922 that spaces with torsion existed, but he achieved this result from an entirely different tack than that used by Finsler. Cartan hypothecated that in the sense that a structural defect of closure in a basis frame implies the existence of curvature 2-forms, a structural defect of closure in the position of the origin could imply the existence of torsion 2-forms. As Brillouin (1959) states

"If one does not admit the symmetry of the (connection) coefficients, one obtains the twisted spaces of Cartan, spaces which scarcely have been used in physics to the present, but which seem to be called to an important role." .

To this day not much has been done with Cartan's spaces with torsion, except in the Kondo theory of dislocation defects in solids. However, Cartan's notions can be made precise and more general in terms of subspaces of a projectivized basis frame. Then not only the position of the origin (the point of perspective in projective geometry), but also its dual polar axis can have closure defects, leading to dislocations and disclinations respectively

(Kiehn 1994)



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