Tuesday, 12 March 2019

Introduction to differential forms in thermodynamics



I've studied differential geometry just enough to be confident with differential forms. Now I want to see application of this formalism in thermodynamics.


I'm looking for a small reference, to learn familiar concepts of (equilibrium?) thermodynamics formulated through differential forms.


Once again, it shouldn't be a complete book, a chapter at max, or an article.



UPD Although I've accepted David's answer, have a look at the Nick's one and my comment on it.



Answer



There are two articles by S.G. Rajeev: Quantization of Contact Manifolds and Thermodynamics and A Hamilton-Jacobi Formalism for Thermodynamics in which he reviews the formulation of thermodynamics in terms of contact geometry and explains a number of examples such as van der Waals gases and the thermodynamics of black holes in this picture.


Contact geometry is intended primarily to applications of mechanical systems with time varying Hamiltonians by adding time to the phase space coordinates. The dimension of contact manifolds is thus odd. Contact geometry is formulated in terms of a basic one form, the contact one form:


$$ \alpha = dq^0 -p_i dq^i$$


($q^0$ is the time coordinate). The key observation in Rajeev's formulation is that one can identify the contact structure with the first law:


$$ \alpha = dU -TdS + PdV$$


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