an:01207008
Zbl 0908.53015
Chen, M.; Tseng, H. C.
On the second law of thermodynamics and contact geometry
EN
J. Math. Phys. 39, No. 1, 329-344 (1998).
00045272
1998
j
53C15 37J99 80A05 58A20
contact structure; thermodynamic phase space; equilibrium; irreversible thermodynamics; Legendre submanifold
\textit{R. Hermann} [``Geometry, physics, and systems'' (Pure Appl. Math. 18, Marcel Dekker, New York) (1973; Zbl 0285.58001)] suggested that classical thermodynamics could be formulated in a geometric setting in terms of a \((2n+1)\)-dimensional contact manifold \(K\) with thermodynamic space \(E_n\) as a base space of \(K\).
\textit{R. Mrugala}, \textit{J. D. Nulton}, \textit{J. C. Sch??n} and \textit{P. Salamon} [Rep. Math. Phys. 29, 109-121 (1991; Zbl 0742.58022)] studied the application of contact geometry to equilibrium thermodynamics, and in the present paper the authors generalize their work to irreversible thermodynamics. They show that the inaccessibility condition of Carath??odory and the assumption of semipositive definite property of the dissipative energy are equivalent to the inequality of Clausius.
The inaccessibility condition gives rise to a generalized Gibbs relation which defines a 1-form \(\omega\) such that its zero reproduces this relation. \(\omega\) is a contact form. The integral surface of the general Gibbs relation is an \(n\)-dimensional 1-graph space \(G\) (Legendre submanifold) of a contact space of a 1-jet space \(J^1(E_n,\mathbb{R})\).
Next, the authors construct an isovector field \(X_f\) such that the inaccessibility condition (thermodynamic law) is invariant under the contact transformation generated by \(X_f\).
A.Bucki (Oklahoma City)
Zbl 0285.58001; Zbl 0742.58022