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On the second law of thermodynamics and contact geometry. (English) Zbl 0908.53015
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 $$K$$.
R. Mrugala, J. D. Nulton, J. C. Schön and 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 Carathéodory 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 $$X_f$$.

MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 80A05 Foundations of thermodynamics and heat transfer 58A20 Jets in global analysis
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References:
 [1] DOI: 10.1016/0034-4877(91)90017-H · Zbl 0742.58022 · doi:10.1016/0034-4877(91)90017-H [2] Clausius R., Philos. Mag. 2 pp 102– (1851) [3] Clausius R., Philos. Mag. 24 pp 201– (1862) [4] Clausius R., Ann. Phys. (Leipzig) 125 pp 313– (1865) [5] DOI: 10.1002/cpa.3160020403 · Zbl 0037.13104 · doi:10.1002/cpa.3160020403 [6] DOI: 10.1007/BF01053791 · Zbl 0893.60073 · doi:10.1007/BF01053791 [7] DOI: 10.1063/1.532072 · Zbl 0901.76071 · doi:10.1063/1.532072 [8] DOI: 10.1063/1.532072 · Zbl 0901.76071 · doi:10.1063/1.532072 [9] DOI: 10.1063/1.532072 · Zbl 0901.76071 · doi:10.1063/1.532072 [10] DOI: 10.1063/1.530117 · Zbl 0786.76071 · doi:10.1063/1.530117
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