Geodesic
Thermodynamics from the differential geometry standpoint
Teoretičeskaâ i matematičeskaâ fizika, Tome 157 (2008) no. 1, pp. 141-148 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study the differential-geometric structure of the space of thermodynamic states in equilibrium thermodynamics. We demonstrate that this space is a foliation of codimension two and find variables in which the foliation fibers are flat. We show that we can introduce a symplectic structure on this space: the external derivative of the $1$-form of the heat source, which has the form of the skew-symmetric product $dT\wedge dS$ in the found variables. The entropy $S$ then plays the role of the Lagrange function (or Hamiltonian) in mechanics, completely determining the thermodynamic system.
Keywords: symplectic structure, space of states, dynamical principle. 
@article{TMF_2008_157_1_a9,
     author = {V. P. Pavlov and V. M. Sergeev},
     title = {Thermodynamics from the~differential geometry standpoint},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {141--148},
     year = {2008},
     volume = {157},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2008_157_1_a9/}
}
TY  - JOUR
AU  - V. P. Pavlov
AU  - V. M. Sergeev
TI  - Thermodynamics from the differential geometry standpoint
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2008
SP  - 141
EP  - 148
VL  - 157
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2008_157_1_a9/
LA  - ru
ID  - TMF_2008_157_1_a9
ER  - 
%0 Journal Article
%A V. P. Pavlov
%A V. M. Sergeev
%T Thermodynamics from the differential geometry standpoint
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2008
%P 141-148
%V 157
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2008_157_1_a9/
%G ru
%F TMF_2008_157_1_a9
V. P. Pavlov; V. M. Sergeev. Thermodynamics from the differential geometry standpoint. Teoretičeskaâ i matematičeskaâ fizika, Tome 157 (2008) no. 1, pp. 141-148. http://geodesic.mathdoc.fr/item/TMF_2008_157_1_a9/

[1] C. Carathéodory, Math. Ann., 67:3 (1909), 355–386 | DOI | MR

[2] M. Born, “Kriticheskie zamechaniya po povodu traditsionnogo izlozheniya termodinamiki”, Razmyshleniya i vospominaniya fizika, Sb. statei, Nauka, M., 1977, 91–121; M. Born, “Kinetische Betrachtungen zur traditionellen Dartstellungen der Thermodynamik”, Phys. Z, 22 (1921), 218–224; 249–254; 282–286

[3] V. V. Kozlov, Teplovoe ravnovesie po Gibbsu i Puankare, Sovremennaya matematika, RKhD, Moskva–Izhevsk, 2002 | MR | Zbl

[4] V. V. Kozlov, Vestn. MGU, 2000, no. 2, 35–40 | MR | Zbl

[5] V. P. Pavlov, V. M. Sergeev, TMF, 153:1 (2007), 18–28 | DOI | MR | Zbl

[6] R. A. Minlos, Vvedenie v matematicheskuyu statisticheskuyu fiziku, MTsNMO, M., 2002 | MR | Zbl

[7] F. A. Berezin, Lektsii po statisticheskoi fizike, MTsNMO, M., 2007

[8] V. M. Sergeev, Predely ratsionalnosti, Fazis, M., 1999