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Phase transitions in geometrothermodynamics. (English) Zbl 1213.83129

Summary: Using the formalism of geometrothermodynamics, we investigate the geometric properties of the equilibrium manifold for diverse thermodynamic systems. Starting from Legendre invariant metrics of the phase manifold, we derive thermodynamic metrics for the equilibrium manifold whose curvature becomes singular at those points where phase transitions of first and second order occur. We conclude that the thermodynamic curvature of the equilibrium manifold, as defined in geometrothermodynamics, can be used as a measure of thermodynamic interaction in diverse systems with two and three thermodynamic degrees of freedom.

MSC:

83E05 Geometrodynamics and the holographic principle
80A10 Classical and relativistic thermodynamics
83C57 Black holes
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
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[1] Gibbs J.W.: The Collected Works, Vol. 1, Thermodynamics. Yale University Press, New Haven (1948) · Zbl 0031.13504
[2] Caratheodory C.: Untersuchungen über die Grundlagen der Thermodynamik. Math. Ann. 67, 355 (1909) · doi:10.1007/BF01450409
[3] Callen H.B.: Thermodynamics and an Introduction to Thermostatics. Wiley, New York (1985) · Zbl 0989.80500
[4] Huang K.: Statistical Mechanics. Wiley, New York (1987)
[5] Rao C.R.: Bull. Calcutta Math. Soc. 37, 81 (1945)
[6] Amari S.: Differential-Geometrical Methods in Statistics. Springer, Berlin (1985) · Zbl 0559.62001
[7] Weinhold, F.: Metric geometry of equilibrium thermodynamics I, II, III, IV, V. J. Chem. Phys. 63:2479, 2484, 2488, 2496 (1975)
[8] Weinhold F.: Metric geometry of equilibrium thermodynamics I, II, III, IV, V. J. Chem. Phys. 65, 558 (1976) · doi:10.1063/1.433136
[9] Ruppeiner G.: Thermodynamics: a Riemannian geometric model. Phys. Rev. A 20, 1608 (1979) · doi:10.1103/PhysRevA.20.1608
[10] Ruppeiner G.: Riemannian geometry in thermodynamic fluctuation theory. Rev. Mod. Phys. 67, 605 (1995) · doi:10.1103/RevModPhys.67.605
[11] Ruppeiner G.: Riemannian geometry in thermodynamic fluctuation theory. Rev. Mod. Phys. 68, 313 (1996) · doi:10.1103/RevModPhys.68.313
[12] Hermann R.: Geometry, Physics and Systems. Marcel Dekker, New York (1973) · Zbl 0285.58001
[13] Quevedo H.: Geometrothermodynamics. J. Math. Phys 48, 013506 (2007) · Zbl 1121.80011 · doi:10.1063/1.2409524
[14] Burke W.L.: Applied Differential Geometry. Cambridge University Press, Cambridge, UK (1987)
[15] Arnold V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1980)
[16] Davies P.C.W.: Thermodynamics of black holes. Rep. Prog. Phys. 41, 1313 (1978) · doi:10.1088/0034-4885/41/8/004
[17] Quevedo H.: Geometrothermodynamics of black holes. Gen. Relativ. Gravit. 40, 971 (2008) · Zbl 1140.83398 · doi:10.1007/s10714-007-0586-0
[18] Álvarez J.L., Quevedo H., Sánchez A.: Unified geometric description of black hole thermodynamics. Phys. Rev. D 77, 084004 (2008) · doi:10.1103/PhysRevD.77.084004
[19] Quevedo H., Sánchez A.: Geometrothermodynamics of asymptotically anti-de Sitter black holes. JHEP 09, 034 (2008) · Zbl 1245.83037 · doi:10.1088/1126-6708/2008/09/034
[20] Quevedo H., Sánchez A.: Geometric description of BTZ black holes thermodynamics. Phys. Rev. D 79, 024012 (2009) · Zbl 1222.83113 · doi:10.1103/PhysRevD.79.024012
[21] Quevedo, H., Taj, S.: Geometrothermodynamics of higher dimensional black holes in Einstein–Gauss–Bonnet theory (2009, in preparation) · Zbl 1243.83050
[22] Akbar, M., Quevedo, H., Sánchez, A., Saifullah, K., Taj, S.: Thermodynamic geometry of charged rotating BTZ black holes (2009, in preparation)
[23] Åman J., Bengtsson I., Pidokrajt N.: Flat information geometries in black hole thermodynamics. Gen. Relativ. Gravit. 38, 1305 (2006) · Zbl 1175.83036 · doi:10.1007/s10714-006-0306-1
[24] Cai R., Cho J.: Thermodynamic curvature of the BTZ black hole. Phys. Rev. D 60, 067502 (1999) · doi:10.1103/PhysRevD.60.067502
[25] Mirza B., Zamaninasab M.: Ruppeiner geometry of RN black holes: flat or curved? J. High Energy Phys. 0706, 059 (2007) · doi:10.1088/1126-6708/2007/06/059
[26] Vázquez, A., Quevedo, H., Sánchez, A.: Thermodynamic systems as bosonic strings (2009); arXiv:hep-th/0805.4819
[27] Landau L.D., Lifshitz E.M.: Statistical Physics. Pergamon Press, London, UK (1980) · Zbl 0080.19702
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