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The Concavity of Entropy and Extremum Principles in Thermodynamics

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Abstract

We revisit the concavity property of the thermodynamic entropy in order to formulate a general proof of the minimum energy principle as well as of other equivalent extremum principles that are valid for thermodynamic potentials and corresponding Massieu functions under different constraints. The current derivation aims at providing a coherent formal framework for such principles which may be also pedagogically useful as it fully exploits and highlights the equivalence between different schemes. We also elucidate the consequences of the extremum principles for the general shape of thermodynamic potentials in relation to first-order phase transitions.

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REFERENCES

  1. H. B. Callen, Thermodynamics and an Introduction to Thermostatistics (Wiley, 1985).

  2. An expanded version of the present article, containing two appendices with some useful technical material, is available at http://www.me.infm.it/∼prestip. In particular, Appendix A of the web version summarizes some properties of concave (and convex) functions that are used in the following, while Appendix B collects a number of relevant results concerning Legendre transforms and their derivatives.

  3. Observe that for a generic quadratic form of two variables, H(ξ1, ξ2)=Aξ2 1+2Bξ1ξ2+Cξ2 2 =A[( ξ1+(B/A) ξ2) 2+(ξ2 2/A2)×(AC-B2)] (where A≠0), H is positive definite iff A>0 and AC-B2>0, and negative definite iff A<0 and AC-B2>0.

  4. Alternatively, one can define ∼F as U(S, V)-TS. In this case, ∼F would be a function of S which parametrically depends on T and V. Anyway, the minimum and the convexity properties of ∼F remain the same (see Appendix B of ref. 2).

  5. See, for instance, I. Ispolatov and E. G. D. Cohen, On first-order phase transitions in microcanonical and canonical non-extensive systems, Physica A 295:475-487 (2001).

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Authors and Affiliations

  1. Unità di Ricerca di Messina, Istituto Nazionale per la Fisica della Materia (INFM), Italy

    Santi Prestipino & Paolo V. Giaquinta

  2. Dipartimento di Fisica, Contrada Papardo, Università degli Studi di Messina, 98166, Messina, Italy

    Santi Prestipino & Paolo V. Giaquinta

Authors
  1. Santi Prestipino
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  2. Paolo V. Giaquinta
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Prestipino, S., Giaquinta, P.V. The Concavity of Entropy and Extremum Principles in Thermodynamics. Journal of Statistical Physics 111, 479–493 (2003). https://doi.org/10.1023/A:1022233814184

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  • DOI: https://doi.org/10.1023/A:1022233814184

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