On subadditivity and convexity properties of thermodynamic functions
It is pointed out that the usual basic postulate of increase of entropy for an isolated system, as stated for example by Tisza and Callen, if mathematically formalized, is expressed as a superadditivity property of entropy. This fact has two kinds of implications: (a) it allows one to deduce in a very direct and mathematically clear way stability properties such as Cv ~ 0 and KT ~ 0 and the equivalence of various thermodynamic schemes as expressed for example by the fact that the "minimum' property of the free energy is a consequence of the 'maximum' property of entropy; (b) it makes it possible to establish a link with foundations research, notably the system developed by Giles, where superadditivity of entropy appears as a consequence of other axioms. INTRODUCTION Recent developments in the foundations of thermodynamics are of two kinds. On the one band l-4 one can find attempts to make rigorous the connection between the possibility of defining en tropy and the classical statements of the second principle. On the other hand56 the concept of entropy is taken for granted and some of its properties are assumed in an axiomatic way, so that attention is turned to deducing rigorous consequences therefrom: these works are on the line of Gibbs. In statistical thermodynamics both attitudes have their counterpart. The second approach has been intensively investigated particularly since 1963, when Ruelle 7 • 8 was able to study in a rigorous way the problern of the so-called 'thermodynamic Iimit' on the basis of the conditions of stability and strong tempering on the intermolecular potentials. In the technical trea tment of the problern the mathematical property of superadditivity was considered and the Iimit thermodynamic functions turned out to have convexity properties. This was just the clue to some improvements in t Work supported in part by CNR (Consiglio Na7ionale delle Ricerche). 229 L. GALGANI AND A. SCOTTI thermodynamics itself, by the realization that superadditivity has a deep physical meaning: it expresses formally the basic postulate assumed by Tisza and Callen • This simple property all ows one to deri ve in a very direct way properties such as continuity and differentiability (almost everywhere) of entropy, stability conditions, extremum properties of thermodynamic potentials and Massieu functions, which had been postulated independently or derived in ·a more complicated or obscure way by those authors. The way this is done is presented in the first three sections. In section 4 the physical meaning of convexity properties is illustrated and the conclusions are given in section 5. 1. ENTROPY SCHEME We consider for definiteness and simplicity a system whose equilibrium states are characterized by the values of three extensive quantities: energy U, volume V and number of moles N; its thermodynamic properties are deduced from the entropy S defined through the functional relation S = .9(U, V, N) (1) The assumed properties of .9 are those of homogeneity, strict monotonicity in U and superadditivity: 9'(2U,2V,2N) = 2Y'(U, V,N) (2real) (2) Y(U 2, V,N) ~ 9'(Ut, V,N)<=> U2 ~ U1 (3) .9'(U1 + U 2, V,+ V2,N1 + N 2) ~ 9'(U1, VbN1) + 9'(U2, V2,N2) (4) Equation 2, commonly employed, is a consequence of the postulate of extensivity of entropy: geometrically it means the 9' is a ruled surface. Equation 3 is related to positivity of temperature; it will be used only for the passage to the energy scheme ( § 2). Equation 4 is the property of superadditivity which expresses physically the increasing property of entropy for isolated systems: for an isolated system of fixed U, V, N the state of unconstrained equilibrium has an entropy greater than all corresponding states of constrained equilibrium. The entropy of the state of constrained equilibrium is represented by the RHS of equation 4, in accordance with the postulate of extensivity. The postulate of increase of entropy, enunciated in words in this form also by Tisza and Callen, was formally exploited only in a way that required consideration of the thermodynamic space of configurations enlarged to include the extensive independent variables of the simple systems constituting the composite constrained one. As it is formalized in equation 4, it just gives a functional relation on the function Y'(U, V, N) itself. The power of this relation is shown by the following immediate consequence: by equations 4 and 2, with ;. = !, one has ( u 1 + u 2 V1 + V2 N 1 + N 2) 2/7 2 , 2 , 2 ~9'(UbVt,Nt)+Y(U2,V2,N2) (5)