The Structure of Thermodynamics

Abstract

Classical Thermodynamics has a beautiful formulation which is rigid, simple, and elegant. It has been formulated in terms of thermodynamic potentials, in particular the internal energy, U, and the entropy, S. Both are extensive variables which scale with the mass of the system. At equilibrium, the constrained potential is a minimum in the energy representation and a maximum in the entropy representation. We discuss the properties of Thermodynamics in the energy representation in detail. In the last section we construct the transformation from the energy to the entropy representation. To each extensive variable, E � (S,V,Nj,···), there corresponds a conjugate intensive vari- able, i� = @U/@E � (T, P,µj,···). The First Law of Thermodynamics expresses the change in energy as dU = PidE � (dU = TdS + ( P)dV + µjdNj + ···). The Second Law of Ther- modynamics is equivalent to the statement that the stability matrix, U�,� = @ 2 U/@E � @E � , is positive definite. The matrix elements of this matrix of mixed second partial derivatives can be measured because they are susceptibilities, giving the linear response of intensive thermody- namic variables to changes in the extensive thermodynamic variables: �i� = PU�,��E � . A large number of equalities and inequalities are obeyed by these linear response functions. Some equalities are consequences of the symmetry relations among the matrix elements U�,� = U�,� (Maxwell Relations). The remainder are a consequence of the relation between the stability matrix and its inverse, U �,� . The inequalities are consequences of the Second Law requirement that U�,� (and its inverse, U �,� ) are positive definite. A systematic way for constructing a change of basis transformation in the tangent plane to the equilibrium manifold is described. This leads directly to a systematic procedure for computing thermodynamic partial derivatives. They are simply expansion coefficients in a preferred basis set. The algorithm is simple and can be carried out in a finite (small) number of steps. A computer implementation of this algorithm is presented in the Appendix. A systematic procedure for constructing new potentials (Gibbs, Helmholtz, Enthalpy) from U is described. These new potentials are not positive-definite. Their metrics are block-diagonal and related to the matrix U�,� in a precisely defined way. The formal structure of classical thermodynamics is simple, elegant, and beautiful. The properties of a physical system in thermodynamic equilibrium, and the fluctuations around thermodynamic equilibrium, are described by a single scalar function. This function can be regarded as a potential. The secrets of this potential can be unlocked by taking its first and second derivatives. In the most useful representations of thermodynamics, the potential is chosen as the total internal energy (U) or the total entropy (S) of the physical system. Classical Thermodynamics was developed in the energy representation. The most important questions at that time concerned the conversion of heat into work. These questions are by now well- understood. The next set of questions to be addressed concern the effect of heat on information.