convert this optimization criterion of the entropy to an equivalent optimization criterion of the free energy, where the free energy is the Legendre transform of the entropy [@Do; @Ca; @Ma]. It is important that the Legendre transform between the entropy and free energy be invertible so the two optimization schemes are equivalent, and this follows from the concavity of the entropy. (See Section 26 of [@Ro] for the mathematics of the Legendre tranform between convex functions.)

Statistical mechanics supplies a model for thermodynamic states, as probability distributions on mechanical multiparticle states. From a given short range particle interaction one can then (in principle) compute the internal energy $E$ and entropy $S$, and prove the above two features of the entropy: its concavity and its optimization role for equilibrium states. To do this one uses the basic Boltzmann/Gibbs ansatz: that the entropy $S(E,N,V)$ is proportional to $$-\sum_j\rho_j\ln(\rho_j)$$ where $\rho_j$ is the probability of multiparticle state $j$, and the equilibrium state is that probability distribution $\{\rho_j\}$ on the set $\Xi(E,N,V)$, of multiparticle states of energy $E$ and particle number $N$ in volume $V$, which maximizes the entropy [@R1; @R2; @Ge; @Wi; @Ma]. (Note that in taking an infinite volume limit, which we 