Which Phases Are Thermodynamically Realizable? A Local Entropy Criterion

In the variational approach to statistical mechanics, equilibrium states are the rigorous analogues of thermodynamic phases; the question of which invariant measures can arise as equilibrium states is therefore the question of which phases are thermodynamically realizable. We prove that for continuous actions of locally compact amenable groups on compact metrizable spaces with finite topological entropy, an ergodic measure $μ$ is an equilibrium state for some continuous potential if and only if the entropy map $h$ is upper semicontinuous at $μ$; equivalently, the unrealizable phases are exactly those hidden behind the convex envelope of the free energy. More generally, the same criterion applies whenever $(X, T)$ has bounded entropy and embeds as an invariant subsystem of a compact metrizable system. As a canonical case, one-point compactification yields a $C_0$-potential realization theorem for locally compact $σ$-compact systems, with applications to countable-state Markov shifts. We also show that the equilibrium-face realization stated by Jenkinson (2006) omits a necessary continuity hypothesis, exhibiting a counterexample on the full shift, and give the sharp corrected statement: a weak-$*$ closed set $\mathcal{E}$ of ergodic measures determines an equilibrium face if and only if $h|_{\mathcal{E}}$ is continuous and $h$ is upper semicontinuous at each point of $\mathcal{E}$.