Water can be ice, liquid, or steam. Physicists call these states phases. In theory, each phase has its own measure — a way to describe how likely it is. But not every measure you can imagine occurs in nature. Some phases are real, while others are just mathematical illusions.
Scientists have found a simple criterion that separates reality from mirage. Take a system that can be in different states and look at its entropy — a measure of disorder. If the entropy changes smoothly, without jumps, as we move from one state to another, then that state is a true phase. If the entropy jumps, that phase cannot exist in equilibrium. It hides behind the convex hull of the free energy, like a mirage behind the horizon.
This criterion works not only for familiar systems but also for very general ones — for instance, Markov shifts, which describe random walks. It turns out that the previous formulation, proposed in 2006, was incomplete. The authors found a counterexample and refined the condition: a phase is real if the entropy is continuous at every point of the set of equilibrium states.
Now physicists can tell in advance which phases are worth looking for in experiments and which are mere mathematical ghosts. This helps avoid wasting time chasing mirages.
as written by the authors arXiv:2604.10225
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 $$\mu$$ is an equilibrium state for some continuous potential if and only if the entropy map $$h$$ is upper semicontinuous at $$\mu$$; 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 $$\sigma$$-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 $$\mathcal{E}$$.
The concept of phase comes from the kitchen: water, ice, and steam are three phases of the same substance. But in statistical mechanics, there are many more phases, and some of them cannot be seen in everyday life.