Entropy geometric construction of a pure substance with normal,   superfluid and supersolid phases
  Using the laws of thermodynamics together with empirical data, we present a qualitative geometric construction of the fundamental relation of a pure substance $S = S(E,N,V)$, with $S$ entropy, $E$ energy, $N$ number of particles and $V$ volume. We analyze two very general type of substances, a "normal" and a "quantum" one, the main difference between them being that the latter presents superfluid phases. It is found that the constant entropy level curves are completely different in both cases, in the normal substances being obtuse while acute in quantum ones. A concomitant signature of the previous result is that the chemical potential can be both positive and negative in quantum substances, but only negative in normal ones. Our results suggest the existence of a region in the quantum substances that may be identified as a supersolid phase. We also make emphasis on the relevance of the present study within the context of superfluidity in ultracold gases. 
Manuel Mendoza-Lopez and Victor Romero-Rochin
cond-mat.stat-mech