The Concept of Entropy and its Concavity for a Finite Protein in its Environment: An exact study on a Square Lattice

We consider a general lattice model of a finite protein in its environment and calculate its Boltzmann entropy SB(E) as a function of its energy E in a microcanonical ensemble, and Gibbs entropy SG(E) as a function of its average energy E in a canonical ensemble by exact enumeration on a square lattice. We find that because of the finite size of the protein, (i) the two are very different and SG(E)>SB(E), (ii) SB(E) need not be concave while SG(E) is, and (iii) SG(E) is relevant for experiments but not SB(E), even though SB(E) is conceptually more useful. We discuss the consequences of these differences. The results are general and applicable to all finite systems.