On Helmholtz free energy for finite abstract simplicial complexes

We prove first that for the Barycentric refinement G1 of a finite abstract simplicial complexG, the Gauss-Bonnet formula χ(G) = ∑ xK (x) holds, where K(x) = (−1)(1−χ(S(x))) is the curvature of a vertex x with unit sphere S(x) in the graph G1. This curvature is dual toK −(x) = (−1) for which GaussBonnet is the definition of Euler characteristic χ(G). Because the connection Laplacian L′ = 1+A′ of an abstract simplicial complex G is unimodular, where A′ is the adjacency matrix of the connection graph G′, the Green function values g(x, y) = (1 + A′)−1 xy are integers and 1 − χ(S(x)) = g(x, x). Gauss-Bonnet for K reads therefore as str(g) = χ(G), where str is the super trace. As g is a time-discrete heat kernel, this is a cousin to McKean-Singer str(e−Lt) = χ(G) for the Hodge Laplacian L = (d+d∗)2 which lives on the same Hilbert space than L′. Both formulas hold for an arbitrary finite abstract simplicial complex G. Writing Vx(y) = g(x, y) for the Newtonian potential of the connection Laplacian, we prove ∑ y Vx(y) = K (x), so that by the new Gauss-Bonnet formula, the Euler characteristic of G agrees with the total potential theoretic energy ∑ x,y g(x, y) = χ(G) of G. The curvature K + now relates to the probability measure p minimizing the internal energy U(p) = ∑ x,y g(x, y)p(x)p(y) of the complex. Since both the internal energy (here linked to topology) and Shannon entropy are natural and unique in classes of functionals, we then look at critical points p the Helmholtz free energy F (p) = βU(p) − TS(p) which combines the energy functional U and the entropy functional S(p) = − ∑ x p(x) log(p(x)). As the temperature T = 1− β changes, we observe bifurcation phenomena. Already for G = K3 both a saddle node bifurcation and a pitchfork bifurcation occurs. The saddle node bifurcation leads to a catastrophe: the function β → F (p(β), β) is discontinuous if p(β) is a free energy minimizer. Date: Mar 18, 2017. 1991 Mathematics Subject Classification. Primary: 05E45, 94A17, 82Bxx, Secondary: 31C20, 58K35 .