On Helmholtz free energy for finite abstract simplicial complexes

Abstract

We prove a Gauss-Bonnet formula X(G) = sum_x K(x), where K(x)=(-1)^dim(x) (1-X(S(x))) is a curvature of a vertex x with unit sphere S(x) in the Barycentric refinement G1 of a simplicial complex G. K(x) is dual to (-1)^dim(x) for which Gauss-Bonnet is the definition of Euler characteristic X. Because the connection Laplacian L'=1+A' of G is unimodular, where A' is the adjacency matrix of of the connection graph G', the Green function values g(x,y) = (1+A')^-1_xy are integers and 1-X(S(x))=g(x,x). Gauss-Bonnet for K^+ reads therefore as str(g)=X(G), where str is the super trace. As g is a time-discrete heat kernel, this is a cousin to McKean-Singer str(exp(-Lt)) = X(G) for the Hodge Laplacian L=dd^* +d^*d which lives on the same Hilbert space than L'. Both formulas hold for an arbitrary finite abstract simplicial complex G. Writing V_x(y)= g(x,y) for the Newtonian potential of the connection Laplacian, we prove sum_y V_x(y) = K(x), so that by the new Gauss-Bonnet formula, the Euler characteristic of G agrees with the total potential theoretic energy sum_x,y g(x,y)=X(G) of G. The curvature K now relates to the probability measure p minimizing the internal energy U(p)=sum_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)=(1-T) U(p)-T S(p) which combines the energy functional U and the entropy functional S(p)=-sum_x p(x) log(p(x)). As the temperature T changes, we observe bifurcation phenomena. Already for G=K_3 both a saddle node bifurcation and a pitchfork bifurcation occurs. The saddle node bifurcation leads to a catastrophe: the function T -> F(p(T),T) is discontinuous if p(T) is a free energy minimizer.