A Statistical Mechanics appoach for a rigidity problem

MESON, ALEJANDRO M.; VERICAT, FERNANDO

JOURNAL OF STATISTICAL PHYSICS

Springer Science

Lugar: Dordrecht; Año: 2006

0022-4715

We focus the problem of establishing when a statistical mechanics system is determined by its free energy.Alattice system,modelled by a directed andweighted graph G(whose vertices are the spins and its adjacency matrix M will be given by the system transition rules), is considered. For a matrix A(q), depending on the system interactions, withentrieswhich are in the ring Z[aq : a ?¸ R+] and such that A(0) equals the integralmatrix M, the system free energy ƒÀA(q) will be defined as the spectral radius of A(q). This kind of free energy will be related with that normally introduced in Statistical Mechanics as proportional to the logarithm of the partition function. Then we analyze under what conditions the following statement could be valid: if two systems have respectively matrices A, B and ƒÀA = ƒÀB then thematrices are equivalent in some sense. Issues of this nature receive the name of rigidity problems. Our scheme, for finite interactions, closely follows that developed, within a dynamical context, by Pollicott and Weiss but now emphasizing their statistical mechanics aspects and including a classification for Gibbs states associated to matrices A(q). Since this procedure is not applicable for infinite range interactions, we discuss a way to obtain also some rigidity results for long range potentials.G(whose vertices are the spins and its adjacency matrix M will be given by the system transition rules), is considered. For a matrix A(q), depending on the system interactions, withentrieswhich are in the ring Z[aq : a ?¸ R+] and such that A(0) equals the integralmatrix M, the system free energy ƒÀA(q) will be defined as the spectral radius of A(q). This kind of free energy will be related with that normally introduced in Statistical Mechanics as proportional to the logarithm of the partition function. Then we analyze under what conditions the following statement could be valid: if two systems have respectively matrices A, B and ƒÀA = ƒÀB then thematrices are equivalent in some sense. Issues of this nature receive the name of rigidity problems. Our scheme, for finite interactions, closely follows that developed, within a dynamical context, by Pollicott and Weiss but now emphasizing their statistical mechanics aspects and including a classification for Gibbs states associated to matrices A(q). Since this procedure is not applicable for infinite range interactions, we discuss a way to obtain also some rigidity results for long range potentials.