Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1071information is contained in the canonical configurational partition function∫ZN C =N∏i=1dq i exp[−βV (q)].Therefore, partition function ZNC is decomposed – in the last term ofequation∫(6.113) – into an infinite summation of geometric integrals,M vdσ /‖∇V ‖, defined on the {M v } v∈R . Once the microscopic interactionpotential V (q) is given, the configuration space of the system is automaticallyfoliated into the family {M v } v∈R of these equipotential hypersurfaces.Now, from standard statistical mechanical arguments we know that, atany given value of the inverse temperature β, the larger the number Nof particles the closer to M v ≡ M uβ are the microstates that significantlycontribute to the averages – computed through Z N (β) – of thermodynamicobservables. The hypersurface M uβ is the one associated with the averagepotential energy computed at a given β,u β = (Z C N ) −1 ∫N∏i=1dq i V (q) exp[−βV (q)].Thus, at any β, if N is very large the effective support of the canonicalmeasure shrinks very close to a single M v = M uβ .Explicitly, the topological hypothesis reads: the basic origin of a phasetransition lies in a suitable topology change of the {M v }, occurring at somev c . This topology change induces the singular behavior of the thermodynamicobservables at a phase transition. By change of topology we meanthat {M v } vvc . In other words,canonical measure should ‘feel’ a big and sudden change of the topology ofthe equipotential hypersurfaces of its underlying support, the consequencebeing the appearance of the typical signals of a phase transition.This point of view has the interesting consequence that – also at finiteN – in principle different mathematical objects, i.e., manifolds of differentcohomology type, could be associated to different thermodynamical phases,whereas from the point of view of measure theory [Yang and Lee (1952)] theonly mathematical property available to signal the appearance of a phasetransition is the loss of analyticity of the grand–canonical and canonicalaverages, a fact which is compatible with analytic statistical measures onlyin the mathematical N → ∞ limit.As it is conjectured that the counterpart of a phase transition is a breakingof diffeomorphicity among the surfaces M v , it is appropriate to choose