Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1075unnormalizable one corresponds to the second–order phase transition of thestatistical model. Various critical exponents can be analytically computedand all applicable scaling laws can be checked. The simplicity of this modelpermits an analytical computation of the largest Lyapunov exponent byexploiting the geometric method proposed in [Caiani et al. (1997)].Mean–Field XY Hamiltonian SystemThe mean–field XY model describes a system of N equally coupled planarclassical rotators (see [Antoni and Ruffo (1995); Casetti et al. (1999)]). Itis defined by a Hamiltonian of the class (6.112) where the potential energyisV (ϕ) =J2NN∑ [1 − cos(ϕi − ϕ j ) ] N∑− h cos ϕ i . (6.120)i,j=1Here ϕ i ∈ [0, 2π] is the rotation angle of the ith rotator and h is an externalfield. Defining at each site i a classical spin vector s i = (cos ϕ i , sin ϕ i )the model describes a planar (XY) Heisenberg system with interactions ofequal strength among all the spins. We consider only the ferromagnetic caseJ > 0; for the sake of simplicity, we set J = 1. The equilibrium statisticalmechanics of this system is exactly described, in the thermodynamic limit,by the mean–field theory [Antoni and Ruffo (1995)]. In the limit h → 0, thesystem has a continuous phase transition, with classical critical exponents,at T c = 1/2, or ε c = 3/4, where ε = E/N is the energy per particle.The Lyapunov exponent λ 1 of this system is extremely sensitive to thephase transition. According to reported numerical simulations (see [Casettiet al. (1999)]), λ 1 (ε) is positive for 0 < ε < ε c , shows a sharp maximumimmediately below the critical energy, and drops to zero at ε c in the thermodynamiclimit, where it remains zero in the whole region ε > ε c , whichcorresponds to the thermodynamic disordered phase. In fact in this phasethe system is integrable, reducing to an assembly of uncoupled rotators.i=16.4.5.3 Euler Characteristics of Hamiltonian SystemsRecall that Euler characteristic χ is a number that is a characterizationof the various classes of geometric figures based only on the topologicalrelationship between the numbers of vertices V , edges E, and faces F , of ageometric Figure. This number, χ = F − E + V, is the same for all figuresthe boundaries of which are composed of the same number of connected