Giulio Pettini — Page 2

arXiv2000

Analytic Lyapunov exponents in a classical nonlinear field equation

It is shown that the nonlinear wave equation $\partial_t^2ϕ- \partial^2_x ϕ-μ_0\partial_x(\partial_xϕ)^3 =0$, which is the continuum limit of the Fermi-Pasta-Ulam (FPU) beta model, has a positive Lyapunov exponent lambda_1, whose analytic energy dependence is given. The result (a first example for field equations) is achieved by evaluating the lattice-spacing dependence of lambda_1 for the FPU model within the framework of a Riemannian description of Hamiltonian chaos. We also discuss a difficulty of the statistical mechanical treatment of this classical field system, which is absent in the dynamical description.

cond-mat.stat-mech↗

arXiv1998

Geometry of dynamics and phase transitions in classical lattice phi^4 theories

We perform a microcanonical study of classical lattice phi^4 field models in 3 dimensions with O(n) symmetries. The Hamiltonian flows associated to these systems that undergo a second order phase transition in the thermodynamic limit are here investigated. The microscopic Hamiltonian dynamics neatly reveals the presence of a phase transition through the time averages of conventional thermodynamical observables. Moreover, peculiar behaviors of the largest Lyapunov exponents at the transition point are observed. A Riemannian geometrization of Hamiltonian dynamics is then used to introduce other relevant observables, that are measured as functions of both energy density and temperature. On the basis of a simple and abstract geometric model, we suggest that the apparently singular behaviour of these geometric observables might probe a major topological change of the manifolds whose geodesics are the natural motions.

hep-th↗