Analytic Lyapunov exponents in a classical nonlinear field equation

It is shown that the nonlinear wave equation $\partial_t^2\phi - \partial^2_x \phi -\mu_0\partial_x(\partial_x\phi)^3 =0$, 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.Comment: 4 pages, 1 figur

On the origin of Phase Transitions in the absence of Symmetry-Breaking

In this paper we investigate the Hamiltonian dynamics of a lattice gauge model in three spatial dimension. Our model Hamiltonian is defined on the basis of a continuum version of a duality transformation of a three dimensional Ising model. The system so obtained undergoes a thermodynamic phase transition in the absence of symmetry-breaking. Besides the well known use of quantities like the Wilson loop we show how else the phase transition in such a kind of models can be detected. It is found that the first order phase transition undergone by this model is characterised according to an Ehrenfest-like classification of phase transitions applied to the configurational entropy. On the basis of the topological theory of phase transitions, it is discussed why the seemingly divergent behaviour of the third derivative of configurational entropy can be considered as the "shadow" of some suitable topological transition of certain submanifolds of configuration space.Comment: 31 pages, 9 figure

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.Comment: REVTeX, 15 PostScript figures, published versio

On the apparent failure of the topological theory of phase transitions

The topological theory of phase transitions has its strong point in two theorems proving that, for a wide class of physical systems, phase transitions necessarily stem from topological changes of some submanifolds of configuration space. It has been recently argued that the $2D$ lattice $\phi^4$-model provides a counterexample that falsifies this theory. It is here shown that this is not the case: the phase transition of this model stems from an asymptotic ($N\to\infty$) change of topology of the energy level sets, in spite of the absence of critical points of the potential in correspondence of the transition energy.Comment: 5 pages, 4 figure

The star formation rate of CaII and damped Lyman-alpha absorbers at 0.4<z<1.3

[abridged] Using stacked Sloan Digital Sky Survey spectra, we present the detection of [OII]3727,3730 nebular emission from galaxies hosting CaII and MgII absorption line systems. Both samples of absorbers, 345 CaII systems and 3461 MgII systems, span the redshift interval 0.4 < z < 1.3; all of the former and half the latter sample are expected to be bona-fide damped Lyman-alpha (DLA) absorbers. The measured star formation rate (SFR) per absorber from light falling within the SDSS fibre apertures (corresponding to physical radii of 6-9 h^-1 kpc) is 0.11-0.14 Msol/yr for the MgII-selected DLAs and 0.11-0.48 Msol/yr for the CaII absorbers. These results represent the first estimates of the average SFR in an absorption-selected galaxy population from the direct detection of nebular emission. Adopting the currently favoured model in which DLAs are large, with radii >9h^-1 kpc, and assuming no attenuation by dust, leads to the conclusion that the SFR per unit area of MgII-selected DLAs falls an order of magnitude below the predictions of the Schmidt law, which relates the SFR to the HI column density at z~0. The contribution of both DLA and CaII absorbers to the total observed star formation rate density in the redshift range 0.4 < z < 1.3, is small, <10% and <3% respectively. The result contrasts with the conclusions of Hopkins et al. that DLA absorbers can account for the majority of the total observed SFR density in the same redshift range. Our results effectively rule out a picture in which DLA absorbers are the sites in which a large fraction of the total SFR density at redshifts z < 1 occurs.Comment: Accepted for publication in MNRAS, 13 pages, 6 figure

Lyapunov exponents from geodesic spread in configuration space

The exact form of the Jacobi -- Levi-Civita (JLC) equation for geodesic spread is here explicitly worked out at arbitrary dimension for the configuration space manifold M_E = {q in R^N | V(q) < E} of a standard Hamiltonian system, equipped with the Jacobi (or kinetic energy) metric g_J. As the Hamiltonian flow corresponds to a geodesic flow on (M_E,g_J), the JLC equation can be used to study the degree of instability of the Hamiltonian flow. It is found that the solutions of the JLC equation are closely resembling the solutions of the standard tangent dynamics equation which is used to compute Lyapunov exponents. Therefore the instability exponents obtained through the JLC equation are in perfect quantitative agreement with usual Lyapunov exponents. This work completes a previous investigation that was limited only to two-degrees of freedom systems.Comment: REVTEX file, 10 pages, 2 figure