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

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

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

Estimating the mass density of neutral gas at z<1z < 1

We use the relationships between galactic HI mass and B-band luminosity determined by Rao & Briggs to recalculate the mass density of neutral gas at the present epoch based on more recent measures of the galaxy luminosity function than were available to those authors. We find $\Omega_{gas}(z=0) \simeq 5 \times 10^{-4}$ in good agreement with the original Rao & Briggs value, suggesting that this quantity is now reasonably secure. We then show that, if the scaling between H I mass and B-band luminosity has remained approximately constant since $z = 1$, the evolution of the luminosity function found by the Canada-France redshift survey translates to an increase of $\Omega_{gas}$ by a factor of $\approx 3$ at $z = 0.5 - 1$ . A similar value is obtained quite independently from consideration of the luminosity function of Mg II absorbers at $z = 0.65$. By combining these new estimates with data from damped \lya systems at higher redshift, it is possible to assemble a rough sketch of the evolution of $\Omega_{gas}$ over the last 90% of the age of the universe. The consumption of H I gas with time is in broad agreement with models of chemical evolution which include the effects of dust, although more extensive samples of damped \lya systems at low and intermediate redshift are required for a quantitative assessment of the dust bias.Comment: LaTeX file, 11 pages, 1 figure, accepted MNRAS pink page

Near-pristine gas at high redshifts: a window on early nucleosynthesis

It has now become recognised that damped Lyman alpha systems play an important role in helping us unravel the origin of chemical elements. In this presentation, we describe the main results of a recently completed survey of the most metal-poor DLAs, aimed at complementing and extending studies of the oldest stars in the Galaxy. The survey has clarified a number of lingering issues concerning the abundances of C, N, O in the low metallicity regime, has revealed the existence of DLA analogues to Carbon-enhanced metal-poor stars, and is providing some of the most precise measures of the primordial abundance of Deuterium.Comment: 11 pages, 7 Figures. Invited presentation at the XII International Symposium on Nuclei in the Cosmos, Cairns, Australia, 5-10 August 2012. To appear in Proceedings of Scienc