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

Dynamic hysteresis from zigzag domain walls

We investigate dynamic hysteresis in ferromagnetic thin films with zigzag domain walls. We introduce a discrete model describing the motion of a wall in a disordered ferromagnet with in-plane magnetization, driven by an external magnetic field, considering the effects of dipolar interactions and anisotropy. We analyze the effects of external field frequency and temperature on the coercive field by Monte Carlo simulations, and find a good agreement with the experimental data reported in literature for Fe/GaAs films. This implies that dynamic hysteresis in this case can be explained by a single propagating domain wall model without invoking domain nucleation.Comment: 10 pages, 13 figures; minor modifications and two figures adde

Phase transitions as topology changes in configuration space: an exact result

The phase transition in the mean-field XY model is shown analytically to be related to a topological change in its configuration space. Such a topology change is completely described by means of Morse theory allowing a computation of the Euler characteristic--of suitable submanifolds of configuration space--which shows a sharp discontinuity at the phase transition point, also at finite N. The present analytic result provides, with previous work, a new key to a possible connection of topological changes in configuration space as the origin of phase transitions in a variety of systems.Comment: REVTeX file, 5 pages, 1 PostScript figur

Topological aspects of geometrical signatures of phase transitions

Certain geometric properties of submanifolds of configuration space are numerically investigated for classical lattice phi^4 models in one and two dimensions. Peculiar behaviors of the computed geometric quantities are found only in the two-dimensional case, when a phase transition is present. The observed phenomenology strongly supports, though in an indirect way, a recently proposed topological conjecture about a topology change of the configuration space submanifolds as counterpart of a phase transition.Comment: REVTEX file, 4 pages, 5 figure

Hamiltonian dynamics and geometry of phase transitions in classical XY models

The Hamiltonian dynamics associated to classical, planar, Heisenberg XY models is investigated for two- and three-dimensional lattices. Besides the conventional signatures of phase transitions, here obtained through time averages of thermodynamical observables in place of ensemble averages, qualitatively new information is derived from the temperature dependence of Lyapunov exponents. A Riemannian geometrization of newtonian dynamics suggests to consider other observables of geometric meaning tightly related with the largest Lyapunov exponent. The numerical computation of these observables - unusual in the study of phase transitions - sheds a new light on the microscopic dynamical counterpart of thermodynamics also pointing to the existence of some major change in the geometry of the mechanical manifolds at the thermodynamical transition. Through the microcanonical definition of the entropy, a relationship between thermodynamics and the extrinsic geometry of the constant energy surfaces $\Sigma_E$ of phase space can be naturally established. In this framework, an approximate formula is worked out, determining a highly non-trivial relationship between temperature and topology of the $\Sigma_E$. Whence it can be understood that the appearance of a phase transition must be tightly related to a suitable major topology change of the $\Sigma_E$. This contributes to the understanding of the origin of phase transitions in the microcanonical ensemble.Comment: in press on Physical Review E, 43 pages, LaTeX (uses revtex), 22 PostScript figure