Classical dynamics on graphs

We consider the classical evolution of a particle on a graph by using a time-continuous Frobenius-Perron operator which generalizes previous propositions. In this way, the relaxation rates as well as the chaotic properties can be defined for the time-continuous classical dynamics on graphs. These properties are given as the zeros of some periodic-orbit zeta functions. We consider in detail the case of infinite periodic graphs where the particle undergoes a diffusion process. The infinite spatial extension is taken into account by Fourier transforms which decompose the observables and probability densities into sectors corresponding to different values of the wave number. The hydrodynamic modes of diffusion are studied by an eigenvalue problem of a Frobenius-Perron operator corresponding to a given sector. The diffusion coefficient is obtained from the hydrodynamic modes of diffusion and has the Green-Kubo form. Moreover, we study finite but large open graphs which converge to the infinite periodic graph when their size goes to infinity. The lifetime of the particle on the open graph is shown to correspond to the lifetime of a system which undergoes a diffusion process before it escapes.Comment: 42 pages and 8 figure

The fractality of the relaxation modes in deterministic reaction-diffusion systems

In chaotic reaction-diffusion systems with two degrees of freedom, the modes governing the exponential relaxation to the thermodynamic equilibrium present a fractal structure which can be characterized by a Hausdorff dimension. For long wavelength modes, this dimension is related to the Lyapunov exponent and to a reactive diffusion coefficient. This relationship is tested numerically on a reactive multibaker model and on a two-dimensional periodic reactive Lorentz gas. The agreement with the theory is excellent

Scaled Spectroscopy of 1Se and 1Po Highly Excited States of Helium

In this paper, we examine the properties of the 1Se and 1Po states of helium, combining perimetric coordinates and complex rotation methods. We compute Fourier transforms of quantities of physical interest, among them the average of the operator cos(theta_12), which measures the correlation between the two electrons. Graphs obtained for both 1Se and 1Po states show peaks at action of classical periodic orbits, either "frozen planet" orbit or asymmetric stretch orbits. This observation legitimates the semiclassical quantization of helium with those orbits only, not just for S states but also for P states, which is a new result. To emphasize the similarity between the S and P states we show wavefunctions of 1Po states, presenting the same structure as 1Se states, namely the "frozen planet" and asymmetric stretch configurations.Comment: revtex 15 pages with 6 figures, 2 figures (large) are available on request at email address [email protected]. to appear in J. Phys. B (April 98

Chaotic and fractal properties of deterministic diffusion-reaction processes

We study the consequences of deterministic chaos for diffusion-controlled reaction. As an example, we analyze a diffusive-reactive deterministic multibaker and a parameter-dependent variation of it. We construct the diffusive and the reactive modes of the models as eigenstates of the Frobenius-Perron operator. The associated eigenvalues provide the dispersion relations of diffusion and reaction and, hence, they determine the reaction rate. For the simplest model we show explicitly that the reaction rate behaves as phenomenologically expected for one-dimensional diffusion-controlled reaction. Under parametric variation, we find that both the diffusion coefficient and the reaction rate have fractal-like dependences on the system parameter.Comment: 14 pages (revtex), 12 figures (postscript), to appear in CHAO

Isometric fluctuation relations for equilibrium states with broken symmetry

We derive a set of isometric fluctuation relations, which constrain the order parameter fluctuations in finite-size systems at equilibrium and in the presence of a broken symmetry. These relations are exact and should apply generally to many condensed-matter physics systems. Here, we establish these relations for magnetic systems and nematic liquid crystals in a symmetry-breaking external field, and we illustrate them on the Curie-Weiss and the $XY$ models. Our relations also have implications for spontaneous symmetry breaking, which are discussed.Comment: 9 pages, 4 figures, in press for Phys. Rev. Lett. to appear there in Dec. 201

Quantum Hall-like effect for cold atoms in non-Abelian gauge potentials

We study the transport of cold fermionic atoms trapped in optical lattices in the presence of artificial Abelian or non-Abelian gauge potentials. Such external potentials can be created in optical lattices in which atom tunneling is laser assisted and described by commutative or non-commutative tunneling operators. We show that the Hall-like transverse conductivity of such systems is quantized by relating the transverse conductivity to topological invariants known as Chern numbers. We show that this quantization is robust in non-Abelian potentials. The different integer values of this conductivity are explicitly computed for a specific non-Abelian system which leads to a fractal phase diagram.Comment: 6 pages, 2 figure

Viscosity in the escape-rate formalism

We apply the escape-rate formalism to compute the shear viscosity in terms of the chaotic properties of the underlying microscopic dynamics. A first passage problem is set up for the escape of the Helfand moment associated with viscosity out of an interval delimited by absorbing boundaries. At the microscopic level of description, the absorbing boundaries generate a fractal repeller. The fractal dimensions of this repeller are directly related to the shear viscosity and the Lyapunov exponent, which allows us to compute its values. We apply this method to the Bunimovich-Spohn minimal model of viscosity which is composed of two hard disks in elastic collision on a torus. These values are in excellent agreement with the values obtained by other methods such as the Green-Kubo and Einstein-Helfand formulas.Comment: 16 pages, 16 figures (accepted in Phys. Rev. E; October 2003

Transport Properties of the Lorentz Gas in Terms of Periodic Orbits

We establish a formula relating global diffusion in a space periodic dynamical system to cycles in the elementary cell which tiles the space under translations.Comment: 8 pages, Postscript, A

On the long-time behavior of spin echo and its relation to free induction decay

It is predicted that (i) spin echoes have two kinds of generic long-time decays: either simple exponential, or a superposition of a monotonic and an oscillatory exponential decays; and (ii) the long-time behavior of spin echo and the long-time behavior of the corresponding homogeneous free induction decay are characterized by the same time constants. This prediction extends to various echo problems both within and beyond nuclear magnetic resonance. Experimental confirmation of this prediction would also support the notion of the eigenvalues of time evolution operators in large quantum systems.Comment: 4 pages, 4 figure