Lyapunov exponents and transport in the Zhang model of Self-Organized Criticality

We discuss the role played by the Lyapunov exponents in the dynamics of Zhang's model of Self-Organized Criticality. We show that a large part of the spectrum (slowest modes) is associated with the energy transpor in the lattice. In particular, we give bounds on the first negative Lyapunov exponent in terms of the energy flux dissipated at the boundaries per unit of time. We then establish an explicit formula for the transport modes that appear as diffusion modes in a landscape where the metric is given by the density of active sites. We use a finite size scaling ansatz for the Lyapunov spectrum and relate the scaling exponent to the scaling of quantities like avalanche size, duration, density of active sites, etc ...Comment: 33 pages, 6 figures, 1 table (to appear

Brane classical and quantum cosmology from an effective action

Motivated by the Randall-Sundrum brane-world scenario, we discuss the classical and quantum dynamics of a (d+1)-dimensional boundary wall between a pair of (d+2)-dimensional topological Schwarzschild-AdS black holes. We assume there are quite general -- but not completely arbitrary -- matter fields living on the boundary ``brane universe'' and its geometry is that of an Friedmann-Lemaitre-Robertson-Walker (FLRW) model. The effective action governing the model in the mini-superspace approximation is derived. We find that the presence of black hole horizons in the bulk gives rise to a complex action for certain classically allowed brane configurations, but that the imaginary contribution plays no role in the equations of motion. Classical and instanton brane trajectories are examined in general and for special cases, and we find a subset of configuration space that is not allowed at the classical or semi-classical level; these correspond to spacelike branes carrying tachyonic matter. The Hamiltonization and Dirac quantization of the model is then performed for the general case; the latter involves the manipulation of the Hamiltonian constraint before it is transformed into an operator that annihilates physical state vectors. The ensuing covariant Wheeler-DeWitt equation is examined at the semi-classical level, and we consider the possible localization of the brane universe's wavefunction away from the cosmological singularity. This is easier to achieve for branes with low density and/or spherical spatial sections.Comment: Shortened to match version accepted by Phys. Rev. D (unabridged text found in version 2), 42 pages, 9 figures, Rextex

The flexible nature of exchange, correlation and Hartree physics: resolving "delocalization" errors in a 'correlation free' density functional

By exploiting freedoms in the definitions of 'correlation', 'exchange' and 'Hartree' physics in ensemble systems we better generalise the notion of 'exact exchange' (EXX) to systems with fractional occupations functions of the frontier orbitals, arising in the dissociation limit of some molecules. We introduce the Linear EXX ("LEXX") theory whose pair distribution and energy are explicitly \emph{piecewise linear} in the occupations $f^{\sigma}_{i}$. {\hi}We provide explicit expressions for these functions for frontier $s$ and $p$ shells. Used in an optimised effective potential (OEP) approach it yields energies bounded by the piecewise linear 'ensemble EXX' (EEXX) energy and standard fractional optimised EXX energy: $E^{EEXX}\leq E^{LEXX} \leq E^{EXX}$. Analysis of the LEXX explains the success of standard OEP methods for diatoms at large spacing, and why they can fail when both spins are allowed to be non-integer so that "ghost" Hartree interactions appear between \emph{opposite} spin electrons in the usual formula. The energy $E^{LEXX}$ contains a cancellation term for the spin ghost case. It is evaluated for H, Li and Na fractional ions with clear derivative discontinuities for all cases. The $p$-shell form reproduces accurate correlation-free energies of B-F and Al-Cl. We further test LEXX plus correlation energy calculations on fractional ions of C and F and again shows both derivative discontinuities and good agreement with exact results