Current Distribution in the Three-Dimensional Random Resistor Network at the Percolation Threshold

We study the multifractal properties of the current distribution of the three-dimensional random resistor network at the percolation threshold. For lattices ranging in size from $8^3$ to $80^3$ we measure the second, fourth and sixth moments of the current distribution, finding {\it e.g.\/} that $t/\nu=2.282(5)$ where $t$ is the conductivity exponent and $\nu$ is the correlation length exponent.Comment: 10 pages, latex, 8 figures in separate uuencoded fil

Constructing and exploring wells of energy landscapes

Landscape paradigm is ubiquitous in physics and other natural sciences, but it has to be supplemented with both quantitative and qualitatively meaningful tools for analyzing the topography of a given landscape. We here consider dynamic explorations of the relief and introduce as basic topographic features ``wells of duration $T$ and altitude $y$''. We determine an intrinsic exploration mechanism governing the evolutions from an initial state in the well up to its rim in a prescribed time, whose finite-difference approximations on finite grids yield a constructive algorithm for determining the wells. Our main results are thus (i) a quantitative characterization of landscape topography rooted in a dynamic exploration of the landscape, (ii) an alternative to stochastic gradient dynamics for performing such an exploration, (iii) a constructive access to the wells and (iv) the determination of some bare dynamic features inherent to the landscape. The mathematical tools used here are not familiar in physics: They come from set-valued analysis (differential calculus of set-valued maps and differential inclusions) and viability theory (capture basins of targets under evolutionary systems) which have been developed during the last two decades; we therefore propose a minimal appendix exposing them at the end of this paper to bridge the possible gap.Comment: 28 pages, submitted to J. Math. Phys -

Electrical conductivity of microemulsions : a case of stirred percolation

The electrical conductivity G of water in oil microemulsions may be described by percolation models [1] : below a critical water concentration Øc, G seems to diverge as (Øc - Ø)- S', while for Ø > Øc G increases as (Ø - Ø c)-T'. This physical situation may be called stirred percolation, referring to the Brownian motion of the medium. The exponents S' and T' are a priori different from the corresponding S and T in the classical situation of frozen percolation. A simple model of stirred percolation accounts fairly well for the measured value of S' = 1.2 ± 0.1 while the accepted value for S is 0.7. The exponent T' (1.4 to 1.6 in our experimental case) is less significant of the difference between stirred and frozen percolation : the values for S' should be about 1.8 and the value accepted for S should be about 1.6.La conductivité électrique G de microémulsions — eau dans l'huile — présente un comportement de percolation [1] : en-dessous d'une concentration d'eau critique Øc, la conductivité semble diverger comme (Øc - Ø)-S', tandis qu'au-dessus de Ø c elle augmente comme (Ø - Øc)T'. Dans cette situation physique que l'on peut qualifier de percolation brassée (par le mouvement Brownien du liquide) les exposants S' et T' n'ont a priori pas de raison d'être identiques aux exposants S et T de la situation classique de percolation gelée. Un modèle simplifié de percolation brassée rend compte de façon satisfaisante de l'exposant S' = 1,2 ± 0,1, alors que la valeur admise pour S est 0,7. L'exposant T', dont la valeur mesurée varie de 1,4 à 1,6, est moins significatif pour la différence entre percolation gelée et brassée : Les valeurs prévues pour S' seraient environ 1,8 contre 1,6 pour S

Electronic and magnetic structure of the undoped two and three-leg ladder cuprates in an itinerant electrons model

We study the electronic and magnetic structure of the undoped ideal two and three-leg ladder cuprates by assuming a moderate on site coulombic repulsion. This analysis is an extension of the Fermi liquids studies proposed for the CuO2 plane in view to explain the high Tc superconductivity and the competition with the antiferromagnetic phase. At zero doping, the quasi-one-dimensionality of the structure results in SDW correlations with different (commensurate) vectors according to the number of legs, which contrasts with the predictions made from the Heisenberg model. At mean field, and for n = 3 (Sr2Cu3O5), we predict a magnetic ordered state, detected by μSr and NMR measurements with critical temperatures consistent with our assumptions on the physical parameters, the modulation vector being $\pi/2$. The presence of several bands at the Fermi level explains why there is no observable gap in the static susceptibility measurements. For n = 2, we predict a gap consistent with experimental Curie susceptibility. But the expected magnetic instability is detected only in La2Cu2O5, where the interladder coupling is stronger. In every case the one-dimensional van Hove singularities are far from the Fermi level, making difficult the obtaining of high Tc superconductivity