A Variational Principle for the Asymptotic Speed of Fronts of the Density Dependent Diffusion--Reaction Equation

We show that the minimal speed for the existence of monotonic fronts of the equation $u_t = (u^m)_{xx} + f(u)$ with $f(0) = f(1) = 0$, $m >1$ and $f>0$ in $(0,1)$ derives from a variational principle. The variational principle allows to calculate, in principle, the exact speed for arbitrary $f$. The case $m=1$ when $f'(0)=0$ is included as an extension of the results.Comment: Latex, postcript figure availabl

Neutron diffraction in a model itinerant metal near a quantum critical point

Neutron diffraction measurements on single crystals of Cr1-xVx (x=0, 0.02, 0.037) show that the ordering moment and the Neel temperature are continuously suppressed as x approaches 0.037, a proposed Quantum Critical Point (QCP). The wave vector Q of the spin density wave (SDW) becomes more incommensurate as x increases in accordance with the two band model. At xc=0.037 we have found temperature dependent, resolution limited elastic scattering at 4 incommensurate wave vectors Q=(1+/-delta_1,2, 0, 0)*2pi/a, which correspond to 2 SDWs with Neel temperatures of 19 K and 300 K. Our neutron diffraction measurements indicate that the electronic structure of Cr is robust, and that tuning Cr to its QCP results not in the suppression of antiferromagnetism, but instead enables new spin ordering due to novel nesting of the Fermi surface of Cr.Comment: Submitted as a part of proceedings of LT25 (Amsterdam 2008

Contest based on a directed polymer in a random medium

We introduce a simple one-parameter game derived from a model describing the properties of a directed polymer in a random medium. At his turn, each of the two players picks a move among two alternatives in order to maximize his final score, and minimize opponent's return. For a game of length $n$, we find that the probability distribution of the final score $S_n$ develops a traveling wave form, ${\rm Prob}(S_n=m)=f(m-v n)$, with the wave profile $f(z)$ unusually decaying as a double exponential for large positive and negative $z$. In addition, as the only parameter in the game is varied, we find a transition where one player is able to get his maximum theoretical score. By extending this model, we suggest that the front velocity $v$ is selected by the nonlinear marginal stability mechanism arising in some traveling wave problems for which the profile decays exponentially, and for which standard traveling wave theory applies

Transport properties in antiferromagnetic quantum Griffiths phases

We study the electrical resistivity in the quantum Griffiths phase associated with the antiferromagnetic quantum phase transition in a metal. The resistivity is calculated by means of the semi-classical Boltzmann equation. We show that the scattering of electrons by locally ordered rare regions leads to a singular temperature dependence. The rare-region contribution to the resistivity varies as $T^\lambda$ with temperature $T,$ where the $\lambda$ is the usual Griffiths exponent which takes the value zero at the critical point and increases with distance from criticality. We find similar singular contributions to other transport properties such as thermal resistivity, thermopower and the Peltier coefficient. We also compare our results with existing experimental data and suggest new experiments.Comment: 4 pages, 1 figur

Macroscopic description of particle systems with non-local density-dependent diffusivity

In this paper we study macroscopic density equations in which the diffusion coefficient depends on a weighted spatial average of the density itself. We show that large differences (not present in the local density-dependence case) appear between the density equations that are derived from different representations of the Langevin equation describing a system of interacting Brownian particles. Linear stability analysis demonstrates that under some circumstances the density equation interpreted like Ito has pattern solutions, which never appear for the Hanggi-Klimontovich interpretation, which is the other one typically appearing in the context of nonlinear diffusion processes. We also introduce a discrete-time microscopic model of particles that confirms the results obtained at the macroscopic density level.Comment: 4 pages, 3 figure

Perturbative Linearization of Reaction-Diffusion Equations

We develop perturbative expansions to obtain solutions for the initial-value problems of two important reaction-diffusion systems, viz., the Fisher equation and the time-dependent Ginzburg-Landau (TDGL) equation. The starting point of our expansion is the corresponding singular-perturbation solution. This approach transforms the solution of nonlinear reaction-diffusion equations into the solution of a hierarchy of linear equations. Our numerical results demonstrate that this hierarchy rapidly converges to the exact solution.Comment: 13 pages, 4 figures, latex2

Erosion waves: transverse instabilities and fingering

Two laboratory scale experiments of dry and under-water avalanches of non-cohesive granular materials are investigated. We trigger solitary waves and study the conditions under which the front is transversally stable. We show the existence of a linear instability followed by a coarsening dynamics and finally the onset of a fingering pattern. Due to the different operating conditions, both experiments strongly differ by the spatial and time scales involved. Nevertheless, the quantitative agreement between the stability diagram, the wavelengths selected and the avalanche morphology reveals a common scenario for an erosion/deposition process.Comment: 4 pages, 6 figures, submitted to PR

Dual Fronts Propagating into an Unstable State

The interface between an unstable state and a stable state usually develops a single confined front travelling with constant velocity into the unstable state. Recently, the splitting of such an interface into {\em two} fronts propagating with {\em different} velocities was observed numerically in a magnetic system. The intermediate state is unstable and grows linearly in time. We first establish rigorously the existence of this phenomenon, called ``dual front,'' for a class of structurally unstable one-component models. Then we use this insight to explain dual fronts for a generic two-component reaction-diffusion system, and for the magnetic system.Comment: 19 pages, Postscript, A