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

The effect of a cutoff on pushed and bistable fronts of the reaction diffusion equation

We give an explicit formula for the change of speed of pushed and bistable fronts of the reaction diffusion equation when a small cutoff is applied at the unstable or metastable equilibrium point. The results are valid for arbitrary reaction terms and include the case of density dependent diffusion.Comment: 7 page

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

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

Validity of the Brunet-Derrida formula for the speed of pulled fronts with a cutoff

We establish rigorous upper and lower bounds for the speed of pulled fronts with a cutoff. We show that the Brunet-Derrida formula corresponds to the leading order expansion in the cut-off parameter of both the upper and lower bounds. For sufficiently large cut-off parameter the Brunet-Derrida formula lies outside the allowed band determined from the bounds. If nonlinearities are neglected the upper and lower bounds coincide and are the exact linear speed for all values of the cut-off parameter.Comment: 8 pages, 3 figure