Recently, it has been shown that when an equation that allows the so-called pulled fronts in the mean-field limit is modeled with a stochastic model with a finite number N of particles per... Show moreRecently, it has been shown that when an equation that allows the so-called pulled fronts in the mean-field limit is modeled with a stochastic model with a finite number N of particles per correlation volume, the convergence to the speed v(*) for N--> infinity is extremely slow-going only as ln(-2)N. Pulled fronts are fronts that propagate into an unstable state, and the asymptotic front speed is equal to the linear spreading speed v(*) of small linear perturbations about the unstable state. In this paper, we study the front propagation in a simple stochastic lattice model. A detailed analysis of the microscopic picture of the front dynamics shows that for the description of the far tip of the front, one has to abandon the idea of a uniformly translating front solution. The lattice and finite particle effects lead to a "stop-and-go" type dynamics at the far tip of the front, while the average front behind it "crosses over" to a uniformly translating solution. In this formulation, the effect of stochasticity on the asymptotic front speed is coded in the probability distribution of the times required for the advancement of the "foremost bin." We derive expressions of these probability distributions by matching the solution of the far tip with the uniformly translating solution behind. This matching includes various correlation effects in a mean-field type approximation. Our results for the probability distributions compare well to the results of stochastic numerical simulations. This approach also allows us to deal with much smaller values of N than it is required to have the ln(-2)N asymptotics to be valid. Furthermore, we show that if one insists on using a uniformly translating solution for the entire front ignoring its breakdown at the far tip, then one can obtain a simple expression for the corrections to the front speed for finite values of N, in which various subdominant contributions have a clear physical interpretation. Show less
Fronts, propagating into an unstable state phi=0, whose asymptotic speed v(as) is equal to the linear spreading speed v* of infinitesimal perturbations about that state (so-called pulled fronts),... Show moreFronts, propagating into an unstable state phi=0, whose asymptotic speed v(as) is equal to the linear spreading speed v* of infinitesimal perturbations about that state (so-called pulled fronts), are very sensitive to changes in the growth rate f(phi) for phi<1. It was recently found that with a small cutoff, f(phi)=0 for phi< epsilon, v(as) converges to v* very slowly from below, as ln(-2) epsilon. Here we show that with such a cutoff and a small enhancement of the growth rate for small phi behind it, one can have v(as)>v*, even in the limit epsilon -->0. The effect is confirmed in a stochastic lattice model simulation where the growth rules for a few particles per site are accordingly modified. Show less
Prigent, A.; Grégoire, G.; Chaté, H.; Dauchot, O.; Saarloos, W. van 2002
We show that turbulent "spirals" and "spots" observed in Taylor-Couette and plane Couette flow correspond to a turbulence-intensity modulated finite-wavelength pattern which in every respect fits... Show moreWe show that turbulent "spirals" and "spots" observed in Taylor-Couette and plane Couette flow correspond to a turbulence-intensity modulated finite-wavelength pattern which in every respect fits the phenomenology of coupled noisy Ginzburg-Landau (amplitude) equations with noise. This suggests the existence of a long-wavelength instability of the homogeneous turbulence regime. Show less
Depending on the growth condition, bacterial colonies can exhibit different morphologies. As argued by Ben-Jacob et al. there is biological and modeling evidence that a nonlinear diffusion... Show moreDepending on the growth condition, bacterial colonies can exhibit different morphologies. As argued by Ben-Jacob et al. there is biological and modeling evidence that a nonlinear diffusion coefficient of the type D(b)=D(0)b(k) is a basic mechanism that underlies almost all of the patterns and generates a long-wavelength instability. We study a reaction-diffusion system with a nonlinear diffusion coefficient and find that a unique planar traveling front solution exists whose velocity is uniquely determined by k and D=D(0)/D(n), where D(n) is the diffusion coefficient of the nutrient. Due to the fact that the bacterial diffusion coefficient vanishes when b-->0, in the front solution b vanishes in a singular way. As a result the standard linear stability analysis for fronts cannot be used. We introduce an extension of the stability analysis that can be applied to singular fronts, and use the method to perform a linear stability analysis of the planar bacteriological growth front. We show that a nonlinear diffusion coefficient generates a long-wavelength instability for k>0 and D0 and k--> infinity the dynamics of the growth zone essentially reduces to that of a sharp interface problem that is reminiscent of a so-called one-sided growth problem where the growth velocity is proportional to the gradient of a diffusion field ahead of the interface. The moving boundary approximation that we derive in these limits is quite accurate but surprisingly does not become a proper asymptotic theory in the strict mathematical sense in the limit D-->0, due to lack of full separation of scales on all dynamically relevant length scales. Our linear stability analysis and sharp interface formulation will also be applicable to other examples of interface formation due to nonlinear diffusion, like in porous media or in the problem of vortex motion in superconductors. Show less
Saarloos, W. van; Frenken, J.W.M.; Gastel, R. van; Albada, S.B. van; Somfai, E. 2002
