Untitled

Abstract

We study the effect of the noise due to microscopic fluctuations on the position of a one dimensional front propagating from a stable to an unstable region in the``linearly marginal stability case.'' By simulating a very simple system for which the effective number N of particles can be as large as N=10 150 , we measure the N dependence of the diffusion constant D N of the front and the shift of its velocity v N . Our results indicate that D N t(log N ) &3 . They also confirm our recent claim that the shift of velocity scales like v min &v N & K(log N ) &2 and indicate that the numerical value of K is very close to the analytical expression K approx obtained in our previous work using a simple cut-off approximation.

Noisy traveling waves: Effect of selection on genealogies

“…if v(γ) has a finite minimum. We think that this prediction agrees with our measurements (4) up to finite size corrections: in fact, for the diffusion constant itself, it already turned out [15,17,34] that the large N asymptotic regime was only observed for much larger systems than the ones studied here.…”

“…Determining quantitatively the effect of a weak noise (ǫ ≪ 1) on the front position is a subject of active research. There is increasing evidence that the dynamics of the position of the front is dominated by the fluctuations near its tip [15,16,17] and that there is a shift in the velocity of the front [8,18,19,20,21,22], logarithmic in the amplitude ǫ of the noise, as predicted by a simple cut-off theory [23].In the present letter, we consider models of an evolving population under selection, which can be described by noisy traveling wave equations. Instead of focusing on the time dependence of the position of the front, we look at the problem from a different perspective: we determine how coalescence times in the genealogy depend on the size of the population.…”

Front propagation up a reaction rate gradient

“…Exactly how to do this is already unclear in the simpler case of the Fisher equation front, where it has proven difficult to come up with a simple explanation for the numerically determined front diffusion constant. [22,23]. The first question to be answered for the gradient case is whether the front can be described as simply diffusing (albeit with an anomalous diffusion constant) or whether the fluctuation effects perhaps lead to even stronger stochasticity.…”

Front propagation and diffusion in theA⇆A+Ahard-core reaction on a chain

“…In addition, in a model that Brunet and Derrida studied in Ref. [8], the front diffusion coefficient D f was numerically shown to vanish only as 1/ ln 3 N . The dominant asymptotic correction to the mean-field result for the front speed in the limit N → ∞ traces simply to the change in the dynamics at ρ = O(1/N ) [7], and as a result appear to be universal.…”