Fluctuating pulled fronts have anomalous diffusive or scaling properties

As we explained in Universal algebraic velocity relaxation of uniformly translating as well as pattern forming pulled fronts, pulled fronts are those particular fronts which propagate into an unstable state and whose asymptotic velocity is simply equal to the linear spreading velocity v* of linear perturbations about the unstable state into which the front propagates. The crucial feature of pulled fronts which makes them so different from the normal types of fronts ("pushed" fronts or fronts between two linearly stable states) is that the dynamically relevant region is actually ahead of the front, not the nonlinear front region that one intuitively associates with the front itself. As discussed in this other writeup, the most striking implication of this is that pulled fronts relax with a universal power law to their asymptotic spead, not exponentially fast. A second implication of this shift of the dynamically important region to the leading edge ahead of the front and of the fact that the weight of this region grows arbitrarily large when a deterministic pulled front relaxes to its asymptotic speed and profile, is that one expects fluctuating (noisy) pulled fronts to behave differently from the usual noisy fronts, e.g., those that separate two linearly stable states.

Indeed, Andrea Rocco, Ute Ebert and myself recently predicted that one dimensional pulled fronts in the presence of multiplicative noise should show subdiffusive behavior: unlike normal fronts which in the presence of multiplicative noise exhibit diffusive wandering (root mean square displacement growing as t1/2), we predict that noisy pulled fronts have their root mean square displacement grow as t 1/4! Our simulations confirm this subdiffusive behavior. The fact that the effective exponent is slightly larger than 1/4 is attributed to slow crossover effects. The derivation of the subdiffusive behavior of pulled fronts is based on mapping the pulled front in the leading edge - which is the region that determines the essential dynamics of a pulled front - onto the 1+1D KPZ equation in 1 space and 1 time dimension (the KPZ equation named after Kardar, Parisi and Zhang).

Secondly, Goutam Tripathy and I have argued that fluctuating pulled fronts in more than one dimension are most likely not obeying the standard KPZ (Kardar-Paris-Zhang) scaling which is believed to be the asymptotic fised point theory for most fluctuating interfaces. We studied this issue by introducing a simple stochastic model whose fronts can be changed from pushed to pulled by tuning some simple reaction rate parameter. Although one should of course always watch out for slow crossovers, our simulations do give support for our assertion that noisy pulled fronts do not obey the standard KPZ scaling.

An interesting twist to the story is a recent discovery of ours: if one combines the insight from the work with Rocco and Ebert with the one from the work with Tripathy, one quickly arrives at the conjecture that fluctuating pulled front in d+1 bulk dimensions should not be in the universality class of the d+1D KPZ equation, but in the universality class of the (d+1)+1D KPZ equation. A scaling plot of the width distribution of Goutam's data supports this conjecture very strongly. Together with Andrea Rocco and Jaume Casademunt, we have recently given a field-theoretic derivation of this result.

Note the interesting dichotomy: while in Universal algebraic velocity relaxation of uniformly translating as well as pattern forming pulled fronts we summarize the remarkable universal 1/t relaxation of deterministic pulled fronts by writing "Some fronts are more equal than others", we now see that for noisy fronts the opposite holds! For, normal fronts relax exponentially and therefore obey diffusive wandering in one dimension and KPZ scaling in more than one dimension (in this sense "all fronts are equal to others"), while noisy pulled fronts are now not equal to others!

Wim van Saarloos
Updated March 22 2001

Relevant papers:
G. Tripathy and W. van Saarloos, Fluctuation and relaxation properties of pulled fronts: a possible scenario for non-KPZ behavior, Phys. Rev. Lett. 85, 3556 (2000) (pdf file).
A. Rocco, U. Ebert and W. van Saarloos, Subdiffusive fluctuations of "pulled" fronts with multiplicative noise, Phys. Rev. E, 62 R13-R16 (2000). (ps file)
G. Tripathy, A. Rocco, J. Casademunt, and W. van Saarloos, The universality class of fluctuating pulled fronts, Phys. Rev. Lett. 86, 5215 (2001) (pdf file)

[Pattern formation] [Wim van Saarloos] [Instituut-Lorentz]