Fluctuating pulled fronts: The origin and the effects of a finite particle cutoff

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 66, Issue 3, 2002, Pages

  Panja, Debabrata ^a   van Saarloos, Wim ^a  

^a UNIVERSITEIT LEIDEN   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

CORRELATION METHODS; ENTROPY; GLASS TRANSITION; MATHEMATICAL MODELS; PHOTONS; PRESSURE EFFECTS; SPECTROSCOPY; THERMAL EFFECTS;

BIOCHEMICALS; CONFIGURATIONAL ENTROPY; GLASS FORMER; RELAXATION PROCESS;

GLASS;

ARTICLE;

EID: 41349097708     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.66.036206     Document Type: Article

References (41)

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25. One should be careful in considering the limits in discussing this issue. Strictly speaking, for any finite particle number N a front is not pulled, but weakly pushed. Hence for any finite N the true asymptotic scaling should be of the KPZ type.
26. The issue is complicated, as there are two effects which may play a role: besides the crossover to KPZ behavior because the finite particle effects make the front weakly pushed, it has been questioned whether the multiplicative noise used in Refs. 1820 is consistent with intrinsic fluctuations, E. Moro, Phys. Rev. Lett. 87, 238303 (2001).
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32. For the FKPP model with a cutoff in the growth term that Brunet and Derrida 9 introduced to model the finite particle effects, it is easy to convince oneself explicitly that the fronts are in fact always weakly pushed, according to the classification of Ref. 17.
33. It is useful to realize that pulled fronts on a lattice without a finite particle cutoff are already somewhat exceptional, in that the lattice effects hardly play a role. Fronts between two linearly stable states are much more sensitive to lattice effects, e.g., they can become “locked” over some nonzero range of driving forces [see, e.g., G. Fáth, Physica D 116, 176 (1998)]. The lattice effects we analyze here are less dramatic, but of a similar nature.
34. We note that several aspects of the analysis in this section show similarities with elements of the analysis of Ref. 9. However, their analysis is simpler because they introduced a cutoff in a phenomenological way into the growth rate in the continuum mean field equation. As a result, they could simply match the two regions where the growth rate was different. Our analysis is beset by the fact that we do attempt to take the lattice effects into account (this already makes matching impossible), as well as the stalling effects and the changes this gives in the profile behind the foremost bin.
35. This means that we start with a guess value of (Formula presented) obtain the theoretical (Formula presented) curve, compare it with the (Formula presented) curve obtained from the computer simulations, and use the nature of discrepancy to make the next guess value of (Formula presented) This process is repeated until one converges to the value of (Formula presented) for which the best agreement between the theoretical and the simulation (Formula presented) curves is achieved. In this recursive feedback method, specifically for (Formula presented) and (Formula presented) one should remember to compare the theoretical and the simulation (Formula presented) curves for (Formula presented) while obtaining the numerical value of (Formula presented) since for (Formula presented) there are other complications that come to play a very significant role. These effects are discussed in Sec. IV C.
36. In actuality, at the instant step (b) is over, the population of the (Formula presented)th bin has grown to a value bigger than 1. As a result, the diffusion of an X particle from the (Formula presented)th bin to the (Formula presented)th bin takes slightly smaller time than the time scale for a new foremost bin creation (Formula presented) A (Formula presented) time scale would be applicable if the population of the (Formula presented)th bin is exactly 1 at the instant step (b) is over. We choose to ignore this, and by choosing to do so, we overestimate the value of (Formula presented) by a small amount.
37. Of course, in an actual process, (Formula presented) is determined probabilistically and not in such a sharp manner. However, we should remember that this is only an estimate, and we will show later that it works reasonably well.
38. For large values of (Formula presented) there can be another kind of event, where the vacant foremost bin event takes place more than once for the same foremost bin. This is more likely for large values of (Formula presented) since the hops from the foremost bin towards the left is much more likely before the number of X particles gets a chance to grow in the foremost bin. In that case, one can simply extend this existing way of estimating (Formula presented) to (Formula presented) (Formula presented) etc.
39. The sequence of random numbers generated by drand48 is random enough for our purposes so long as we use any large enough (Formula presented) initial seed. There is nothing special about choosing 123 456, it is as arbitrarily chosen as any other initial seed (Formula presented) We have also run the simulations for (Formula presented) and (Formula presented) with two other random seeds, namely, (Formula presented) and (Formula presented) For both of these two cases, the average speeds calculated by means of Eq. (5.7) fell within the error bars of Eq. (5.8). Thus, the chance of our results being affected by inherent correlations of the random number generator provided in the standard C library is ruled out.
40. We have verified that at (Formula presented) the front shape is still given by the solution of the linear equation, Eq. (4.6).
41. W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Veterling, Numerical Recipes (Cambridge University Press, Cambridge, 1986).