Fronts with a growth cutoff but speed higher than v*
Saarloos, W. van; Panja, D.
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Saarloos, W. van, & Panja, D. (2002). Fronts with a growth cutoff but speed higher than v*.
Physical Review E, 66, 15206-15206. Retrieved from https://hdl.handle.net/1887/5523
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Fronts with a growth cutoff but with speed higher than the linear spreading speed
Debabrata Panja and Wim van SaarloosInstituut-Lorentz, Universiteit Leiden, Postbus 9506, 2300 RA Leiden, The Netherlands
共Received 5 April 2002; published 24 July 2002兲
Fronts, propagating into an unstable state⫽0, whose asymptotic speed vasis equal to the linear spreading
speedv*of infinitesimal perturbations about that state共so-called pulled fronts兲, are very sensitive to changes in the growth rate f () for Ⰶ1. It was recently found that with a small cutoff, f ()⫽0 for ⬍, vas
converges to v* very slowly from below, as ln⫺2. Here we show that with such a cutoff and a small enhancement of the growth rate for small behind it, one can have vas⬎v*, even in the limit→0. The effect
is confirmed in a stochastic lattice model simulation where the growth rules for a few particles per site are accordingly modified.
DOI: 10.1103/PhysRevE.66.015206 PACS number共s兲: 05.45.⫺a, 05.70.Ln, 47.20.Ky Pulled fronts are those fronts that propagate into a linearly
unstable state, and whose asymptotic front speedvasequals
the linear spreading speed v* of infinitesimal perturbations about the unstable state关1–3兴. The name pulled front refers to the picture that in the leading edge of these fronts, the perturbation about the unstable state grows and spreads with speedv*, while the rest of the front gets ‘‘pulled along’’ by the leading edge. That this notion is not merely an intuitive picture but can be turned into a mathematically precise analysis is illustrated by the recent derivation of exact results for the general power law convergence of the front speed to the asymptotic value v* 关3兴. Fronts that propagate into a
linearly unstable state and whose asymptotic speed vas⬎v*
are referred to as pushed, as it is the nonlinear growth in the region behind the leading edge that pushes their front speed to higher values. If the state is not linearly unstable, thenv*
is trivially zero; in such cases the front propagation is always dominated by the nonlinear growth in the front region itself, and hence fronts in this case are in a sense ‘‘pushed’’ too.
For the field(x,t), the dynamics of fronts that we con-sider in this paper is given by the usual nonlinear diffusion equation
t ⫽
2
x2⫹ f共兲. 共1兲
In the standard case, the growth function f () has the form
f ()⫽⫺n, with n⬎1. Equation 共1兲 has two stationary states for (x,t): (x,t)⫽0 and (x,t)⫽1. Of these,
(x,t)⫽1 is stable and (x,t)⫽0 is unstable. The asymptotic speed of 共pulled兲 fronts propagating from
(x,t)⫽1 into(x,t)⫽0 in Eq. 共1兲 is v*⫽2.
The sensitivity of pulled fronts to the precise dynamics for small perturbations about the unstable state has recently surfaced in a remarkable way 关4兴. Often, in equations like Eq. 共1兲, the field(x,t) is the density of particles in a con-tinuum description. If one then considers fronts in stochastic particle model versions of Eq.共1兲, the linear growth term in
f () implies that for small particle density, the rate at which new particles are created is proportional to the density itself. Brunet and Derrida 关4兴 were the first to realize the fact that for new particles to be created in any given realization, the
density must be at least one ‘‘quantum’’ of particle density strong, and that this provides a natural lower cutoff for the growth that strongly affects the front speed. Indeed, to mimic this effect, they considered a deterministic front of the type in Eq.共1兲 with n⫽3, and by hand introduced a cutoff of the type sketched in Fig. 1共a兲 in the growth function at ⫽
Ⰶ1. In this paper, we denote their growth function by
f (,)⬅关⫺3兴⌰(⫺), where ⌰ is the unit step
func-tion. For small, the asymptotic front speed vas() was then
found to be关4兴
vas共兲⯝v*⫺ 2
共ln 兲2⫹•••. 共2兲
Brunet and Derrida subsequently identified with 1/N, where N is the average number of particles at the saturation state of the front, corresponding to the stable state (x,t)
⫽1 of the density field. The slow logarithmic convergence to
the asymptotic front speed from below as a function of N, implied by Eq.共2兲, has been confirmed in various studies of stochastic lattice models 关4–9兴. Note that for ⬍, the growth function f vanishes, and as a result, strictly speaking,
the state ⫽0 is not linearly unstable; hence fronts in this model are always weakly pushed for any nonzero value of
关10兴.
In this paper, we demonstrate an even more surprising aspect of the sensitivity to small changes in the growth func-tion f of the ‘‘pulled’’ fronts that we have at ⫽0: if f is sufficiently enhanced in a range of of the order of, the asymptotic front speed vas can become larger than v* and
not converge tov* as→0. For fluctuating fronts, this
im-plies that if the stochastic growth rates for small occupation densities ni are somewhat enhanced over a linear behavior ⬃ni, then such stochastic fronts may move faster thanv*
and never converge to their naive mean field limit for N →⬁. This effect may be of relevance for the coarse-grained field theory for diffusion-limited aggregation, as it is empiri-cally known to be essential to modify the growth function for small cluster densities关11兴.
We now discuss our results first, and then summarize their derivation.
To be specific, we consider the nonlinear diffusion equa-tion 共1兲 with the growth function sketched in Fig. 1共b兲,
f共,,r兲⫽ f共,兲 for ⬍ and
⫽/r for ⭐⭐/r, 共3兲
with r⬍1. We show that while for any fixed value of r lim
→0
f共,,r兲⫽ f共兲, 共4兲
the asymptotic front speedvas(,r) has the property that
lim →0 vas共,r兲⫽v* for r⬎rc, lim →0 vas共,r兲⬎v* for r⬍rc, 共5兲 where rc⫽ 1⫹e⫺(v*2⫺2) v*2 ⫽0.283 833 . . . . 共6兲
Hereafter, for simplicity, we denote vas(,r) simply by v.
For→0, the asymptotic speed at a given value of r⭐rcin
our model is given by the relation
r⫽ 1 v2
冋
1⫹ 1⫺ 2 v冑
v2 4 ⫺1 1⫹ 2 v冑
v2 4 ⫺1 ⫻exp冉
⫺再
v2⫺ 2 1⫹ 2 v冑
v2 4 ⫺1冎冊
册
, 共7兲from which the value of rc, given by Eq.共6兲, follows.
These expressions show that the limits do not commute for r⬍rc: taking the limit→0 first in f yields a front speed
v* but the limit vas(→0,r)⬎v*. The reason is that for r
⬍rc there is always a little tail of the front that runs faster
than v* and makes nonzero. Once is nonzero, growth continues and the region behind it just has to follow it with the same asymptotic speed.
Our analysis is corroborated by numerical results obtained by solving Eq. 共1兲 forward in time, 共with Gaussian initial conditions兲. The data for v vs r at ⫽2⫻10⫺5are shown as solid dots in Fig. 2. Note that for r⬍rc, the solid dots fall on
top of our prediction 共7兲 drawn with a solid line, while for
r⬎rc, they systematically fall below the solid line v⫽v*.
The reason for it is the difference between the rates of con-vergence as →0, which is illustrated in the inset of Fig. 2 by means of the schematically drawn dashed line. The ar-rows in the inset indicate the rate of convergence of the dashed r-v curve towards the limiting one, given by Eq.共7兲. For r⬎rc, the convergence is ⬃ln⫺2 as in the case for r ⫽1, analyzed in Ref. 关4兴; but for r⬍rc the convergence is
much faster, n⫺1. This latter behavior is illustrated for r
⫽0.2 and n⫽2,3 in Fig. 3—note the fine scale on the
verti-cal axis.
The fact that the effect of increasing asymptotic speed with decreasing r below rc is a real effect for stochastic
fronts too is illustrated by the crosses in Fig. 2: these repre-sent the data for the average speed of fronts in a reaction-diffusion system X 2X, for discrete X particles on a lattice with N⫽104 关12兴, where the growth rates have been modi-fied when the number of particles nion a lattice site i is less
than 1/r. In accord with the shape of the growth function f illustrated in Fig. 1共b兲, the rate at which particles are created at a lattice site i with 1⭐ni⬍1/r particles is simply taken to
be the same as the rate for ni⫽1/r 共corresponding to the FIG. 2. Comparison of simulation data for vas(,r) with the
analytical prediction共7兲, which is plotted as the solid line. The solid dots represent the numerical data for Eq.共3兲 with ⫽2⫻10⫺5and
n⫽3. The crosses are the data points for fronts in the stochastic
growth model described in the text. Inset: illustration of the leading order rate of convergence of thevas(,r) curve to the →0 limit,
by means of the schematic dashed curve.
DEBABRATA PANJA AND WIM van SAARLOOS PHYSICAL REVIEW E 66, 015206共R兲 共2002兲
integral values 1/r⫽1, 2, 3, 4, 5, 7, and 10, due to the dis-creteness of particles兲. As one can see from Fig. 2, already when r⫽0.5, i.e., when only the growth rate at lattice sites with one particle is increased by a factor 2, the asymptotic growth speed is above the valuev*⫽2.
In the remainder of this paper, we derive the analytical results for the nonlinear diffusion equation with the growth function共3兲. Our analysis is based on the following observa-tion: for ⫽0, it is well known that the nonlinear diffusion equation allows a continuous family of front solutions with v⭓v*. When such fronts solutions are parametrized by their velocity v, and when the growth rate is modified to allow a transition to a ‘‘pushed’’ front with velocity v†, it is also
known 关2,3兴 that solutions with v⬍v† are unstable to a lo-calized mode. In our analysis, we therefore consider a front with a given fixed velocity v and, for small , determine when upon decreasing r a localized mode of the stability operator crosses the eigenvalue zero. In the limit →0 this marks the selected pushed front in the r-v diagram.
To carry out the linear stability analysis of the front solu-tion, it is convenient to follow the standard route of trans-forming the linear eigenvalue equation into a Schro¨dinger eigenvalue problem 关1,3兴. We consider a function (x,t), which is infinitesimally different from the asymptotic front solution as() in the comoving frame ⫽x⫺vt, i.e.,
(x,t)⫽as()⫹(,t). Upon linearizing Eq.共1兲 in the co-moving frame, one finds that the function (,t) obeys the following equation: t ⫽v ⫹ 2 2⫹ ␦f共兲 ␦
冏
⫽ as . 共8兲 Since this equation is linear in, the question of stability can be answered by studying the spectrum of the temporal eigen-values. To this end, we express(,t) as共,t兲⫽e⫺Ete⫺v/2E共兲, 共9兲
which converts Eq. 共8兲 to a one-dimensional Schro¨dinger equation for a particle in a potential withប2/2m⫽1,
冋
⫺ d 2 d2⫹ v2 4 ⫺ ␦f共兲 ␦冏
⫽ as册
E共兲⫽EE共兲. 共10兲In Eq.共10兲, the quantity
V共兲⫽
冋
v 2 4 ⫺ ␦f共兲 ␦冏
⫽ as册
plays the role of the potential. It is easily obtained explicitly from the expression 共3兲 for f (,,r) as
V共兲⫽
冋
v 2 4 ⫺1⫹nas n⫺1共兲册
⌰共1⫺兲⫹v 2 4 ⌰共⫺1兲 ⫺ 1 rv␦共⫺0兲, 共11兲 where(0)⫽ and(1)⫽/r. The form of the potential for v⬎v* and small is sketched in Fig. 4. Keep in mind that as() is a monotonically increasing function from/r at1towards the left, and thatas(→⫺⬁)⫽1. As a result, in Fig. 4, V() also increases monotonically towards the left for⬍1. On the right of1, V() is constant atv2/4, and at0, there is an attractive ␦-function potential of strength (rv)⫺1 关13兴. The crucial feature for the stability analysis below is the fact that V() stays remarkably flat at a value 2/r over a distance (1⫺2)⯝兩ln /r兩 关10兴, and on the left of2, it increases to the value ofv2/4⫹n⫺1, over a distance
of order unity.
If there exist negative eigenvalues of the Schro¨dinger equation 共10兲, then according to Eq. 共9兲, (,t) grows in time in the comoving frame, i.e., the front solutionas() is unstable. For our purpose, therefore, we look for the value of
r at which there is a bound state of Eq.共10兲 with eigenvalue E, such that E→0⫺ for the potential sketched in Fig. 4. This
is a problem in elementary quantum mechanics. For →0, FIG. 3. Numerical data for vas(,r) as a function of for n
⫽2 and 3, at r⫽0.2. The graph demonstrates the insensitivity of v
to for small values of 共note the fine scale on the vertical axis兲, as well as the convergence asn⫺1to its→0 value ⯝2.246, given by Eq.共7兲, for two different values of n.
FIG. 4. The potential V() for v⭓v*and infinitesimally small
in the Schro¨dinger operator that determines the temporal
eigen-values of the stability analysis.2marks the position of the region
of finite width where the potential crosses over from the asymptotic value on the left where as⬇1 to the value in the well whereas
Ⰶ1, 1the position of the step and0the position of the␦-function
the potential V() is essentially constant in the left neighbor-hood of1, and hence forv⬎v* and E→0⫺,E() can be
written as
E共兲⫽A⫺e1(⫺1) for ⭐1, ⫽Ae2(⫺0)⫹Be2(0⫺) for 1⭐⭐0,
⫽A2e⫺2(⫺0) for ⬎0,
共12兲
where1⫽
冑
v2/4⫺1 and 2⫽v/2. The functionE() mustbe continuous at 1 and0, while its slope is continuous at
1, but not at0. Matching of these boundary conditions to determine the value of r, where the bound state eigenvalue E crosses zero, also requires an expression for the distance 0
⫺1. To this end, we divide the range ofvalues between 0 and 1 into the three regions marked in Fig. 4: 共i兲 region I, where as⬍, 共ii兲 region II, where ⭐as⬍/r, and 共iii兲 region III, where as⭓/r. In the comoving frame, the asymptotic shape as() of the front is the solution of the
differential equation as
⬙
⫹vas⬘
⫹ f (as,,r)⫽0, where aprime denotes a derivative with respect to. The solutions of
as() in the regions I and II that satisfy the continuity of as() andas
⬘
() are, respectively, given byas共兲⫽e⫺v(⫺0) and as共兲⫽
冋
⫺ rv2册
e v(0⫺)⫹共0⫺兲 rv ⫹ rv2. 共13兲 The length 0⫺1 of region II is obtained by equatingas(1) from the second line of Eq.共13兲 to /r. After
divid-ing out a factor of /r, this condition becomes
冋
r⫺ 1 v2册
e v(0⫺1)⫹0⫺1 v ⫹ 1 v2⫽1. 共14兲Thereafter, using Eqs.共12兲 and 共13兲, one arrives at Eq. 共7兲. The above analysis yields the relation betweenv and the critical value of r in the limit→0. The convergence with , i.e., the rate of approach with of the dashed curve to the solid one in Fig. 2, can be obtained by considering the effect
of the nasn⫺1() term of V() on the eigenfunctions and eigenvalues. For v⬎v*, this term is simply a correction of order n⫺1 to the finite bottom value of the potential. This term can be included perturbatively, and accordingly it leads to a shift of order n⫺1 in the critical value of r. As Fig. 3 illustrates, this prediction is confirmed numerically. The case v⫽v* calls for a more detailed analysis, since the bottom value of the potential vanishes in the limit →0. In this case, it is known 关3,4兴 that as()⬃(C⫹D)e⫺, so
V()⯝n(/r)n⫺1(/1)n⫺1e(n⫺1)(1⫺) in the leading order of . In dominant order, we need to keep only the exponen-tial behavior, and the solution of E() is then given by the Bessel function A⫺K0(2
冑
nn⫺1e⫺(n⫺1)(1⫺)/rn⫺1) in theleft neighborhood of 1. The 兩ln⫺1兩 scaling for the asymptotic approach of the dashed curve to the solid one is then easily obtained once the boundary conditions at1 and
0 are matched using Eq.共14兲.
The logarithmic convergence of v to v* from below for
r⬎rccan be understood from an argument along the lines of
that for r⫽1 关4兴. For v⬍v*, the front profile as() in re-gion III is of the form as()⬃C sin关k(⫺2)⫹兴e⫺. For r
⫽1, region II is absent; in that case, the matching to the
profile in region I and the divergence of the width 0⫺2
⯝兩ln 兩 implies k⯝兩ln兩⫺1. For r
c⬍r⬍1, the matching to
region II will change the prefactor, but k will still scale as
兩ln兩⫺1because the width of region III still diverges
logarith-mically. As for rc⭐r⬍1, this translates into a scaling of
v*⫺v as 兩ln 兩⫺2, with a prefactor that depends on r. Note that this scaling is nicely consistent with the convergence of the r-v curve towards the point (rc,v*) from the left, due to
the fact that the slope of this curve vanishes at this point, and the convergence from below to this point scales as the square of the convergence from the left.
We finally end this paper with the note that if the 共nonne-gative兲 growth rate is bounded from above by f (,) in the interval⭐/r, but is equal to f (,) for ⬎/r, then as
→0, the asymptotic front speed converges to v* with the
same logarithmic convergence of Eq.共2兲 for any r. It simply
follows from the inequality vas(/r)⬍v⬍vas(), where
vas() is given by Eq. 共2兲.
D.P. wishes to acknowledge financial support from ‘‘Fun-damenteel Onderzoek der Materie’’共FOM兲.
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关12兴 This corresponds to ␥˜⫽1 and N⫽104
for the model consid-ered in Ref.关9兴.
关13兴 The␦-function in Eq.共11兲 appears from the functional deriva-tive of f in V, since there is a discontinuity of magnitude in f (as,,r) atas⫽. This discontinuity contributes an amount
equal tor⫺1d⌰(as⫺)/das⫽关r兩as⬘(0)兩兴⫺1␦(⫺0) to
V(). Asas⬘⫽r according to Eq. 共13兲, the prefactor 1/(rv).
DEBABRATA PANJA AND WIM van SAARLOOS PHYSICAL REVIEW E 66, 015206共R兲 共2002兲
