Shift in the velocity of a front due to a cut-off

Eric Brunete-mail: eric.brunet@ens.fr and Bernard Derridae-mail: bernard.derrida@ens.fr

Physical Review E 1997, 56 (3), 2597--2604






Abstract: We consider the effect of a small cut-off epsilon on the velocity of a traveling wave in one dimension. Simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cut-off epsilon is to select a single velocity which converges when epsilonto0 to the one predicted by the marginal stability argument. For small epsilon, the shift in velocity has the form K(logepsilon)-2 and our prediction for the constant K agrees very well with the results of our simulations. A very similar logarithmic shift appears in more complicated situations, in particular in finite size effects of some microscopic stochastic systems. Our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the Fisher-Kolmogorov or similar equations.
02.50.Ey, 03.40.Kf, 47.20.Ky

1   Introduction

Equations describing the propagation of a front between a stable and an unstable state appear [1, 2, 3, 4, 5, 6, 7] in a large variety of situations in physics, chemistry and biology. One of the simplest equations of this kind is the Fisher-Kolmogorov [1, 2] equation



which describes the evolution of a space and time dependent concentration h(x,t) in a reaction-diffusion system. This equation, originally introduced to study the spread of advantageous genes in a population [1], has been widely used in other contexts, in particular to describe the time dependence of the concentration of some species in a chemical reaction [8, 9].

For such an equation, the uniform solutions h=1 and h=0 are respectively stable and unstable and it is known [3, 7, 10, 11, 12] that for initial conditions such that h(x,0) to 1 as x to -infty and h(x,0) to 0 as x to + infty there exists a one parameter family Fv of traveling wave solutions (indexed by their velocity v) of the form



with Fv decreasing, Fv(z) to 1 as z to - infty and Fv(z) to 0 as z to infty. The analytic expression of the shape Fv is in general not known but one can determine the range of velocities v for which solutions of type (2) exist. If one assumes an exponential decay



it is easy to see by replacing (2) and (3) into (1) that the velocity v is given by



As gamma is arbitrary, this shows the well known fact that the range of possible velocities is v ge 2. The minimal velocity v0=2 is reached for gamma0=1 and for steep enough initial conditions h(x,0) (which decay faster than e-gamma0 x), the solution selected [3, 4, 6, 7, 10, 11, 12] for large t is the one corresponding to this minimal velocity v0.

Equations of type (1) are obtained either as the large scale limit [5, 8, 13, 14, 15, 16] or as the mean field limit [17] of physical situations which are discrete at the microscopic level (particles, lattice models, etc.) As the number of particles is an integer, the concentration h(x,t) could be thought as being larger than some epsilon, which would correspond to the value of h(x,t) when a single particle is present. Equations of type (1) appear then as the limit of the discrete model when epsilonto0. Several authors [8, 13, 14] have already noticed in their numerical works that the speed vepsilon of the discrete model converges slowly, as epsilon tends to 0, towards the minimal velocity v0. We believe that the main effect of having epsilonne0 is to introduce a cut-off in the tail of the front, and that this changes noticeably the speed.

The speed of the front is in general governed by its tail. In the present work, we consider equations similar to (1), which we modify in such a way that whenever h(x,t) is much smaller than a cut-off epsilon, it is replaced by 0. The cut-off epsilon can be introduced by replacing (1) by



with



For example, one could choose a(h)=1 for hgeepsilon and a(h)=h/epsilon for hleepsilon. Another choice that we will use in section iv is simply a(h)=1 if h>epsilon and a(h)=0 if hleepsilon.

The question we address here is the effect of the cut-off epsilon on the velocity vepsilon of the front. We will show that the velocity vepsilon converges, as epsilon to 0, to the minimal velocity v0 of the original problem (without cut-off) and that the main correction to the velocity of the front is



for an equation of type (1) for which the velocity is related to the exponential decay gamma of the shape (2) by some relation v(gamma). (Everywhere we note by v0 the minimal velocity and gamma0 the corresponding value of the decay gamma.) In the particular case of equation (1), where v(gamma) is given by (4), this becomes



In section ii we describe an equation of type (1) where both space and time are discrete, so that simulations are much easier to perform. The results of the numerical simulations of this equation are described in section iii: as epsilonto0, the velocity is seen to converge like (logepsilon)-2 to the minimal velocity v0, and the shape of the front appears to take a scaling form.

In section iv we show that for equations of type (1) in presence of a small cut-off epsilon as in (5), one can calculate both the shape of the front and the shift in velocity. The results are in excellent agreement with the numerical data of section iii.

In section v we consider a model defined, for a finite number N of particles, by some microscopic stochastic dynamics which reduces to the front equation of sections iii and iv in the limit Ntoinfty. Despite the presence of noise, our simulations indicate that in this case too, the velocity dependence of the front decays slowly (as (log N)-2) to the minimal velocity v0 of the front.

2   A discrete front equation

To perform numerical simulations, it is much easier to study a case where both time and space are discrete variables. We consider here the equation



where



Time is a discrete variable and if initially the concentration h(x,0) is only defined when x is an integer, h(x,t) remains so at any later time. Because t and x are both integers, the cut-off epsilon can be introduced as in (9) in the crudest way using a Heaviside Theta function. (We have checked however that other ways of introducing the cut-off epsilon as in (5, 6) do not change the results.)

Equation (9) appears naturally (in the limit epsilon=0) in the problem of directed polymers on disordered trees [17, 18] (where the energy of the bonds is either 1 with probability p or 0 with probability 1-p). At this stage we will not give a justification for introducing the cut-off epsilon. This will be discussed in section v.

We consider for the initial condition a step function



Clearly for such an initial condition, h(x,t)=1 for x < 0 at all times. As h(x,t)simeq1 behind the front and h(x,t)simeq0 ahead of the front, we define the position Xt of the front at time t by



The velocity of the front vepsilon can then be calculated by



where the average is taken over time. (Note that as h(x,t) is only defined on integers, the difference Xt+1-Xt is time dependent and has to be averaged as in (12).)

When epsilon=0, the evolution equation (9) becomes



As for (1), there is a one parameter family of solutions Fv of the form (2) indexed by the velocity v which is related (3) to the exponential decay gamma of the shape by



(This relation is obtained as (4) by considering the tail of the front where h(x,t) is small and where therefore (13) can be linearized.)

One can show that for p<1/2, v(gamma) reaches a minimal value v0 smaller than 1 for some gamma0, whereas for pge1/2, v(gamma) is a strictly decreasing function of gamma, implying that the minimal velocity is v0= limgammatoinftyv(gamma)=1.

We will not discuss here this phase transition and we assume from now on that p<1/2. Table i gives some values of v0 and gamma0 obtained from (14).
p 0.05 0.25 0.45
gamma0 2.751 111... 2.553 244... 4.051 851...
v0 0.451 818... 0.810 710... 0.979 187...

Table 1: Values of gamma0 and v0 for some p when epsilon =0.
It is important to notice that for p<1/2, the function v(gamma) has a single minimum at gamma0. Therefore, there are in general two choices gamma1 and gamma2 of gamma for each velocity v. For vne v0, the exponential decay of Fv(z) is dominated by min(gamma1,gamma2). As v to v0, the two roots gamma1 and gamma2 become equal, and the effect of this degeneracy gives (in a well chosen frame)



where A is a constant. This large z behavior can be recovered by looking at the general solution of the linearized form of equation (13)



3   Numerical determination of the velocity

We iterated numerically (9) with the initial condition (10) for several choices of p < 1/2 and for epsilon varying between 0.03 and 10-17. We observed that the speed is usually very easy to measure because, after a short transient time, the system reaches a periodic regime for which



for some constants T and Y. The speed vepsilon of the front is then simply given by



For example, for p=0.25 and epsilon=10-5, we find T=431 and Y=343 so that vepsilon=343/431. The emergence of this periodic behavior is due to the locking of the dynamical system of the h(x,t) on a limit cycle. Because Y and T are integers, our numerical simulations give the speed with an infinite accuracy.

For each choice of p and epsilon, we measured the speed of the front, as defined by (12) and its shape. Figure 1 is a log-log plot of the difference v0-vepsilon versus epsilon (varying between 0.03 to 10-17) for three choices of the parameter p. The solid lines on the plot indicate the value predicted by the calculations of section iv.





We see on this figure that the velocity vepsilon converges slowly towards the minimal velocity v0 as epsilonto0. Our simulations, done over several orders of magnitude (here, fifteen), reveal that the convergence is logarithmic: v0 - vepsilon~ (logepsilon)-2.

As the front is moving, to measure its shape, we need to locate its position. Here we use expression (11) and we measure the shape sepsilon(z) of the front at a given time t relative to its position Xt by



When the system reaches the limit cycle (17), the shape sepsilon(z) becomes roughly independent of the time chosen. (In fact it becomes periodic of period T, but the shape sepsilon has a smooth envelope.) We have measured this shape at some arbitrary large enough time to avoid transient effects. As we expect sepsilon(z) to look more and more like Fv0(z) as epsilon tends to 0, we normalize this shape by dividing it by e-gamma0z. The result sepsilon(z) egamma0z is plotted versus z for p=0.25 and epsilon=10-9, 10-11, 10-13, 10-15 and 10-17 in figure 2.





On the left part of the graph, our data coincide over an increasing range as epsilon decreases, indicating that far from the cut-off, the shape converges to expression (15) of Fv0(z). On the right part, the curves increase up to a maximum before falling down to some small value which seems to be independent of epsilon. When epsilon is multiplied by a constant factor (here 10-2), the maximum as well as the right part of the curves are translated by a constant amount. This indicates that for epsilon small enough, the shape sepsilon(z) in the tail (that is for z large) takes the scaling form



We will see that our analysis of section iv does predict this scaling form. As one expects this shape to coincide with the asymptotic form (15) of Fv0(z) for 1 ll z ll |log epsilon|, the scaling function G(y) should be linear for small y.

4   Calculation of the velocity for a small cut-off

The first remark we make is that as soon as we introduce a cut-off through a function a(h) which is everywhere smaller than 1, the velocity vepsilon of the front is lowered compared to the velocity obtained in the absence of a cut-off. This is easy to check by comparing a solution hepsilon(x,t) of (5) where a(h) is present and a solution h0(x,t) of (1). If initially hepsilon(x,0) < h0(x,0), the solution hepsilon will never be able to take over the solution h0. Indeed, would the two functions hepsilon(x,0) and h0(x,0) coincide for the first time at some point x, we would have at that point partial2 hepsilon/ partial x2 le partial2 h0 / partial x2 and together with the effect of a(h) this would bring back the system in the situation where hepsilon(x,t) < h0(x,t) [3, 7]. This shows that vepsilonle v0.

For the calculation of the velocity vepsilon, we will consider first the modified Fisher-Kolmogorov equation (5) when the cut-off function a(h) is simply given by



In this section we will calculate the leading correction to the velocity when epsilon is small and we will obtain the scaling function G which appears in (20). Then we will discuss briefly how our analysis could be extended to more general forms of the cut-off function a(h) or to other traveling wave equations such as (9).

As vepsilon is the velocity of the front, its shape sepsilon(z)=h(z+vepsilont,t) in the asymptotic regime satisfies



When epsilon is small, with the choice (21) for a(h), we can decompose the range of values of z into three regions:

Region I
where sepsilon(z) is not small compared to 1.
Region II
where epsilon < sepsilon(z) ll 1.
Region III
where sepsilon(z) <epsilon.
In region I, the shape of the front sepsilon looks like Fv0 whereas in regions II and III, as sepsilon is small, it satisfies the linear equations



These linear equations (22,23) can be solved easily. The only problem is to make sure that the solution in region II and its derivative coincides with Fv0 at the boundary between  I and II and with the solution valid in region III at the boundary between II and III. If we call Delta the shift in the velocity



and if we note gammar ± i gammai the two roots of the equation v(gamma) = vepsilon, the shape sepsilon is given in the three regions by



and we can determine the unknown quantities C, D, z0 and vepsilon by using the boundary conditions.

For large z we know from (15) that Fv0(z) simeq A z e-gamma0 z for some A. Therefore, as gamma0- gammar ~ Delta and gammai ~ Delta1/2, the boundary conditions between regions I and II impose, to leading order in Delta1/2, that C= A/ gammai and D=0.

At the boundary between regions II and III, we have sepsilon(z)=epsilon and z=z0. If we impose the continuity of sepsilon and of its first derivative at this point, we get



and



Taking the ratio between these two relations leads to



When Delta is small, gammar simeq gamma0 = 1, vepsilonsimeq v0 = 2 and gammai ~ Delta1/2. Thus the only way to satisfy (27) is to set gammai z0 simeq pi and pi - gammai z0 simeq gammai ~ Delta1/2. Therefore, (26) implies to leading order that z0 simeq -(log epsilon)/ gamma0 and the condition gammai z0 simeq pi gives



Then, as gammai is small, the difference Delta=v0-vepsilon is given by



which is the result announced in (7) and (8).

A different cut-off function a(h) should not affect the shape of sepsilon in the region II or the size z0 of region II. Only the precise matching between regions II and III might be modified and we do not think that this would change the leading dependency of z0 in epsilon which controls everything. In fact there are other choices of the cut-off function a(h) (piecewise constant) for which we could find the explicit solution in region III, confirming that the precise form of a(h) does not change (28). The generalization of the above argument to equations other than (1) (and in particular to the case studied in sections ii and iii) is straightforward. Only the form of the linear equation is changed and the only effect on the final result (7) is that one has to use a different function v(gamma).

When expression (7) is compared in figure 1 to the results of the simulations, the agreement is excellent. Moreover, in region II, one sees from (25) and (28) that



which also agrees with the scaling form (20).

Recently, for a simple model of evolution [19, 20] governed by a linear equation, the velocity was found to be the logarithm of the cut-off to the power 1/3. This result was obtained by an analysis which has some similarities to the one presented in this section.

5   A stochastic model

Many models described by traveling wave equations originate from a large scale limit of microscopic stochastic models involving a finite number N of particles [13, 14, 15, 16]. Here we study such a microscopic model, the limit of which reduces to (13) when Ntoinfty. Our numerical results, presented below, indicate a large N correction to the velocity of the form vNsimeq v0 - a (log N)-2 with a coefficient a consistent with the one calculated in section iv for epsilon=1/N.

The model we consider in this section appears in the study of directed polymers [14] and is, up to minor changes, equivalent to a model describing the dynamics of hard spheres [15]. It is a stochastic process discrete both in time and space with two parameters: N, the number of particles, and p, a real number between 0 and 1. At time t (t is an integer), we have N particles on a line at integer positions x1(t), x2(t), ..., xN(t). Several particles may occupy the same site. At each time-step, the N positions evolve in the following way: for each i, we choose two particles ji and j'i at random among the N particles. (These two particles do not need to be different.) Then we update xi(t) by



where alphai and alpha'i are two independent random numbers taking the value 1 with probability p or 0 with probability 1-p. The numbers alphai, alpha'i, ji and j'i change at each time-step. Initially (t=0), all particles are at the origin so that we have xi(0)=0 for all i.

At time t, the distribution of the xi(t) on the line can be represented by a function h(x,t) which counts the fraction of particles strictly at the right of x.



Obviously h(x,t) is always an integral multiple of 1/N. At t=0, we have h(x,0)=1 if x<0 and h(x,0)=0 if xge0. One can notice that the definition of the position Xt of the front used in (11) coincides with the average position of the N particles



Given the positions xi(t) of all the particles (or, equivalently¸ given the function h(x,t)), the xi(t+1) become independent random variables. Therefore, given h(x,t), the probability for each particle to have at time t+1 a position strictly larger than x is given by



The difficulty of the problem comes from the fact that one can only average h(x,t+1) over a single time-step. On the right hand side of (34) we see terms like h2(x,t) or h(x-1,t)h(x,t) and one has to calculate all the correlations of the h(x,t) in order to find langle h(x,t+1)rangle. This makes the problem very difficult for finite N. However, given h(x,t), the xi(t+1) are independent and in the limit Ntoinfty, the fluctuations of h(x,t+1) are negligible. Therefore, when Ntoinfty, h(x,t) evolves according to the deterministic equation (13). As the initial condition is a step function, we expect the front to move, in the limit Ntoinfty, with the minimal velocity v0 of (14).

For large but finite N, we expect the correction to the velocity to have two main origins. First, h(x,t) takes only values which are integral multiples of 1/N, so that 1/N plays a role similar to the cut-off epsilon of section ii. Second, h(x,t) fluctuates around its average and this has the effect of adding noise to the evolution equation (13). In the rest of this section we present the results of simulations done for large but finite N and we will see that the shift in the velocity seems to be very close to the expression of section iv when epsilon=1/N.

With the most direct way of simulating the model for N finite, it is difficult to study systems of size much larger than 106. Here we use a more sophisticated method allowing N to become huge. Our method, which handles many particles at the same time, consists in iterating directly h(x,t).



Knowing the function h(x,t) at time t, we want to calculate h(x,t+1). We call respectively xmin and xmax the positions of the leftmost and rightmost particles at time t and l=xmax-xmin+1. In terms of the function h(x,t), one has 0<h(x,t)<1 if and only if xminle x < xmax. Obviously, all the positions xi(t+1) will lie between xmin and xmax+1. The probability pk that a given particle i will be located at position xmin+k at time t+1 is



with langle h(x,t+1)rangle given by (34). Obviously, pkne0 only for 0 le k le l.

The probability to have, for every k, nk particles at location xmin+k at time t+1 is given by



Using a random number generator for a binomial distribution, expression (36) allows to generate random nk. This is done by calculating n0 according to the distribution



then n1 with



and so on. This method can be iterated to produce the l+1 numbers n0, n1, ..., nl distributed according to (36). Then we construct h(x,t+1) by



As the width l of the front is roughly of order log N, this method allows N to be very large.

Using this method with the generator of random binomial numbers given in [21], we have measured the velocity vN of the front for several choices of p (0.05, 0.25 and 0.45) and for N ranging from 100 to 1016. We measured the velocities with the expression



Figure 3 is a log-log plot of the difference v0-vN versus 1/N compared to the prediction (7) for epsilon=1/N. The variation of vN when using longer times or different random numbers were not larger than the size of the symbols.





We see on figure 3 that the speed vN of the front seems to be given for large N by



where the coefficient K is not too different from the prediction (7).

The agreement is however not perfect. The shift v0-vN seems to be proportional to (log N)-2, but the constant looks on figure 3 slightly different from the one predicted by (7). A possible reason for this difference could have been the discretization of the front: instead of only cutting off the tail as in sections iii and iv, here the whole front h(x,t) is constrained to take values multiple of 1/N. One might think that this could explain this discrepancy. However, we have checked numerically (the results are not presented in this paper) that equation (13) with h(x,t) constrained to be a multiple of a cut-off epsilon does not give results significantly different from the simpler model of sections iii and iv with only a single cut-off. So we think that the full discretization of the front can not be responsible for a different constant K. The discrepancy observed in figure 3 is more likely due to the effect of the randomness of the process. It is however not clear whether this mismatch would decrease for even larger N. It would be interesting to push further the numerical simulations and check the N-dependence of the front velocity for very large N.

6   Conclusion

We have shown in the present work that a small cut-off epsilon in the tail of solutions of traveling wave equations has the effect of selecting a single velocity vepsilon for the front. This velocity vepsilon converges to the minimal velocity v0 when epsilonto0 and the shift v0-vepsilon is surprisingly large (7, 8).

Very slow convergences to the minimal velocity have been observed in a number of cases [8, 13, 14, 15] as well as the example of section v. As the effect of the cut-off on the velocity is large, it is reasonable to think that it would not be much affected by the presence of noise. The example of section v shows that the cut-off alone gives at least the right order of magnitude for the shift and it would certainly be interesting to push further the simulations for this particular model to see whether the analysis of section iv should be modified by the noise. The numerical method used in section v to study a very large (N~ 1016) system was very helpful to observe a logarithmic behavior. We did not succeed to check in earlier works [13, 14, 15, 22] whether the correction was logarithmic, mostly because the published data were usually too noisy or obtained on a too small range of the parameters. Still even if the cut-off was giving the main contribution to the shift of the velocity, other properties would remain very specific to the presence of noise like the diffusion of the position of the front [16].

Our approach of section iv shows that the effect of a small cut-off is the existence of a scaling form (20, 30) which describes the change in the shape of the front in its steady state. The effect of initial conditions for usual traveling wave equations (with no cut-off) leads to a very similar scaling form for the change in the shape of the front in the transient regime. This is explained in the appendix where we show how the logarithmic shift of the position of a front due to initial conditions [10, 23] can be recovered.

A   Effect of initial conditions on the position and on the shape of the front

In this appendix we show that ideas very similar to those developed in section iv allow one to calculate the position and the shape at time t of a front evolving according to (1), or to a similar equation, given its initial shape. The main idea is that in the long time limit, there is a region of size (t)1/2 ahead of the front which keeps the memory of the initial condition. We will recover in particular the logarithmic shift in the position of the front due to the initial condition [10, 23], namely that if the initial shape is a step function



then the position Xt of the front at time t increases like



More generally, if initially



we will show that for nu>-2



whereas the shift is given by (A2) for nu<-2. Here, there is no cut-off but the transient behavior in the long time limit gives rise to a scaling function very similar to the one discussed in section iv.

If we write the position of the front at time t as



we observed numerically (as in figure 2 of section iii) and we are going to see in the following that the shape of the front takes for large t the scaling form



very similar to (20, 30).

If we use (A5) and (A6) into the linearized form of equation (1), we get using, the fact that v0=2 and gamma0=1,



where z=(x-Xt)t-alpha. By writing that the leading orders of the different terms of (A7) are comparable, we see that we must have



for some beta, and that the right hand side of (A7) is negligible. Therefore, the equation satisfied by G is



and the position of the front is given by



As in section iv, we expect that as ttoinfty, the front will approach its limiting form and therefore that for z small, the shape will look like (15). Therefore we choose the solution Gbeta(z) of (A10) which is linear at z=0. This solution can be written as an infinite sum



(The second expression is not valid when beta is a non-positive integer.)

To determine beta, one can notice that the scaling form (A6) has to match the initial condition when x is large and t of order 1. We thus need to calculate the asymptotic behavior of G(z) when z is large.

For certain values of beta, there exist closed expressions of the sum (A12). For instance,



One can check directly on (A10) that Gbeta has a symmetry



For any beta, one can obtain the large z behavior of G(z). To do so, we note that for beta>0, one can rewrite (A12) as



For 0<beta<1, the second integral in (A15) has a non zero limit and this gives the asymptotic behavior of Gbeta(z)



From (A12), one can also show that



implying that (A16) remains valid for all beta except for beta=3/2, 5/2, 7/2, etc., where the amplitude in (A16) vanishes. For these values of beta, Gbeta(z) decreases faster than a power law (see (A13)).

The functions Gbeta calculated so far are acceptable scaling functions for the shape of the front only for betale3/2. Indeed, one can see in (A16) that for 3/2<beta<5/2 the function Gbeta(z) is negative for large z. In fact, for all beta>3/2, this function changes its sign at least once, so that the scaling form (A6) is not reachable for an initial h(x,0) which is always positive. It is only for betale3/2 that Gbeta remains positive for all z>0.

Looking at the asymptotic form (A16), we see that if initially h(x,0)=xnue-gamma0 x, the only function Gbeta(z) which has the right large z behavior is such that 1-2beta=nu, and this gives, together with (A11), the expression (A4) for the shift of the position. As the cases beta>3/2 are not reachable, all initial conditions corresponding to nu<-2 or steeper (such as step functions) give rise to G3/2 and the shift in position given by (A2).

All the analysis of this appendix can be extended to other traveling wave equations such as (13), with more general functions v(gamma) (having a non-degenerate minimum at gamma0) as in (14). Then the expressions (A2, A4) of the shift become



and





We thank C. Appert, V. Hakim and J.L. Lebowitz for useful discussions.

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