Dynamical features of reaction-diffusion fronts in fractals - Core

PHYSICAL REVIEW E 69, 016613 共2004兲

Dynamical features of reaction-diffusion fronts in fractals Vicenc¸ Me´ndez,1 Daniel Campos,2,* and Joaquim Fort3,† 1

Departamento de Medicina, Universitat Internacional de Catalunya, c./Gomera s/n, 08190-Sant Cugat del Valle´s, Barcelona, Spain 2 Grup de Fı´sica Estadı´stica, Departamento de Fı´sica, Universitat Auto`noma de Barcelona, E-08193 Bellaterrra, Spain 3 Departamento de Fı´sica, Universitat de Girona, Campus de Montilivi, 17071 Girona, Catalonia, Spain 共Received 12 September 2003; published 29 January 2004兲 The speed of front propagation in fractals is studied by using 共i兲 the reduction of the reaction-transport equation into a Hamilton-Jacobi equation and 共ii兲 the local-equilibrium approach. Different equations proposed for describing transport in fractal media, together with logistic reaction kinetics, are considered. Finally, we analyze the main features of wave fronts resulting from this dynamic process, i.e., why they are accelerated and what is the exact form of this acceleration. DOI: 10.1103/PhysRevE.69.016613

PACS number共s兲: 05.45.Df, 05.40.⫺a, 87.10.⫹e

I. INTRODUCTION

Reaction-diffusion models are based in general on the partial differential equation

⳵ P 共 x,t 兲 ⳵ 2 P 共 x,t 兲 ⫽D ⫹ f 共 P 兲, ⳵t ⳵x2

共2兲

where d w ⭓2 is the random-walk dimension of the fractal 共for d w ⫽2 one recovers the classical case兲. The main advances in the field of transport in fractals have been focused on finding a transport equation for the probability density P(x,t) of finding a random walker at time t at a distance x

*Email address: [email protected] †

Email address: [email protected]

1063-651X/2004/69共1兲/016613共7兲/$22.50

冋 冉 冊册

共1兲

where P(x,t) is the density of particles at time t at the position x, D is the diffusion coefficient, and f ( P) is the growth 共reaction兲 function. The characteristic wave front solutions that arise from these models make them suitable for many different applications which cover biological 关1兴 and human 关2兴 invasions, forest fires 关3兴, epidemics 关4兴, tumor growth 关5兴, etc. In spite of this, reaction-diffusion equations are often criticized, by arguing that such simple models cannot account for the complexity of real systems. Specifically, how the intricate features of spatial systems must be modeled is still a current problem. Some efforts have been made in the last years both to show the need for models able to predict spatial complexity 关6兴 and propose some possible solutions 关7兴. Among these proposals, one of the most attractive and recurrent ones consists of assuming that the spatial structures exhibit self-similarity properties at a certain range of scales, so fractal scaling formalism may be considered. In this work, we try to show some of the main concepts involved in adapting reaction-diffusion to self-similar spatial systems. It is well known that the mean-square displacement of a random walker in a fractal object fulfills the subdiffusive behavior 关8兴

具 x 2 典 ⬃t 2/d w ,

from its starting point. At present, it is known that for all particles starting from the same origin and for sufficiently large times and distances, the following form of the probability density 关9,10兴 P 共 x,t 兲 ⬃exp ⫺c

u⫽

x

u

t 1/d w

共3兲

,

d w d min d w ⫺d min

共4兲

is valid for a large class of fractals. In Eq. 共4兲, d min is the fractal dimension of the minimum distance between points of the fractal 关11兴. The classical, Euclidean case corresponds to d w ⫽2,d min ⫽1 and, therefore, u⫽2 so that the solution is a Gaussian. Transport equations proposed to date have tried to reproduce results 共2兲–共4兲 somehow. The first attempt was made by O’Shaugnessy and Procaccia 共SP兲 关12兴. They derived from scaling and renormalization arguments the SP equation

冋

册

⳵P ⳵ ⳵P 1 D 共 x 兲 x d f ⫺1 , ⫽ d ⫺1 ⳵t x f ⳵x ⳵x

共5兲

where d f is the geometric fractal dimension of the fractal, D(x)⫽D * x 2⫺d w , and the constant D * is a kind of diffusion coefficient. The exact solution for this equation is known and its second moment behaves like t 2/d w , in agreement with Eq. 共2兲. However, this solution lacks the scaling 共3兲 and 共4兲 predicted by numerical simulations. Later on, Giona and Roman 共GR兲 constructed the fractional GR equation 关13兴

⳵ 1/d w P ⳵t

1/d w

⫽⫺A

冉

冊

⳵P ␬ ⫹ P , ⳵x x

共6兲

where ␬ ⫽(d f ⫺1)/2. The exact solution of Eq. 共6兲 is also known and yields again the behavior t 2/d w for the second moment of P, Eq. 共2兲. However, the exact solution for P and Eq. 共6兲 do not recover the Gaussian solution and the d f -dimensional classical diffusion equation, respectively.

69 016613-1

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ME´NDEZ, CAMPOS, AND FORT

PHYSICAL REVIEW E 69, 016613 共2004兲

To overcome these difficulties, more recently a generalization of the SP equation including a temporal fractional derivative

⳵␥P ⳵t

␥

⫽

1 x

d f ⫺1

冋

⳵ ⳵P D 共 x 兲 x d f ⫺1 ⳵x ⳵x

册

⫺

⫹␧ d w ⫺2 D * x 2⫺d w 共7兲

has been proposed 关14兴. Finally, very recently we have proposed a partial differential equation 关the Campos-Me´ndez-Fort 共CMF兲 equation兴 from a stronger physical justification than the previous models, which also improves the results obtained by previous approaches 关9兴. Our goal is to find the analytical relationship for the speed of fronts when the above transport equations couple to a reaction process modeled by logistic-KPP 共KolmogorovPetrovskii-Piskunov兲 kinetics. We employ the method of reduction of the reaction-transport equation to a HamiltonJacobi. Also, we are able to derive in some cases the speed of the fronts by using the local-equilibrium 共LE兲 approach 关15兴 when the spatial correlations are small.

First of all we consider that the reaction-diffusion process in a fractal is described by the equation for the probability density of O’Shaughnessy and Procaccia 关Eq. 共5兲兴 coupled to a KPP kinetic term

冉

冊

In order to find the asymptotic speed for the traveling wave fronts in Eq. 共8兲 we will first make use of HamiltonJacobi dynamics 关16兴. The starting point is the hyperbolic scaling procedure t→t/␧, x→x/␧, and the representation of the rescaled probability density function P ␧ (x,t) ⫽ P(x/␧,t/␧) in WKB form

冉

P ␧ 共 x,t 兲 ⫽exp ⫺

冊

G ␧ 共 x,t 兲 , ␧

G ␧ 共 x,t 兲 ⭓0,

冉冊

⳵G ␧ ⫹ ⳵t x

⳵ 2G ␧

2

⫹r.

⳵x2

共10兲

d w ⫺2

D*

冉 冊 ⳵G ⳵x

2

⫹r⫽0,

共11兲

where G(x,t)⫽lim␧→0 G ␧ (x,t). In order to find the solution

for G(x,t) in Eq. 共11兲 we make use of the Hamilton equations

冋 册

⳵H ␧ dx 共 ␶ 兲 ⫽ ⫽2 d␶ ⳵ p共 ␶ 兲 x共 ␶ 兲

d w ⫺2

D*p共 ␶ 兲,

⳵H ␧ d w ⫺2 d p共 ␶ 兲 ⫽⫺ ⫽ 共 d w ⫺2 兲 D * p 共 ␶ 兲 2 , 共12兲 d␶ ⳵x共 ␶ 兲 x 共 ␶ 兲 d w ⫺1 where H⬅⫺ ⳵ G/ ⳵ t and p( ␶ )⬅ ⳵ G/ ⳵ x( ␶ ) and ␶ stands for the temporal coordinate. The solution of the Hamilton-Jacobi equation 共29兲 can be written as G 共 x,t 兲 ⫽min x(•)

再冕

t

0

冎

L 共 x, ␶ 兲 d ␶ : x 共 0 兲 ⫽0, x 共 t 兲 ⫽x , 共13兲

where L(x, ␶ )⫽p( ␶ ) 关 dx( ␶ )/d ␶ 兴 ⫺H is the Lagrangian associated with H. Integrating Eq. 共12兲, one has x( ␶ ) ⫽x( ␶ /t) 2/d w under the boundary conditions x(0)⫽0, x(t) ⫽x, and L 共 x, ␶ 兲 ⫽

x dw t 2 D * ␧ d w ⫺2 d w2

⫺r

and from Eq. 共13兲 G 共 x,t 兲 ⫽

共9兲

where the action functional G ␧ has to be found. It follows from Eq. 共9兲 that, as long as the function G(x,t) ⫽lim␧→0 G ␧ (x,t) is positive, the rescaled field P ␧ (x,t)→0 as ␧→0. The boundary of the set where G(x,t)⬎0 can be regarded as a reaction front. Therefore, we may argue that the reaction front position x(t) can be determined from the equation G(x(t),t)⫽0. Substituting Eq. 共9兲 into the rescaled equation for P ␧ (x,t) we find the equation for G ␧ (x,t)

⳵G␧ ⳵x

The first and third terms in the right-hand side of Eq. 共10兲 have the same order of magnitude and in the asymptotic limit (␧→0) both terms may be neglected in front of the second term, and in consequence the Hamilton-Jacobi for the front propagation in a fractal media is

共8兲

A. Reduction to a Hamilton-Jacobi equation

冉 冊

⫺␧ d w ⫺1 D * x 2⫺d w

II. SP EQUATION WITH REACTION

⳵P ⳵ ⳵P 1 ⫽ d ⫺1 D * x d f ⫺d w ⫹1 ⫹r P 共 1⫺ P 兲 . ⳵t x f ⳵x ⳵x

⳵G␧ ⳵G␧ ⫽⫺␧ d w ⫺1 D * 共 d f ⫺d w ⫹1 兲 x d w ⫹1 ⳵t ⳵x

x dw tD * ␧ d w ⫺2 d w2

⫺rt.

The speed of the front is computed from G(x,t)⫽0, and after inverting the hyperbolic scaling one finally has the exact expression for the speed of the front v共 t 兲 ⫽2

冋

D *r d ⫺2 d w ⫺2

d ww

t

册

1/d w

⬃t (2/d w )⫺1 ,

共14兲

which describes a decelerated front, and we note that the acceleration depends on the parameter d w .

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PHYSICAL REVIEW E 69, 016613 共2004兲

DYNAMICAL FEATURES OF REACTION-DIFFUSION . . . B. Local-equilibrium approach

The local-equilibrium hypothesis has been very successful in thermodynamics 共where one assumes a Gibbs equation with the local values of temperature, pressure, etc.兲, radiative transfer, astrophysics 共where one assumes a radiative emission-absorption equilibrium at the local temperature兲, etc. Here, we will apply such a heuristic approximation to the problem of front propagation. Equation 共8兲 can be rewritten as

⳵P ⳵P ⳵2P 1 1 * * D 共 d f ⫺d w ⫹1 兲 D ⫽ ⫹ ⳵ t x d w ⫺1 ⳵ x x d w ⫺2 ⳵x2 ⫹r P 共 1⫺ P 兲 .

共15兲

This equation has the form

⳵P ⳵2P ⳵P ⫽D 共 x 兲 2 ⫺V 共 x 兲 ⫹r P 共 1⫺ P 兲 , ⳵t ⳵x ⳵x

共16兲

which is an advection-reaction-diffusion equation in a medium moving at speed V(x). For homogeneous values of D and V, the speed of fronts is obviously v ⫽2 冑rD⫹V,

共17兲

which is nothing but Fisher’s speed 2 冑rD 共which holds for an observer attached to the moving medium兲, as seen by an observer moving with speed ⫺V. This result v ⫽2 冑rD⫹V may be also derived by using the linearization and variational techniques in Ref. 关17兴. We propose that, if inhomogeneities are sufficiently smooth, this equation should hold locally, so that we obtain the local-equilibrium prediction v LE ⯝2 冑rD 共 x 兲 ⫹V 共 x 兲 ,

共18兲

FIG. 1. Comparison between numerical results for the speed of fronts of Eq. 共8兲 for d f ⫽1 and for d f ⫽0.01 and the analytical results from the HJ method given in Eq. 共14兲 共solid line兲 and the numerical result for the LE method obtained from Eq. 共21兲 with d f ⫽1. All magnitudes are dimensionless. We observe a very good agrement between analytical and numerical solutions in the asymptotic regime. It is clear that d f does not affect the value of the speed for large times. We have taken D * ⫽r⫽1 and d w ⫽2.32.

something surprising at first sight. In order to further explore this point, we have solved numerically Eq. 共8兲, under the initial condition P(x,0)⫽1 for x⬍0 and P(x,0)⫽0 for x ⬎0, for two very different values of d f and we have observed, as we show in Fig. 1, that the speed depends weakly on d f only at the very initial transient and loses this dependence as time grows. Moreover, in Fig. 1 we compare the speed given in Eq. 共14兲 or 共20兲 with the speed obtained from Eq. 共19兲. To do this we must first solve Eq. 共19兲. Under x(0)⫽0 we obtain

which for Eq. 共15兲 yields

冑rD * D * 共 d f ⫺d w ⫹1 兲 dx . v LE ⫽ ⯝2 (d ⫺2)/2 ⫺ dt x w x d w ⫺1

共19兲

Note that if one takes the hyperbolic scaling x→x/␧, t →t/␧ with ␧→0 in Eq. 共19兲, one observes that the second term on the right-hand side 共rhs兲 of 共19兲 is negligible, compared to the first one, and Eq. 共19兲 can be written as

冑rD * dx ⯝2 (d ⫺2)/2 dt x w

for x,t→⬁,

which may be integrated to yield x(t)⫽ 关 d w 冑rD * t 兴 2/d w , where we have assumed the initial condition x(0)⫽0. Finally, the speed may be obtained as

冋

D *r dx v LE ⫽ ⯝2 dt 共 d w t 兲 d w ⫺2

册

t⫽

y d w 冑rD *

⫹

冊

where x⫽y 2/d w . In Fig. 1 we have solved numerically the transcendent equation 共21兲 and evaluated dx/dt to obtain the speed. Note that the effect of d f 关which appears in Eq. 共21兲 but not in Eq. 共20兲兴 on the asymptotic speed of the front is inappreciable. This shows that it is justified to neglect the second term in the rhs of Eq. 共19兲, as done above. III. GR EQUATION WITH REACTION

In this section we study the speed of fronts for the fractional advection equation GR 共6兲 with reaction

1/d w

⬃t (2/d w )⫺1 for t→⬁,

⳵ 1/d w P 共20兲

which coincides with Eq. 共14兲. The asymptotic speed does not depend on the fractal dimension d f , which could be

冉

2y 冑r/D * 共 d f ⫺d w ⫹1 兲 , 共21兲 ln 1⫺ 2d w r d f ⫺d w ⫹1

⳵t where

016613-3

1/d w

⫽⫺A

冉

冊

⳵P ␬ ⫹ P ⫹r P 共 1⫺ P 兲 , ⳵x x

共22兲

ME´NDEZ, CAMPOS, AND FORT

⳵ 1/d w P ⳵t

⫽

1/d w

⫽

冉

1

冉

1

1 ⌫ 1⫺ dw 1 ⌫ 1⫺ dw

冊 冊

PHYSICAL REVIEW E 69, 016613 共2004兲

⳵ ⳵t

冕

⳵ ⳵t

冕

P 共 x,t ⬘ 兲

t

共 t⫺t ⬘ 兲

0

1/d w

⳵␥P

dt ⬘

⳵t

⳵ t 1/d w

⫽

␧

冉

⌫ 1⫺ ⫽

1 dw

冊

冕

冊

冕

␧

冉

1 ⌫ 1⫺ dw

z

⫺1/d w

0

P 共 x,t⫺z 兲 dz

共23兲

⳵ t 1/d w

z

t/␧

z ⫺1/d w

0

冊

␧ /␧

冕

冋

册

⳵ ⳵P D * x d f ⫺d w ⫹1 ⫹r P 共 1⫺ P 兲 , ⳵x ⳵x

共24兲

冉

t/␧

冉 冊

冊

z

e

0

⫽

⳵G ⫺ ⳵t

⳵ P␧ ⳵2P␧ ⫺ 2 ␧z⫹••• , ⳵t ⳵t

⫺1/d w z ⳵ t G ␧

␥

⳵ 1 ⌫ 共 1⫺ ␥ 兲 ⳵ t

冕

t

P 共 x,t ⬘ 兲

0

共 t⫺t ⬘ 兲 ␥

dt ⬘ .

Taking the hyperbolic scaling and Eq. 共9兲 in the limit ␧→0 one finds the Hamilton-Jacobi equation

⳵ ␧ P 共 x,t⫺␧z 兲 dz ⳵t

⫺1/d w

0

冉

x

⳵t

t/␧

⫺ 共 ⳵ t G ␧ 兲 e ⫺G ⫽ 1 ⌫ 1⫺ dw

1 d f ⫺1

⳵␥P

␥

⫽D * x 2⫺d w ␧ d w ⫺2

From the Hamilton ⫽Bx( ␶ ) ⫺1⫹d w /2 and

equations

冉 冊 ⳵G ⳵x

2

one

⫹r. obtains

p( ␶ )

x 共 ␶ 兲 d w /2 2D * B d ⫺2 ␧ w 共 r⫹D * ␧ d w ⫺2 B 2 兲 (1/␥ )⫺1 ␶ , ⫽ d w /2 ␥

where P ␧ (x,0)⫽0 for x⬎0, and making use of Eq. 共9兲 one finds

⳵ 1/d w P ␧

⫽

where the fractional time derivative is defined as 关14兴

t

and z⫽t⫺t ⬘ . As in the preceding section we first take the asymptotic limit for large space and time by employing the hyperbolic scaling. The left-hand side of Eq. 共6兲 is

⳵ 1/d w P ␧

␥

共25兲

where B is an integration constant to be determined from the condition x( ␶ ⫽t)⫽x, and therefore

dz⫹O 共 ␧ 兲 .

x d w /2 2D * B d ⫺2 ␧ w 共 r⫹D * ␧ d w ⫺2 B 2 兲 (1/␥ )⫺1 t. ⫽ d w /2 ␥

The right-hand side of Eq. 共6兲 is transformed into

冉

On the other hand, the Lagrangian function is

冊

⳵ P␧ ␬ ⫺A ␧ ⫹␧ P ␧ ⫹r P ␧ 共 1⫺ P ␧ 兲 ⳵x x

L 共 x,t 兲 ⫽ 共 r⫹D * ␧ d w ⫺2 B 2 兲 ⫺1⫹1/␥

⳵ G ␧ ⫺G ␧ /␧ ␧ ⫽A ⫹re ⫺G /␧ ⫹O 共 ␧ 兲 . e ⳵x

冋冉 冊

From G(x,t)⫽0 we have

Taking ␧→0 one gets the Hamilton-Jacobi equation

冉 冊 ⫺

⳵G ⳵t

1/d w

⫽A

D * ␧ d w ⫺2 B 2 ⫽

⳵G ⫹r. ⳵x

冉 冊

冋

x x L 共 x,t 兲 ⫽ 共 d w ⫺1 兲 Atd w Atd w

r␥ , 2⫺ ␥

and inserting this into Eq. 共25兲 one has

From the Hamilton equations one has the Lagrangian 1/(d w ⫺1)

册

2 ⫺1 D * ␧ d w ⫺2 B 2 ⫺r . ␥

册

x d w /2⫽d w 冑D * / ␥ ␧ (d w ⫺2)/2

⫺d w r ,

冉 冊 r 2⫺ ␥

(2⫺ ␥ )/2␥

2 (1⫺ ␥ )/ ␥ t,

and taking dx/dt the speed is and the speed of the front will be v ⫽Ad w

冉 冊 rd w d w ⫺1

冋

冉 冊

2 2(1⫺ ␥ )/ ␥ r v共 t 兲 ⫽2 ␥ 2⫺ ␥

d w ⫺1

,

which is time independent. IV. FRACTIONAL SP EQUATION WITH REACTION

Another equation proposed is the fractional version of the SP equation 共7兲 which with reaction takes the form

(2⫺ ␥ )/ ␥

D* 共 d w t 兲 d w ⫺2

册

1/d w

⬃t (2/d w )⫺1 共26兲

once the hyperbolic scaling is inverted. Note that for ␥ ⫽1, Eq. 共14兲 is recovered. However, the scaling law for the speed of the front in time is the same as for ␥ ⫽1 in Eq. 共20兲 so that the fractional derivative, although affecting the mean-square displacement, does not affect this scaling law.

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DYNAMICAL FEATURES OF REACTION-DIFFUSION . . .

d p共 ␶ 兲 ⳵H ⫽⫺ d␶ ⳵x共 ␶ 兲

V. CMF EQUATION WITH REACTION

We start from the CMF equation

⳵P 4D 0 ⳵ ⫽ ⳵ t d w2 x d f ⫺1 ⳵ x

冋冉 冊 x

d w ⫺u

t 1/d w

册

⫽⫺

⳵P x d f ⫺d w ⫹1 ⫹r P 共 1⫺ P 兲 , ⳵x 共27兲

L 共 x, ␶ 兲 ⫽

After taking into account the hyperbolic scaling and the field G ␧ (x,t)⫽⫺␧ ln P(x/␧,t/␧) one has from Eq. 共27兲 ␧

⳵G ⳵G 4D 0 ⫺ 2 ␧ ␣ ⫹1 t (u/d w )⫺1 x ⫺u⫹1 共 d f ⫹1⫺u 兲 ⳵t ⳵x dw

⫽

d w2

4D 0 d w2

␧

␧ ␣ t (u/d w )⫺1 x ⫺u⫹2

冉 冊

␣ ⫹1 (u/d w )⫺1 ⫺u⫹2

t

x

⳵G␧ ⳵x

⳵ 2G ␧ ⳵x2

Finally, from Eq. 共13兲 we obtain G 共 x,t 兲 ⫽

2

⫺r⫹re

⫺G ␧ /␧

G 共 x,t 兲 ⫽

,

␧ /␧

冉

冊

The speed of the front is computed from G(x,t)⫽0, and after inverting the hyperbolic scaling one finally has

冉

冊

1 ⫺1. dw

冊冉 冊 u dw

1/u

共 4rD 0 兲 1/u t (1/d w )⫺1⫹1/u

⬃t (1/d min)⫺1 .

B. Local-equilibrium approach

Following the same method as in previous sections, we find for Eq. 共27兲 v LE ⫽

O 共 ␧ ␣ ⫹1 兲 ⰆO 共 ␧ ␣ 兲 ⰆO 共 ␧ 0 兲 .

4 冑D 0 r x ⫺(u/2)⫹1 4D 0 x ⫺u⫹1 ⫺ d ⫺u⫹1 , 兲 共 f d w t (1/2)⫺(u/2d w ) d w2 t 1⫺u/d w 共33兲

which leads us to the front speed

By keeping the terms up to O(␧ ␣ ), one has

冉 冊

共32兲

It is important to note that the scaling law in Eq. 共32兲 does not depend on d f and d w but only on d min

)⫽0, provided G ␧ (x,t)⬎0 and

⳵ G 4D 0 ⫺(u/d )⫹u⫺1 (u/d )⫺1 ⫺u⫹2 ⳵ G w ⫹ 2 ␧ t w x ⳵t ⳵x dw

共31兲

d w x u t ⫺(u/d w ) ⫺rt ␧. 4D 0 u

1 dx 1 ⫽ ⫹ v HJ 共 t 兲 ⫽ dt u dw

The first term in the left-hand side 共lhs兲 of Eq. 共28兲 and the second term in the rhs of Eq. 共28兲 are O(␧ 0 ). The second term in the lhs and the first term in the rhs of Eq. 共28兲 are O(␧ ␣ ⫹1 ) while the third term in the lhs is O(␧ ␣ ). For fractals d min⭓1, so that u⭓d w /(d w ⫺1) or ␣ ⬎0. Therefore, in the limit ␧→0 one has G(x,t)⫽lim␧→0 G ␧ (x,t), lim␧→0 f (e ⫺G

d w ␧ (u/d w )⫺u⫹1 x u t ⫺(u/d w ) ⫺rt. 4D 0 u

If one inverts the hyperbolic scaling in Eq. 共31兲, by taking x→␧x and t→␧t, one has

where

冉

共30兲

1 (u/d )⫺u⫹1 u (u/d )⫺1 ⫺(2u/d ) w ⫺r. ␧ w x ␶ w t 4D 0

␧

共28兲

␣ ⫽u 1⫺

共 2⫺u 兲 ␧ ⫺(u/d w )⫹u⫺1 ␶ (u/d w )⫺1

⫻x 共 ␶ 兲 ⫺u⫹1 p 2 共 ␶ 兲 .

A. Reduction to a Hamilton-Jacobi equation

4D 0

d w2

Integrating Eq. 共30兲, one has x( ␶ )⫽x( ␶ /t) 2/d w under the boundary conditions x(0)⫽0,x(t)⫽x, and

which has been derived very recently 关9兴.

⫹

4D 0

v LE ⫽

2

⫹r⫽0. 共29兲

Equation 共29兲 is the Hamilton-Jacobi equation for the problem 共27兲 and may be solved by using Hamilton’s equations

⳵H 8D 0 dx 共 ␶ 兲 ⫽ ⫽ 2 ␧ ⫺(u/d w )⫹u⫺1 ␶ (u/d w )⫺1 x 共 ␶ 兲 ⫺u⫹2 p 共 ␶ 兲 , d␶ ⳵ p共 ␶ 兲 dw

冉

1 1 ⫹ u dw

冊冉

⬃t (1/d min)⫺1

4 dw 1⫹ u

冊

2/u

for t→⬁.

共 rD 0 兲 1/u t (1/d w )⫺1⫹1/u

共34兲

It is interesting to note that although the LE prediction 共34兲 has the same scaling law as the HJ result 共32兲, the factor is different unless d w ⫽2d min . However, we have checked numerically that for the typical ranges of d w 共from 2 to 4) and d min 共from 1 to 2) in fractals, the differences between the two models are negligible 共see Fig. 2兲.

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FIG. 2. Comparison between numerical results for the speed of fronts of Eq. 共27兲 and the analytical results from the HJ method given in Eq. 共32兲 共solid line兲 and the numerical result for the LE method obtained from Eq. 共34兲 with d f ⫽1. All magnitudes are dimensionless. There is good agrement between analytical and numerical solutions in the asymptotic regime. We have taken D * ⫽r ⫽1 and d w ⫽2.32, d min ⫽1.1.

VI. DISCUSSION

After all the mathematical formalism, the main conclusion we obtain is that, despite all these equations seeking to describe the features of transport on fractals, the characteristics of the fronts predicted in the four cases shown are clearly different. First of all, we should note that a diffusion equation on fractals must agree with the results 共2兲–共4兲. We have shown 关9兴 previously that equation CMF is the only one that can exactly reproduce them, so we expect that the fronts predicted by this equation correspond to the real case. One may wonder why fronts on fractals should be accelerated. To answer this we need to introduce in our discussion the ‘‘chemical distance’’ l 关11兴, which is defined as the shortest path between two points belonging to the fractal 共Fig. 3兲.

Usually, random walks 共or diffusion兲 on a fractal is described as follows. A particle at point s 共see Fig. 3兲 can jump to every one of its first neighbors with the same probability. At the next time step, it may again jump to the new first neighbors with the same probability, and so on. We can see that the possible points reached after two jumps forward have in common that their chemical distance to the origin s is the same 共but not their Euclidean distance兲. Hence, we see that transport on fractals take place through the chemical distance space and in this space the fractal fronts are expected to show constant speed, as in homogeneous media. Nevertheless, we are usually interested in the results for the Euclidean space, so we need the well-known relationship between r and l 关11兴, l⬃r d min,

共35兲

where the condition d min ⭓1 comes directly from the fact that l is always greater than r, as seen in Fig. 3. From this expression we can conclude that, assuming that fronts advance at a constant speed in the chemical distance space, they cannot do so in the Euclidean space. Moreover, we observe that this behavior is due to the parameter d min 共in the nonfractal case we have d min ⫽1, so the behavior in both spaces will then be the same兲; it agrees with the CMF equation 共27兲, which predicts an acceleration dependence only on d min 关see Eq. 共32兲 or 共34兲兴, while the other equations analyzed 共Secs. II, III, and IV兲 do not take into account this essential parameter. In fact, the form of the time exponent in Eqs. 共32兲 and 共34兲 can be justified. We may define l f r and r f r as the chemical and Euclidean distances of the wave front position, respectively. As the front speed is constant in the chemical distance space, we can assume that l f r grows linearly with t. This relation, in addition to Eq. 共35兲, leads to r f r ⬃t 1/d min,

共36兲

and the time dependence expected for the fronts in the Euclidean space is then v⬃

FIG. 3. Sierpinski gasket as an example of random walk on a fractal. The bold lines show the difference between the Euclidean distance r and the ‘‘chemical distance’’ l between two points of the fractal s and s ⬘ 共note that l⭓r for any couple of points兲.

r f r (1/d )⫺1 ⬃t min , t

共37兲

in agreement with the acceleration predicted by the CMF equation 关Eq. 共32兲 or 共34兲兴. This result, which had been already predicted for the speed of propagation of fronts in percolation clusters 关18兴, should be valid for all those transport processes on fractals which, as it happens in most cases, take place through the chemical distance space 共as in Fig. 3兲. In conclusion, we have shown that the CMF equation not only describes the features of diffusion on fractals better than previous ones 关9兴, but it also predicts some essential features of the propagative processes on those heterogeneous media, i.e., the speed and acceleration of the wave fronts derived when a reaction 共logistic兲 process, widely used in biophysics 关19兴, is considered. In consequence, we think that CMF equation is the best analytical approach proposed to date for description of transport on fractals.

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DYNAMICAL FEATURES OF REACTION-DIFFUSION . . .

One of the main results presented here is the wave front speed 共32兲, which shows that the acceleration is determined by the parameter d min , as argued theoretically above. It is interesting to note that this parameter is not the same as that responsible for anomalous diffusion, namely, d w 关see Eq. 共2兲兴, contrary to what one could expect. We think that an exhaustive analysis of the meaning of the parameter d w ,

which has been traditionally defined just from Eq. 共2兲, is still needed. This and many other questions which have not been explained theoretically yet 关18兴 show very clearly that there is still a lot of work to do on fractal dynamics. Although theoretical research on fractals decreased after a boom in the early 1980s, we consider that efforts in this field, as the one presented here, are still strongly useful.

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