Anomalous diffusion and fractional diffusion equation: anisotropic media and external forces

Physics Letters A 347 (2005) 160–169 www.elsevier.com/locate/pla

Anomalous diffusion and fractional diffusion equation: anisotropic media and external forces M.F. de Andrade, E.K. Lenzi ∗ , L.R. Evangelista, R.S. Mendes, L.C. Malacarne Departamento de Física, Universidade Estadual de Maringá, Av. Colombo 5790, 87020-900 Maringá-PR, Brazil Received 17 May 2005; received in revised form 22 July 2005; accepted 26 July 2005 Available online 18 August 2005 Communicated by C.R. Doering

Abstract We analyze a fractional diffusion equation by taking an anisotropic case into account. In our analysis, we also consider a spatial time dependent diffusion coefficient and the presence of external forces in the system. For the cases analyzed here, we obtain exact solutions and show that the solutions have an anomalous spreading. In addition, we discuss a rich class of diffusive processes, including normal and anomalous ones described by this equation.  2005 Elsevier B.V. All rights reserved. PACS: 82.20.Db; 66.10.Cb; 05.60.+w; 05.40.+j

1. Introduction Nowadays, the anomalous diffusion has been extensively investigated due to the broadness of its physical applications. In fact, it is present in several physical situations such as modelling of non-Markovian dynamical processes in protein folding, relaxation to equilibrium in a system (such as polymer chains and membranes) with long temporal memory [1–4], anomalous transport in disordered systems [5], transport in the stochastic layer of AC-driven Hamiltonian systems [6], modelling dispersive transport under the influence of an external force field [7]. In this context, the fractional diffusion equations [8–14] play an important role in describing these phenomena with the advantage, to other models, that they include external fields in a straightforward way when calculating boundary value problems. For these reasons, the formal aspects of the fractional diffusion equations have also been investigated. For instance, the behavior of fractional diffusion at the origin was analyzed in [15], a fractional Fokker–Planck equation was derived from a generalized master equation, boundary value problems for fractional diffusion equa* Corresponding author.

E-mail address: [email protected] (E.K. Lenzi). 0375-9601/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.07.090

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tions were studied by considering a finite and a semi-infinite intervals [16] and in [17] a harmonic analysis of random fractional diffusion-wave equations was done. From the above discussion, we note the relevance of these investigations concerning the fractional diffusion equations and the importance of extending these analyzes to incorporate other situations. For example, an anomalous diffusion in anisotropic media in a crystal with randomly distributed topological defects [18] and a diffusion on two-dimensional percolation of anisotropic clusters [19]. In this direction, we dedicate this work to obtain new classes of solutions for a fractional diffusion equation taking an anisotropic medium into account and accomplishing the presence of external forces. More precisely, we focus our attention on the following anisotropic fractional diffusion equation: 
 N t N
 ∂  ∂γ  ∂  ∂  Fi (x)ρ( ρ( x, ¯ t) = dt ( x; ¯ t − t ) ρ( x, ¯ t ) − ¯ x, ¯ t) , D ij ∂t γ ∂xi ∂xj ∂xi i,j =1 0

(1)

i=1

with x¯ = (x1 , x2 , . . . , xN ). Fi (x) ¯ is an external force and Dij (x; ¯ t) are the diffusion coefficients. We use the Caputo operator for the time derivative with 0 < γ < 1. Also, we employ, in general, the boundary condition ∞  ρ(x, ¯ t) → 0. For Eq. (1), one can prove that −∞ N ¯ t) is time independent (hence, if ρ is lim|x|→∞ ¯ i=1 dxi ρ(x, γ normalized at t = 0, it will remain so forever). Indeed, if we write Eq. (1) as ∂t ρ = −∇J and, for simplicity,  ∞ N assume the boundary condition lim|x|→∞ J (±∞, t) → 0, it can be shown that −∞ i=1 dxi ρ(x, ¯ t) is a constant ¯ of motion. Note also that Eq. (1) has a kernel which takes a memory effect into account, besides the fractional derivative. We investigate, in Section 2, the solutions of Eq. (1) for the N -dimensional case. We start by considering a time dependent diffusion coefficient, i.e., Dij = Di δij t α−1 /Γ (α) (Di = const) and after we incorporate a spatial dependence on the diffusion coefficient, i.e., Dij = Di δij t α−1 |xi |−θi /Γ (α). Both cases are solved without external forces. Next, we consider Dij = Di |xi |−θi δij δ(t) (Di = const) taking the external force Fi = (2 + θi )(Ki /xi )|xi |νi with νi = −θi into account. In this context, the solutions are expressed in terms of the Fox H function. Following, we consider the external force Fi = −ki xi + (Ki /xi )|xi |νi and, in particular, we also analyze the effects produced by the presence of a source (absorbent) term in the diffusion equation. Finally, we present our conclusions in Section 3.

2. Fractional diffusion equation ¯ t) = Di (t)δij in the absence of external forces. For this case Let us start by considering the case given by Dij (x, Eq. (1) is given by N


∂γ ρ( x, ¯ t) = ∂t γ

t

i=1 0

dt  Di (t − t  )

∂2 ρ(x, ¯ t  ). ∂xi2

(2)

Note that Eq. (2) can be useful to describe an anomalous diffusive process, since the waiting time distribution function associated to this process has a non-usual behavior. This feature is due to the presence of the fractional time derivative and a time dependent diffusion coefficient. In particular, by employing the formalism of continuous time random walk [20], one can verify this statement. Eq. (2) can also be useful to investigate dichotomous random process [21]. By using Eq. (1), a well-known limitation of the description of diffusion processes with the diffusion equation, i.e., the infinite velocity of information propagation inherent to a parabolic equation, can be avoided by choosing a suitable kernel [22]. Before analyzing the solutions for the above equation, it is interesting to study the second moment xi2  for this equation. The second moment contains information about the time scaling and the spreading behavior of Eq. (2) for a given boundary condition. After some calculations it is possible to show that

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2

xi =

2 Γ (1 + γ )

t

dt  (t − t  )γ Di (t  )

(3)

0

 for the initial condition ρ(x, ¯ 0) = N i=1 δ(xi ). Eq. (3) suggests that the moments for the above equation are not coupled, indicating that the spreading of Eq. (2) occurs independently for each direction. However, as we discuss below, the solution of Eq. (2) has all its directions coupled, in contrast with the usual case corresponding to γ = 1 and Di (t) = Di δ(t). To investigate the solutions for Eq. (2), we apply the Laplace transform in order to reduce Eq. (2) to ¯ s) − s γ −1 ρ(x, ¯ 0) = s γ ρ(x,

N


Di (s)

i=1

∂2 ρ(x, ¯ s). ∂xi2

(4)

The solution for Eq. (4) is γ

s 4 (2+N )−1 ρ(x, ¯ s) = N (2π D¯ 1···N (s)) 2

1−N 1   N

 N 2 4 2
x2
x2 γ i i K N −1 s 2 2 Di (s) Di (s) i=1

(5)

i=1

N  1 ¯ N and Kν (x) is a modified Bessel for the initial condition ρ(x, ¯ 0) = N i=1 δ(xi ), where D1···N (s) = i=1 [Di (s)] function. Notice that obtaining the inverse Laplace transform of Eq. (5) for a general diffusion coefficient is a difficult task. However, for some cases it is possible to obtain the inverse of Laplace transform. For instance, Di (s) = const (Di (t) = Di δ(t)) recovers the result presented in [8] for the one-dimensional case and Di (s) = Di s −α (Di (t) = Di t α−1 /Γ (α)) leads us to 1− N (γ +α),γ +α  

N 2  2
x 1  i 20 ρ(x, ¯ t) = (6) H ,  12 N 4Di t γ +α  N  (4π D¯ t α+γ ) 2 1···N

i=1

1−

2

,1 (0,1)

  (a ,A ),...,(ap ,Ap )  is the Fox H function [23]. From Eq. (6), we can verify an anomalous spreading. where Hm n x  1 1 pq

(b1 ,B1 ),...,(bq ,Bq )

This anomalous behavior is also manifested in the second moment which is given by xi2  ∝ t γ +α (γ + α < 1, = 1 or > 1 characterize sub, normal or superdiffusive process, respectively). In addition, it is important to note that this distribution cannot be written as the usual one, i.e., ρ(x, ¯ t) = ρ1 (x1 , t) · · · ρN (xN , t). Now, we extend the above result by incorporating a spatial dependence on the diffusion, i.e., Dij = Di t α−1 δij |xi |−θi /Γ (α). This spatial dependence has been applied in several situations such as diffusion on fractals [24], turbulence [25], describing fast electrons in a hot plasma in the presence of a dc electric field [26] and studying mean first passage time [27]. For this case, we have that N

1
∂γ ρ(x, ¯ t) = γ ∂t Γ (α)

t

i=1 0



 α−1

dt (t − t )

  ∂ −θi ∂  ρ(x, ¯ t) . |xi | ∂xi ∂xi

(7)

By using the previous procedure, we found that the solution for the above equation is given by (1−ξ(γ +α),γ +α) 

N   1 N 
|xi |2+θi  2+θi 2 + θi 1 ρ(x, ¯ t) = (8) H21 02 ,  1   2 D t γ +α 4Di t γ +α  (2 + θ ) 2Γ i i 2+θi i=1 i=1 (1−ξ,1)(0,1)  where ξ = N i=1 1/(2 + θi ) (see Figs. 1 and 2). Eq. (8) extends the results present in [28] by considering the presence of anisotropy and a time dependence on the diffusion coefficient and the results found in [29] for the two-dimensional case. It is interesting to note that the solution given by Eq. (8) can manifest an anomalous behavior at the origin as reported in [15] depending on the dimension N and on the parameters θi in consid-

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Fig. 1. Behavior of t (γ +α)ξ ρ(x, ¯ t)/N¯ versus r 2 /t γ +α for typical values of ξ by considering, for simplicity, γ + α = 1/3 where 1   2+θi /(4D ) and N ¯ = N (2 + θi )/[2((2 + θi )2 Di ) 2+θi Γ ( 1 )]. r2 = N i i=1 |xi | i=1 2+θ i

eration. In fact, Eq. (8) near the origin can be expanded by using the series of the Fox H function that results in  2+θi /[(2 + θ )2 D t γ ], which presents a divergent beρ(x, ¯ t) ∼ r −2(ξ −1) /t (γ +α)ξ for ξ > 1, where r 2 = N i i i=1 |xi | havior at the origin (r = 0). Similar behavior is found in [15] for the radial diffusion equation for the N -dimensional ρ(x, ¯ t) = 0 to the case. Another interesting case appears when we incorporate the boundary condition lim|x|→0 ¯  above analysis. In this case, the solution for Eq. (7) subjected to the initial condition ρ(x, ¯ 0) = N i=1 δ(xi − ξi ) is given by ρ(x, ¯ t) =

1+θ ∞ N  2(xi ξi ) 2

i=1

2 + θi

0

···

∞ N 0 i=1

 dki ki J 1+θi

2+θi

2+θi

2ki ξi 2 2 + θi

2+θi  N   
2ki xi 2 2 γ +α ki Di t Eγ +α − J 1+θi 2+θ 2+θi

i=1

(9) (see Fig. 3) where Jν (x) is the Bessel function and Eγ¯ (x) is the Mittag-Leffler function [9]. This function is an extension of the usual exponential one and is related to the changes produced in the waiting time distribution by the fractional derivative associated to this process of diffusion. For the particular case γ + α = 1 the above equation can be reduced to 2+θi 2+θi 1+θi 2+θi   +ξi x N  − i (ξi xi ) 2 2(ξi xi ) 2 2 e (2+θi ) Di t I 1+θi , ρ(x, ¯ t) = (10) (2 + θi )Di t (2 + θi )2 Di t 2+θi i=1

where Iν (x) is a modified Bessel function.

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Fig. 2. Behavior of ρ(x, ¯ t)/N¯ versus r 2 for typical values of t by considering, for simplicity, γ + α = 1/3 where r 2 = 1 N 1 )]. ξ = 1/2 and N¯ = i=1 (2 + θi )/[2((2 + θi )2 Di ) 2+θi Γ ( 2+θ

N

i=1 |xi |

2+θi /(4D ), i

i

We may also incorporate the external force Fi = (2 + θi )(Ki /xi )|xi |νi with νi = −θi in Eq. (1) and consider the ¯ t) = Di |xi |−θi δ(t)δij . It is interesting to note that the potential related to this external diffusion coefficient Dij (x, force extends the logarithmic potential used, for instance, to establish a connection between the fractal diffusion coefficient and the generalized mobility [30]. After these considerations, Eq. (1) reads 
   N N
∂γ ∂ ∂ Ki −θi ∂ −θi ρ( x, ¯ t) = |x | ρ( x, ¯ t) − ) |x | ρ( x, ¯ t) . D (2 + θ i i i i ∂t γ ∂xi ∂xi ∂xi xi i=1

(11)

i=1

The above equation has no stationary solutions. This fact can be verified from the second moment which is given by xi2  ∝ t 2γ /(2+θi ) and implies in ρ(x, ¯ t → ∞) → 0. The solution for Eq. (11) is given by N 

2 + θi



1 ρ(x, ¯ t) =  1 Ki  (2 + θ )2 D t γ i i i=1 2Γ 2+θi + Di (1−γ ∆,γ ) 

N
|xi |2+θi  20 × H1 2 ,  4Di t γ  i=1

(1−∆,1)(0,1)



1 2+θi



|xi |2+θi (2 + θi )2 Di t γ

 Ki

Di

(12)

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Fig. 3. Behavior of ρ(x, ¯ t) versus |x1 | for typical values of t by considering, for simplicity, the one-dimensional case of Eq. (9), γ = 1/2, θ1 = 0, ξ1 = 1 and D1 = 1.

where ∆ =

N

i=1 Ki /Di

ρ(x, ¯ t) ∼

N  i=1

×

2Γ



+ ξ . The asymptotic behavior for Eq. (12) is 2 + θi

1 2+θi

1 Ki  (2 + θ )2 D t γ i i + Di

(γ −1)∆

 N
|xi |2+θi i=1

4Di t γ





2−γ

exp −(2 − γ )γ

1 2+θi

γ 2−γ



|xi |2+θi (2 + θi )2 Di t γ

 Ki

Di

 N 1 
|xi |2+θi 2−γ i=1

4Di t γ

.

(13)

In particular, Eq. (13) shows that the solution cannot be expressed as a product of one dimension solutions, i.e., ρ(x, ¯ t) = ρ(x1 , t) · · · ρ(xN , t). Another interesting external force is Fi = −ki xi + (Ki /xi )|xi |νi with νi = −θi which is derived from potential that contains a quadratic and a power law terms. For this external force, the diffusion equation is given by 
    N N
∂ ∂ Ki ∂γ −θi ∂ −θi ρ( x, ¯ t) = |x | ρ( x, ¯ t) − x + |x | D −k ρ( x, ¯ t) . i i i i i ∂t γ ∂xi ∂xi ∂xi xi i=1

(14)

i=1

In order to solve this equation, we consider the following solution: ρ(x, ¯ t) =

∞
n1 =0

···

∞
nN =0

Ψn1 ···nN (x)φ ¯ n1 ···nN (t),

(15)

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where Ψn1 ···nN (x) ¯ is the eigenfunction to be found and φn1 ···nN (t) is a time dependent function. By substituting the above equation in Eq. (14), we obtain dγ φn ···n (t) = −λn1 ···nN φn1 ···nN (t) dt γ 1 N

(16)

    N
∂ ∂ Ki Ψ (x) ¯ − −ki xi + |xi |−θi Ψ (x) ¯ Di |xi |−θi ¯ = −λn1 ···nN Ψ (x). ∂xi ∂xi xi

(17)

and

i=1

The solution for Eq. (16) is given in terms of the Mittag-Leffler function as follows:   φn1 ···nN (t) = φn1 ···nN (0)Eγ −λn1 ···nN t γ

(18)

with λn1 ···nN = (2 + θ1 )k1 n1 + · · · + (2 + θN )kN nN . The solution for Ψn1 ···nN (x) ¯ is given by   i i  N 2+θi  ki |xi |2+θi Ki  (2+θi )Di ki (2 + θi )Γ (ni + 1) (α¯ i ) ki |xi | Di − (2+θi )Di ¯ = |xi | e Lni Ψn1 ···nN (x) ,   Ki +Di (2 + θi )Di (2 + θi )Di i=1 2Γ (2+θ)Di + ni K +D

(19)

(α) ¯

where α¯ i = {(Ki + Di )/[(2 + θi )Di ]} − 1 and Ln (x) is the associated Laguerre polynomial. Thus, by considering  the initial condition ρ(x, ¯ 0) = N i=1 δ(xi − ξi ), the solution for Eq. (14) is given by   Ki +Di ∞  N ki |xi |2+θi Ki
(2+θi )Di ki (2 + θi )Γ (ni + 1) Di − (2+θi )Di ··· |xi | e ρ(x, ¯ t) =   Ki +Di (2 + θi )Di n1 =0 nN =0 i=1 2Γ (2+θ)Di + ni       ki |xi |2+θi ki |ξi |2+θi × L(nα¯i i ) L(nα¯i i ) Eγ −λn1 ···nN t γ (2 + θi )Di (2 + θi )Di ∞


(20)

(see Figs. 4 and 5). For this case we have the stationary solution ρ(x) ¯ =

N  i=1

2Γ

2 + θi  Ki +Di  (2+θ)Di



ki (2 + θi )Di



Ki +Di (2+θi )Di

|xi |

Ki Di

e

−

ki |xi |2+θi (2+θi )Di

(21)

and, in particular, it is equal to the usual one. The last statement can be verified from the above equation taking the limit t → ∞ into account. This fact is also verified in [9] for a quadratic potential. We may also incorporate an absorbent (source) term in the diffusion equation, such as α|xi |ηi ρ(x, ¯ t), with ηi = 2 + θi . The reaction term like the previous one may be useful to investigate catalytic processes in regular, heterogeneous, or disordered systems [31]. It may also contain an irreversible first-order reaction of transported substance so that the rate of removal is αρ [32] and may appear in heat flow involving heat production [33]. For this case the diffusion equation reads 
    N N
∂ ∂ Ki ∂γ −θi ∂ −θi ρ(x, ¯ t) = ρ(x, ¯ t) − |xi | Di |xi | −ki xi + ρ(x, ¯ t) ∂t γ ∂xi ∂xi ∂xi xi i=1

−

N


i=1

α|xi |2+θi ρ(x, ¯ t).

(22)

i=1

This equation may be used to investigate subdiffusion-limited reactions [34] and also by a suitable change it may correspond to a Schrödinger-like equation for γ = 1 with a mass dependent on the position [35]. By using the

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Fig. 4. Behavior of ρ(x, ¯ t) versus |x1 | for typical values of t by considering, for simplicity, the one-dimensional case of Eq. (20), γ = 1/2, K1 = 0, θ1 = 1, ξ1 = 1, k1 = 1 and D1 = 1. Note that Eq. (20) evolves to Eq. (21) for long times indicating that the time fractional derivative produces an anomalous relaxation to the equilibrium situation.

previous procedure employed to find the solution for Eq. (14), we verify that the solution of Eq. (22) is given by √2 ∞ ∞  N ki +4αDi ki (|xi |2+θi −|ξi |2+θi ) Ki

− − (|x |2+θi +|ξi |2+θi ) 2(2+θi )Di ··· |xi | Di e e 2(2+θi )Di i ρ(x, ¯ t) = n1 =0



nN =0 i=1

ki2 + 4αDi



Ki +Di (2+θi )Di

(2 + θi )Γ (ni + 1)   Ki +Di (2 + θi )Di 2Γ (2+θ + n i ) D i i   2  k 2 + 4αD  ki + 4αDi i   i (α¯ i ) 2+θi (α¯ i ) 2+θi |xi | |ξi | × Lni (23) Lni Eγ −λ¯ n1 ···nN t γ , (2 + θi )Di (2 + θ )Di    2 k ki2 + 4αDi ]} + 4αD (2+θ ){n +[K +D ]/[2D (2+θ )]−k [K +D ]/[2(2+θ )D where λ¯ n1 ···nN = N i i i i i i i i i i i i i=1 i N and the initial condition is given by ρ(x, ¯ 0) = i=1 δ(xi − ξi ). ×

3. Summary and conclusion We have worked an anisotropic fractional diffusion equation by considering the N -dimensional case. We have first analyzed the free case by taking a time and spatial dependent diffusion coefficient. Afterwards, we have incorporated external forces in our analysis. In particular, we have analyzed the external force Fi = (2 + θi )(Ki /xi )|xi |νi

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Fig. 5. Behavior of ρ(x, ¯ t) versus |x1 | for typical values of t by considering, for simplicity, the one-dimensional case of Eq. (20), γ = 1/2, K1 = 4, θ1 = 1, ξ1 = 1, k1 = 1 and D1 = 1. As in Fig. 4 note that Eq. (20) for this set of parameters also evolves to Eq. (21) for long times.

and the external force Fi = −ki xi + (Ki /xi )|xi |νi in both cases νi = −θi . In addition, we have also investigated the effects produced by a source (absorbent) term α|xi |ηi ρ with ηi = 2 + θi . For these cases, we have obtained exact solutions given in terms of the Fox H function or the associated Laguerre polynomial and the Mittag-Leffler function. The presence of these functions, Fox H function and the Mittag-Leffler function, are due to the fractional derivative present in the diffusion equation. In fact, the presence of a fractional derivative in the diffusion equation changes the waiting time probability density function. Therefore, we have an anomalous relaxation for this case that differs from the usual case characterized by an exponential relaxation. Another interesting feature of these cases is the behavior at the origin which, depending on the parameters considered, can be divergent as the results found in [15]. We have point out that the stationary solutions are equal to the usual one. This feature is very interesting since the solutions have their spatial variables coupled for the time dependent problem and they are not coupled in the stationary case. In particular, this result is in agreement with the results presented in [36] concerning the fractional diffusion equations and thermodynamics. We have extended the results presented in [28] for a fractional diffusion equation, the two-dimensional results obtained for an anisotropic fractional diffusion equation [29], the Rayleigh process [37] and the asymptotic results reported in [38] for homogeneous and isotropic random walk models. Finally, we expect the results presented herein will be useful to discuss the situations where the anisotropy and the anomalous diffusion are present. Acknowledgements We thank CNPq, CAPES and Fundação Araucária (Brazilian agencies) for the financial support.

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