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An anomalous diffusion model in an external force fields on fractals
Physics Letters A 312 (2003) 187–197 www.elsevier.com/locate/pla
An anomalous diffusion model in an external force fields on fractals ✩ Fu-Yao Ren a,∗ , Jin-Rong Liang b , Wei-Yuan Qiu a , Xiao-Tian Wang c , Y. Xu a , R.R. Nigmatullin d a Department of Mathematics, Fudan University, Shanghai 200433, PR China b Department of Mathematics, East China Normal University, Shanghai 200062, PR China c Department of Mathematics, Wuhan University, Wuhan 430072, PR China d Theoretical Physics Department, Kazan State University, Kazan 420008, Tatarstan, Russia
Received 1 January 2003; received in revised form 29 March 2003; accepted 8 April 2003 Communicated by C.R. Doering
Abstract We present a fractional diffusion equation involving external force fields for transport phenomena in random media. It is shown that this fractional diffusion equation obey generalized Einstein relation, and its stationary solution is the Boltzmann distribution. It is proved that the asymptotic behavior of its solution is stretched Gaussian and that its solution can be expressed in the form of a function of a dimensionless similarity variable, not only for constant potentials but also for logarithm, harmonic, analytic and generic potentials. A comparison with the fractional Fokker–Planck equation is given. 2003 Elsevier Science B.V. All rights reserved.
1. Introduction In connection with the growing interest in the physics of complex systems, anomalous transport properties and their description have received considerable interest. One found application in a wide field ranging from physics and chemistry to biology and medicine [1–5]. Anomalous diffusion in one dimension is characterized by the occurrence of a mean square displacement of the form 2 X (t) = x2 (t) =
2Kγ tγ , Γ (1 + γ )
0<γ <1
(1.1)
which deviates from the linear Brownian dependence on time [3]. In Eq. (1.1), the anomalous diffusion coefficient Kγ is introduced, which has the dimension [Kγ ] = cm2 s−γ . ✩
Project supported by NSFC (No. 10271031) and by the Shanghai Priority Academic Discipline.
* Corresponding author.
E-mail address: [email protected] (F.-Y. Ren). 0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00644-3
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Diffusion on fractals exhibits many anomalous features due to geometrical constraints imposed by the complex structure on a diffusion process. These constraints represent spatial correlations of the structure (self-similarity) which persist on all length scales leading to an anomalous behavior of the mean square displacement of a random walker on all time scales. Recently in the literature, in order to describe diffusion processes in disordered media, some authors proposed an extension of the Fokker–Planck equation, which is called fractional Fokker–Planck equation [6–12]. A fractional Fokker–Planck equation (FFPE) describing anomalous transport close to thermal equilibrium was presented recently [6]. Since it describes subdiffusion in the force-free case, it involves a strong, i.e., slowly decaying memory. In Ref. [8] Metzler et al. gave a seminal framework and investigated the anomalous diffusion and relaxation involving external fields with the one-dimensional fractional Fokker–Planck equation 1−γ W˙ (x, t) = 0 Dt LFP W
(1.2)
in respect to its physical properties, where W (x, t) is the probability density function at position x at time t and the FP operator ∂ V (x) ∂ LFP = (1.3) + Kγ , ∂x mηγ ∂x with the external potential V (x) [13], contains the anomalous diffusion constant Kγ and the anomalous friction coefficient ηγ with the dimension [ηγ ] = sγ −2 , herein m denotes the mass of the diffusion particle, and 1−γ W 0 Dt
1 ∂ = Γ (γ ) ∂t
t dτ 0
W (x, τ ) , (t − τ )1−γ
0 < γ < 1.
(1.4) 1−γ
The right-hand side of the FFPE (1.2) is equivalent to the fractional expression −0 Dt
V (x) ∂ S(x, t) = − + Kγ mηγ ∂x
∂S(x, t)/∂x, where (1.5)
is the probability current. It has been shown that as in the standard FPE, the FFPE (1.2) in the force-free case possesses a scaling variable, i.e., WF =0 (x, t) = [Kγ t γ ]−1/2 f (z) with the dimensionless similarity variable z = x/(Kγ t γ )1/2 and that the asymptotic shape of WF =0 is stretched Gaussian [14]. But for arbitrary external potentials V (x), such simple scaling behavior is not found yet. In Ref. [24], the fractional Fokker–Planck equation α −θ
LFP W (x, t) 0 Dt W (x, t) = Gx
(1.6)
has been presented, where G > 0 is to be determined, θ 0 is a parameter, 0 < α < 1, if θ = 0 then α = γ and Eq. (1.6) reduces to γ 0 Dt W (x, t) = GLFP W (x, t),
(1.7)
which leads to the FFPE (1.2). It is proved that for the FFPE (1.6), its solution has the asymptotic behavior log W (x, t) ∼ −cξ u , where ξ ≡ x/t
γ /2
1,
(1.8) γ u = 1/ 1 − , 2
(1.9)
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and possesses a scaling variable for constant potential, linear potentials, logarithm potentials and harmonic potentials. We naturally ask the following problem: Can we establish a reasonable model for describing anomalous transport processes in an external force fields on random fractal structure for which the stretched Gaussian asymptotic behavior and scaling variable are expected to be universal? This question is interesting and important. In this Letter, we generalize a fractional diffusion equation of Giona and Roman [15] to the case in external force fields and then discuss the diffusion in external force fields. Unlike fractional Fokker–Planck equation, such a fractional diffusion equation involves only a first order differential with respect to the space variable x. We prove that the solutions of the fractional diffusion equations in external force fields possess a scaling variable for arbitrary external potentials V (x) and obtain that their asymptotic solutions are stretched Gaussian. In addition, it is proved that the necessary and sufficient condition of its stationary solution being the Boltzmann distribution is that the Stokes–Einstein–Smoluchowski relation holds. The Letter is organized as follows. In Section 2 we derive fractional diffusion equations involving external potentials. In Section 3 we respectively give the solutions of fractional diffusion equations in the cases of constant potentials, logarithm potentials, harmonic potentials, analytic potentials and generic potentials.
2. Fractional diffusion equations involving external potentials In this section, we will derive the fractional diffusion equation involving the external potential V (x) by using a heuristic argument of Giona and Roman. For the anomalous diffusion, we assume that diffusion sets are underlying fractals (underlying fractals denote self-similar sets in [19] or net fractals in [20,21]). The relationship between the total probability current S(x, t) up to time t and the average probability density W (x, t) should be (cf. [15,16,18]) t S(x, τ ) dτ = x
df −1
0
t K(t − τ )W (x, τ ) dτ,
(2.1)
0
where df is the fractal dimension of the system considered, the diffusion kernel on the underlying fractal should behave as Aα K(t − τ ) = (2.2) (t − τ )α with 0 < α < 1, where α is a diffusion exponent and Aα is a constant that can be determined [21]. On the other hand, stimulated by the work of Metzler et al. on the probability current, the probability current on the underlying fractal structure is defined by
∂ df −1 −θ V (x) x + Kγ W (x, t), S(x, t) = −Bx (2.3) mηγ ∂x where B is a positive constant to be determined, the parameter θ is necessarily positive, θ 0. If B = df = 1 and θ = 0, it reduces to (1.5). Eq. (2.3) together with (2.1) and (2.2) yield a fractional diffusion equation on the fractal involving external potentials,
∂ α −θ V (x) W (x, t), (2.4) + Kγ 0 Dt W (x, t) = −Gx mηγ ∂x where G =
B Aα Γ (1−α)
> 0.
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From (2.4) and (1.1), the simple scaling considerations give that γ 1 + θ . α= (2.5) 2 Furthermore, we assume that the probability density W (x, t) satisfies the following normalization condition ∞
dx x df −1 W (x, t) = 1,
(2.6)
0
and the extraction of moments (x)n is defined by (x)n =
∞
dx x df −1 x n W (x, t).
(2.7)
0
If df = 1, they reduce to the expressions of ordinary normalization condition and moments, respectively. Before we discuss the solution of Eq. (2.4), let us consider its stationary solution. It is readily seen that if a stationary state is reached, S(x, t) in (2.3) must be a constant. Thus, if S(x, t) = 0 for any x, it vanished for all x [13], and the stationary solution is given by V (x)Wst /[mηγ ] + Kγ Wst = 0, i.e., Wst (x) = exp V (0) − V (x) /(mηγ Kγ ) . Comparing this expression with the required Boltzmann distribution Wst ∝ exp{−V (x)/[kB T ]}, we find a generalization of the Einstein relation, also referred to as Stokes–Einstein–Smoluchowski relation, Kγ =
kB T , mηγ
(2.8)
for the anomalous coefficients Kγ and ηγ . Thus, the process described by Eq. (2.4) fulfills the linear relation between anomalous friction and diffusion coefficients, reflecting the fluctuation–dissipation theorem. Contrarily, if (2.8) holds, the its stationary solution must be the Boltzmann distribution.
3. Solutions of fractional diffusion equations In this section we respectively give the solutions of the fractional diffusion equation (2.4) in the cases of constant potentials, logarithm potentials, harmonic potentials, analytic potentials and generic potentials. It can be shown that the Laplace transform of the fractional derivative (1.4): L 0 Dtα W (x, t) = s α W (x, s). Using this result, from (2.4) we get
∂W (x, s)
V (x) , s α W (x, s) = −Gx −θ W (x, s) + Kγ mηγ ∂x
(3.1)
where W (x, s) denotes the Laplace transform of W (x, t). 3.1. Case I: constant potentials If the external field is force-free, i.e., V (x) = const, then (3.1) degenerates to
s α W (x, s) = −Gx −θ Kγ
∂W (x, s) , ∂x
(3.2)
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191
which has the solution W (x, s) = q0 (s) exp −
s α x θ +1 , (θ + 1)GKγ
(3.3)
where q0 (s) = G1 s ξ ensures the normalization condition. Herein
G1 =
(θ + 1)1−df /(θ +1)
Γ (df /(θ + 1))(GKγ )df /(θ +1)
,
(3.4)
and ξ = αdf /(θ + 1) − 1. It follows from (1.1) and (2.7) (when n = 2) that α=
γ 1 + θ , 2
B = g(df ),
(3.5)
where α/γ
Γ (γ (df / 2α) γ α/γ −1 . g(df ) = Aα Γ (1 − α) (2Kγ ) α Γ (γ (df + 2)/ 2α)
(3.6)
Noting 0 < α < 1 and (3.7) we have α γ /2 and θ < 2/γ − 1. Let us discuss the asymptotic behavior of W (x, t) as predicted by (3.3). We expect that W (x, t) t −γ df /2
x X
δ
u x , exp −b X
(3.7)
where x/X 1 and t → ∞. The Laplace transform of (3.7) can be evaluated by applying the method of steepest descent and the result compared with (3.3). This yields (cf. Ref. [15]) u
(ds − 1), (3.8) 2 where ds = γ df is called the fraction or spectral dimension [22,23]. Thus the asymptotic shape of W (x, t) is stretched Gaussian. This result coincides with that in the force-free case [14]. Now we give the solution of the fractional diffusion equation (2.5) in Case I. Using the inversion theorem of Laplace transform, from (3.3) we have (see Appendix A) −1/2 W (x, t) = Kγ t γ df (3.9) f (z), u = 2α/γ (1 − α),
δ =
where f (z) =
∞ 1/2
G1 Kγ π
Cn (ξ )zn(2α/γ ) ,
z = x/t γ /2 ,
ξ=
n=0
1 γ df − 1. 2
Putting df = 1 this result coincides with that in Ref. [14]. 3.2. Case II: logarithm potentials If the external field is a logarithm potential, i.e., V (x) = b ln x, then (3.1) becomes
κ
∂W (x, s) + W (x, s) , S α W (x, s) = −G x −θ ∂x x where G = GKγ > 0, κ = b/(mηγ Kγ ). Eq. (3.10) is the fractional diffusion equation on fractals in [15].
(3.10)
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A simple integration of (3.10) yields
q(s) s α x θ +1 W (x, s) = κ exp −
, x (θ + 1)G
(3.11)
where
(df −κ)/(θ +1) 1 sα θ + 1 , q(s) = Γ ((df − κ)/(θ + 1)) s (θ + 1)G
(3.12)
if df − κ > 0, ensures the normalization condition of W (x, t). Thus, it follows from (3.11) that W (x, s) = G1 s ζ x −κ exp −G2 s α ,
(3.13)
where
ξ = α(df − κ)/ θ + 1 − 1
(3.14)
and
G1 =
(θ + 1)1−(df −κ)/(θ +1) Γ ((df − κ)/(θ + 1))(GKγ )(df
, −κ)/(θ +1)
G2 =
It follows from (1.1) and (2.7) (when n = 2) that γ α= B = g(df − κ). 1 + θ , 2 Using the same methods in Ref. [15], from (3.18) we have δ
u x x W (x, t) t −γ df /2 exp −const × X X when x/X 1 and t → ∞, where u = 2α/γ (1 − α),
δ = u
γ df − 1 −κ . 2
As in Case I, we can show that −1/2 −κ · z f (z), W (x, t) = Kγ t γ df
x θ +1 . (θ + 1)GKγ
(3.15)
(3.16)
(3.17)
where f (z) =
∞ 1/2
G1 Kγ π
Cn (ξ )zn(2α/γ ) ,
n=0
where z = x/t γ /2 is a dimensionless similarity variable, {Cn (ξ )} are the same as in (3.15) but herein ξ = γ 2 (df − κ) − 1. 3.3. Case III: harmonic potentials In this paragraph we deal with the special and physically important case of an harmonic potential field, i.e., V (x) = λ2 x 2 , leading to the Hookean force F (x) = −λx directed at the original. Similar consideration may be found in Refs. [7,12]. In this case, Eq. (3.1) reduces to
λx ∂W (x, s)
s α W (x, s) = −Gx −θ (3.18) . + Kγ mηγ ∂x
F.-Y. Ren et al. / Physics Letters A 312 (2003) 187–197
A simple integration of (3.18) yields
(θ +1) κλ
W (x, s) = q(s) exp − x 2 − xs α/(θ +1) /G3 2
193
(3.19)
where q(s) =
c0 s αz0 −1 ∞ ,
1 + n=1 r2n s −2nα/(θ +1)
ensures the normalization condition of W (x, t). Herein, κ=
1 mηγ Kγ
r2n =
,
G3 = Gγ1/(θ +1) ,
c2n Γ (z2n )G2n γ Γ (z0 )
,
zn =
Gγ = (θ + 1)GKγ ,
df + n , θ + 1
cn =
c0 =
1 λκ n , − n! 2
θ + 1 z , Γ (z0 )Gγ0
n = 1, 2, . . . .
(3.20)
It follows from (1.1) and (2.7) (when n = 2) that γ 1 + θ , B = g(df ). α= (3.21) 2 Let us discuss the asymptotic behavior of W (x, t) predicted by (3.25). Noting (3.27), (3.25) can be written as follows ∞ (θ +1) γ /2 −(1−γ df /2) −nγ , W (x, s) = φ0 (x)s (3.22) · 1+ exp − xs /G3 d2n s n=1
where
κλ 2 φ0 (x) = c0 exp − x , d2 = −r2 , 2 d2n = −(d2(n−1)r2 + d2(n−2)r4 + · · · + d2 r2(n−1) + r2n ).
(3.23)
θ +1 W (x, s) ≈ φ0 (x)s −(1−γ df /2) exp − xs γ /2 /G3
(3.24)
Thus
when |s| 1. Using the same methods as in Refs. [15,17], we have δ
u x x W (x, t) ∼ t −γ df /2 φ0 (x) exp −const × X X
(3.25)
when x/X 1 and t → ∞, where δ and u are the same as in (3.8). To determine the solution of (2.4) associated to Case III, taking the Laplacian inversion transform on both sides of (3.22) we have W (x, t) = φ0 (x)
∞
d2n f2n (x, t),
(3.26)
n=0
where f2n (x, t) = L−1 F2n (x, s) , θ +1 , F2n (x, s) = s ξ2n exp − xs γ /2 /Gγ
d0 = 1,
ξn =
γ (df − n) − 1. 2
(3.27)
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Since ξ2n+1 + 1 < 0 for n 1, we can show that f2n (x, t) = 0. Since ξ0 + 1 > 0, then as in Case I we have −1/2 φ0 (x)f (z), W (x, t) = Kγ t γ df (3.28) where z = x/t γ /2 , f (z) =
1/2
G1 K γ π
∞
n=0 Cn (ξ0 )z
n(2α/γ )
and Cn (ξ0 ) is defined as in Appendix A.
3.4. Case IV: generic potentials In this paragraph we discuss the generic external potential V (x) = b log x +
∞
ak x k .
(3.29)
k=1
In this case, the solution of the fractional diffusion equation (3.1) is as follows
W (x, s) = q(s) exp −κV (x) · exp −s α x θ +1 /Gγ ,
(3.30)
where κ = 1/(mηγ Kγ ), Gγ = (θ + 1)GKγ , and ∞ αz0 −1 −αn/(θ +1) 1+ ηn s q(s) = c0 s
(3.31)
n=1
if df > bκ, ensures the normalization condition of W (x, t). Herein, c0 = (θ + 1)/Γ (z0 )Gzγ0 ,
G3 = Gγ1/(θ +1) ,
ηn = βn Γ (ξn )Gn3 ,
ξn = (df − bκ + n)/(θ + 1), n 0, n κ β0 = 1, βn = − kak βn−k , n 1. n k=1
It follows from (1.1) and (2.7) (when n = 2) that γ 1 + θ , B = g(df − bκ). α= 2 Thus, it follows from (3.30) that W (x, s) ≈ φ1 (x)s when |s| 1, where
φ1 (x) = c0 exp −κ
γ /2 /G
3) γ /2 bκ (xs )
−(1− 21 df ) exp{−(xs
∞
(θ +1) }
,
(3.32)
an x n ,
+1) . G3 = G1/(θ γ
(3.33)
n=1
Using the same methods in Refs. [15,17], from (3.32) we have δ
u x x −γ df /2 W (x, t) ∼ t φ1 (x) exp −const × X X
(3.34)
when x/X 1 and t → ∞, where δ and u are the same as in (3.16). We now turn to discuss the expression of dimensionless similarity variable of W (x, t). It follows from (3.30) that ∞ W (x, s) = c0 exp −κV (x) (3.35) dn L−1 Fn (s) , n=0
F.-Y. Ren et al. / Physics Letters A 312 (2003) 187–197
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where d0 = 1, d1 = κa1G3 Γ (z1 )/Γ (z0 ), dn = −(dn−1 η1 + dn−2 η2 + · · · + d1 ηn−1 + ηn ) (n = 2, 3, . . . , ), γ Fn (s) = s ξn exp −G2 s α , ξn = (df − bκ − n) − 1, n = 0, 1, . . . . 2 If b = 0 and 0 < df − bκ < 1. Similar to Case III, since ξ0 + 1 > 0 but ξn + 1 < 0 for n 1, we have −1/2 −bκ z f (z), W (x, t) = φ1 (x) Kγ t γ df
(3.36) (3.37)
(3.38)
1/2 G K ∞ n(2α/γ ) . where z = x/t γ /2 , and f (z) = 1 π γ n=0 Cn (ξ0 )z If b = 0 and df − bκ > 1, then ξi + 1 > 0 (i = 0, 1), but ξn + 1 < 0 for n 2. Thus
−1/2 −bκ W (x, t) = φ1 (x) Kγ t γ df f (z) + d1 t γ /2 f1 (z) , z
(3.39)
1/2 G K ∞ n(2α/γ ) . where f1 (z) = 1 π γ n=0 Cn (ξ1 )z Furthermore, if a1 = 0 we still have the same results as in case of 0 < df − bκ < 1. On the other hand, if b = 0, i.e., V (x) is analytic with form
V (x) =
∞
an x n .
(3.40)
n=1
Similar consideration can be found in Ref. [7]. In this case, we have δ
u x x −γ df /2 W (x, t) ∼ t φ1 (x) exp −const × X X
(3.41)
when x/X 1 and t → ∞, and −1/2 W (x, t) = φ1 (x) Kγ t γ df f (z) + t γ /2 d1 f1 (z) ,
(3.42)
where δ and u are the same as in (3.8). In particular, if a1 = 0, then −1/2 W (x, t) = c0 exp −κV (x) Kγ t γ df f (z).
(3.43)
4. Conclusion and discussion In order to describe anomalous diffusion processes involving external potential fields on fractal structures, we introduce a fractional diffusion equation involving external potentials. This equation is different from the fractional Fokker–Planck equation. The solution of this fractional diffusion equation possesses following properties: The necessary and sufficient condition of its stationary solution being the Boltzmann distribution exp{−V (x)/ [kB T ]} is that the generalization of the Einstein relation, also preferred to as stokes-Einstein–Smoluchowski relation holds Kγ = kB T /mηγ , for the anomalous coefficients Kγ and ηγ . This result is consistent with that in Ref. [8]. The asymptotic shape of solution of Eq. (2.5) is stretched Gaussian not only for constant potentials but also for logarithm, harmonic, analytic and generic potentials.
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Solutions of (2.5) can be expressed in the form of −1/2 −κ −1/2 −κ Kγ t γ df z f (z) or Kγ t γ df z φ(x) f (z) + t γ /2 d1 f1 (z) with the dimensionless similarity variable z = x/t γ /2 for any potentials mentioned above, where κ and d1 are constants but may be zero, f (z) and f1 (z) are functions of z, and df is the fractal dimension of the underlying structure. It is surprising that the presence of f1 (z) is due to the presence of term a1 x in the external potentials. In each case (constant, logarithm, harmonic, analytic and generic potentials) there exists an intrinsic relationship, 2α = γ (1 + θ ). We know that α and γ are structural parameter of the underlying fractal structure and α can be determined explicitly (see Refs. [19–21]). It is interesting that as same as the fractional Fokker–Planck equation (1.6), the solution of fractional diffusion
equation (2.5) also has the asymptotic behavior log W (x, t) ∼ −cξ u where ξ ≡ x/t γ /2 1, u = 2α/γ (1 − α),
only u > u = 1/(1 − γ /2) of FFPE for 1/2 < ρ = α/γ < 1/γ which means that the asymptotic behavior decays quickly than that ones of FFPE, and u = u if and only if ρ = 1/2 which means that both of FFPE and Eq. (2.4) have same decay.
Appendix A Using the inversion theorem of Laplace transform, from (3.3) we have G0 W (x, t) = π
∞
τ ξ exp − tτ + G2 τ α cos απ sin G2 τ α sin απ − ξ π dτ
0
G0 −ξ −1 t = π
∞
y ξ exp − y + G2 (y/t)α cos απ sin G2 (y/t)α sin απ − ξ π dy,
(A.1)
0
where
x θ +1 . G2 =
(θ + 1)GKγ Using the expressions of power series of entire-functions ez , cos z and sin z and the integral representation of Γ function, a simple integration of (A.1) yields the following expression with the dimensionless similarity variable z = x/t γ /2, −1/2 W (x, t) = Kγ t γ df (A.2) f (z), where f (z) =
∞ 1/2
G1 Kγ π
Cn (ξ )zn(2α/γ ) ,
n=0
C0 = a Γ (γ df /2) > 0, Cn (ξ ) = a bn + b an−1 , a 2k+1bm an (ξ ) = C n+1 Γ α(ξ + 2 + n) , (−1)k+m (2k + 1)!m! k,m0,2k+m=n
bn (ξ ) =
k,m0,2k+m=n
(−1)k+m
a 2k bm n C Γ α(ξ + 1 + n) , (2k)!m!
F.-Y. Ren et al. / Physics Letters A 312 (2003) 187–197
a = − sin ξ π > 0, b = cos ξ π, a = sin απ, 1−γ γ (2α/γ ) df /2α . C= , G1 = 2αGKγ (GKγ )γ df /2α Γ (γ df /2α)
b = cos απ,
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(A.3)
An anomalous diffusion model in an external force fields on fractals ✩ Fu-Yao Ren a,∗ , Jin-Rong Liang b , Wei-Yuan Qiu a , Xiao-Tian Wang c , Y. Xu a , R.R. Nigmatullin d a Department of Mathematics, Fudan University, Shanghai 200433, PR China b Department of Mathematics, East China Normal University, Shanghai 200062, PR China c Department of Mathematics, Wuhan University, Wuhan 430072, PR China d Theoretical Physics Department, Kazan State University, Kazan 420008, Tatarstan, Russia
Received 1 January 2003; received in revised form 29 March 2003; accepted 8 April 2003 Communicated by C.R. Doering
Abstract We present a fractional diffusion equation involving external force fields for transport phenomena in random media. It is shown that this fractional diffusion equation obey generalized Einstein relation, and its stationary solution is the Boltzmann distribution. It is proved that the asymptotic behavior of its solution is stretched Gaussian and that its solution can be expressed in the form of a function of a dimensionless similarity variable, not only for constant potentials but also for logarithm, harmonic, analytic and generic potentials. A comparison with the fractional Fokker–Planck equation is given. 2003 Elsevier Science B.V. All rights reserved.
1. Introduction In connection with the growing interest in the physics of complex systems, anomalous transport properties and their description have received considerable interest. One found application in a wide field ranging from physics and chemistry to biology and medicine [1–5]. Anomalous diffusion in one dimension is characterized by the occurrence of a mean square displacement of the form 2 X (t) = x2 (t) =
2Kγ tγ , Γ (1 + γ )
0<γ <1
(1.1)
which deviates from the linear Brownian dependence on time [3]. In Eq. (1.1), the anomalous diffusion coefficient Kγ is introduced, which has the dimension [Kγ ] = cm2 s−γ . ✩
Project supported by NSFC (No. 10271031) and by the Shanghai Priority Academic Discipline.
* Corresponding author.
E-mail address: [email protected] (F.-Y. Ren). 0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00644-3
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Diffusion on fractals exhibits many anomalous features due to geometrical constraints imposed by the complex structure on a diffusion process. These constraints represent spatial correlations of the structure (self-similarity) which persist on all length scales leading to an anomalous behavior of the mean square displacement of a random walker on all time scales. Recently in the literature, in order to describe diffusion processes in disordered media, some authors proposed an extension of the Fokker–Planck equation, which is called fractional Fokker–Planck equation [6–12]. A fractional Fokker–Planck equation (FFPE) describing anomalous transport close to thermal equilibrium was presented recently [6]. Since it describes subdiffusion in the force-free case, it involves a strong, i.e., slowly decaying memory. In Ref. [8] Metzler et al. gave a seminal framework and investigated the anomalous diffusion and relaxation involving external fields with the one-dimensional fractional Fokker–Planck equation 1−γ W˙ (x, t) = 0 Dt LFP W
(1.2)
in respect to its physical properties, where W (x, t) is the probability density function at position x at time t and the FP operator ∂ V (x) ∂ LFP = (1.3) + Kγ , ∂x mηγ ∂x with the external potential V (x) [13], contains the anomalous diffusion constant Kγ and the anomalous friction coefficient ηγ with the dimension [ηγ ] = sγ −2 , herein m denotes the mass of the diffusion particle, and 1−γ W 0 Dt
1 ∂ = Γ (γ ) ∂t
t dτ 0
W (x, τ ) , (t − τ )1−γ
0 < γ < 1.
(1.4) 1−γ
The right-hand side of the FFPE (1.2) is equivalent to the fractional expression −0 Dt
V (x) ∂ S(x, t) = − + Kγ mηγ ∂x
∂S(x, t)/∂x, where (1.5)
is the probability current. It has been shown that as in the standard FPE, the FFPE (1.2) in the force-free case possesses a scaling variable, i.e., WF =0 (x, t) = [Kγ t γ ]−1/2 f (z) with the dimensionless similarity variable z = x/(Kγ t γ )1/2 and that the asymptotic shape of WF =0 is stretched Gaussian [14]. But for arbitrary external potentials V (x), such simple scaling behavior is not found yet. In Ref. [24], the fractional Fokker–Planck equation α −θ
LFP W (x, t) 0 Dt W (x, t) = Gx
(1.6)
has been presented, where G > 0 is to be determined, θ 0 is a parameter, 0 < α < 1, if θ = 0 then α = γ and Eq. (1.6) reduces to γ 0 Dt W (x, t) = GLFP W (x, t),
(1.7)
which leads to the FFPE (1.2). It is proved that for the FFPE (1.6), its solution has the asymptotic behavior log W (x, t) ∼ −cξ u , where ξ ≡ x/t
γ /2
1,
(1.8) γ u = 1/ 1 − , 2
(1.9)
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and possesses a scaling variable for constant potential, linear potentials, logarithm potentials and harmonic potentials. We naturally ask the following problem: Can we establish a reasonable model for describing anomalous transport processes in an external force fields on random fractal structure for which the stretched Gaussian asymptotic behavior and scaling variable are expected to be universal? This question is interesting and important. In this Letter, we generalize a fractional diffusion equation of Giona and Roman [15] to the case in external force fields and then discuss the diffusion in external force fields. Unlike fractional Fokker–Planck equation, such a fractional diffusion equation involves only a first order differential with respect to the space variable x. We prove that the solutions of the fractional diffusion equations in external force fields possess a scaling variable for arbitrary external potentials V (x) and obtain that their asymptotic solutions are stretched Gaussian. In addition, it is proved that the necessary and sufficient condition of its stationary solution being the Boltzmann distribution is that the Stokes–Einstein–Smoluchowski relation holds. The Letter is organized as follows. In Section 2 we derive fractional diffusion equations involving external potentials. In Section 3 we respectively give the solutions of fractional diffusion equations in the cases of constant potentials, logarithm potentials, harmonic potentials, analytic potentials and generic potentials.
2. Fractional diffusion equations involving external potentials In this section, we will derive the fractional diffusion equation involving the external potential V (x) by using a heuristic argument of Giona and Roman. For the anomalous diffusion, we assume that diffusion sets are underlying fractals (underlying fractals denote self-similar sets in [19] or net fractals in [20,21]). The relationship between the total probability current S(x, t) up to time t and the average probability density W (x, t) should be (cf. [15,16,18]) t S(x, τ ) dτ = x
df −1
0
t K(t − τ )W (x, τ ) dτ,
(2.1)
0
where df is the fractal dimension of the system considered, the diffusion kernel on the underlying fractal should behave as Aα K(t − τ ) = (2.2) (t − τ )α with 0 < α < 1, where α is a diffusion exponent and Aα is a constant that can be determined [21]. On the other hand, stimulated by the work of Metzler et al. on the probability current, the probability current on the underlying fractal structure is defined by
∂ df −1 −θ V (x) x + Kγ W (x, t), S(x, t) = −Bx (2.3) mηγ ∂x where B is a positive constant to be determined, the parameter θ is necessarily positive, θ 0. If B = df = 1 and θ = 0, it reduces to (1.5). Eq. (2.3) together with (2.1) and (2.2) yield a fractional diffusion equation on the fractal involving external potentials,
∂ α −θ V (x) W (x, t), (2.4) + Kγ 0 Dt W (x, t) = −Gx mηγ ∂x where G =
B Aα Γ (1−α)
> 0.
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From (2.4) and (1.1), the simple scaling considerations give that γ 1 + θ . α= (2.5) 2 Furthermore, we assume that the probability density W (x, t) satisfies the following normalization condition ∞
dx x df −1 W (x, t) = 1,
(2.6)
0
and the extraction of moments (x)n is defined by (x)n =
∞
dx x df −1 x n W (x, t).
(2.7)
0
If df = 1, they reduce to the expressions of ordinary normalization condition and moments, respectively. Before we discuss the solution of Eq. (2.4), let us consider its stationary solution. It is readily seen that if a stationary state is reached, S(x, t) in (2.3) must be a constant. Thus, if S(x, t) = 0 for any x, it vanished for all x [13], and the stationary solution is given by V (x)Wst /[mηγ ] + Kγ Wst = 0, i.e., Wst (x) = exp V (0) − V (x) /(mηγ Kγ ) . Comparing this expression with the required Boltzmann distribution Wst ∝ exp{−V (x)/[kB T ]}, we find a generalization of the Einstein relation, also referred to as Stokes–Einstein–Smoluchowski relation, Kγ =
kB T , mηγ
(2.8)
for the anomalous coefficients Kγ and ηγ . Thus, the process described by Eq. (2.4) fulfills the linear relation between anomalous friction and diffusion coefficients, reflecting the fluctuation–dissipation theorem. Contrarily, if (2.8) holds, the its stationary solution must be the Boltzmann distribution.
3. Solutions of fractional diffusion equations In this section we respectively give the solutions of the fractional diffusion equation (2.4) in the cases of constant potentials, logarithm potentials, harmonic potentials, analytic potentials and generic potentials. It can be shown that the Laplace transform of the fractional derivative (1.4): L 0 Dtα W (x, t) = s α W (x, s). Using this result, from (2.4) we get
∂W (x, s)
V (x) , s α W (x, s) = −Gx −θ W (x, s) + Kγ mηγ ∂x
(3.1)
where W (x, s) denotes the Laplace transform of W (x, t). 3.1. Case I: constant potentials If the external field is force-free, i.e., V (x) = const, then (3.1) degenerates to
s α W (x, s) = −Gx −θ Kγ
∂W (x, s) , ∂x
(3.2)
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191
which has the solution W (x, s) = q0 (s) exp −
s α x θ +1 , (θ + 1)GKγ
(3.3)
where q0 (s) = G1 s ξ ensures the normalization condition. Herein
G1 =
(θ + 1)1−df /(θ +1)
Γ (df /(θ + 1))(GKγ )df /(θ +1)
,
(3.4)
and ξ = αdf /(θ + 1) − 1. It follows from (1.1) and (2.7) (when n = 2) that α=
γ 1 + θ , 2
B = g(df ),
(3.5)
where α/γ
Γ (γ (df / 2α) γ α/γ −1 . g(df ) = Aα Γ (1 − α) (2Kγ ) α Γ (γ (df + 2)/ 2α)
(3.6)
Noting 0 < α < 1 and (3.7) we have α γ /2 and θ < 2/γ − 1. Let us discuss the asymptotic behavior of W (x, t) as predicted by (3.3). We expect that W (x, t) t −γ df /2
x X
δ
u x , exp −b X
(3.7)
where x/X 1 and t → ∞. The Laplace transform of (3.7) can be evaluated by applying the method of steepest descent and the result compared with (3.3). This yields (cf. Ref. [15]) u
(ds − 1), (3.8) 2 where ds = γ df is called the fraction or spectral dimension [22,23]. Thus the asymptotic shape of W (x, t) is stretched Gaussian. This result coincides with that in the force-free case [14]. Now we give the solution of the fractional diffusion equation (2.5) in Case I. Using the inversion theorem of Laplace transform, from (3.3) we have (see Appendix A) −1/2 W (x, t) = Kγ t γ df (3.9) f (z), u = 2α/γ (1 − α),
δ =
where f (z) =
∞ 1/2
G1 Kγ π
Cn (ξ )zn(2α/γ ) ,
z = x/t γ /2 ,
ξ=
n=0
1 γ df − 1. 2
Putting df = 1 this result coincides with that in Ref. [14]. 3.2. Case II: logarithm potentials If the external field is a logarithm potential, i.e., V (x) = b ln x, then (3.1) becomes
κ
∂W (x, s) + W (x, s) , S α W (x, s) = −G x −θ ∂x x where G = GKγ > 0, κ = b/(mηγ Kγ ). Eq. (3.10) is the fractional diffusion equation on fractals in [15].
(3.10)
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A simple integration of (3.10) yields
q(s) s α x θ +1 W (x, s) = κ exp −
, x (θ + 1)G
(3.11)
where
(df −κ)/(θ +1) 1 sα θ + 1 , q(s) = Γ ((df − κ)/(θ + 1)) s (θ + 1)G
(3.12)
if df − κ > 0, ensures the normalization condition of W (x, t). Thus, it follows from (3.11) that W (x, s) = G1 s ζ x −κ exp −G2 s α ,
(3.13)
where
ξ = α(df − κ)/ θ + 1 − 1
(3.14)
and
G1 =
(θ + 1)1−(df −κ)/(θ +1) Γ ((df − κ)/(θ + 1))(GKγ )(df
, −κ)/(θ +1)
G2 =
It follows from (1.1) and (2.7) (when n = 2) that γ α= B = g(df − κ). 1 + θ , 2 Using the same methods in Ref. [15], from (3.18) we have δ
u x x W (x, t) t −γ df /2 exp −const × X X when x/X 1 and t → ∞, where u = 2α/γ (1 − α),
δ = u
γ df − 1 −κ . 2
As in Case I, we can show that −1/2 −κ · z f (z), W (x, t) = Kγ t γ df
x θ +1 . (θ + 1)GKγ
(3.15)
(3.16)
(3.17)
where f (z) =
∞ 1/2
G1 Kγ π
Cn (ξ )zn(2α/γ ) ,
n=0
where z = x/t γ /2 is a dimensionless similarity variable, {Cn (ξ )} are the same as in (3.15) but herein ξ = γ 2 (df − κ) − 1. 3.3. Case III: harmonic potentials In this paragraph we deal with the special and physically important case of an harmonic potential field, i.e., V (x) = λ2 x 2 , leading to the Hookean force F (x) = −λx directed at the original. Similar consideration may be found in Refs. [7,12]. In this case, Eq. (3.1) reduces to
λx ∂W (x, s)
s α W (x, s) = −Gx −θ (3.18) . + Kγ mηγ ∂x
F.-Y. Ren et al. / Physics Letters A 312 (2003) 187–197
A simple integration of (3.18) yields
(θ +1) κλ
W (x, s) = q(s) exp − x 2 − xs α/(θ +1) /G3 2
193
(3.19)
where q(s) =
c0 s αz0 −1 ∞ ,
1 + n=1 r2n s −2nα/(θ +1)
ensures the normalization condition of W (x, t). Herein, κ=
1 mηγ Kγ
r2n =
,
G3 = Gγ1/(θ +1) ,
c2n Γ (z2n )G2n γ Γ (z0 )
,
zn =
Gγ = (θ + 1)GKγ ,
df + n , θ + 1
cn =
c0 =
1 λκ n , − n! 2
θ + 1 z , Γ (z0 )Gγ0
n = 1, 2, . . . .
(3.20)
It follows from (1.1) and (2.7) (when n = 2) that γ 1 + θ , B = g(df ). α= (3.21) 2 Let us discuss the asymptotic behavior of W (x, t) predicted by (3.25). Noting (3.27), (3.25) can be written as follows ∞ (θ +1) γ /2 −(1−γ df /2) −nγ , W (x, s) = φ0 (x)s (3.22) · 1+ exp − xs /G3 d2n s n=1
where
κλ 2 φ0 (x) = c0 exp − x , d2 = −r2 , 2 d2n = −(d2(n−1)r2 + d2(n−2)r4 + · · · + d2 r2(n−1) + r2n ).
(3.23)
θ +1 W (x, s) ≈ φ0 (x)s −(1−γ df /2) exp − xs γ /2 /G3
(3.24)
Thus
when |s| 1. Using the same methods as in Refs. [15,17], we have δ
u x x W (x, t) ∼ t −γ df /2 φ0 (x) exp −const × X X
(3.25)
when x/X 1 and t → ∞, where δ and u are the same as in (3.8). To determine the solution of (2.4) associated to Case III, taking the Laplacian inversion transform on both sides of (3.22) we have W (x, t) = φ0 (x)
∞
d2n f2n (x, t),
(3.26)
n=0
where f2n (x, t) = L−1 F2n (x, s) , θ +1 , F2n (x, s) = s ξ2n exp − xs γ /2 /Gγ
d0 = 1,
ξn =
γ (df − n) − 1. 2
(3.27)
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Since ξ2n+1 + 1 < 0 for n 1, we can show that f2n (x, t) = 0. Since ξ0 + 1 > 0, then as in Case I we have −1/2 φ0 (x)f (z), W (x, t) = Kγ t γ df (3.28) where z = x/t γ /2 , f (z) =
1/2
G1 K γ π
∞
n=0 Cn (ξ0 )z
n(2α/γ )
and Cn (ξ0 ) is defined as in Appendix A.
3.4. Case IV: generic potentials In this paragraph we discuss the generic external potential V (x) = b log x +
∞
ak x k .
(3.29)
k=1
In this case, the solution of the fractional diffusion equation (3.1) is as follows
W (x, s) = q(s) exp −κV (x) · exp −s α x θ +1 /Gγ ,
(3.30)
where κ = 1/(mηγ Kγ ), Gγ = (θ + 1)GKγ , and ∞ αz0 −1 −αn/(θ +1) 1+ ηn s q(s) = c0 s
(3.31)
n=1
if df > bκ, ensures the normalization condition of W (x, t). Herein, c0 = (θ + 1)/Γ (z0 )Gzγ0 ,
G3 = Gγ1/(θ +1) ,
ηn = βn Γ (ξn )Gn3 ,
ξn = (df − bκ + n)/(θ + 1), n 0, n κ β0 = 1, βn = − kak βn−k , n 1. n k=1
It follows from (1.1) and (2.7) (when n = 2) that γ 1 + θ , B = g(df − bκ). α= 2 Thus, it follows from (3.30) that W (x, s) ≈ φ1 (x)s when |s| 1, where
φ1 (x) = c0 exp −κ
γ /2 /G
3) γ /2 bκ (xs )
−(1− 21 df ) exp{−(xs
∞
(θ +1) }
,
(3.32)
an x n ,
+1) . G3 = G1/(θ γ
(3.33)
n=1
Using the same methods in Refs. [15,17], from (3.32) we have δ
u x x −γ df /2 W (x, t) ∼ t φ1 (x) exp −const × X X
(3.34)
when x/X 1 and t → ∞, where δ and u are the same as in (3.16). We now turn to discuss the expression of dimensionless similarity variable of W (x, t). It follows from (3.30) that ∞ W (x, s) = c0 exp −κV (x) (3.35) dn L−1 Fn (s) , n=0
F.-Y. Ren et al. / Physics Letters A 312 (2003) 187–197
195
where d0 = 1, d1 = κa1G3 Γ (z1 )/Γ (z0 ), dn = −(dn−1 η1 + dn−2 η2 + · · · + d1 ηn−1 + ηn ) (n = 2, 3, . . . , ), γ Fn (s) = s ξn exp −G2 s α , ξn = (df − bκ − n) − 1, n = 0, 1, . . . . 2 If b = 0 and 0 < df − bκ < 1. Similar to Case III, since ξ0 + 1 > 0 but ξn + 1 < 0 for n 1, we have −1/2 −bκ z f (z), W (x, t) = φ1 (x) Kγ t γ df
(3.36) (3.37)
(3.38)
1/2 G K ∞ n(2α/γ ) . where z = x/t γ /2 , and f (z) = 1 π γ n=0 Cn (ξ0 )z If b = 0 and df − bκ > 1, then ξi + 1 > 0 (i = 0, 1), but ξn + 1 < 0 for n 2. Thus
−1/2 −bκ W (x, t) = φ1 (x) Kγ t γ df f (z) + d1 t γ /2 f1 (z) , z
(3.39)
1/2 G K ∞ n(2α/γ ) . where f1 (z) = 1 π γ n=0 Cn (ξ1 )z Furthermore, if a1 = 0 we still have the same results as in case of 0 < df − bκ < 1. On the other hand, if b = 0, i.e., V (x) is analytic with form
V (x) =
∞
an x n .
(3.40)
n=1
Similar consideration can be found in Ref. [7]. In this case, we have δ
u x x −γ df /2 W (x, t) ∼ t φ1 (x) exp −const × X X
(3.41)
when x/X 1 and t → ∞, and −1/2 W (x, t) = φ1 (x) Kγ t γ df f (z) + t γ /2 d1 f1 (z) ,
(3.42)
where δ and u are the same as in (3.8). In particular, if a1 = 0, then −1/2 W (x, t) = c0 exp −κV (x) Kγ t γ df f (z).
(3.43)
4. Conclusion and discussion In order to describe anomalous diffusion processes involving external potential fields on fractal structures, we introduce a fractional diffusion equation involving external potentials. This equation is different from the fractional Fokker–Planck equation. The solution of this fractional diffusion equation possesses following properties: The necessary and sufficient condition of its stationary solution being the Boltzmann distribution exp{−V (x)/ [kB T ]} is that the generalization of the Einstein relation, also preferred to as stokes-Einstein–Smoluchowski relation holds Kγ = kB T /mηγ , for the anomalous coefficients Kγ and ηγ . This result is consistent with that in Ref. [8]. The asymptotic shape of solution of Eq. (2.5) is stretched Gaussian not only for constant potentials but also for logarithm, harmonic, analytic and generic potentials.
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Solutions of (2.5) can be expressed in the form of −1/2 −κ −1/2 −κ Kγ t γ df z f (z) or Kγ t γ df z φ(x) f (z) + t γ /2 d1 f1 (z) with the dimensionless similarity variable z = x/t γ /2 for any potentials mentioned above, where κ and d1 are constants but may be zero, f (z) and f1 (z) are functions of z, and df is the fractal dimension of the underlying structure. It is surprising that the presence of f1 (z) is due to the presence of term a1 x in the external potentials. In each case (constant, logarithm, harmonic, analytic and generic potentials) there exists an intrinsic relationship, 2α = γ (1 + θ ). We know that α and γ are structural parameter of the underlying fractal structure and α can be determined explicitly (see Refs. [19–21]). It is interesting that as same as the fractional Fokker–Planck equation (1.6), the solution of fractional diffusion
equation (2.5) also has the asymptotic behavior log W (x, t) ∼ −cξ u where ξ ≡ x/t γ /2 1, u = 2α/γ (1 − α),
only u > u = 1/(1 − γ /2) of FFPE for 1/2 < ρ = α/γ < 1/γ which means that the asymptotic behavior decays quickly than that ones of FFPE, and u = u if and only if ρ = 1/2 which means that both of FFPE and Eq. (2.4) have same decay.
Appendix A Using the inversion theorem of Laplace transform, from (3.3) we have G0 W (x, t) = π
∞
τ ξ exp − tτ + G2 τ α cos απ sin G2 τ α sin απ − ξ π dτ
0
G0 −ξ −1 t = π
∞
y ξ exp − y + G2 (y/t)α cos απ sin G2 (y/t)α sin απ − ξ π dy,
(A.1)
0
where
x θ +1 . G2 =
(θ + 1)GKγ Using the expressions of power series of entire-functions ez , cos z and sin z and the integral representation of Γ function, a simple integration of (A.1) yields the following expression with the dimensionless similarity variable z = x/t γ /2, −1/2 W (x, t) = Kγ t γ df (A.2) f (z), where f (z) =
∞ 1/2
G1 Kγ π
Cn (ξ )zn(2α/γ ) ,
n=0
C0 = a Γ (γ df /2) > 0, Cn (ξ ) = a bn + b an−1 , a 2k+1bm an (ξ ) = C n+1 Γ α(ξ + 2 + n) , (−1)k+m (2k + 1)!m! k,m0,2k+m=n
bn (ξ ) =
k,m0,2k+m=n
(−1)k+m
a 2k bm n C Γ α(ξ + 1 + n) , (2k)!m!
F.-Y. Ren et al. / Physics Letters A 312 (2003) 187–197
a = − sin ξ π > 0, b = cos ξ π, a = sin απ, 1−γ γ (2α/γ ) df /2α . C= , G1 = 2αGKγ (GKγ )γ df /2α Γ (γ df /2α)
b = cos απ,
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