Confined Anomalous Dynamics: A Fractional Diffusion Approach
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where K., with dimension [K,,] = cm 2 sec-,,, is a generalised diffusion coefficient. For -Y< 1 one speaks of subdiffusion, and -y > 1 denotes superdiffusion. Anomalous diffusion is often closely related to the validity of the generalised central limit theorem, and thus to Levy statistics [1,2]. In what follows, we restrict our considerations to the subdiffusive regime, and to one dimension, with obvious generalisations to the multidimensional case. For the description of subdiffusion, the fractional diffusion equation (FDE)
W(xt) =t)
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has been recently established [3]. The fractional Riemann-Liouville operator oD'-' introduces a slowly decaying memory, and it is defined through the convolution [4] oDt-W(x,t) =
1 fot 1 fdr n(y) Jo
W(x,'r) Wx r (t - r)1_-Y
3
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The extension to fractional differentiation is given by oD'-"W(x,t) = -LoDV"W, with 0 < - < 1. A fundamental property that distinguishes fractional from integer derivatives, is
Dat# =
t
F(1 +
)
t_•
o°Dtl =
r(1 - a +±3)
1
t-"'
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r()-a)
i.e. the fractional differentiation [a non-integer] of a constant is an inverse power-law [4]. Fractional diffusion equations provide a powerful tool for the description of anomalous diffusion, and can be derived from the probabilistic continuous time random walk scheme [5,6]. Recently, fractional Fokker-Planck equations have been derived and applied to anomalous diffusion in external fields [7-10]. In a broad range of systems which display anomalous transport, boundary value problems might become relevant. These usually enter as a discrete set of eigenvalues, or as an additional reaction term. It is our aim here to investigate general solution techniques for fractional equations of the diffusion type. 281
Mat. Res. Soc. Symp. Proc. Vol. 543 c 1999 Materials Research Society
II. FRACTIONAL DIFFUSION AND BOUNDARY VALUE PROBLEMS A. The method of images It has been shown that the solution W(x,t) of the FDE (2) for natural boundary conditions limlxl_, 0 W(x, t) = 0 and an initial delta distribution W(O, x) = S(x), i.e. the propagator, suffices to determine the boundary value problem of two absorbing or two reflecting boundaries which we assume to lie at x = ±a [11]. Accordingly, the free solution is successively folded at the boundaries [i.e. repeatedly reflected at ±a], to result in the boundary value solution e(x,t)=--
[W(x+4rna,t)T:FW(4ma-x±2a,t)],
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m=-0c
where the minus sign stands for absorbing, the plus sign for reflecting boundaries at x = ±a [11,12]. We note that the solution for the mixed condition of one absorbing and one reflecting boundary is obtained via a final folding in the origin of the solution for two absorbing boundaries, speaking in terms of the method of images [11]. Employing the relation
_
e -ikm
=
e-ik/2,•00(--1)t
(m + ±
), we rewrite Eq.
(5), after a Laplace
transform [indicated through the u-dependence], as g(x,u)
= (4a)f-' 1
eiirmx/(2a)
[W (k =
-,
uF(-1)
m
W (k
=
a,
.
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1. Absorbing boundaries
For absorbing boundaries in ±a, defined via the D
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