8762 J. Phys. Chem. B, Vol. 104, No. 36, 2000
Additions and Corrections
ADDITIONS AND CORRECTIONS 2000, Volume 104B G. Margolin and B. Berkowitz*: Application of Continuous Time Random Walks to Transport in Porous Media Following are two corrections to the paper. The analysis, results, and conclusions remain unchanged. 1. Section 4.3 should read as follows. 4.3. The FPTD for β > 2. We now demonstrate that when β > 2, Gaussian behavior arises, in accordance with the central limit theorem. Indeed, in this case the leading terms of ψ*(u) for small u (i.e., for long times) are ψ*(u) ≈ 1 - 〈t〉u + 〈t2〉u2/2 ≈ exp{-〈t〉u + σt2u2/2}, where σt2 ≡ 〈t2〉 - 〈t〉2. Comparing this expansion to (5), we see that to obtain the FPTD we can set β ) 2 in the formulas (12)-(14). Recalling that g ) lbβ and l ) L /〈l〉, and that in this case bβ ) σt2/(2〈t〉2), (12) will transform into
{
}
(l - τ)2 FPTD ) 1/2 exp 4g g 〈t〉2xπ 1
)
x
V〈t〉
{
exp 2
2πLσt
(L - Vt)2 2LVσt2/〈t〉
}
(19)
FPTD )
2π〈t〉xlbβ
2π〈t〉xlbβ )
x
h2m sin ∑ Γ(2m + 1) m)0
∞
1
)
Γ(m + 1/2)
∞
V〈t〉
2πLσt
2
(-h2)m ∑ m)0
{
exp -
bβ>2 ≡
2
1 2πiν〈t〉
}
2LVσt /〈t〉
while eq 14 is identically zero (i.e., no long tail).
1 2πiν〈t〉
)
2m(2m)!
const 〈l〉
(21)
(22)
1
c+i∞ dr er+γr ∫c-i∞
∫
c+i∞ r e c-i∞
∞
dr
β
∞
(γrβ)n
n)0
n!
∑
γn
∑ ν〈t〉 n)0 Γ(n + 1)Γ(-βn)
)(20)
)
where now 〈l〉 is any distance where the FPTD is measured and σt2 is the variance of this FPTD distribution. At longer distances the relative time spread σt2/〈t〉2 decreases as 1/〈l〉. We mention that the ratio σt2/〈t〉 is constant, independent of our choice of 〈l〉, as must also follow from (19) and (20). This is analogous to the behavior of the spatial moments with time. 2. There is a misprint in eq 10. The correct version is
)
(2m - 1)!!xπ
(L - Vt)2
2〈t〉
2
σt2∼〈l〉
π(2m + 1) 2
σt2
or
FPTD )
and (13) will give the same result:
1
The solutions (19) and (20) represent Gaussian behavior. We note here that the FPTD curves cannot be compared directly to concentration distributions over space: integration of a concentration distribution over space gives unity, while integration over time of an FPTD gives unity. Finally, from the scaling relation (16), and with β ) 2, it follows that
1 ∞ n Γ(βn + 1) γ sin πβn πν〈t〉 n)0 Γ(n + 1)
∑
10.1021/jp002470l Published on Web 08/22/2000
10.1021/jp002470l CCC: $19.00 © 2000 American Chemical Society Published on Web 08/22/2000
