Time moment analysis of first passage time, time lag and residence time problems via Taylor expansion of transmission matrix

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Time moment analysis of first passage time, time lag and residence time problems via

Taylor expansion of transmission matrix

Jenn-Shing Chen and Wen-Yih Chang

Citation: The Journal of Chemical Physics 112, 4723 (2000); doi: 10.1063/1.481028

View online: http://dx.doi.org/10.1063/1.481028

View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/112/10?ver=pdfcov

Published by the AIP Publishing

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Time moment analysis of first passage time, time lag and residence time

problems via Taylor expansion of transmission matrix

Jenn-Shing Chena)and Wen-Yih Chang

Department of Applied Chemistry, National Chiao-Tung University, Hsin-Chu, 30500, Taiwan

共Received 28 October 1999; accepted 15 December 1999兲

Taylor expansion共with respect to the Laplace variable, s兲 of the transmission matrix, T(s), has been developed for the diffusion transport with position-dependent diffusivity, D(x) and partition coefficient, K(x). First, we find the relation between the expansion coefficients of the matrix elements and the moments of the first passage times by connecting them to Jˆ(s), the Laplace transform of the escaping flux, J(t). The moments can be formulated by repeated integrals of K(x) and关D(x)K(x)兴⫺1from solving the backward diffusion equation subject to appropriate initial and boundary conditions. In this way, Taylor expansion coefficients of T11(s), T21(s), and T22(s) are

expressed in terms of the repeated integrals. Further application of the identity det关T(s)兴⫽1 leads to the Taylor expansion T12(s). With the knowledge of the Taylor expansion of T(s), the formulation

of the time moments for diffusion problems with position dependent D(x) and K(x) subject to various initial and boundary conditions is then just a simple, algebraic manipulation. Application of this new method is given to the membrane permeation transport and mean residence time problem. © 2000 American Institute of Physics.关S0021-9606共00兲50810-1兴

INTRODUCTION

Diffusion is a ubiquitous process in the physical world. It is of great theoretical importance with a multiplicity of applications in such diverse fields as chemical reaction,1,2 electrochemistry,3 colloidal science,4 solid state physics,5 semiconductor-device fabrication and operation,6 physical ceramics,7 biophysics,8 drug delivery,9 and environmental science.10 One way to characterize a diffusion system in which a particle initially located at x⫽x0 within a finite

do-main is by means of the probability density of the time re-quired for the particle escaping from this domain for the first time, i.e., the distribution of the first-passage time.11,12 Com-plete information of the probability distribution can be ob-tained only for some particular cases. Thus, one is usually forced to resort to the time moments. Of the most important among them is the first moment, i.e., the mean first passage time. The latter is often related to the reciprocal of a 共first-order兲 rate constant if a chemical reaction is modeled by diffusion over a potential.12 In order to have more informa-tion about the distribuinforma-tion, higher moments are required. For example, without the second moment the dispersion of the distribution cannot be estimated.13

For a diffusion with initial condition of Dirac delta-function type, the first and higher moments are obtainable from solving the backward diffusion equation with appropri-ate boundary conditions.12 Another approach proposed by Deutch14is the use of repeated integration over the original diffusion equation. He obtained the mean first passage time for a heterogeneous domain with initial distributions of ei-ther Dirac delta-function type or of saturated equilibrium. However, the results for the second moment is not given.

Now turn our attention to membrane diffusion transport.

Of them the absorptive permeation is the commonest prac-tice. The experiment is set up under a zero initial activity within the whole membrane, and a constant and a zero ac-tivity at the upstream and downstream faces, respectively. Permeability, P, and time lag, tL, are crucial parameters to

estimate the total release Q(t) as a function of time through the asymptotic linear equation Q(t)⫽P(t⫺tL).

15,16

tL can

be expressed by tL⫽兰0⬁t关(d/dt)Jd(t)/Jd,ss兴dt,

17

with Jd(t)

the time-dependent flux at the downstream face and Jd,ssthe

steady-state flux. Mathematically tL is the first moment of

the (d/dt)Jd(t)/Jd,ss distribution. Various mathematical

techniques have been employed to formulate the first mo-ment, i.e., the time lag, for diffusion with position-dependent partition coefficient and diffusivity. However, up to date, we have not found the formulation for the higher moments. Re-cently, the matrix theoretical analysis in the Laplace domain on the diffusion transport problem has been put forth.17–22 This analysis allows us to formulate the time lag and mean first passage time in terms of the derivative 共with respect to the Laplace variable, s兲 of the elements of the transmission matrix.17We will extend the analysis to the treatment on the higher moments for first passage time and membrane trans-port problems as well. We shall see that Taylor expansion of the transmission matrix, T(s), plays an important role in the analysis. We found that the coefficients of expansion appear to be in the forms of repeated integrals of K(x) and 关D(x)K(x)兴⫺1. With this as an instrument, the moment analysis for the afore-mentioned diffusion problems can be reduced to a simple, algebraic manipulation.

TRANSMISSION MATRIX FOR DIFFUSION TRANSPORT

Traditionally the membrane permeation transport, due to whose underlying process is diffusion, described by the Fick’s diffusion equation, or Smoluchowski equation if the

a兲_{Author to whom correspondence should be addressed.}

4723

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membrane is inhomogeneous and/or is subject to external force. Alternative to this description is through the vehicle of transmission matrix.17–22In this approach the membrane per-meation system is treated as an electrical network system with two input ports and two output ports. The penetrant activity and flux at the upstream face are then assigned to the input ports as an ordered pair, and those at the downstream face to output ports as another ordered pair. The transmission matrix which links two ordered pairs appears to be charac-teristic to the properties of the membrane, including its thick-ness, the diffusivity and partition coefficient of the penetrant in the membrane. The magnitude of the last two entities may be position dependent.

冋

册

, 共1兲

where aˆu(s) and Jˆu(s) are the Laplace transform of the

pen-etrant activity, au(t), and flux Ju(t), respectively, at the face x⫽0, or at upstream face. Their counterparts at the face x

⫽h, or at downstream face, are aˆd(s) and Jˆd(s),

respec-tively. For convenience, the face at x⫽0 and the upstream face are used interchangeably, so are the face at x⫽h and downstream face.

If the initial condition is of Dirac delta-function type located at x0, such that 0⬍x0⬍h, the transport equation is

given by17

冋

aˆd共s兲 Jˆd共s兲

册

⫽T共s兲

冋

册

冋

cosh共qh兲 ⫺sinh共qh兲

DqK

⫺DqK sinh共qh兲 cosh共qh兲

册

, 共4兲

where q⫽

冑

s/D and h is the thickness of the domain. For

diffusion through a heterogeneous domain T(s) takes on the form17

T共s兲⫽T共n兲共s兲T共n⫺1兲共s兲¯T共1兲共s兲. 共5兲

This is a consequence of the fact that the heterogeneous do-main is look upon as an assembly of many共say n兲 thin sub-domains, with the first connected to the upstream face at x ⫽0, followed by the second, etc., up to the last (nth) which adjoins the downstream face at x⫽h. Here T(i)(s), the trans-mission matrix for the ith subdomain, takes the same form of Eq.共4兲, except that D, K, q, h, are replaced, respectively, by

Di,Ki,qi,hi, the corresponding entities for the ith

subdo-main.

TAYLOR EXPANSION OF TRANSMISSION MATRIX

The coefficients of the sn terms (n⫽0,1,2,...) of T11(s)

and T22(s) will be derived by the use of the backward

dif-fusion equation11,12,23 1 K d dxDK d dx␮n⫽⫺n␮n⫺1, n⫽1,2,3,..., with ␮0⫽1, 共6兲

where␮n⫺1 and␮n are the (n⫺1)th and nth moments,

spectively. We consider a particle initially located at a re-flecting face x⫽0, and the other boundary at x⫽h, from which the particle escapes, is absorbing. The moments are solved in Eqs. 共A1兲–共A3兲 in the Appendix. The escaping probability as a function of time is represented by J_d(t),17 whose Laplace transform is related by

Jˆd共s兲⫽

1

T11共s兲

, 共7兲

which is obtained by a substitution of aˆ_d(s)⫽0, Jˆ_u(s)⫽0, and T*(s)⫽T(s) into Eq. 共2兲. To proceed, we expand T₁₁(s) and Jˆd(s) in terms of Taylor series about s⫽0

T11共s兲⫽T11共0兲⫹ s 1_slim_→0 dT11共s兲 ds ⫹ s2 2!_slim_→0 d2T11共s兲 ds2 ⫹¯ ⫽␣0⫹␣1s⫹␣2s2⫹¯ , 共8兲 Jˆd共s兲⫽Jˆd共0兲⫹ s 1!_slim_→0 dJˆ_d共s兲 ds ⫹ s2 2!_slim_→0 d2_Jˆ d共s兲 ds2 ⫹¯ ⫽␮0 d_⫺ s 1!␮1 d_⫹s 2 2!␮2 d_{¯ .} _共9兲

The definition of ␣_n, n⫽0,1,2,3,..., is self-explanatory as seen from Eq.共8兲. The use of the superscript d is to specify that the particle exits from the downstream face. The equal-ity in Eq.共9兲 is due to the fact that

␮n d_⫽

冕

0 ⬁ tnJ_d共t兲dt⫽共⫺1兲nlim s→0 dnJˆd共s兲 dsn , 共10兲

since Jd(t) is look upon as the escaping probability density.

Inserting Eqs. 共8兲 and 共9兲 into Eq. 共7兲, we obtain the Taylor expansion of T11(s) in terms of repeated integrals over K(x)

and关D(x)K(x)兴⫺1 as

T11共s兲⫽␣0⫹␣1s⫹␣2s2⫹¯ , 共11兲

with

4724 J. Chem. Phys., Vol. 112, No. 10, 8 March 2000 J. S. Chen and W. Y. Chang

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been used.

If a particle is initially at a reflecting face at x⫽h, and the exit x⫽0 is absorbing, also from Eq. 共2兲 we have

Jˆu共s兲⫽⫺

1

T₂₂共s兲, 共15兲

which is given by substituting T*(s)⫽I and Jˆd(s)⫽0, aˆu(s)⫽0. Exactly following the procedure for obtaining the

Taylor expansion of T11(s), we arrive at the Taylor

With respect to the Taylor series of T21(s), we consider

an initial activity of saturated equilibrium at a constant level

a0 throughout the whole diffusion domain. The face at x

⫽0 is reflecting, while the exit face at x⫽h is absorbing. The Laplace transform of J_de(t) becomes

Jˆ_de共s兲⫽⫺T21共s兲 T11共s兲

a0

s , 共20兲

which is given from Eq.共3兲 by substitution of aˆ_d(s)⫽0 and

Jˆu(s)⫽0.22The superscript e in Eq.共20兲 is to specify that the

initial condition is of saturated equilibrium distribution. The Taylor expansion of Jˆ_de(s) for saturated equilibrium distribu-tion can be expressed by

Jˆd e_共s兲⫽

冉

_␮ 0 e_⫺_␮ 1 e s⫹␮2 e 2!s 2_⫺¯

冊

冕

0 h a0K共x0兲dx0 共21兲

whose expansion coefficients ␮_ie(i⫽0,1,2) are taken from Eqs.共A7兲–共A9兲. Putting Eqs. 共11兲 and 共21兲 into Eq. 共20兲, we obtain the Taylor expansion of T21(s) as

all␥i positive.

As far as the Taylor expansion of T12(s) is concerned,

the identity19

T11共s兲T22共s兲⫺T12共s兲T21共s兲⫽1, 共26兲

is employed. Substitution of Eqs. 共11兲, 共16兲, and 共22兲, re-spectively for T₁₁(s), T₂₂(s), and T₂₁(s) with their coeffi-cients expressed by repeated integrals into Eq. 共26兲, we ob-tain after some algebra the Taylor expansion of T₁₂(s) as

dx2dx. 共30兲 Again the negative sign before ␤i(i⫽0,1,2,...) is to render

␤i positive.

Extended calculation to higher terms enables us to gen-eralize the Taylor expansion of the transmission matrix,

y h K˜␦n_⫺1共z兲dz dy, ␦n⫽˜␦n共0兲, n⭓1. 共35兲

It is interesting to note that if the integrands K and 1/DK are exchanged in ␣n,␦n is obtained, and vice versa; if they are

exchanged in␤n then we obtain␥n⫹1, and vice versa.

APPLICATION

Equation共31兲 with coefficients expressed by Eqs. 共32兲– 共35兲 is the principal result of our derivation. It will be ap-plied to the time moment analysis of the membrane perme-ation transport and the residence time problem. But before doing this, the classical problem of first passage time for a particle initially located at a point x0within a finite diffusion

domain will be worked out. We consider quite a general case that both the boundaries at x⫽0 and at x⫽h are partially absorbing with radiative constant ␬_u and ␬_d, respectively. Substituting Jˆu(s)⫽␬uaˆu(s) and Jˆd(s)⫽␬daˆd(s) into Eq.

共2兲, one obtains Jˆu共s兲⫽␬u T₂₂*_共s兲⫺␬_dT₁₂*_共s兲 共␬dT11共s兲⫺T21共s兲兲⫹␬u共␬dT12共s兲⫺T22共s兲兲 , 共36兲 Jˆ_d共s兲⫽␬_dT11共s兲T22*共s兲⫺T21共s兲T*12共s兲⫹␬u共T12共s兲T22*共s兲⫺T22共s兲T12*共s兲兲 共␬dT11共s兲⫺T21共s兲兲⫹␬u共␬dT12共s兲⫺T22共s兲兲 . 共37兲

T_{i j}*(s) in Eqs. 共36兲 and 共37兲 is the matrix element of

T*(s), which is the transmission matrix for the subdomain from x⫽x0 to x⫽h. The coefficients of Ti j*(s) are identical

to those of Ti j(s), except that 0 for the lower limit of the last

integration in Eqs.共32兲–共35兲 is replaced by x₀, for example,

␣2*⫽兰x₀ h_K_兰 x₂ h _(1/DK)_兰 x 2 ⬘ h K兰_x 1 h_(1/DK)dx 1

⬘

dx₁dx₂

⬘

dx₂ and ␥2*⫽兰x₀ h K兰_x 2 h (1/DK)兰_x 2 ⬘ h K dx1dx2

⬘

dx2.

To proceed, the Taylor expansion of Jˆd(s)⫺Jˆu(s) is

cal-culated by Eqs. 共36兲 and 共37兲 to be

Jˆd共s兲⫺Jˆu共s兲⫽ A₀ B0⫹ A₁⫺B₁ B0 s ⫹A2B0⫺B0B2⫺A1B1⫹B1 2 B₀2 s 2_{⫹¯ ,} 共38兲 where A0⫽B0⫽␬d⫺␬u⫺␬u␬d␤0, 共39兲 A1⫽⫺␬u␦1*⫹␬d共␣1⫹␦1*⫺␥1␤0*兲 ⫺␬u␬d共␤1⫹␤0␦1*⫺␦1␤0*兲, 共40兲 A2⫽⫺␬u␦2*⫹␬d共␣2⫹␦2*⫹␣1␦1*⫺␥1␤1*⫺␥2␤0*兲 ⫺␬u␬d共␤0␦2*⫹␤1␦1*⫺␦1␤1*⫹␤2⫺␦2␤0*兲, 共41兲 B1⫽␬d␣1⫺␬u␦1⫺␬u␬d␤1⫹␥1, 共42兲 B₂⫽␬_d␣₂⫺␬_u␦₂⫺␬_u␬_d␤₂⫹␥₂. 共43兲 Since the moments correspond to the expansion coefficients in Eq.共38兲, it readily follows that

␮0⫽1, 共44兲 ␮1⫽ B1⫺A1 B₀ , 共45兲 ␮2⫽2 A2B0⫺B0B2⫺A1B1⫹B1 2 B₀2 . 共46兲

In the case that the face x⫽0 is radiative, and the face

x⫽h is reflecting, i.e.,␬d⫽0, we have from Eq. 共45兲 the first

moment ␮1⫽ lim ␬d→0 B1⫺A1 B₀ ⫽␦1⫺␦1*⫺ ␥1 ␬u ⫽

冕

0 x₀ ₁ DK

冕

x h K d y dx⫺兰0 h K dx ␬u , 共47兲

and from Eq.共46兲 the second moment

␮2⫽ lim ␬d→0 2A2B0⫺B0B2⫺A1B1⫹B1 2 B₀2 ⫽2 ␬u共⫺␬u␦2⫹␥2⫹␬u␦2*兲⫹共␥1⫺␬u␦1兲共␥1⫹␬u␦1*⫺␬u␦1兲 ␬u 2 , 共48兲

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whose explicit expression in terms of integrals is a little bit complicated and will not be given here. Equation 共47兲 after some algebraic manipulation is in agreement with the result 关Eq. 共8兲 in Ref. 14兴 derived by Deutch by means of the method of repeated integration.

Although our method provides another instrument in dealing with the classical problem of first passage time. We would not like to claim that it is superior over the established methods such as solving backward diffusion equation or re-peated integration on the original diffusion equation. How-ever, up to date, we found the corresponding backward dif-fusion equation for the time moments of membrane permeation and for the residence time problem is of not avail, and the method of repeated integration appears to be applicable with the limitation to first moment 共time lag兲 analysis. In what follows we will demonstrate that the time moment analysis of these problems can be solved by our new method.

For a absorptive permeation experiment, the system is stipulated with a zero initial activity throughout the whole membrane and the boundary conditions at the downstream face aˆd(s)⫽0, and at the upstream face aˆu(s)⫽␳0/Kus

where Ku is the partition coefficient, ␳0 is a constant

pen-etrant concentration at the upstream face. Solution for Jˆd(s)

from Eq. 共1兲 is found to be

Jˆd共s兲⫽⫺ 1 T12共s兲 a ˆu共s兲⫽⫺ 1 T12共s兲 ␳0 Kus ⫽

再

_␤1 0⫺ ␤1 ␤0 2s⫹

冉

␤1 2 ␤0 3⫺ ␤2 ␤0 2

冊

s2⫹¯

冎

⫻

冉

␳0 Kus

冊

. 共49兲

The first moment共time lag兲 then is given by

tL⫽

冕

0 ⬁td/dt J_d共t兲 Jd,ss dt ⫽ lim s→0 ⫺共d/ds兲sJˆd共s兲 Jd,ss ⫽␤1 ␤0 ⫽兰0 h_{共1/DK兲兰} x h_K_兰 x₁ h _共1/DK兲dx 1

⬘

dx₁dx 兰0 h_共1/DK兲dx ⫽兰0 h_K_兰 0 x_{共dy/DK兲兰} x h_{共dy/DK兲dx} 兰0 h_共1/DK兲dx . 共50兲

The last equality is due to the Dirichlet formula.24Equation 共50兲 is also derived by repeated integration.25,26

For the second moment, t_L(2), we have

t_L共2兲⫽

冕

0 ⬁ t2d dtJd共t兲dt/Jd,ss ⫽ lim s→0 共d2_/ds2_兲sJˆ d共s兲 sJˆ_d共s兲 ⫽2共共␤1 2_/_␤ 0 3_兲⫺共_␤ 2/␤0 2_兲兲共1/␤0兲 ⫽2

冉

␤1 2 ␤0 2⫺ ␤2 ␤0

冊

⫽2

冉

t_L2⫺␤2 ␤0

冊

, 共51兲

where the explicit expressions for t_L,␤₀, and␤₂and shown in Eqs.共50兲, 共28兲, and 共30兲, respectively.

Another important membrane transport is desorptive per-meation, which is stipulated with the same boundary condi-tions as absorptive permeation except for a constant initial activity a0 throughout the whole membrane. Solving for

Jˆd(s) using Eq. 共3兲, we obtain

Jˆd共s兲⫽⫺ T₂₂共s兲 T12共s兲 a₀ s ⫽

冋

_␤1 0 ⫹␦1␤0⫺␤1 ␤0 2 s ⫹␤0 2_␦ 2⫺␤0␤2⫺␤1␦1␤0⫹␤1 2 ␤0 3 s2⫹¯

册

a0 s . 共52兲

Again we have the first moment, i.e., the time lead, t_⫹

t_⫹⫽ lim s→0 ⫺共d/ds兲sJˆd共s兲 Jd,ss ⫽␤1⫺␦1␤0 ␤0 . 共53兲

Substituting the Taylor expansion coefficients in the form of repeated integers followed by using Dirichlet formula, we have t_⫹⫽⫺兰0 h K兰0 x_{共dy/DK兲兰} 0 x_{共dy/DK兲dx} 兰0 h_共1/DK兲dx . 共54兲

The second moment (t_⫹(2)) expression in terms of repeated integrals appears to be a little bit cumbersome, and only that in terms of expansion coefficients is given

t_⫹共2兲⫽2

冋

␦2⫺ ␤2 ␤0⫹ ␤1 ␤0 t_⫹

册

⫽2

冋

␦2⫺ ␤2 ␤0⫹tL t_⫹

册

, 共55兲 where Eq.共50兲 for tL has been used.

As the last example of application, we consider the mean residence time at position x for a particle initially located at

x0, 0⬍x, x0⬍h. Both end boundaries are absorbing. The

mean residence time is the average of the total time that the particle has resided at position x if the observation time is taken up to a time t.27–29Mathematically it can be defined in terms of the integration of Green’s function␳(x,t

⬘

兩x0) with

respect to t

⬘

from 0 to t. The Green’s function is a solution of the diffusion equation under the same boundary conditions

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with the initial condition␦(x⫺x0). The Green’s functionlike

penetrant activity has been solved in terms of the transmis-sion matrix element elsewhere17to be

aˆ共x,s兲⫽

冦

⫺T12 共3兲_共s兲T 12 共1兲_共s兲 T12共s兲 x⬍x0 ⫺T共32兲12 共s兲T12共21兲共s兲 T12共s兲 ⫹T共2兲12共s兲 x⬎x0 , 共56兲

where T(1)(s), T(2)(s), T(3)(s), T(21)(s), and T(32)(s) are the transmission matrix for the subregions from 0 to x, x to

x0, x0 to h, 0 to x0, and x to h, respectively, if x0⬎x. If

x0⬍x, then the coordinates x0 and x are interchanged. Thus

the mean residence time at x for infinite observation time, corresponding to t→⬁, is found to be ␶共x兩x0兲⫽

冕

0 ⬁ ␳共x,t

⬘

兩x0兲dt

⬘

⫽ lim s→0 ␳ ˆ共x,s兩x0兲 ⫽ lim s→0 a ˆ共x,s兲K共x兲, 共57兲

where the last equality is due to the definition17 of activity

a(x,t)⫽␳(x,t)/K(x). Substitution Eq.共56兲 into Eq. 共57兲, we find ␶共x兩x0兲 ⫽

冦

lim s_→0 ⫺T12共3兲共s兲T12共1兲共s兲 T12共s兲 K共x兲 x⬍x0 lim s→0

冋

⫺T12共32兲共s兲T12共21兲共s兲 T12共s兲 ⫹T12共2兲共s兲

册

K共x兲 x⬎x0 . 共58兲 In terms of repeated integrals it becomes

␶共x兩x0兲⫽

冦

K共x兲

冕

0 x ₁ D共x

⬘

兲K共x

⬘

兲dx

⬘

冕

x₀ h ₁ D共x兲K共x兲dx

再

DISCUSSION AND CONCLUSION

Frisch et al. also studied the time moment problems for membrane absorptive permeation, desorptive permeation and sorption transports.30They theoretically investigated the time moments of the amount of penetrant in an absorbing 共or desorbing兲 membrane composed of a linear laminated me-dium. They have shown that these time moments can be obtained via a precursor B_n(x) by ␮_n⫽兰₀hK(x)B_n(x)dx, where B_n(x) itself can be obtained from an integral with

Bn_⫺1(x) as a part of integrand. Thus, with B0(x) known,

other Bn(x) can be recursively calculated. Similar scheme is

also applied to the calculation of the time moments for the difference between the instantaneous and asymptotic flows up to a given time through a membrane. As a comparison, our new method appears to be simple and straightforward.

Recently Zwanzig23has elucidated the effect of potential roughness on the effective diffusivity of a particle under the influence of the potential U0(x)⫹U1(x), where U0(x) is the

spatially varying part and U1(x) is the fluctuating part. The

latter is responsible for the potential roughness. It is found that the effective diffusivity D* is related to the original diffusivity D by

D*⫽ D

具

exp关共U1共x兲/kT兲兴

典具

exp关⫺共U1共x兲/kT兲兴

典

, 共60兲 where k is the Boltzmann constant and具典 denotes the spatial average. As a first example, if the roughness is simply

U1(x)⫽⑀cos(qx), then23

D*⫽ D

关I0共⑀/kT兲兴2

, 共61兲

where I₀ is the modified Bessel function of the zeroth kind and⑀/kT is its argument. If the amplitude of the roughness is a Gaussian distribution, with a probability proportional to exp(⫺U2_/2_⑀2_{) in which}_⑀2_⫽

_具

_U

1

2

_典

_{, then}23

D*⫽D exp关⫺共⑀/kT兲2兴. 共62兲

In the above examples, it is interesting to note that the parameters␣n,␤n, ␥n, and␦nare modified by being

mul-tiplied by a factor关I0(⑀/kT)兴2n for the first example and by

兵exp关⫺(⑀/kT)2兴其nfor the second. Since both I0(⑀/kT)⬎1 and

exp(⫺⑀/kT)⬍1 for⑀⬎0, we assert that the effect of potential roughness in this two cases is to increase the magnitude of

D*, and hence the time moments of orders⭓1.

In conclusion, we have given an alternative approach to the time moment analysis for diffusion problems. First we calculate in the Laplace domain the escaping flux with ap-propriate boundary conditions using one of the transport equations关Eqs. 共1兲–共3兲兴. Expand the flux into a power series of s, we found the coefficients of expansion, which corre-spond to the time moments, is a combination of the coeffi-cients of the Taylor series of the matrix elements. With the latter expressed by repeated integrals as shown in Eqs.共32兲– 共35兲, the time moments can be also represented by the re-peated integrals as well. As compared to the traditional itera-tive Green’s function,11 solving the backward diffusion equations11,12,23and repeated integration,14our method gains

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some advantage in the sense that it can be accomplished in a simple, straightforward way, involving only algebraic opera-tion.

APPENDIX

The boundary conditions for a reflecting boundary at x ⫽0 and an absorbing boundary at x⫽h are ␮n

d_(h)_⫽0,

⫽0. The superscript u is to specify that the flux exits from the upstream face. With the initial position at x⫽h the time moments for escaping probability are

ACKNOWLEDGMENTS

We are grateful to the National Science Council of Tai-wan for the financial support of this work under Grant Nos. 88-2113-M-009-005 and 89-2113-M-009-008. We also thank Professor C. Y. Mou, Department of Chemistry, Na-tional Taiwan University for drawing our attention to the Zwanzig’s paper 共Ref. 23兲.

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