December 2020
EPL, 132 (2020) 50007
doi: 10.1209/0295-5075/132/50007

www.epljournal.org

Lateral diffusion on a frozen random surface
Takao Ohta1(a) and Shigeyuki Komura2(b)1Center for Integrative Medicine and Physics, Institute for Advanced Study, Kyoto University
Kyoto 606-8302, Japan2
Department of Chemistry, Graduate School of Science, Tokyo Metropolitan University
Tokyo 192-0397, Japan
received 30 September 2020; accepted in final form 22 December 2020
published online 24 February 2021
05.40.Jc – Brownian motion
87.16.dp – Transport, including channels, pores, and lateral diffusion
PACS 02.50.Ey – Stochastic processes
PACS

PACS

Abstract – The lateral diffusion coefficient of a Brownian particle on a two-dimensional random
surface is studied in the quenched limit for which the surface configuration is time independent. We
start with the stochastic equation of motion for a Brownian particle on a fluctuating surface, which
has been derived by Naji and Brown. The mean square displacement of the particle projected ona base plane is calculated exactly under the condition that the surface with a constant shape has
no spatial correlation. We prove that the obtained lateral diffusion coefficient is in between the
known upper and lower bounds.
c 2021 EPLA
Copyright !

Introduction. – Brownian motion on fluctuating surfaces has been of interest for over the past four decades.
The major reason is to elucidate the transport of biomolecules on cell membranes, which plays a crucial rolefor biological functions and signal processing in living
cells [1]. Advances in experimental techniques such as fluorescence photobleaching recovery [2,3] and single-particle
tracking [4,5] motivated theoretical studies of diffusion ona restricted geometry. Safman and Delbrück introduced amodel of Brownian motion in fluid membranes to calculate
a translational mobility of a Brownian particle [6]. Thisapproach has been developed, together with experimental
studies, to describe the diffusion in flat membranes [7].
Lateral diffusion on curved geometries has been investigated theoretically both on static surfaces [8,9] anddynamically fluctuating membranes [9–13]. The diffusion
coefficient due to the geometrical effect of a curved surface
and that arising from the interaction between the Brownian particle and membrane has been obtained. Most of
the dynamical studies are concerned with the annealedcase, in which the motion of the Brownian particle is muchslower than the shape relaxation of the membrane. Oneof the present authors [14] has formulated diffusion on afluctuating surface starting with the bulk Brownian motion with the constraint that the particle is always on the(a) E-mail: takaoohta1949@gmail.com(b) E-mail: komura@tmu.ac.jp

surface. It has been shown that the diffusion coefficient in
the annealed case has a new contribution arising from thetime correlation of the lateral components of the surface
velocity as well as the ordinary geometrical part [14].
The quenched case where the shape relaxation is slow
compared with the motion of a Brownian particle has been
investigated less intensively. In one dimension, it has beenshown that diffusion on a static surface is equivalent toBrownian motion in potential fields [9,15]. The formulafor the diffusion constant has been derived for both periodic [16–18] and random potentials [9,15]. However, tothe authors’ knowledge, no exact theory for the diffusion
coefficient has been available so far for two-dimensionalsurfaces. We address this problem in the present study.
The quenched situation is expected in soft matter inthermal equilibrium, where interconnected structures withcomplex curved surfaces are often observed. Diffusion
on such structures is highly non-trivial and the surface
dynamics is not always relevant. Typical examples areorganelles in living cells [19] and the sponge phase in microemulsions [20]. Actually simulations of diffusion on
static curved biological surfaces have been conducted [19].
One may also expect the quenched situation in nonequilibrium systems. For example, the migrating velocityof an active Brownian particle which undergoes selfpropulsion consuming energy produced inside or on thesurface of the particle [21,22] can be larger than the velocity caused only by thermal fluctuations. Quite recently,

50007-p1

Takao Ohta and Shigeyuki Komura
such studies in confined geometries have been carried out
both experimentally [23] and theoretically [24]. When amembrane is subjected to non-thermal noise out of equilibrium [25], it might also increase the migration velocity.
The application of the present theory is not limited tothe systems mentioned above. It is possible, for example, that the result is useful for understanding chemical
kinetics of adatoms on rough solid surfaces [26].
Langevin equation. – We consider a Brownian particle migrating on a surface in a simple model system as
schematically shown in fig. 1. The thickness of the surface is regarded as infinitesimal by assuming that theparticle radius is much larger. Let us suppose that the!
particle is located at R(t)= (Rx (t), Ry (t), Rz (t)) with⊥!
!
Rz = h(R (t), t) and R⊥ = (Rx , Ry ) where h(!r , t) is theheight of the surface at !r on the base plane. Naji and
Brown have formulated a theory for Brownian motion ona fluctuating surface to derive the stochastic equation forthe particle [11]. The random force acting on the particle is defined on the tangent plane at each point on thesurface. They solved numerically the Langevin equation
coupled with the time evolution equation for the heightvariable to investigate the crossover from the annealed toquenched cases.
The stochastic equation of motion for a Brownian particle in the Naji-Brown theory [11] takes the form
! ⊥ = Z!ν ,∂t R

Fig. 1: The height variable h(x, y) for a random surface indicated by the red mesh. The position of the Brownian par! = (Rx , Ry , Rz ) withticle (drawn in light blue) is given by R
Rz = h(Rx , Ry ).

Here we describe an alternative derivation of the tensor
Z, which has an intuitive implication. Let us introduce thetangent unit vector !t, normal unit vector !n, and vector m!
which is orthogonal to the former two vectors as depicted
in fig. 2. These unit vectors are defined, respectively, as(1)

where Z is a 2 × 2 matrix whose components are given by(∂i h)(∂j h)
√ ,
g+ g

(2)

g = 1 + (∂x h)2 + (∂y h)2 .

(3)

Zij = δij −withThe argument !r in the height variable should be replaced! ⊥ after taking the spatial derivative. The random
by R
force !ν in eq. (1) obeys Gaussian statistics with zero meanand!t = √ √1(∂x h, ∂y h, g − 1)g g−1
= (cos θ cos φ, cos θ sin φ, sin θ),1!n = √ (−∂x h, −∂y h, 1)g= (− sin θ cos φ, − sin θ sin φ, cos θ),!t × !n
1
m
! = −
= √
(−∂y h, ∂x h, 0)
g−1|!t × !n|= (− sin φ, cos φ, 0),

(7)

(8)

(9)

(4) where θ is the angle between !t and the x-y planeand φ the angle√ between the projected vector !s =√The positive constant D0 is the diffusion coefficient on a(∂x h, ∂y h, 0)/( g g − 1) and the x-axis as shown in fig. 2! ⊥ ), the noise
flat plane. Since Z depends on !ν through h(R& 1). The unit vectors !t, m! and !n constiterm in eq. (1) is generally multiplicative. We employ (notice that |!s| =
tute
a
set
of
orthogonal
coordinatesateachpoint on thethe Stratonovich interpretation of the multiplicative noise,
curvedsurface.
while Naji and Brown take the Ito prescription with an
The velocity of the Brownian particle on the left-handadditive drift term in eq. (1). This difference does not
sideof eq. (1) is defined on the base plane, whereas the ranaffect the diffusion coefficient [27]. Naji and Brown havedomforce on the right-hand side is parallel to the tangent
shown that [11]plane[11]. Keeping this fact in mind, we consider a map(5) ping from the tangent to base planes. Let us introduce
Zik Zjk = (g −1 )ij ,! with arbitrary
where the repeated indices imply summation. The tensor a vector on the tangent plane ξt!t + ξm m
−1
and
ξ
.
The
projectionof
thisvector on theconstants
ξtm
g is the inverse of the metric tensor gij = δij +(∂i h)(∂j h)
x-y
plane
can
be
writtenasand is given by$νi (t)νj (t" )% = 2D0 δij δ(t − t" ).

(g −1 )ij = δij −(∂i h)(∂j h).
g(6)50007-p2

ηi = ξt ti + ξm mi = Qij ξj ,(10)

Lateral diffusion on a frozen random surface
Mean square displacement and lateral diffusion
coefficient. – Now we solve eq. (1) in the case of a random time-independent surface. The solution can be written formally as)
d!r Z(h(!r ))ij
Ri⊥ (t) − Ri⊥ (0) =) t! ⊥ (s))νj (s),ds δ(!r − R
(16)
×
0

from which one obtains))! ⊥ (t) − R! ⊥ (0))2 =(R
d!r1 d!r2 Z(h(!r1 ))ij Z(h(!r2 ))ik) t) t
! ⊥ (s1 ))νj (s1 )×ds1
ds2 δ(!r1 − R
0Fig. 2: The blue vector !n is the normal unit vector, the red
vector !t the tangent unit vector, and the green vector m! theunit vector perpendicular to both !n and !t. The vector !s (|!s| !=1) is a projection of !t to the x-y plane. The vector m! is onthe x-y plane. The tangent plane touches the curved surface
at x = y = z = 0.

∂x h
√ √
 g g−1
Q=

∂y h
√ √
g g−1

−∂y h √g−1 
.
∂x h √g−1! ⊥ (t) − R! ⊥ (0))2 %$(R))= d!r1 d!r2 $Z(h(!r1 ))ij Z(h(!r2 ))ik %h) t) t! ⊥ (s1 ))δ(!r2 − R! ⊥ (s2 ))×
ds1
ds2 $δ(!r1 − R

P = QA(12)0×νj (s1 )νk (s2 )%!ν ,

'

∂x h

∂y h

−∂y h

∂x h

(

=

'

cos φ

sin φ

− sin φ

cos φ

(

(13)

is the only choice in terms of the angle φ to satisfy thesymmetry requirement. In fact, it is readily proved from
eqs. (11) and (13) that P is equal to Z given by eq. (2).
The relation in eq. (5) is also verified by noting that
Qik Qjk = (g −1 )ij

(14)

(18)

where $· · · %h and $· · · %!ν indicate the average over h
and !ν , respectively. Since we have assumed that the
randomness of the height does not have any spatial
correlation, the decoupling $Z(h(!r1 ))ij Z(h(!r2 ))ik %h =$Z(h(!r1 ))ij %h $Z(h(!r2 ))ik %h is allowed and each part is constant in space because of the translational invariance.
Therefore, one has! ⊥ (0))2 % = $Zij %h $Zij %h 2D0 t = 2dDt,! ⊥ (t) − R
$(R

with
1
A= √
g−1

0

(11)

However this is inadequate to enter directly in theLangevin equation (1) since the off-diagonal elements of
Q are not symmetric under the interchange x ↔ y, whichis required by the isotropy of the base plane. We note that

(17)

The mean square displacement on the base plane canbe obtained by averaging eq. (17) over the randomness ofthe surface height and the random force. Since these are
generally independent of each other for a frozen surface,
one may write eq. (17) as

where the third component is excluded, i.e., i, j = 1, 2 andξ1 = ξt and ξ2 = ξm . From eqs. (7) and (9), we find thatthe matrix Q takes the form0! ⊥ (s2 ))νk (s2 ).
× δ(!r2 − R(19)where d is the dimensionality of the surface and we haveused eq. (4). The effective diffusion coefficient D will beobtained shortly. The expression in eq. (19) implies thattaking an average of Z in the Langevin equation (1) withrespect to the surface randomness is justified. By using theisotropy of space, the effective lateral diffusion coefficient
D is given for d = 2 by
* +
,-2
1
D
1
.
(20)
= $Z11 %2h =
1+ √
D0
2
g
h

This is the main result of this letter. When the surface isand Aji = (A−1 )ij . The mapping in eq. (10) is then writ- a curved line embedded in two dimensions, on the other√ten ashand, only the element Z11 = 1/ g exists and one obtains
for d = 1
!η = P ζ!
(15)
*
-2
1
D
=
.
(21)
√
!
with ζ! = A−1 ξ.
D0g h50007-p3

Takao Ohta and Shigeyuki Komura
This is the known exact result [9,11] and has also beenobtained by the present method [14] (see footnote 1 ).
The ensemble average can be replaced by the surface
average (the contour average in one dimension) definedas [10]) L) L√dx dy gA,
(22)
$A%s ≡ )0 L )0 L√
dx dy g00where L is the system size. Hereafter we shall use thisnotation of the average $· · · %s .
The upper and lower bounds for the diffusion coefficientin two dimensions have been obtained, respectively, as [10]
+
* - ,
DU
1
1
=
,
(23)
1+
D0
2
g s
√
2$1/ g%2s
Fig. 3: The scaled diffusion coefficient D/D0 (solid line), theDL=.
(24) upper and lower bounds DU /D0 (dotted line), and DL /D0
D0
1 + $1/g%s
In order to prove DL ≤ D ≤ DU , the inequality$A%2 ≤ $A2 %(25)

is useful. Applying this to eq. (20), one has
+* +
,- ,2
1
D
1
=
1+ √
D0
2
g
s
.+ +
,,2 /
1
1
≤
.
1+ √
2
g

(dashed line) given by eqs. (29), (30), and (31), respectively, as! 2 $1/2 for a quenched Gaussian random
a function of w = "(∇h)
surface.

2∞2where erfc(x) = (2/π) x dt e−t . On the other hand, theexplicit forms of the upper and lower bounds given byeqs. (23) and (24), respectively, have been obtained for
Gaussian surfaces as [10]
(26)

s

4
DU
13
1 + (1/w2 ) exp(1/w2 )E1 (1/w2 ) ,
=
D0
2
DL
2π[(1/w) exp(1/w2 ) erfc(1/w)]2,=
D0
1 + (1/w2 ) exp(1/w2 )E1 (1/w2 )(30)

It is readily shown that the last expression in eq. (26) is(31)
smaller than DU so that D ≤ DU .
On the other hand, the lower bound eq. (24) satisfies
2∞
where E1 (x) = x dt e−t /t. In fig. 3, eqs. (29), (30),0 √ 120 √ 12and (31) are plotted as a function of w. The diffusion co2 1/ g s
2 1/ g s
DL=≤(27) efficient D/D0 decreases monotonically as w is increased,
0 √ 12 .
D0
1 + $1/g%s
1 + 1/ g s
approaching the asymptotic value of 1/4.
It is easy to see that D is larger than the last expression
Discussion. – Gustafsson and Halle [10] have arguedof eq. (27) leading to DL ≤ D.
that there are two candidates for the diffusion coefficient
In the above, we have presented a mathematically rigoron the frozen disordered surface. Although our result inous proof for DL ≤ D ≤ DU . It might be useful, however,eq. (20) is exact, we compare it with other approximate
to show the relative magnitude and all over behavior offormulas. One is the result obtained by the area scaling.
these three quantities. This is possible if the probability
Gustafsson and Halle employ their previous result in one! is Gaussian [10],distribution of p! = ∇hdimension [9], where the diffusion coefficient is given by+,eq. (21) which is the square of the contour average of thep! 22(28) projected contour length. On dimensional consideration,f (p) = 2 exp − 2 ,ww
they replace the square of the projected contour length! 2 %1/2 is the root-mean-square fluctuation. by the ratio of the base area to the surface area in two
where w = $(∇h)
2∞
The function f (p) is normalized as 0 dp pf (p) = 1. The dimensions. Taking the surface average, one obtains [10]exact diffusion coefficient D in eq. (20) turns out to be
*
1
DA
.
(32)
=
√
√
D
1
D0
g s
(29)
= [1 + π(1/w) exp(1/w2 ) erfc(1/w)]2 ,D0
4
1 The spatial average in eq. (A.13) in ref. [14] should read the
The accuracy of this result cannot be judged a priori, as
noted by the authors themselves of ref. [10].
surface (contour) average.
50007-p4Lateral diffusion on a frozen random surface
The other is obtained by the effective medium approxiSummary. – In summary, the present theory promation [10]vides an exact prediction for the diffusion coefficient on
a two-dimensional frozen surface, which satisfies the reDM
1
(33)
= √ .
striction due to the upper and lower bounds. We startD0$ g% swith the Langevin equation for a Brownian particle on a
Gustafsson and Halle start with the Smoluchowski equa- random surface formulated by Naji and Brown [11]. Thetion for non-interacting Brownian particles whose position lateral diffusion coefficient is obtained by averaging theis defined on the base plane and derive the steady state coefficient multiplied by the random force, which is valid
solution approximately. The proportionality constant be- in the quenched limit for random surfaces without any
tween the average current of the particles and the imposed spatial correlation.
linear macroscopic gradient of the particle concentration
In the present theory, we have employed the Monge
gives us the diffusion coefficient apart from a geometrical representation with the single-valued height function. Exfactor. In order to determine the microscopic current in tending the theory to interconnected random and isotropic
the Smoluchowski equation, Gustafsson and Halle consider surfaces embedded in three dimensions is an interestinga small circular area on the base plane in which the current future problem.
is constant in space, whereas the current outside the circle is approximated by the macroscopic one. In this way,∗∗∗
the problem becomes equivalent to the dielectric boundary value problem of an infinite cylinder with an external
TO is indebted to Dr. Kazuhiko Seki for bringing
electric field. The dielectric constant inside the circle is his attention to ref. [15]. SK thanks Yuto Hosaka fordifferent from that of the outside. From the relation be- useful discussion. SK acknowledges support by a Granttween the induction and the applied field [28] and noting in-Aid for Scientific Research (C) (Grant No. 18K03567
that the eigenvalues of the tensor (g −1 )ij in eq. (6) are and Grant No. 19K03765) from the Japan Society for
given by 1 and 1/g, one obtains a closed equation for the the Promotion of Science and support by a Grant-in-Aid
diffusion coefficient, whose solution is given by eq. (33). for Scientific Research on Innovative Areas “Information
See ref. [10] for more details.
Physics of Living Matters” (Grant No. 20H05538) from
The requirement DL ≤ DA , DM ≤ DU has been verified the Ministry of Education, Culture, Sports, Science and! 2 [10].
for small values of (∇h)
Technology of Japan.
One of the most distinct differences between the present
exact result in eq. (20) and the approximants in eqs. (32)
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