VOLUME 87, N
UMBER 1 PHYSICAL REVIEW LETTERS 2J
ULY 2001
Giant Acceleration of Free Diffusion by Use of Tilted Periodic Potentials
P. Reimann,
1
C. Van den Broeck,
2
H. Linke,
3,4
P. Hänggi,
1
J. M. Rubi,
5
and A. Pérez-Madrid
5
1
Universität Augsburg, Institut für Physik, Universitätsstrasse 1, D-86135 Augsburg, Germany
2
Limburgs Universitair Centrum, B-3590 Diepenbeek, Belgium
3
School of Physics, University of New South Wales, Sydney 2051, Australia
4
Department of Physics, University of Oregon, Eugene, Oregon 97403-1274
5
Departament de Fisica Fonamental, Facultat de Fisica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain
(Received 27 February 2001; published 18 June 2001)
The effective diffusion coefficient for the overdamped Brownian motion in a tilted periodic potential is
calculated in closed analytical form. Universality classes and scaling properties for weak thermal noise
are identified near the threshold tilt where deterministic running solutions set in. In this regime the dif-
fusion may be greatly enhanced, as compared to free thermal diffusion with, for a realistic experimental
setup, an enhancement of up to 14 orders of magnitude.
DOI: 10.1103/PhysRevLett.87.010602 PACS numbers: 05.40.– a, 02.50.Ey, 05.60. –k
Thermal diffusion in a tilted periodic potential plays
a prominent role in Josephson junctions [1], rotating
dipoles in external fields [2], superionic conductors [3],
charge density waves [4], synchronization phenomena
[5], diffusion on crystal surfaces [6], particle separation
by electrophoresis [7], and biophysical processes such as
intracellular transport [8], to name just a few [9]. Also
the Brownian motion in a “traveling periodic potential”
V 共x 2yt兲 can be readily mapped onto a static tilted peri-
odic potential [10].
In many cases of interest, the diffusion can be modeled
as overdamped Brownian motion in 1D:
h
ᠨ
x共t兲 苷 2V
0
共共共x 共t兲兲兲兲 1 F 1
p
2hkT j共t兲 , (1)
where h is the viscous friction coefficient (static mobility),
V 共x兲 is a periodic potential,
V 共x 1 L兲 苷 V 共x兲 , (2)
F is a static “tilting force,” and k is Boltzmann’s constant.
The thermal fluctuations at temperature T are modeled
[9,11] by the unbiased d-correlated Gaussian noise j共t兲.
The first basic quantity of interest is the particle cur-
rent 具
ᠨ
x典 :苷 lim
t!`
具x共t兲典兾t. Its analytical solution [see
Eq. (7) below] goes back to Stratonovich [12] and has sub-
sequently been rederived many times [9]. In this Letter,
the quantity of foremost interest is the effective diffusion
coefficient
D :苷 lim
t!`
具x
2
共t兲典 2 具x共t兲典
2
2t
. (3)
For V
0
共x兲⬅0 and arbitrary F, the diffusion coefficient
is given by Einstein’s result D
0
:苷 kT兾h, whereas for
F 苷 0 and arbitrary V 共x兲, an analytical prediction for D is
due to [13]. In this Letter we derive an analytical formula
for D when both V 共x兲 and F are arbitrary, analogous to
Stratonovich’s landmark result for 具
ᠨ
x典. Specifically, near
the threshold tilt where deterministic running solutions set
in, we find that diffusion is greatly enhanced and that it
obeys a specific universal scaling relation.
Our starting point is the following exact expressions for
the particle current and for the diffusion coefficient:
具
ᠨ
x典 苷
L
具t共x
0
! x
0
1 L兲典
, (4)
D 苷
L
2
2
具t
2
共x
0
! x
0
1 L兲典 2 具t共x
0
! x
0
1 L兲典
2
具t共x
0
! x
0
1 L兲典
3
, (5)
where x
0
is an arbitrary reference point, 具t
n
共a ! b兲典 is
the nth moment of the first passage time from a to b . a
for a stochastic trajectory obeying (1), and where it is as-
sumed that F . 0 in order to keep those moments finite.
Such relations have been previously proposed for certain
random processes in discrete space [14,15] and have been
anticipated without proof in [16] also for the present con-
tinuous dynamics (1). A proof will be given at the end of
this Letter (see also [17]).
For the dynamics (1), the moments of the first passage
time are given by the well-known closed analytical recur-
sion (see, e.g., section 7 in [11], and further references
therein)
具t
n
共a ! b兲典 苷
Z
b
a
dx
Z
x
2`
dy
D
0
n具t
n21
共 y ! b兲典
3 exp兵关V 共x兲 2 V共 y兲 2 共x 2 y兲F兴兾kT其
(6)
for n 苷 1, 2, . . . , with 具t
0
共 y ! b兲典 :苷 1. By introducing
(6) into (4) and (5), one finds, after somewhat tedious
manipulations, the result
具
ᠨ
x典 苷
1 2 e
2LF兾kT
R
x
0
1L
x
0
dx
L
I
1
共x兲
, (7)
D 苷 D
0
R
x
0
1L
x
0
dx
L
I
2
1
共x兲I
2
共x兲
关
R
x
0
1L
x
0
dx
L
I
1
共x兲兴
3
, (8)
where we have introduced
I
6
共x兲 :苷
Z
L
0
dy
D
0
e
兵6V 共x兲7V共x7y兲2yF其兾kT
. (9)
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VOLUME 87, N
UMBER 1 PHYSICAL REVIEW LETTERS 2J
ULY 2001
The so far excluded case F , 0 can be readily reduced
to an equivalent dynamics with F . 0, yielding exactly
the same result [(7) and (8)]. Finally, the case F
苷 0
follows by continuation, in agreement with [13]. Note that
(2) implies I
6
共x兲 苷 I
6
共x 1 L兲 and hence the choice of
x
0
is indeed irrelevant in (7) and (8). Further, Einstein’s
relation D 苷 D
0
is readily recovered by observing that
I
6
共x兲 苷 const in the special case V
0
共x兲⬅0.
Equation (7) is Stratonovich’s exact expression for the
particle current [12], while the corresponding general for-
mula for the diffusion coefficient (8) is the first central
result of this Letter. Figure 1 exemplifies its excellent
agreement with accurate numerical simulations, while an
approximate result from [18] captures only the qualitative
behavior. The most interesting feature in Fig. 1 is the peak
near the “critical tilt” F
c
, which gets more and more pro-
nounced as kT (or D
0
苷 kT兾h) decreases. This brings
us to our second main preoccupation, namely, the case of
weak noise 共kT
ø LF
c
兲 in combination with a tilt F close
to the “critical” threshold value F
c
beyond which deter-
ministically running solutions appear in (1). The resulting
diffusion depends crucially on the form of the potential
around its “dynamical bottleneck” x
c
. Without loss of gen-
erality, we focus on x
c
苷 0 and we assume the following
small x behavior:
V 共x兲 2 xF 苷 2mxjxj
q21
2ex , (10)
with m.0, q . 1 (to guarantee differentiability at
x 苷 0), and small e :苷 F 2 F
c
. In the remainder of the
0
5
10
15
20
0,6 0,8 1 1,2 1,4
D/D
0
F
FIG. 1. Diffusion coefficient (3) versus the tilt F for the sto-
chastic dynamics (1) with a sinusoidal periodic potential V共x兲 苷
U
0
sin共2px兾L兲. Using dimensionless units, the parameter val-
ues are h 苷 U
0
苷 1, L 苷 2p, and kT 苷 D
0
苷 0.1. Note that
the critical tilt [onset of deterministically running solutions in
(1)] occurs at F 苷 F
c
苷 1. Solid line: analytical prediction
(8). Filled dots: numerical simulations with an estimated rela-
tive uncertainty of 0.01. Dashed line: analytical approximation
D 苷 kTd具
ᠨ
x典兾dF from [18]. Dash-dotted line, filled squares,
and dotted line: same as solid line, filled dots, and dashed line,
respectively, but now for kT 苷 D
0
苷 0.01.
interval 关2L兾2, L兾2兴, the “total potential” V 共x兲 2 xF in
(1) is assumed to be strictly monotonically decreasing (i.e.,
“tilted to the right” with F
c
. 0). Outside 关2L兾2, L兾2兴
the behavior of V 共x兲 is fixed by the periodicity (2). Note
that q
苷 3 in the generic case [e.g., an analytic V 共x兲 such
as in Fig. 1]. Other q values can be readily realized experi-
mentally by tailoring the form of V 共x兲.
To get an intuitive feeling about the role of the exponent
q, it is instructive to study the deterministic motion (1) in
a critical 共e 苷 0兲 potential (10) extending over the entire
x axis, i.e.,
ᠨ
x共t兲 苷 a
jx共t兲j
q21
with a :苷 m q兾h.0.
A straightforward calculation then shows that it takes
an infinite amount of time to travel from any x
i
, 0 to
x
f
苷 0 when q $ 2, while a finite time is sufficient for
1 , q , 2. On the other hand, finite time suffices to go
from x
i
苷 2` to any x
f
, 0 for q . 2, while this trav-
eling time diverges for q # 2. Complementary results are
found for the traveling times in the region x $ 0. We thus
expect that for q . 2, with sufficiently small (but finite)
kT and sufficiently small e, the motion of the particle is
dominated by the passage through the bottleneck region in
the vicinity of x 苷 0, where the potential is well approxi-
mated by the form given in (10). On the other hand, for
2 $ q . 1, the form of V共x兲 outside a small neighbor-
hood of x 苷 0 is also expected to come into play.
With this insight in mind, we turn to the evaluation of
(8) when q . 2. First, it is convenient to make the special
choice x
0
苷 2L兾2. Second, in the vicinity of x 苷 0 the
approximation (10) can be introduced into (9). Third, due
to our above considerations we can extend this approxima-
tion to the entire interval 关2L兾2, L兾2兴 and finally extend
this integration domain to 关2`, `兴 without notably chang-
ing the values of the integrals in (8). Closer inspection
shows that this approximation in fact becomes asymptoti-
cally exact as e and kT tend to zero. In this way, one ob-
tains for q . 2 the result
D 苷 D
0
µ
L
q
m
kT
∂
2兾q
R
dx K
2
共x, g兲K共2x, g兲
关
R
dx K共x, g兲兴
3
, (11)
where integration limits 6` have been omitted and
g :苷 e兾关m
1兾q
共kT兲
121兾q
兴 , (12)
K共x, g兲 :苷
Z
`
0
dy e
兵2xjxj
q21
1共x2y兲 jx2yj
q21
2gy其
. (13)
Note that both fractions in (11) as well as the “scaled
tilt” g are dimensionless and that K共x, g兲 is a dimension-
less function of its (dimensionless) arguments x and g.
Hence, (11) has the form of a scaling law with a universal
scaling function and specific critical exponents depending
on q . 2. The most remarkable feature is the divergence
of D兾D
0
when kT tends to zero for any fixed g value, i.e.,
we recover (cf. Fig. 1) a huge enhancement of thermal dif-
fusion.Forq 苷 3, the scaling function appearing in (11) is
depicted in Fig. 2. From this plot it follows that the asymp-
totic scaling form (11) is already rather well satisfied for
moderately small kT values and that the enhancement of
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0,02
0,04
0,06
0,08
-5 0 5 10
G(
γ)
γ
FIG. 2. Bold solid line: dimensionless scaling function
G共g兲 :苷
R
dx K
2
共x, g兲K共2x, g兲兾关
R
dx K共x, g兲兴
3
in (11) for
q 苷 3 versus its dimensionless argument g from (12). Solid
and dash-dotted lines: same as the respective lines in Fig. 1,
but now plotted in the form 共D兾D
0
兲共kT 兾L
q
m兲
2兾q
[cf. (11)]
versus g [cf. (12)] with q 苷 3.
diffusion is most pronounced for jgj # O 共1兲. Similar be-
havior is recovered for any other q . 2. In other words,
the diffusion coefficient as a function of F exhibits a pro-
nounced “resonance” peak at F 苷 F
c
.
The basic physical mechanism behind this effect may
be explained as follows. As discussed above, the noisy
dynamics (1) is, for e 苷 0 and small kT, dominated by
the passage through the dynamical bottleneck at x 苷 0
[cf. (10)]. Since e 苷 0, a very small perturbation due
to thermal noise is already sufficient to kick the particle
across the point x 苷 0. This small variation in compari-
son with an unperturbed particle is subsequently greatly
enhanced by the further dynamical evolution. The result
is a huge dispersion for a statistical ensemble of particles
subjected to different realizations of the noise.
Finally, we briefly turn to the case q # 2. The region
outside a small neighborhood of x 苷 0 is then no longer
negligible for the passage time from 2L
兾2 to L兾2, ren-
dering the analysis more complicated. For simplicity, we
restrict ourselves to the interesting situation of a poten-
tial given by (10) in the entire interval 关2L兾2, L兾2兴 with
2 . q . 4兾3, yielding the result
D 苷 D
0
µ
L
q
m
kT
∂
324兾q
R
dx K
2
共x, g兲K共2x, g兲
关共
kT
L
q
m
兲
2兾q21
S共g兲 1
2
q21
q共22q兲
兴
3
,
(14)
where S共g$0兲 :苷 0 and
S共g,0兲 :苷
2pjg兾qj
共22q兲兾共q21兲
q共q 2 1兲
3 exp兵2共q 2 1兲 jg兾qj
q兾共q21兲
其 . (15)
Technical details as well as the discussion of other q val-
ues and more complicated potentials V 共x兲 will be given
elsewhere.
The salient difference of (14) in comparison with (11)
is a competition between the two terms in the denomina-
tor on the right-hand side: for any fixed g value, the sec-
ond term dominates when kT becomes sufficiently small.
Thus D兾D
0
increases proportional to 共kT兲
4兾q23
, i.e., we
find again a huge enhancement of thermal diffusion. More
subtle is the behavior of (14) as a function of g for a small
but fixed kT value. For arbitrary positive as well as for
moderately negative g values, it is still the second term
in the denominator which dominates, and thus the g de-
pendence of D is governed by
R
dx K
2
共x, g兲K共2x, g兲.On
the other hand, for large negative g values we can evaluate
the latter integral by means of a saddle point approxima-
tion, yielding the result S
2
共g兲兾2. Since S共g兲 from (15) in-
creases very fast with decreasing g, the right-hand side of
(14) increases very fast as long as
R
dx K
2
共x, g兲K共2x, g兲
governs the g dependence. However, again due to this fast
increase, the first summand in the denominator starts to
compete with the second summand and ultimately takes
over, leading to a decrease of D proportional to 1兾S共g兲.
Thus a peak appears at a (negative) g value for which both
terms in the denominator are of the same order of magni-
tude. The detailed quantitative calculation is straightfor-
ward and leads, for 2 . q . 4兾3, to the result
D共g
max
兲 苷
2
22q
q共2 2 q兲
27
L
q
m
h
, (16)
g
max
⯝ 2q
∑
2 2 q
2q共q 2 1兲
ln
µ
L
q
m
kT
∂∏
121兾q
. (17)
Note that the maximal diffusion coefficient in (16) is in-
dependent of kT. In other words, the maximal enhance-
ment of diffusion is even stronger than for q . 2 [cf. (11)].
These predictions are confirmed by direct numerical evalu-
ation of the exact formula (8) in Fig. 3.
These results can be applied, for instance, to the ther-
mally induced diffusion of a particle that moves in a liq-
uid under the action of gravitation along the rigid surface
of a critically tilted periodic geometrical profile. For a
spherical iron particle of 1 mm radius in water and a criti-
cally tilted profile that decreases by 1.5 cm per spatial
period L 苷 10 cm, one finds, at room temperature for
the thermal diffusion coefficient, D
0
⯝ 2 3 10
212
cm
2
兾s.
From (11), with q 苷 3, one finds at the critical tilt D ⯝
5 3 10
23
cm
2
兾s, i.e., an enhancement by about 9 orders
of magnitude. For a specially tailored profile of the form
(10) with q 苷 3兾2, one finds from (16) that D ⯝ 2 3
10
2
cm
2
兾s, i.e., the enhancement of thermal diffusion is
improved by another 5 orders of magnitude as compared to
the case q 苷 3 and should be easily observable on macro-
scopic length and time scales. An actual experimental re-
alization is presently under construction.
We conclude with the promised proof of (4) and (5) for
F . 0 and an arbitrary x
0
. To this end, we denote by c
an arbitrary point between a and b . a. Then the first
passage time t共a ! b兲 can be decomposed into the time to
travel from a to c, plus the time to travel from c to b.For
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0,05
0,1
0,5 0,6 0,7 0,8 0,9 1
D
F
FIG. 3. Diffusion coefficient (8) versus the tilt F for a potential
V共x兲 苷 V 共x 1 L兲 defined by (10) for x [ 关2L兾2, L兾2兴. Using
dimensionless units, the parameter values are q 苷 3兾2, h 苷 1,
L 苷 2, m 苷 1, F
c
苷 1. The five curves, with sharper peaks
corresponding to lower temperatures, represent the following
values of kT 苷 D
0
: 3 3 10
22
, 10
22
, 10
23
, 10
24
, 10
25
. The
theoretically predicted peak height for asymptotically small kT
from (16) is 0.
1.
a white noise driven process (1), the latter two times are
statistically independent of each other [19] and all statisti-
cal properties of t共a ! b兲 are exactly the same as those of
t共a ! c兲 1 t共c ! b兲. As a consequence, 具t共a ! b兲典 苷
具t共a ! c兲典 1 具t共c ! b兲典 and analogously for the first
passage time dispersion 具Dt
2
共a ! b兲典 :苷 具t
2
共a ! b兲典 2
具t共a ! b兲典
2
. Further, t共x
0
! x
0
1 lL兲 is statistically
equivalent to a sum of l independent, random variables,
t共x
0
! x
0
1 L兲,...,t共共共x
0
1 共l 2 1兲L ! x
0
1 lL兲兲兲, and,
due to the periodicity (2), they are identically distributed.
Invoking the central limit theorem, the distribution of the
first passage times t共x
0
! x
0
1 lL兲 thus approaches for
large l a Gaussian distribution with mean value l具t共x
0
!
x
0
1 L兲典 and variance l具Dt
2
共x
0
! x
0
1 L兲典.
Next, we introduce “coarse-grained states” 兵x
m
:苷 x
0
1
mlL其
`
m苷2`
, where l is a large but fixed integer [20]. The
process x共t兲 is said to be in a certain “state” from the in-
stant it hits the associated point x
m
until the moment it hits
one of the adjacent points x
m61
. It follows that the diffu-
sion coefficient D is identical for the original process x共t兲
and its coarse-grained counterpart, due to the long-time
limit in (3), and similarly for the current 具
ᠨ
x典. Next we note
that “backward transitions” x
m
哫 x
m21
are suppressed by
a Boltzmann factor exp兵2lLF兾kT其 compared to x
m
哫
x
m11
and therefore are negligible for sufficiently large l.
The remaining “forward transitions” between neighboring
“states” x
m
and x
m11
are identically distributed random
events with a probability distribution which is identical
to the first passage time distribution for the original pro-
cess x共t兲. On the other hand, we have seen above that for
sufficiently large l this distribution is completely fixed by
具t共x
0
! x
0
1 L兲典 and 具Dt
2
共x
0
! x
0
1 L兲典. Thus, if the
latter two quantities are the same for two processes (1) then
具
ᠨ
x典 and D will also be the same in the two cases. Conse-
quently, it is sufficient to prove (4) and (5) for the special
case that V
0
共x兲⬅0. In this case, 具
ᠨ
x典 苷 F兾h, D 苷 D
0
,
and the evaluation of 具t共x
0
! x
0
1 L兲典 and 具Dt
2
共x
0
!
x
0
1 L兲典 according to (6) is straightforward. As a result,
one sees that (4) and (5) are indeed fulfilled.
This work was supported by the exchange program of
the Deutscher Akademischer Austauschdienst (P. R., P. H.)
and the Acciones Integradas Hispano-Alemanas under
HA1999-0081 (J. M. R., A. P.-M.), the Program in Inter-
University Attraction Poles of the Belgian Government
(C. V.d.B.), the Australian Research Council (H. L.), DFG-
Sachbeihilfe HA1517/13-2, and the Graduiertenkolleg
GRK283 (P. R., P. H.).
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[17] B. Lindner, M. Kostur, and L. Schimansky-Geier, Fluct.
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[18] G. Constantini and F. Marchesoni, Europhys. Lett. 48
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[19] This is so because (1) is a strong Markov process, and first
passage times are Markov stopping times.
[20] The t ! ` limit [e.g., in (3)] is associated with m ! `,
not with l ! `.
010602-4 010602-4
![FIG. 2. Bold solid line: dimensionless scaling function G g : R dx K2 x,g K 2x,g R dx K x,g 3 in (11) for q 3 versus its dimensionless argument g from (12). Solid and dash-dotted lines: same as the respective lines in Fig. 1, but now plotted in the form D D0 kT Lqm 2 q [cf. (11)] versus g [cf. (12)] with q 3.](/figures/fig-2-bold-solid-line-dimensionless-scaling-function-g-g-r-73lx4ohe.webp)
![FIG. 1. Diffusion coefficient (3) versus the tilt F for the stochastic dynamics (1) with a sinusoidal periodic potential V x U0 sin 2px L . Using dimensionless units, the parameter values are h U0 1, L 2p , and kT D0 0.1. Note that the critical tilt [onset of deterministically running solutions in (1)] occurs at F Fc 1. Solid line: analytical prediction (8). Filled dots: numerical simulations with an estimated relative uncertainty of 0.01. Dashed line: analytical approximation D kTd x dF from [18]. Dash-dotted line, filled squares, and dotted line: same as solid line, filled dots, and dashed line, respectively, but now for kT D0 0.01.](/figures/fig-1-diffusion-coefficient-3-versus-the-tilt-f-for-the-3tbaqeck.webp)
