PHYSICAL REVIEW E 80, 021903 共2009兲
Noisy swimming at low Reynolds numbers
Jörn Dunkel* and Irwin M. Zaid
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom
共Received 30 March 2009; published 5 August 2009兲
Small organisms 共e.g., bacteria兲 and artificial microswimmers move due to a combination of active swimming and passive Brownian motion. Considering a simplified linear three-sphere swimmer, we study how the
swimmer size regulates the interplay between self-driven and diffusive behavior at low Reynolds number.
Starting from the Kirkwood-Smoluchowski equation and its corresponding Langevin equation, we derive
formulas for the orientation correlation time, the mean velocity and the mean-square displacement in three
space dimensions. The validity of the analytical results is illustrated through numerical simulations. Tuning the
swimmer parameters to values that are typical of bacteria, we find three characteristic regimes: 共i兲 Brownian
motion at small times, 共ii兲 quasiballistic behavior at intermediate time scales, and 共iii兲 quasidiffusive behavior
at large times due to noise-induced rotation. Our analytical results can be useful for a better quantitative
understanding of optimal foraging strategies in bacterial systems, and they can help to construct more efficient
artificial microswimmers in fluctuating fluids.
DOI: 10.1103/PhysRevE.80.021903
PACS number共s兲: 87.17.Jj, 05.40.Jc, 47.63.Gd, 47.63.mf
I. INTRODUCTION
Biological 关1–3兴 and artificial microswimmers 关1,4–7兴
move through a fluid by performing a series of self-induced
shape changes 关8,9兴. Handicapped by their tiny size 共typically a few micrometers for a bacterium 关10兴兲, they are
forced to swim at very low Reynolds numbers R Ⰶ 1
关11–13兴. Hence, in order to account for the resulting lack of
inertia, the swimming strategies of microorganisms are very
different from those operative at human length scales. More
precisely, since the fluid flow is reversible at low Reynolds
number, locomotion in this regime is possible only if the
swimming stroke violates certain time-reversal symmetries
关8,13–16兴.
Stimulated by experimental advances 关1–4,17,18兴, in recent years considerable progress was achieved in understanding the dynamics of deterministic microswimmers models
关19–28兴. Yet, comparatively little is known quantitatively
about the complex interplay between active self-motion, hydrodynamic interactions, and thermal fluctuations in the surrounding fluid 关29–32兴. Very recently, first steps toward clarifying these issues were made by Howse et al. 关6兴, who
measured in their experiments the mean square displacement
of chemically driven colloidal spheres, and by Lobaskin et
al. 关33兴, who studied the Brownian motion of a triangular
microswimmer at intermediate Reynolds numbers R ⬃ 1 by
combining Lattice Boltzmann simulations with a Langevin
description of the swimmer in phase space. In the present
paper, we would like to complement these investigations by
concentrating on the diffusive properties of mechanically
driven microswimmers at low Reynolds numbers R Ⰶ 1.
This limit case is most relevant for bacterial motions and
allows one to treat diffusive effects within configuration
space.
Specifically, we will focus on the following questions:
how does the size of the swimmer affect its effective mobil-
*jorn.dunkel@physics.ox.ac.uk
1539-3755/2009/80共2兲/021903共11兲
ity in a noisy fluid? Which details govern the transition from
quasiballistic self-motion to the diffusive regime? To shed
light on these issues, we shall consider simplified quasilinear
p-sphere swimmers similar to those proposed by Najafi and
Golestanian 关20兴. More precisely, we will assume that internal forces, which generate the swimming strokes, are mediated by interaction potentials. This approach permits us to
treat thermal diffusion effects within the KirkwoodSmoluchowski scheme, originally developed to describe the
diffusion of polymers in a fluctuating medium 关34–37兴.
Starting from the Kirkwood-Smoluchowski equation
共KSE兲 ensures that hydrodynamic and stochastic forces are
consistently coupled on the level of the Fokker-Planck description in configuration space 关34兴. Moreover, as discussed
in Sec. III, the corresponding Langevin equation can be used
to derive closed analytical formulas for the orientation correlation time, the mean velocity and the mean square displacement of a single swimmer in three space dimensions.
Although the analytical and numerical results in this paper
refer to the case of a quasilinear three-sphere swimmer
共p = 3兲, the formalism can be easily generalized to more complex models 共e.g., flexible p-sphere swimmer chains兲. Therefore, this approach can be generally very useful for studying
Brownian motion effects and hydrodynamic interactions in
active biological systems at low Reynolds numbers. Furthermore, since it is straightforward to implement an external
confinement 共e.g., tweezer or lattice potentials兲, the
Kirkwood-Smoluchowski scheme can help to construct and
optimize arrays 关38兴 of, e.g., micropumps that work efficiently on those scales where thermal fluctuations in fluid
become non-neglible.
Thus, purpose and content of the present paper can be
summarized as follows: first, we will discuss a convenient
formalism that allows to simulate active microswimming by
means of Langevin equations and interaction potentials 共Sec.
II兲. Subsequently, we derive analytic results for the diffusion
of a single swimmer 共Sec. III兲, thereby extending recent
work of Golestanian and Ajdari 关25兴 on deterministic swimmers. A thorough analysis of the single swimmer case is
instructive for a number of reasons: 共i兲 Exact analytical re-
021903-1
©2009 The American Physical Society
PHYSICAL REVIEW E 80, 021903 共2009兲
JÖRN DUNKEL AND IRWIN M. ZAID
sults provide a useful test of numerical simulations, cf. Sec.
IV. 共ii兲 Recently, Leoni et al. 关4兴 were able to experimentally
realize an individual three-sphere system similar to those
considered here. 共iii兲 Understanding the noise-induced behavior of a single swimmer is a prerequisite for understanding complex behavior and pattern formation in, e.g., selfassembling bacterial systems 关10兴. 共iv兲 Dynamical
calculations as those presented below may provide a “microscopic” justification for purely probabilistic models of bacterial motility 关39兴. 共v兲 Depending on the swimmer size, we
find a rather sharp transition from purely Brownian to quasiballistic motions. From a 共bio兲 physical perspective, it is
remarkable that the transition occurs when the three-sphere
swimmer model is tuned to bacterial parameters. Hence,
loosely speaking, exploiting the interplay between Brownian
motion and active swimming may indeed represent a useful
strategy 共not only兲 in nature.
II. GENERAL THEORETICAL BACKGROUND
We consider an ensemble of N microswimmers, each consisting of p spheres. Neglecting inertia, the state of the system at time t is described by a set of coordinates 共X␣兲
= 兵X共␣i兲共t兲其, where ␣ = 1 , . . . , pN is a sphere index, and i
= 1 , 2 , 3 labels the space dimension. Our subsequent analysis
rests on the assumption that the stochastic dynamics of the
swimmers in the fluid can be described, at least approximately, by the KSE 关34–37兴, representing evolution for the
N-particle probability density f共t , 兵x共␣i兲其兲. We begin by recalling how the KSE can be translated into a Langevin equation
for numerical simulations 关36,37兴. Details of the swimming
mechanism will be discussed in Sec. II B.
Here, we are interested in the effects of hydrodynamic
interactions, corresponding to H共␣i兲共␤ j兲 ⫽ 0. A simple approximation for H共␣i兲共␤ j兲, obtained by solving the Stokes equation
for a pointlike source, is the Oseen tensor 关40,41兴
H共O␣i兲共␤ j兲 =
冉
冊
1
r r
␦ij + ␣␤2i ␣␤ j ,
8␲␮r␣␤
r␣␤
␣ ⫽ ␤,
共2b兲
where r␣␤i ª x␣i − x␤i and r␣␤ ª 兩x␣ − x␤兩. However, the associated diffusion tensor DO ª kBTHO is not necessarily positive definite, leading to unphysical behavior if sphere separations become too small 关42,43兴. In our numerical
simulations we shall therefore use the improved approximation
H共M␣i兲共␤ j兲 = H共O␣i兲共␤ j兲 +
共a␣2 + a␤2 兲
3
24␲␮r␣␤
冉
␦ij − 3
冊
r␣␤ir␣␤ j
, 共2c兲
2
r␣␤
which was derived by Mazur 关44兴. The additional term on
the right-hand side of Eq. 共2c兲 can be understood as the
next-order correction in a radius-over-distance expansion of
the mobility tensor for two spheres 关45兴. For spheres of equal
size 共a␣ = a␤兲, the tensor H = HM defined by Eqs. 共2a兲 and
共2c兲, reduces to the Rotne-Prager-Yamakawa tensor 关43,46兴.
While Eq. 共2c兲 gives a more accurate description than Eq.
共2b兲 at moderate densities, both expressions become invalid
if the distance between spheres becomes very small. At very
high densities, when sphere-sphere collisions dominate the
dynamics, near-field hydrodynamics and lubrication effects
must be modeled more carefully 关47兴. In the present paper,
however, we focus on systems that can be described by Eq.
共2c兲.
Unlike the Oseen tensor HO from Eq. 共2b兲, the tensor
H = HM is positive definite for r␣␤ ⬎ a␣ + a␤ and thus can be
Cholesky decomposed in the form
A. Kirkwood-Smoluchowski and Langevin equation
Considering a fluid of viscosity ␮ and temperature T, the
KSE reads 关34兴
1
H共␣i兲共␤ j兲 = C共␣i兲共␥k兲C共␤ j兲共␥k兲 .
2
⳵t f = ⳵共␣i兲H共␣i兲共␤ j兲兵关⳵共␤ j兲U兴f + kBT⳵共␤ j兲 f其.
The decomposition Eq. 共3兲 is crucial if one wishes to find a
Langevin representation for the stochastic process 兵X共␣i兲共t兲其
described by the KSE Eq. 共1兲: upon noting that 关36,48兴
共1兲
Here, kB denotes the Boltzmann constant, ⳵共␣i兲 ª ⳵ / ⳵x共␣i兲, and
a sum is performed over equal double indices 共␣i兲. The 共effective 关34兴兲 potential U governs all the internal and external
swimmer interactions 共see examples in Sec. II B兲, apart from
the hydrodynamic interactions mediated by the fluid. The
latter are included in the tensor H. Considering spherical
particles of radius a␣, the “diagonal” components of H are
given by
␦ij
H共␣i兲共␣ j兲 = ,
␥␣
␥␣ = 6␲␮a␣ ,
共2a兲
where ␦ij denotes the Kronecker symbol, and ␥␣ is the
Stokes friction coefficient. The hydrodynamic interactions
between different spheres are encoded in the “off-diagonal”
components H共␣i兲共␤ j兲, ␣ ⫽ ␤. If these hydrodynamic interactions are neglected, corresponding to the 共infinitely dilute兲
limit case H共␣i兲共␤ j兲 = 0, Eq. 共1兲 reduces to an “ordinary”
Smoluchowski equation with a diffusion constant D␣
= kBT / ␥␣ for each sphere.
⳵共␣i兲H共␣i兲共␤ j兲 ⬅ 0
共3兲
共4兲
holds 共for both HO and HM 兲, one finds that the KSE Eq. 共1兲
corresponds to the following Ito-Langevin equation 关49兴
dX共␣i兲共t兲 = H共␣i兲共␤ j兲F共␤ j兲dt + 共kBT兲1/2C共␣i兲共␥k兲dB共␥k兲共t兲. 共5兲
Here, 兵F共␤ j兲其 ª 兵−⳵共␤ j兲U其 comprises the deterministic forces
acting on the spheres, and 兵B共␥k兲共t兲其 is a collection of standard Wiener processes; i.e., the increments dB共␥k兲共t兲 ª B␥k
共t + dt兲 − B共␥k兲共t兲 are independent GAUSSIAN random numbers
with distribution
2
P兵dB共␥k兲共t兲 苸 关u,u + du兴其 =
e−u /共2dt兲
du,
共2␲dt兲1/2
共6兲
and, according to the Ito scheme 关49兴, the coefficients
C共␣i兲共␥k兲 are to be evaluated at time t. For completeness, we
still note that, upon formally dividing by dt, the stochastic
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PHYSICAL REVIEW E 80, 021903 共2009兲
NOISY SWIMMING AT LOW REYNOLDS NUMBERS
differential Eq. 共5兲 can be rewritten as a “standard” Langevin
equation
Ẋ共␣i兲共t兲 = H共␣i兲共␤ j兲F共␤ j兲 + 共kBT兲1/2C共␣i兲共␥k兲␰共␥k兲共t兲,
共7a兲
where Ẋ共␣i兲共t兲 ª dX共␣i兲共t兲 / dt denotes the velocity, and
␰共␥k兲共t兲 ª dB共␥k兲共t兲 / dt represents GAUSSIAN white noise, i.e.,
具␰共␣i兲共t兲典 = 0,
共7b兲
具␰共␣i兲共t兲␰共␤ j兲共t⬘兲典 = ␦␣␤␦ij␦共t − t⬘兲.
共7c兲
B. Swimming mechanism
Having discussed how to implement fluctuations, we still
need to specify the swimming mechanism. To this end, consider two spheres ␣ and ␤ that form the leg of a swimmer.
We assume that the internal forces between beads ␣ and ␤,
which generate the swimming stroke, can be derived from a
time-dependent potential of the form
Uleg共t,d␣␤兲 =
k0
兵d␣␤ − 关ᐉ + ␭␣␤ sin共␻t + ␸␣␤兲兴其2 , 共8a兲
2
with d␣␤共t兲 ª 兩X␣共t兲 − X␤共t兲兩 denoting the distance between
the spheres, ␭␣␤ the approximate amplitude of the stroke, and
ᐉ ⬎ a␣ + a␤ + ␭␣␤ the mean length of the leg. The potential
Uleg gives rise to two characteristic time scales: the driving
period T␻ ª 2␲ / ␻ and, for a sphere of mass M ␣, the oscillator period T0 ª 2␲ / 冑k0 / M ␣. Since we are interested in the
overdamped regime, these time scales must be long compared to the characteristic damping time T␥ ª M ␣ / ␥␣. More
precisely, we have to impose
T␥ Ⰶ T0 Ⰶ T␻
n␤␥ ª d␤␥/d␤␥ .
In order to ensure that the swimmer moves quasilinearly
关25兴, we introduce a stiffness potential
K 2
ᐉ 共n␣␤ · n␤␥ − 1兲2 ,
2
Q
共␦ik − n␤␥in␤␥k兲n␣␤i ,
d␤␥
F␤link = − 共F␣link + F␥link兲
共10b兲
共10c兲
where n␣␤i ª d␣␤i / d␣␤, and
Q ª Kᐉ2共n␣␤ · n␤␥ − 1兲.
共10d兲
Equations 共8兲 and 共9兲 provide a convenient way of modeling
and simulating rigid or flexible p-sphere swimmers by means
of potentials. We note that, by construction, the sum over the
internal swimmer forces is zero. The total potential U appearing in the KSE Eq. 共1兲 is obtained by summing over all
effective interaction potentials Eqs. 共8兲 and 共9兲.
III. ANALYTICAL RESULTS
We next summarize formulae for the correlation time of
the orientation vector, the mean swimmer velocity and the
spatial mean square displacement of an isolated three-sphere
swimmer. These analytical results can be obtained from the
Langevin Eq. 共5兲 by using the Oseen approximation H
⯝ HO, and their explicit derivation is discussed in the Appendix.
A swimmer’s motion can be characterized by its geometric center
1
R共t兲 ª 共X1 + X2 + X3兲
3
共11a兲
and the orientation vector
N共t兲 ª
共8b兲
corresponding to slow driving and fast relaxation. The constraint Eq. 共8b兲 ensures that our swimmers behave similar to
a shape-driven swimmer 关25兴.
In the remainder, we shall focus on three-sphere swimmers, representing the smallest self-swimming system within
our approach 共two-sphere swimmers can achieve active locomotion only due to collective effects 关16兴兲. We consider
three spheres 共␣ , ␤ , ␥兲 forming a swimmer with central
sphere ␤, e.g., in the case of a single swimmer 共␣ , ␤ , ␥兲
= 共1 , 2 , 3兲 with middle sphere ␤ = 2. The legs are given by
d␣␤ ª X␤ − X␣ and d␤␥ ª X␥ − X␤, and we still define normalized connectors
n␣␤ ª d␣␤/d␣␤,
F␥link =
X3 − X1
.
兩X3 − X1兩
共11b兲
We are interested in determining the mean square displacement
DR共t兲 ª 具关R共t兲 − R共0兲兴2典,
共12a兲
and the correlation function
DN共t兲 ª 具N共t兲N共0兲典,
共12b兲
where the average is taken over fluctuations in the fluid 共i.e.,
over all realizations of the Wiener process兲.
It is convenient to discuss DN共t兲 first. For a deterministic
initial state N共0兲 = 关Nk共0兲兴, we can write DN共t兲
= 具Nk共t兲典Nk共0兲 with a summation over equal indices. To obtain an analytical formula for 具Nk共t兲典, we assume that Eq.
共8b兲 holds true and that bending is neglible, K Ⰷ k0. Then the
swimmer behaves like a stiff, shape-driven NajafiGolestanian 关20兴 swimmer and we can approximate
共9兲
d12 ª X2 − X1 ⯝ Nd12 ,
共13a兲
which, for K Ⰷ k0, penalizes bending. The resulting force
components F共lin␣i兲 ª −⳵共␣i兲Ulin read explicitly
d23 ª X3 − X2 ⯝ Nd23 ,
共13b兲
d13 ª X3 − X1 ⯝ N共d12 + d23兲,
共13c兲
Ulin =
F␣link = −
Q
共␦ik − n␣␤in␣␤k兲n␤␥i ,
d␣␤
共10a兲
where, cf. Eq. 共8a兲,
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PHYSICAL REVIEW E 80, 021903 共2009兲
JÖRN DUNKEL AND IRWIN M. ZAID
d12 = ᐉ + ␭12 sin共␻t + ␸12兲,
共13d兲
d23 = ᐉ + ␭23 sin共␻t + ␸23兲.
共13e兲
Adopting the Oseen approximation H ⯝ H , one can derive
from the Langevin Eq. 共5兲 the following linear evolution
equation 共see App. A1兲
B3 ª
O
具Ṅk典 = −
k BT
3
2␲␮d13
冋冉
冊 册
2 d13 d13
+
− 1 具Nk典,
3 a1
a3
共14兲
where 具Ṅk典 ª 具dNk共t兲 / dt典. The 1 / a␣-parts are contributions to
the rotation rate due to noise on the spheres, whereas the
3
contribution is a correction due to hydrodynamic inter1 / d13
actions. Equation 共14兲 can be solved exactly. The solution
exhibits an exponentially decaying oscillatory behavior due
to the periodicity of the swimming stroke d13. However, for
ᐉ Ⰷ max兵␭␣␤其 it suffices to approximate d13 ⯝ 2ᐉ, yielding an
exponential decay DN共t兲 ⯝ exp共−t / ␶N兲 with orientation correlation time
␶N ⯝
再 冋 冉 冊 册冎
k BT
16␲␮ᐉ3
4 ᐉ
ᐉ
+
−1
3 a1 a3
.
共15兲
共16a兲
where
B3具ḋ12k典 + C具ḋ23k典
,
B 1B 3 − C 2
共16b兲
共16c兲
are the noise-averaged internal forces on the first and third
sphere 共the force on the central sphere can be eliminated by
virtue of F共1k兲 + F共2k兲 + F共3k兲 ⬅ 0兲, respectively, and
具ḋ12k典 = 具Ṅk典d12 + 具Nk典ḋ12 ,
共16d兲
具ḋ23k典 = 具Ṅk典d23 + 具Nk典ḋ23
共16e兲
the mean change in the leg vectors due stochastic rotations
and swimming strokes, with abbreviations
冊
冊
1
1
1
1
1
A1 ª
−
+
−
,
␥1 ␥2 4␲␮ d13 d23
1
1
1
1
1
−
+
−
,
A3 ª
␥3 ␥2 4␲␮ d13 d12
1
1
1
+
−
,
␥1 ␥2 2␲␮d12
共16j兲
具Ṙ共t兲典 = V具N共t兲典.
共17a兲
Here, V denotes the stroke-averaged velocity 共i.e, over an
interval 关t , t + T␻兴兲 of the corresponding deterministic swimmer 关25兴. By using the approximation Eq. 共17a兲 instead of
the exact results Eq. 共16兲 one neglects mean velocity oscillations on small time scales. For example, when considering
equal-sized beads with a␣ = a and ᐉ Ⰷ max兵a , ␭␣␤其, then
冉 冊
7
␭12␭23
sin ⌬␸ ,
a␻
24
ᐉ2
共17b兲
where ⌬␸ ª ␸12 − ␸23 is the phase difference of the leg contractions, and higher-order terms in a and ␭␣␤ are neglected.
Integrating Eq. 共17兲 with 具N共t兲典 ⯝ N共0兲exp共−t / ␶N兲, we obtain
for the position mean value of the swimmer the simple approximate result
具R共t兲典 ⯝ R共0兲 + V␶N关1 − exp共− t/␶N兲兴N共0兲;
C具ḋ12k典 + B1具ḋ23k典
具F共3k兲典 =
,
B 1B 3 − C 2
B1 ª
冊
Since the quantities 具Nk典, 具Ṅk典, d␣␤, and ḋ␣␤ are known, Eqs.
共16兲 provide a closed analytical result for the mean velocity
具Ṙk典 of a shape-driven swimmer 共within the Oseen approximation兲. In particular, Eqs. 共16兲 generalize the corresponding
velocity formulas for a deterministic swimmer, recently obtained by Golestanian and Ajdari 关25兴, to the “noisy swimming” regime.
For realistic swimmer parameters the orientation correlation time ␶N is typically much larger than the driving period
T␻ = 2␲ / ␻. In this case, the rather lengthy result Eq. 共16兲 can
be considerably simplified 共see last part of App. A2a for
details兲 to read
V=
A1
A3
具Ṙk典 = 具F共1k兲典 + 具F共3k兲典,
3
3
冉
冉
冉
共16i兲
1
1
1
1
1
−
+
−
.
␥2 4␲␮ d12 d23 d13
−1
The time parameter ␶N not only determines the temporal
correlation of the orientation vector, it also plays an important role for the dynamics of the geometric center R共t兲. As
shown in Appendix, the mean swimmer velocity 具Ṙ共t兲典 is
governed by the equation
具F共1k兲典 = −
Cª
1
1
1
+
−
,
␥2 ␥3 2␲␮d23
i.e., in the asymptotic limit t → ⬁,
具R共⬁兲典 = R共0兲 + V␶NN共0兲.
共19兲
Finally, let us still consider the mean-square displacement
DR共t兲 ª 具关R共t兲 − R共0兲兴2典 for a stiff, shape-driven three-sphere
swimmer described by Eqs. 共13兲. As discussed in App. A2b,
by starting from the Langevin equation for R共t兲, one can
show that DR共t兲 decomposes into the form
DR共t兲 = DRp 共t兲 + DRa 共t兲,
where the first part
DRp 共t兲 =
共16f兲
共16g兲
冉
冊
冊
1 k BT 1
1
1
2 k BT
+ +
t+
9 ␲␮ a1 a2 a3
9 ␲␮
⫻ds
共16h兲
共18兲
冉
1
1
1
+
+
d12 d23 d13
共20a兲
冕
t
0
共20b兲
comprises passive Brownian motion contributions due to
thermal diffusion of the spheres 共first line兲 and hydrodynamic Oseen interactions between them 共second line兲, while
the second part
021903-4
PHYSICAL REVIEW E 80, 021903 共2009兲
NOISY SWIMMING AT LOW REYNOLDS NUMBERS
冕 冕
t
DRa 共t兲 ⯝ V2
t
ds
du具N共s兲N共u兲典
共20c兲
0
0
is the contribution due to active self-swimming. Similar to
Eq. 共17兲, the expression Eq. 共20c兲 is valid if the orientation
correlation time is much larger than the stroke period, ␶N
Ⰷ T␻. Inserting the above result 具N共t兲N共s兲典 ⯝ exp共−t / ␶N兲 and
approximating d12 = d23 = d13 / 2 ⯝ ᐉ, we obtain for the spatial
mean-square displacement
DR共t兲 ⯝
冉
冊
1 k BT 1
1
1 5
+ + + t
9 ␲␮ a1 a2 a3 ᐉ
+ 2V2␶N关t + ␶N共e−t/␶N − 1兲兴.
共21兲
The approximate result Eq. 共21兲 provides a coarse-grained
stroke-averaged description of the translational diffusion 共details of the swimming stroke are encoded in V兲. An analogous formula can be used to describe the diffusion of a
spherical, chemically driven microswimmer 关6兴. By virtue of
Eq. 共21兲, we can readily distinguish three distinct regimes:
共i兲 For t Ⰶ ␶N we can expand the exponential term to linear
order and find
DR共t兲 ⯝
DRp 共t兲,
IV. SWIMMER TUNING & NUMERICAL SIMULATIONS
We simulate a single three-sphere swimmer described by
the interaction potentials Eqs. 共8兲 and 共9兲 and governed by
the Langevin Eq. 共5兲. We consider identical spheres of radius
a␣ = a, mass M ␣ = M, and equal stroke amplitudes ␭12 = ␭23
= ␭. The density of the spheres is chosen as ␳ = 103 kg/ m3
共water兲, and the fluid is water at room temperature 关␮
= 10−3 kg/ 共ms兲 , T = 300 K兴. By fixing the parameters of
the spring and bending potentials as k0 = 10−4 kg/ s2, ␻
= 103 Hz, and K = 10k0, we satisfy the time scale condition
Eq. 共8b兲 while ensuring that the swimmer behaves approximately stiff and shape-driven. The velocity V of the swimmer
is optimized by choosing ⌬␸ = ␲ / 2, cf. Eq. 共23a兲.
We are primarily interested in understanding how a
change in the swimmer size may affect the diffusive behavior. We therefore fix the ratios ᐉ / a = ᐉ / ␭ = 10 and only vary
the leg length ᐉ from 1 to 10 ␮m in our simulations 共i.e., the
mean swimmer length is ⌳ = 2ᐉ兲. Put differently, we scale the
swimmer proportionally by varying ᐉ. Having specified all
parameters, it is useful to summarize the relevant formulae
for our choice
V⯝
共22a兲
i.e., passive Brownian motion dominates on very short time
scales.
共ii兲 For t ⱗ ␶N we need to include terms quadratic in t / ␶N
and obtain
Vt
␶N ⯝
DRp
t
2 2
DR共t兲 ⯝ DRp 共t兲 +
␶N
,
共22b兲
i.e., ballistic motion can dominate on intermediate time
scales provided V2 / ␶N is large enough 共cf. examples below兲.
共iii兲 For t Ⰷ ␶N, we recover diffusive behavior
lim
t→⬁
DR共t兲
t
=
DRp 共t兲
t
+ 2V2␶N .
共22c兲
If the swimmer is constructed such that 2V2␶N Ⰷ DRp 共t兲 / t,
then its diffusive behavior at large times is due to noiseinduced rotation 共with persistence time ␶N兲.
In a sense, the above results provide a “microscopic” justification for the assumptions made by Lovely and Dahlquist
关39兴, who studied purely probabilistic models of bacterial
motion. We also note that the asymptotic behavior for t
Ⰷ ␶N and 2V2␶N Ⰷ DRp 共t兲 / t agrees qualitatively with results
reported by Lobaskin et al. 关33兴 for triangular swimmers in
the moderate Reynolds number regime R ⬃ 1. In this context, we also mention recent work by Golestanian et al. 关50兴,
who discuss similar scaling relations for the diffusion of
phoretic swimmers.
In the remainder, we are going to compare the analytical
predictions with results of computer simulations, based on a
direct numerical integration of the Langevin Eq. 共5兲 for the
spheres. Our main focus is on the transition from the passive
Brownian motions to the active swimming regime.
冉 冊
冉 冊
冉 冊
7
␭2
ᐉ
a␻ 2 sin ⌬␸ = 0.292 ,
24
ᐉ
s
16␲␮ᐉ3 8ᐉ
−1
k BT
3a
⯝
−1
= 0.473
ᐉ 3s
,
␮m3
1 k BT 3 5
␮m3
= 5.127
+
,
9 ␲␮ a ᐉ
ᐉs
共23a兲
共23b兲
共23c兲
and
2V2␶N = 0.080
ᐉ5
.
␮ m 3s
共23d兲
It is remarkable that increasing the swimmer size by one
order of magnitude increases the orientation correlation time
␶N by three orders of magnitude.
Figures 1 and 2 depict the results of numerical simulations 共symbols兲 of the Langevin Eq. 共5兲 and also the corresponding theoretical predictions 共dashed lines兲. The numerical data points represent averages over 100 trajectories with
identical initial conditions. More precisely, at time t = 0 the
swimmer is pointing along the x3 axis with the first sphere
being located at the origin, i.e., X1共0兲 = 0, X2共0兲 = ᐉN共0兲, and
X3共0兲 = 关2ᐉ − ␭ sin共⌬␸兲兴N共0兲, where N共0兲 = 共0 , 0 , 1兲.
As evident from the diagrams in Figs. 1 and 2, the results
of the numerical simulations are very well matched by the
theoretical curves over several orders of magnitude in time.
In particular, for a leg length in the range ᐉ ⬃ 5 ␮m 共red
“+”/green “⫻” symbols兲 one readily observes the three
aforementioned regimes: 共i兲 Brownian diffusion at small
time scales t Ⰶ ␶N, 共ii兲 ballistic behavior at intermediate time
scales, and 共iii兲 quasidiffusive behavior due to noise-induced
rotation for t Ⰷ ␶N.
We conclude the discussion of the numerical results by
addressing a few technical aspects that might be relevant and
helpful for future simulations. When considering ensembles
021903-5
PHYSICAL REVIEW E 80, 021903 共2009兲
DN (t) = N (t)N (0)
JÖRN DUNKEL AND IRWIN M. ZAID
displacement and other statistical observables agree with
those obtained for very small time steps dt = 0.1T␥. Generally, a satisfactory resolution of the bending and relaxation
dynamics of the legs/spheres would require dt Ⰶ 共M / K兲1/2
and dt Ⰶ 共M / k0兲1/2, respectively.
1
0.5
◦ = 1.0 µm
+ = 2.5 µm
× = 5.0 µm
= 10.0 µm
0
0.01
0.1
1
10
100
V. CONCLUSIONS
1000
t [s]
FIG. 1. 共Color online兲 Orientation correlation function for four
different swimmer sizes 2ᐉ. Symbols represent averages over 100
trajectories, numerically calculated from the Langevin Eq. 共5兲 using
parameters as described in the text. The dashed lines depict the
theoretically predicted exponential decay DN共t兲 ⯝ exp共−t / ␶N兲 with
orientation correlation time ␶N determined by Eq. 共15兲. It is remarkable that changing the swimmer size by one order of magnitude
increases the correlation time by three orders of magnitude.
with N ⬎ 1 swimmers the computationally most expensive
step is the Cholesky decomposition of the diffusion tensor,
see Eq. 共3兲, which is approximately of the order O关共pN兲3兴
关36兴. It is also worthwhile to briefly comment on the choice
of the time step dt in the Langevin simulations. Ideally, one
would like to choose dt smaller than the smallest dynamical
time scale in the system, which for our model is given by the
damping time T␥ = M / 共6␲␮a兲. For the swimmer parameters
considered here we find T␥ ⬃ 2 ⫻ 10−9共ᐉ / ␮m兲2s, which
means that adopting dt ⬃ T␥ would not allow us to simulate
experimentally accessible time scales in the seconds range.
Since we are not interested in the dynamical details at very
short times scales, we choose in our simulations the time step
larger than T␥, but much smaller than the period T␻ of a
swimming stroke by fixing dt = 10−3T␻ for ᐉ ⬍ 5 ␮m and
dt = 10−2T␻ if ᐉ ⱖ 5 ␮m. We verified, however, that for intermediate time scales 共of the order of a few stroke periods
T␻ ⬃ 6 ⫻ 10−3 s兲 the numerical results for the mean square
DR (t)/t [µm2 /s]
104
103
102
◦ = 1.0 µm
+ = 2.5 µm
× = 5.0 µm
= 10.0 µm
101
Understanding the interplay between Brownian motion,
hydrodynamic interactions, and self-propulsion is a prerequisite for understanding the dynamics of bacteria and artificial
swimming devices at the microscale. In the first part, we
discussed how one can model these phenomena by means of
stochastic processes 共overdamped Langevin equations兲. Subsequently, as a first application, we focused on the size dependence of diffusive behavior at low Reynolds numbers
共R Ⰶ 1兲 for a quasilinear three-sphere swimmer model. Our
theoretical analysis complements a recent experimental study
by Howse et al. 关6兴, who investigated the diffusion of chemically driven, spherical colloids, and theoretical work by Lobaskin et al. 关33兴, who considered the Brownian dynamics of
an artificial triangular microswimmer at moderate Reynolds
numbers 共R ⬃ 1兲.
Starting from the Kirkwood-Smoluchowski equation
关34–37兴, we derived analytical results for the orientation correlation time, the mean velocity, and the mean-square displacement of an overdamped, quasilinear three-sphere swimmer 关20,25兴. Analytical formulae as derived here are useful
for testing numerical simulations and 共in兲 validating simplified probabilistic models 关39兴. Moreover, they provide detailed insight into the size-regulated transition from predominantly random to quasiballistic motions.
The proposed method of modeling swimmers by effective
potentials within a Langevin scheme can be readily extended
to study complex behavior in larger swimmer ensembles.
However, at high swimmer densities, collisions, and nearfield hydrodynamics affect the diffusive behavior 关47,51,52兴
and it will be necessary to modify the hydrodynamic interaction tensor accordingly. Generally, a useful dimensionless
quantifier for the efficiency of active swimming relative to
diffusion is given by the ratio
⑀=
100
10
−1
0.001 0.01
0.1
1
10
100
1000
t [s]
FIG. 2. 共Color online兲 Mean-square displacement DR共t兲
ª 具关R共t兲 − R共0兲兴2典 divided by time t for the same set of parameters
as in Fig. 1. The dashed lines correspond to the analytical formula
共21兲. The dynamics of small swimmers 共ᐉ = 1 ␮m, blue circles兲 is
dominated by Brownian motion on all time scales, whereas big
swimmers 共ᐉ = 10 ␮m, black diamonds兲 can move ballistically for
several minutes. In the intermediate region 共ᐉ ⬃ 5 ␮m, red
“+“/green “⫻“兲 we observe a ballistic transition from ordinary
Brownian motion at small times t Ⰶ ␶N to noise-induced rotational
diffusion at large times t Ⰷ ␶N. Since ␶N ⬃ ᐉ3, the transition from
Brownian to quasiballistic motion is very sharp. Interestingly
enough, typical sizes of bacteria lie in or near this niche 关10兴.
␶N
.
⌳/V
共24兲
The denominator corresponds to the time needed by a unperturbed swimmer of velocity V to move one body length ⌳,
and the swimmer geometry is encoded in the orientation correlation time ␶N. For ⑀ Ⰷ 1 共⑀ Ⰶ 1兲 self-propulsion is effective
共noneffective兲. For nonisolated swimmers, ␶N is not only determined by thermal effects, but also by collisions with other
swimmers 关51,52兴.
With regard to future studies we note that the combination
of thermal fluctuations and hydrodynamic coupling might
also lead to interesting behavior in simple arrangements of
microswimmers. For example, experiments have shown that
colloidal spheres localized in an array of optical traps create
memory effects 关53兴 and driven vibrations 关54兴. Our formalism provides a starting point for the investigation of many
021903-6
PHYSICAL REVIEW E 80, 021903 共2009兲
NOISY SWIMMING AT LOW REYNOLDS NUMBERS
self-propelled bodies in separate potentials, which could be
helpful for interpreting experimental data of trapped bacteria
or for constructing pumps from a collection of microswimmers.
To summarize, the above results may provide guidance
for constructing artificial microswimmers or pumps that
work efficiently in the critical transition region that separates
Brownian from quasideterministic motions. In particular, by
tuning the parameters to the narrow cross-over region one
could construct swimmers that explore with high probability
a maximized volume fraction within a given period of time.
The fact that many bacteria live near or exactly in this niche
关10兴 suggest that this may indeed be a useful strategy. Thus,
exploiting the interplay between noise and active self-motion
could lead to novel applications 关55兴, e.g., with regard to the
controlled transport 关56兴 and distribution of chemical and
biological substances in small scale technical devices or even
within the human body.
H共O␣i兲共␤ j兲 ª
⯝
The authors would like to thank Matthieu Dufay, Bortolo
Mognetti, Olivier Pierre-Louis, Victor Putz, and Julia Yeomans for helpful discussions. This work was supported by
the EPSRC grant no. EP/D050952/1 共J.D.兲.
APPENDIX: CALCULATIONS
This appendix provides derivations of analytical results
for the orientation correlation function, the mean velocity,
and the spatial mean square displacement from the Langevin
Eq. 共5兲. Our calculations are based on the following simplifying assumptions:
共i兲 The motion of the three-sphere swimmer is approximately stiff and shape-driven, i.e.,
共A1a兲
d23 ª X3 − X2 ⯝ Nd23 ,
共A1b兲
d13 ª X3 − X1 ⯝ N共d12 + d23兲,
共A1c兲
d12 = ᐉ + ␭12 sin共␻t兲,
共A1d兲
d23 = ᐉ + ␭23 sin共␻t − ⌬␸兲.
共A1e兲
共A2a兲
and, with F3i + F2i + F1i = 0,
F2i = − 共g1 + g3兲Ni ,
共A2b兲
where g␣ is the force amplitude, and Ni denotes a component
of the orientation vector
N共t兲 ª
X3 − X1
.
兩X3 − X1兩
1
共␦ij + NiN j兲.
8␲␮d␣␤
共A3兲
1. Orientation correlation function
Given a deterministic initial state N共0兲 = 关Nk共0兲兴, we are
interested in DN共t兲 ª 具N共t兲N共0兲典 = 具Nk共t兲典Nk共0兲.
The first step is to find the stochastic differential equation
共SDE兲 for the orientation vector N共t兲. This can be achieved
by virtue of the Ito formula 关49,57兴, yielding
共A2c兲
共A4兲
where dX共␣i兲共t兲 is governed by Eq. 共5兲, and the diffusion
tensor is given by D ª kBTH ⯝ kBTHO.
To determine 具Nk共t兲典, we use that 具C共␣i兲共␥k兲dB共␥k兲共t兲典 = 0 for
an Ito SDE, and therefore
d具X共␣i兲共t兲典 = 具H共␣i兲共␤ j兲F共␤ j兲典dt.
共A5兲
Taking the average of Eq. 共A4兲 and inserting Eq. 共A5兲, we
obtain
具Ṅk共t兲典 = 具H共␣i兲共␤ j兲F共␤ j兲⳵共␣i兲Nk典 + 具D共␣i兲共␤ j兲⳵共␣i兲⳵共␤ j兲Nk典,
共A6兲
where d具Ṅk共t兲典 ª 具dNk共t兲 / dt典. We next evaluate the two
terms the right-hand side of Eq. 共A6兲 separately. To this end
we note that
⳵共␣ j兲Nk =
1
共␦3␣ − ␦1␣兲共␦ jk − N jNk兲,
d13
共A7兲
yielding for the first term
H共␣i兲共␤ j兲F共␤ j兲⳵共␣i兲Nk = 关H共3i兲共␤ j兲 − H共1i兲共␤ j兲兴
⫻
F共␤ j兲
d13
共␦ik − NiNk兲.
Using Oseen approximation Eq. 共A3兲, we find
In this case the internal forces, which generate the swimming
strokes, point along the swimmer’s axis and we may write
F3i = g3Ni
冊
The second line follows from assumption Eq. 共A1兲.
where
F1i = g1Ni,
冉
1
d d
␦ij + ␣␤i2 ␣␤ j ,
8␲␮d␣␤
d␣␤
dNk共t兲 = ⳵共␣i兲NkdX共␣i兲 + D共␣i兲共␤ j兲⳵共␣i兲⳵共␤ j兲Nkdt,
ACKNOWLEDGMENTS
d12 ª X2 − X1 ⯝ Nd12 ,
共ii兲 We adopt the Oseen approximation, i.e., H ⯝ HO
where
H共3i兲共␤ j兲F共␤ j兲 =
冋
共A8兲
册
共g3 + g1兲
g3
g1
+
−
N,
␥3 4␲␮d31 4␲␮d32 i
and, similarly, H共1i兲共␤ j兲F共␤ j兲 ⬀ Ni. Taking into account that
Ni共␦ik − NiNk兲 ⬅ 0,
共A9兲
the first term on the right-hand side of Eq. 共A6兲 vanishes, and
Eq. 共A6兲 reduces to
具Ṅk共t兲典 = 具D共␣i兲共␤ j兲⳵共␣i兲⳵共␤ j兲Nk典.
By virtue of Eq. 共A7兲, we find
021903-7
共A10兲
PHYSICAL REVIEW E 80, 021903 共2009兲
JÖRN DUNKEL AND IRWIN M. ZAID
1
⳵共␣i兲⳵共␤ j兲Nk =
2
d13
3
1
具Ṙi共t兲典 = 兺 具H共␣i兲共␤ j兲F共␤ j兲典
3 ␣=1
共␦3␤ − ␦1␤兲共␦3␣ − ␦1␣兲
⫻ 共3N jNkNi − N j␦ik − Nk␦ij − Ni␦ jk兲.
3
共A11兲
To obtain the right-hand side of Eq. 共A10兲, we still need to
contract with the diffusion tensor D共␣i兲共␤ j兲. This results in two
contributions: From the diagonal part we get
1
共␦3␣ − ␦1␣兲共␦3␣ − ␦1␣兲
␥
共␣ j兲 ␣
k BT 兺
⫻
1
2
d13
= − k BT
冊
1 2Nk
1
+
2 ,
␥1 ␥3 d13
共A12a兲
3
2␲␮d13
冓冉
1
g1 g2 g3
g2
g3
+
+
+
+
Ni
3
␥1 ␥2 ␥3 4␲␮d12 4␲␮d13
冊
g1
g3
g1
g2
+
+
+
4␲␮d12 4␲␮d23 4␲␮d13 4␲␮d23
冋
⫻
册
k BT
1 2
1
+
具Nk典,
2 +
3
␥1 ␥3 d13 2␲␮d13
¬
2. Motion of the geometric center
冋
冉
1
1
1
1
1
−
+
−
␥3 ␥2 4␲␮ d13 d12
冊册
冊册
1
+ 具Nig3典
3
A1
A3
具Nig1典 + 具Nig3典.
3
3
Following a similar procedure, we can derive analytical
expression for the mean velocity 具Ṙ共t兲典 and the mean-square
displacement DR共t兲.
a. Mean velocity
First, we would like to determine the mean velocity 具Ṙ共t兲典
of the swimmer’s geometric center
共A17兲
具ḋ12k典 = 具Ṅk典d12 + 具Nk典ḋ12 ,
共A18a兲
具ḋ23k典 = 具Ṅk典d23 + 具Nk典ḋ23 .
共A18b兲
On the other hand, from the definition of the vectors d␣␤ and
the Langevin equations for dX␣, we have
共A13兲
具ḋ12k典 = 具H共2k兲共␤ j兲F共␤ j兲 − H共1k兲共␤ j兲F共␤ j兲典,
共A19a兲
具ḋ23k典 = 具H共3k兲共␤ j兲F共␤ j兲 − H共2k兲共␤ j兲F共␤ j兲典.
共A19b兲
Inserting the explicit expressions for H共␣i兲共␤ j兲 and F共␤ j兲, Eqs.
共A19兲 can be rewritten as
Averaging the stochastic differential equation
1
dR共t兲 ª 共dX1 + dX2 + dX3兲
3
冊冔
Consequently, to find 具Ṙi共t兲典, we still need to determine
the mean forces 具F共␤i兲典 = 具Nig␤典 on the first and last sphere,
␤ = 1 , 3. This can be achieved as follows: Eq. 共A1兲 implies
that
which for ␥␣ = 6␲␮a␣ gives Eq. 共14兲.
1
R共t兲 ª 共X1 + X2 + X3兲.
3
冉
1
1
1
1
1
1
具Ṙi共t兲典 = 具Nig1典
−
+
−
3
␥1 ␥2 4␲␮ d13 d23
Combining the two contributions we find
冋 冉
共␦ij + NiN j兲
g␤
g ␤N j = N i 兺
.
8␲␮d␣␤
␤⫽␣ 4␲␮d␣␤
Since the internal swimming forces sum to zero, we may
eliminate g2 by using g2 = −共g1 + g3兲 to obtain
共A12b兲
具Ṅk共t兲典 = − kBT
冔
+ 共1 − ␦␣␤兲H共␣i兲共␤ j兲F共␤ j兲 .
Inserting this into Eq. 共A15兲 gives
+
Nk .
␥␣
Considering as before a stiff, shape-driven swimmer and
Oseen interactions H = HO, we have F共␤i兲 ⯝ g␤Ni and, therefore,
具Ṙi共t兲典 =
共␦ij + NiN j兲 1
⫻
2 共3NiNkN j − N j␦ik − Nk␦ij − Ni␦ jk兲
8␲␮d␣␤ d13
=
F共␣i兲
共A16兲
兺 共1 − ␦␣␤兲共␦3␣ − ␦1␣兲共␦3␤ − ␦1␤兲
共␣i兲共␤ j兲
k BT
冓
共A15兲
while the hydrodynamic off-diagonal terms give
k BT
1
兺
3 ␣=1
兺 H共␣i兲共␤ j兲F共␤ j兲 ⯝ ␤兺
␤⫽␣
⫽␣
共3N jNkN j − N j␦ jk − Nk␦ jj − N j␦ jk兲
冉
=
共A14兲
具ḋ12k典 = −
with respect to the underlying Wiener process and dividing
by dt, we obtain
−
021903-8
冋
冋
册
1
1
1
+
−
具Nkg1典
␥1 ␥2 2␲␮d12
冉
1
1
1
1
1
−
+
−
␥2 4␲␮ d12 d23 d13
冊册
具Nkg3典
PHYSICAL REVIEW E 80, 021903 共2009兲
NOISY SWIMMING AT LOW REYNOLDS NUMBERS
¬− B1具Nkg1典 − C具Nkg3典
and
具ḋ23k典 =
冋
冉
共A20a兲
1
1
1
1
1
−
+
−
␥2 4␲␮ d12 d23 d13
冋
册
冊册
1
1
1
+
−
具Nkg3典
+
␥2 ␥3 2␲␮d23
共A20b兲
具ḋ12k典 = − B1具Nkg1典 − C具Nkg3典,
共A21a兲
具ḋ23k典 = C具Nkg1典 + B3具Nkg3典,
共A21b兲
With left-hand side given by Eqs. 共A19兲. This is easily done
and we may summarize the result for the mean velocity
A1
A3
具Nkg1典 + 具Nkg3典,
3
3
共A22a兲
where
B3具ḋ12k典 + C具ḋ23k典
,
具Nkg1典 = −
B 1B 3 − C 2
dsV共s兲,
共A25兲
具Ṙk共t兲典 ⯝ V具Nk共t兲典.
共A26兲
t
For example, when considering equal-sized beads with a␣
= a and ᐉ Ⰷ max兵a , ␭12 , ␭23其, then
Hence, in order to obtain the unknown expectation values
具Nkg1典, we have to solve the linear system
具Ṙk共t兲典 =
V共t兲 =
a␻␭23
a␻␭12
cos共␻t + ␸12兲 +
cos共␻t + ␸23兲 + V,
ᐉ
ᐉ
where
冉 冊
7
␭12␭23
a␻
sin ⌬␸
24
ᐉ2
V=
C具ḋ12k典 + B1具ḋ23k典
,
B 1B 3 − C 2
b. Spatial diffusion
Using the result for dR共t兲 from above, we may rewrite the
mean square displacement DR共t兲 as
DR共t兲 ª 具关R共t兲 − R共0兲兴2典
共A22b兲
=
冓冕
t
dRk共u兲
=
冕 冕 冓冋
t
ds
0
冋
⫻
共A22d兲
du
0
共A22e兲
Since the quantities 具Nk典, 具Ṅk典, d␣␤, ḋ␣␤, A␣, B␣, and C are
known we have thus obtained a closed analytical result for
the mean swimmer velocity within the Oseen approximation.
Analogous calculations can be performed for HM , but do not
yield much additional insight 共for a single swimmer兲.
Additional simplifications. If the orientation correlation
time ␶N is larger than the driving period T = 2␲ / ␻ then
具ḋ12k典 ⯝ 具Nk典ḋ12,
具ḋ23k典 ⯝ 具Nk典ḋ23 .
⬘
冉
A1
A3
N kg 1 + N kg 3
3
3
冕
册冔
册
s
u
具关C共␣k兲共␥n兲兴sdB共␥n兲共s兲
共A28兲
Here, we have again used that 具兰f共X␣兲dB共␥n兲共t兲典 = 0 holds for
Ito integrals. We consider the two remaining integrals in Eq.
共A28兲 separately, starting with the second one. We find
3
DRp 共t兲 ª
共A23兲
1
兺
9 ␣,␣ =1
⬘
冕
具关C共␣k兲共␥n兲兴sdB共␥n兲共s兲
⫻ 关C共␣⬘k兲共␥⬘n⬘兲兴udB共␥⬘n⬘兲共u兲典
具Ṙk共t兲典 ⯝ V共t兲具Nk共t兲典,
V共t兲 = −
冔
⫻ 关C共␣⬘k兲共␥⬘n⬘兲兴udB共␥⬘n⬘兲共u兲典.
In this case, we may simplify
where
1
兺
9 ␣,␣ =1
+
dRk共s兲
A1
A3
N kg 1 + N kg 3
3
3
3
具ḋ23k典 = 具Ṅk典d23 + 具Nk典ḋ23 .
冕
t
0
t
共A22c兲
with
具ḋ12k典 = 具Ṅk典d12 + 具Nk典ḋ12 ,
共A27兲
and ⌬␸ ª ␸12 − ␸23, and higher-order terms have been neglected.
0
具Nkg3典 =
t+T␻
so that
具Nkg1典
¬C具Nkg1典 + B3具Nkg3典.
冕
Vª
冊 冉
A3 Cḋ12 + B1ḋ23
A1 B3ḋ12 + Cḋ23
+
3
3
B 1B 3 − C 2
B 1B 3 − C 2
3
共A24a兲
=
2
兺
9 ␣,␣ =1
⬘
冊
2
= k BT
9
共A24b兲
is a periodic function, V共t兲 = V共t + T␻兲. Since we assumed
␶N Ⰷ T␻, we can achieve further simplification by replacing
V共t兲 with its stroke average
冕
t
ds具D共␣k兲共␣⬘k兲典
0
冕 冋
t
ds
0
3
3
3
3
+
+
␥1 ␥2 ␥3
2
+ kBT 兺 共1 − ␦␣␣⬘兲
9
␣,␣ =1
⬘
where
021903-9
册
冕
t
0
O
dsH共␣k兲共␣⬘k兲 ,
PHYSICAL REVIEW E 80, 021903 共2009兲
JÖRN DUNKEL AND IRWIN M. ZAID
O
H共␣k兲共␣⬘k兲 =
1
␦kk + NkNk
=
.
8␲␮d␣␣⬘ 2␲␮d␣␣⬘
共A29兲
For spherical particles we have ␥␣ = 6␲␮a␣ and, therefore,
冉
冊
1 k BT 1
1
1
+ +
t
DRp 共t兲 =
9 ␲␮ a1 a2 a3
2 k BT
9 ␲␮
+
冕 冉
t
ds
0
relation functions. However, instead of calculating these correlation functions exactly, we may approximate, for ␶N Ⰷ T␻,
the integrand by 关cf. Eqs. 共A17兲 and 共A26兲兴
具关 . . . 兴s关 . . . 兴u典 ⯝ V2具Nk共s兲Nk共u兲典,
where V is the stroke-averaged velocity of the corresponding
deterministic swimmer, cf. Eqs. 共A25兲 and 共A26兲. Adopting
this approximation we find
冊
1
1
1
+
+
.
d12 d23 d13
t
For ᐉ Ⰷ max兵␭␣␤其, the integrand in the second line can be
approximated by 5 / 共2ᐉ兲 yielding
DRp 共t兲
冉
冊
1 k BT 1
1
1 5
⯝
+ + + t.
9 ␲␮ a1 a2 a3 ᐉ
冕 冕 冓冋
t
DRa 共t兲 ª
ds
0
冋
du
0
A1
A3
N kg 1 + N kg 3
3
3
A1
A3
N kg 1 + N kg 3
⫻
3
3
册冔
册
t
ds
0
du具Nk共s兲Nk共u兲典
0
t
= V2
t
ds
0
共A30兲
It remains to determine the first 共double兲 integral in Eq.
共A28兲, reading
t
冕 冕
冕 冕
DRa 共t兲 ⯝ V2
du exp共− 兩u − s兩/␶N兲
0
= 2V2␶N关t + ␶N共e−t/␶N − 1兲兴,
and thus the final result
DR共t兲 = DRp 共t兲 + DRa 共t兲
s
⯝
.
u
冉
冊
1
1 5
1 k BT 1
+ + + t
9 ␲␮ a1 a2 a3 ᐉ
+ 2V2␶N关t + ␶N共e−t/␶N − 1兲兴.
共A31兲
The subscripts indicate the time arguments in the bracketed
expressions, respectively. Upon recalling that F共␣i兲 = g␣Ni is
the internal force acting on sphere ␣, we see that the contribution DRa 共t兲 is essentially determined by the force-force cor-
The first part represents passive 共thermal兲 diffusion; the second part is due to active swimming 共note that ␶N is temperature dependent as well兲.
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