PHYSICAL REVIEW E 80, 021111 共2009兲
Subdiffusive behavior in a trapping potential: Mean square displacement
and velocity autocorrelation function
M. A. Despósito1,2,* and A. D. Viñales1
1
Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
2
Consejo Nacional de Investigaciones Científicas y Técnicas, Buenos Aires, Argentina
共Received 16 February 2009; published 14 August 2009兲
A theoretical framework for analyzing stochastic data from single-particle tracking in viscoelastic materials
and under the influence of a trapping potential is presented. Starting from a generalized Langevin equation, we
found analytical expressions for the two-time dynamics of a particle subjected to a harmonic potential. The
mean-square displacement and the velocity autocorrelation function of the diffusing particle are given in terms
of the time lag. In particular, we investigate the subdiffusive case. Using a power-law memory kernel, exact
expressions for the mean-square displacement and the velocity autocorrelation function are obtained in terms
of Mittag-Leffler functions and their derivatives. The behaviors for short-, intermediate-, and long-time lags are
investigated in terms of the involved parameters. Finally, the validity of usual approximations is examined.
DOI: 10.1103/PhysRevE.80.021111
PACS number共s兲: 02.50.⫺r, 05.40.⫺a, 05.10.Gg, 05.70.Ln
I. INTRODUCTION
The viscoelastic properties of complex fluids, such as
polymers, colloids, and biological materials, can be derived
from the dynamics of individual spherical particles embedded in it 关1,2兴. Particle tracking microrheology experiments
关1–8兴 are based on the observation of the motion of individual tracer particles. In a typical microrheology experiment, particle positions are recorded in the form of a time
sequence and information about the dynamics is essentially
extracted by measuring the mean-square displacement
共MSD兲 of the probe particles 关9–11兴.
On the other hand, during the last 2 decades, optical traps
共or “optical tweezers”兲 have been increasingly used for position detection, covering a wide range of applications in
physics and biology 关3,12–17兴. In an optical trap, the interaction between the laser and the trapped particle can be approximated by a harmonic potential 关18兴. If the particle is
embedded in a Newtonian fluid, the temporal behavior of its
position X共t兲 is described by the standard Langevin equation
mẌ共t兲 + ␥Ẋ共t⬘兲 + ␬X = F共t兲,
共1兲
where m is the mass of the particle, ␥ is the friction coefficient, ␬ is the trap stiffness, and F共t兲 represents the random
thermal force, which is zero centered and has a flat power
spectrum 关19兴. From an experimental point of view, calibration of the trap stiffness is necessary to determine the trapping force at any position where the trapping potential is
harmonic. A usually employed method for calibration in normal viscous fluids is the power spectrum method 关19兴.
Quantitative measurements with optical traps in normal
viscous fluids have been applied in numerous microrheology
experiments. Optical tweezers microrheological technique
was extended to measure the viscoelastic properties of complex fluids 关20–22兴. However, it has been recently noted that
the Langevin equation 共1兲 fails to describe the stochastic mo-
tion of the trapped bead when it is embedded in a viscoelastic medium 关20,21兴. In this case, the dynamics of the particle
differs from that in a pure viscous media because the stochastic process exhibits an anomalous diffusive behavior
关2,11,22–24兴. As a consequence, the use of a trapping complicates the analysis of the obtained data, since the interactions with the viscoelastic environment overlap with the influence of the trapping force 关1,22,25兴. Moreover, calibration
by power spectrum method becomes invalid because it based
on the assumption that a bead is trapped in a purely viscous
fluid. For example, in the application of optical traps in living cells, it is necessary to take into account the a priori
unknown viscoelastic properties of the medium to properly
calibrate the trap 关20兴.
When particles diffuse through a soft complex fluids or
biological materials, the mean-square displacement exhibits
a slow relaxation with a power-law decay in the range of
large times. In absence of external forces, a theoretical description is now well established. A quantitative analysis of
the resulting subdiffusive behavior enables one to extract the
rheological properties of the material 关1兴. Based on a generalized Langevin equation 共GLE兲, Mason and co-workers
关5,6兴 obtained a direct relation in the Laplace domain between the mean-square displacement of free tracer particles
and the viscoelastic parameters of the medium.
On the other hand, if the anomalously diffusing particle is
subjected to an external harmonic well, the decay of the position autocorrelation function is strongly nonexponential
关17,26,27兴. For example, the corresponding GLE for a particle in a viscoelastic medium and subjected to a harmonic
potential was recently investigated by us in Ref. 关27兴, where
analytical expressions for the evolution of mean values and
variances in terms of Mittag-Leffler functions were obtained
for any temporal range and involved parameters. Nevertheless, from an experimental point of view, it is necessary to
get expressions for the two-time correlation functions. For
instance, the mean-square displacement can be expressed as
␳共␶兲 = lim 具关X共t + ␶兲 − X共t兲兴2典,
t→⬁
*[email protected]
1539-3755/2009/80共2兲/021111共7兲
共2兲
where 兩X共t + ␶兲 − X共t兲兩 is the particle displacement between
021111-1
©2009 The American Physical Society
PHYSICAL REVIEW E 80, 021111 共2009兲
M. A. DESPÓSITO AND A. D. VIÑALES
two time points, t denotes the absolute time, while ␶ is the
so-called lag time 关28兴. Alternative information about the
experimentally observed diffusive behavior can be extracted
from the normalized velocity autocorrelation function
共VACF兲 关29兴 defined as 关27,30兴
CV共␶兲 = lim
t→⬁
具V共t + ␶兲V共t兲典
.
具V2共t兲典
共3兲
Then, to calculate the MSD and the VACF, one must know
the behavior of the two-time correlations 具X共t + ␶兲X共t兲典 and
具V共t + ␶兲V共t兲典.
The aim of this paper is to obtain analytical expressions
for the MSD 共2兲 and VACF 共3兲 for a harmonically bounded
particle immersed in a viscoelastic environment. This enables us to provide a theoretical framework to analyze the
data obtained by particle tracking microrheology experiments in viscoelastic media and using a trapping potential.
For this purpose, in Sec. II, we present the corresponding
GLE. The two-time dynamics is obtained, which enables us
to calculate the MSD and VACF for arbitrary memory kernels. Section III is devoted to the study of the subdiffusive
case. The complete analytical solutions are obtained and approximate expressions for different temporal ranges are
given. Moreover, the validity of the overdamped approximation is analyzed. Finally, a summary of our results is presented in Sec. IV.
of the general form f = −共␬XX + ␬Y Y + ␬ZZ兲 and assuming that
the local environment surrounding the sphere is isotropic.
However, our model could also be used to investigate the
case of a trapped particle close to a limiting wall, where the
motion becomes anisotropic but the drag term can be separated into components parallel and perpendicular to the surface 关32兴.
The GLE 共4兲 can be formally solved by means of the
Laplace transformation. Taking into account the deterministic initial conditions x0 = X共0兲 and v0 = Ẋ共0兲, the evolution of
the Laplace transform of the position X共t兲 reads
X̂共s兲 = x0
冉
冊冉
冊
1
1
− ␻20Î共s兲 + v0 + F̂共s兲 Ĝ共s兲,
s
m
共7兲
where F̂共s兲 is the Laplace transform of the noise. The relaxation function G共t兲 is the Laplace inversion of
Ĝ共s兲 =
1
,
ˆ
s + s␥共s兲 + ␻20
2
共8兲
where ␥ˆ 共s兲 is the Laplace transform of the damping kernel
and
Ĝ共s兲
s
Î共s兲 =
共9兲
is the Laplace transform of
I共t兲 =
II. DIFFUSION IN A HARMONIC WELL
冕
t
dt⬘G共t⬘兲.
共10兲
0
A. Formal solution for the GLE
In what follows, we consider the dynamics of a spherical
bead of mass m immersed in a complex or viscoelastic environment and simultaneously trapped in a harmonic potential.
The resulting motion can be described by the following
GLE:
mẌ共t兲 + m
冕
t
dt⬘␥共t − t⬘兲Ẋ共t⬘兲 +
m␻20X
= F共t兲,
where ␻0 = 冑␬ / m is the frequency of the trap, ␥共t兲 is the
dissipative memory kernel representing the viscoelastic friction, and the thermal force F共t兲 is a zero-centered and stationary random force with a correlation function of the form
共11兲
where
ĝ共s兲 = sĜ共s兲.
共12兲
From Eqs. 共7兲 and 共11兲, a formal expression for the displacement X共t兲 and the velocity V共t兲 can be written as
X共t兲 = 具X共t兲典 +
V共t兲 = 具V共t兲典 +
1
m
冕
1
m
冕
t
dt⬘G共t − t⬘兲F共t⬘兲,
共13兲
dt⬘g共t − t⬘兲F共t⬘兲,
共14兲
0
t
0
where
共6兲
where T is the absolute temperature and kB is the Boltzmann
constant.
Although we consider the one-dimensional GLE 共4兲, our
results can be extrapolated to the two- or three-dimensional
case, assuming that the resulting motion can be described by
two or three uncoupled equations for independent coordinates. This can be achieved if the trap exerts an elastic force
冊
1
F̂共s兲 ĝ共s兲 − x0␻20Ĝ共s兲,
m
共5兲
The integral term in Eq. 共4兲 represents the dependence of
the viscous force on the velocity history and the memory
kernel ␥共t兲 is related to the noise correlation function C共t兲
via the second fluctuation-dissipation theorem 关31兴
C共t兲 = kBTm␥共t兲,
冉
V̂共s兲 = v0 +
共4兲
0
具F共t兲F共t⬘兲典 = C共兩t − t⬘兩兲.
On the other hand, the Laplace transform of the velocity
V共t兲 = Ẋ共t兲 satisfies that
具X共t兲典 = x0关1 − ␻20I共t兲兴 + v0G共t兲,
共15兲
具V共t兲典 = v0g共t兲 − x0␻20G共t兲
共16兲
are the position and velocity mean values evolution, respectively.
B. Expressions for the MSD and VACF
To calculate the two-time properties of the dynamical
variables involved in the expressions of the MSD 共2兲 and
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PHYSICAL REVIEW E 80, 021111 共2009兲
SUBDIFFUSIVE BEHAVIOR IN A TRAPPING…
VACF 共3兲, we will make use of the double Laplace transform
technique 关33兴. Then, from Eqs. 共7兲 and 共11兲, we have
具V共t + ␶兲V共t兲典 =
冉
具X̂共s兲X̂共s⬘兲典 = x20␹ˆ 共s兲␹ˆ 共s⬘兲 + v20Ĝ共s兲Ĝ共s⬘兲 + x0v0关␹ˆ 共s兲Ĝ共s⬘兲
+ ␹ˆ 共s⬘兲Ĝ共s兲兴 +
冉
+ ␻20 x20␻20 −
1
Ĝ共s兲Ĝ共s⬘兲具F̂共s兲F̂共s⬘兲典,
m2
冊
k BT
k BT
g共␶兲 + v20 −
g共t + ␶兲g共t兲
m
m
冊
k BT
G共t + ␶兲G共t兲
m
− x0v0␻20关g共t + ␶兲G共t兲 + g共t兲G共t + ␶兲兴.
共23兲
共17兲
具V̂共s兲V̂共s⬘兲典 = v20ĝ共s兲ĝ共s⬘兲 + x20␻40Ĝ共s兲Ĝ共s⬘兲
− x0v0␻20关ĝ共s兲Ĝ共s⬘兲 + ĝ共s⬘兲Ĝ共s兲兴
+
1
ĝ共s兲ĝ共s⬘兲具F̂共s兲F̂共s⬘兲典,
m2
共18兲
where
␹ˆ 共s兲 =
1
− ␻20Î共s兲
s
共19兲
lim ␥共t兲 = lim s␥ˆ 共s兲 = 0.
t→⬁
is the Laplace transform of ␹共t兲 = 1 − ␻20I共t兲.
In the Appendix, we show how the last term of Eqs. 共17兲
and 共18兲 can be calculated. Inserting expressions 共A4兲 and
共A5兲 into Eqs. 共17兲 and 共18兲 and making a double Laplace
inversion, we arrive at
冉
It is important to notice that the analytical expressions
共22兲 and 共23兲 are valid for all absolute times t and time lags
␶. They enable us to obtain the two-time dynamics for an
arbitrary memory kernel provided that the fluctuationdissipation theorem 共6兲 is satisfied. Nevertheless, to evaluate
the relevant experimental magnitudes 共2兲 and 共3兲, we must
take the limit t → ⬁. In this case, these expressions could be
simplified as follows. Taking into account the usual assumption that the time-dependent frictional coefficient ␥共t兲 goes to
zero when t → ⬁ 关34兴 and using the final value theorem 关35兴,
one gets
冊
Noticing that the Laplace transform of the relaxation function I共t兲 defined through Eq. 共9兲 is
Î共s兲 =
共20兲
冊
冊
共21兲
CV共␶兲 = g共␶兲.
␳共⬁兲 =
具关X共t + ␶兲 − X共t兲兴 典
2kBT
I共␶兲 − 2x0v0␻20关G共t + ␶兲 − G共t兲兴关I共t + ␶兲 − I共t兲兴
m
冉
冊
k BT
关G共t + ␶兲 − G共t兲兴2
m
+ ␻20 x20␻20 −
冊
k BT
关I共t + ␶兲 − I共t兲兴2 ,
m
共28兲
共22兲
共29兲
Taking into account Eqs. 共26兲 and 共27兲, the equilibrium
value of the MSD is given by
2
冉
2kBT
I共␶兲
m
and
Finally, by considering time lags ␶ ⬎ 0, from Eqs. 共20兲 and
共21兲 we have
+ v20 −
共27兲
Applying these conditions in order to take the limit t
→ ⬁ in Eqs. 共22兲 and 共23兲 and using the definitions 共2兲 and
共3兲, one finally obtains the simpler expressions
␳共␶兲 =
k BT
G共t兲G共t⬘兲
m
− x0v0␻20关g共t兲G共t⬘兲 + g共t⬘兲G共t兲兴.
=
共26兲
G共⬁兲 = g共⬁兲 = 0.
k BT
k BT
g共t兲g共t⬘兲
g共兩t − t⬘兩兲 + v20 −
m
m
冉
共25兲
and using Eqs. 共9兲 and 共12兲 gives
k BT
k BT 2
+
关I共t兲 + I共t⬘兲 − I共兩t − t⬘兩兲兴 −
␻ I共t兲I共t⬘兲,
m
m 0
+ ␻20 x20␻20 −
s−1
,
s + s␥ˆ 共s兲 + ␻20
2
I共⬁兲 = 1/␻20
+ x0v0关␹共t兲G共t⬘兲 + ␹共t⬘兲G共t兲兴
冉
共24兲
the application of the final value theorem and the use of
condition 共24兲 yield 关34兴
k BT
G共t兲G共t⬘兲
具X共t兲X共t⬘兲典 = x20␹共t兲␹共t⬘兲 + v20 −
m
具V共t兲V共t⬘兲典 =
s→0
2kBT
m␻20
,
共30兲
while, as expected, the VACF decays to zero, i.e., CV共⬁兲 = 0.
It is worth pointing out that in experimental realizations,
the time lag is ␶min ⱕ ␶ ⱕ ␶max, being ␶min the acquisition time
interval and ␶max the measurement time. Moreover, if N is the
number of steps n taken at intervals ␶min, only small values
of n共n ⬍ N / 10兲 are used. Therefore, it is important to obtain
valid expressions for all observational time scales instead of
getting only its behavior to large times.
021111-3
PHYSICAL REVIEW E 80, 021111 共2009兲
M. A. DESPÓSITO AND A. D. VIÑALES
⬁
To conclude this section, we will find the extension for a
trapped particle of the widely used Mason formula 关5,6兴.
Taking the Laplace transform of Eq. 共28兲 and using the definition 共25兲 of the relaxation function I共t兲, one gets
s␥ˆ 共s兲 =
2kBT 1
− s2 − ␻20 ,
m s␳ˆ 共s兲
共32兲
where R is the radius of the spherical tracer particle, ␩ˆ 共s兲 is
the bulk frequency-dependent viscosity of the medium, and
Ĝ共s兲 is the Laplace-transformed shear modulus of the viscoelastic fluid. Equation 共32兲 is based on the implicit assumption that Stokes drag for viscous fluids 共no-slip boundary conditions兲 can be generalized to viscoelastic fluids at all
s 关6兴.
共− 1兲k
2kBT
共k兲
共␻0␶兲2k␶2E2−␭,3+␭k
共− ␥␭␶2−␭兲,
兺
m k=0 k!
共37兲
⬁
共31兲
which gives a direct relation between the mean-square displacement of the particle and the memory kernel. Finally,
expression 共31兲 can be related with the viscoelastic parameters of the complex fluid using the generalized StokesEinstein equation 关1,5,6兴
s␥ˆ 共s兲 = 6␲Rs␩ˆ 共s兲 = 6␲RĜ共s兲,
␳共␶兲 =
C V共 ␶ 兲 = 兺
k=0
C共t兲 = C␭
t−␭
,
⌫共1 − ␭兲
共33兲
where ⌫共z兲 is the gamma function 关38兴. The exponent ␭ for a
viscoelastic medium is taken as 0 ⬍ ␭ ⬍ 1. The proportionality coefficient C␭ is independent of time but depends on the
exponent ␭.
Using the fluctuation-dissipation relation 共6兲, the memory
kernel ␥共t兲 can be written as
␥␭
␥共t兲 =
t−␭ ,
⌫共1 − ␭兲
⬁
E␣,␤共y兲 = 兺
j=0
s−1
.
s2 + ␥␭s␭ + ␻20
共39兲
共40兲
Using the series expansions 共39兲 and 共40兲, one can realize
that the short-time behavior of the MSD reads
␳共␶兲 ⬇
再
冎
␻2
2␥␭
k BT 2
␶ −
␶4−␭ − 0 ␶4 ,
12
m
⌫共5 − ␭兲
共41兲
where the first term shows that the particle undergoes ballistic motion when time is very small 关39兴. The second term
comes from the influence of the viscoelastic medium while
the third term corresponds to the fact that the particle begins
to “see” the trap. The short-time behavior of the VACF can
be obtained in a similar way. In this case, we get
␻2
␥␭
␶2−␭ − 0 ␶2 .
2
⌫共3 − ␭兲
C V共 ␶ 兲 ⬇ 1 −
共42兲
On the other hand, for ␥␭␶2−␭ Ⰷ 1, the MSD and VACF
can be obtained introducing the asymptotic behavior of the
Mittag-Leffler function 关38兴
E␣,␤共− y兲 ⬃
1
,
y⌫共␤ − ␣兲
y⬎0
共43兲
into Eqs. 共37兲 and 共38兲. After some calculations, we have
␳共␶兲 ⬇
2kBT
m␻20
再 冉 冊冎
1 − E␭ −
冉
␻20 ␭
␶
␥␭
共44兲
,
冊
␻20 ␭
1 d2
E
−
␶ ,
␭
␥␭
␻20 d2␶
共45兲
where E␭共y兲 = E␭,1共y兲 denotes the one-parameter MittagLeffler function 关38兴.
It is worth pointing out that these expressions can be also
obtained discarding the inertial term s2 in Eq. 共36兲. In this
case, we get
共36兲
Î共s兲 =
The complete temporal behavior of the relaxation functions I共t兲, G共t兲, and g共t兲 was previously obtained by us in
Ref. 关27兴. Using those results in Eqs. 共28兲 and 共29兲, we have
␤⬎0
共j + k兲 ! y j
dk
.
= k E␣,␤共y兲 = 兺
dy
j=0 j ! ⌫共␣共j + k兲 + ␤兲
C V共 ␶ 兲 ⬇ −
共35兲
␣ ⬎ 0,
⬁
E␣共k兲,␤共y兲
共34兲
In this situation, the Laplace transform of the relaxation
function Î共s兲 reads
Î共s兲 =
yj
,
⌫共␣ j + ␤兲
and E␣共k兲,␤共y兲 is the derivative of the Mittag-Leffler function
where ␥␭ = C␭ / kBT. Then, its Laplace transform reads
␥ˆ 共s兲 = ␥␭s␭−1 .
共38兲
where E␣,␤共y兲 is the generalized Mittag-Leffler function 关38兴
defined by the series expansion
III. SUBDIFFUSIVE BEHAVIOR
Notice that the previous results are valid for any memory
kernel that satisfies condition 共24兲. On the other hand, it is
well known that in the absence of active transport, the dynamics of the particle in a viscoelastic fluid or complex media is subdiffusive and thus the stochastic process presents a
long-time tail noise. The most utilized model to reproduce a
subdiffusive behavior is characterized by a noise correlation
function exhibiting a power-law time decay 关27,36,37兴
共− 1兲k
共k兲
共␻0␶兲2kE2−␭,1+␭k
共− ␥␭␶2−␭兲,
k!
冉
冊
1 1
s−1
s␭−1
=
,
−
␥␭s␭ + ␻20 ␻20 s s␭ + ␻20/␥␭
共46兲
and using that the Laplace transform of the Mittag-Leffler
function 关38兴
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PHYSICAL REVIEW E 80, 021111 共2009兲
SUBDIFFUSIVE BEHAVIOR IN A TRAPPING…
1
1.4
0.75
1.2
0.5
CV Τ
ΡΤΡ
1
0.8
0.6
0.25
0
0.25
0.4
0.5
0.2
0.75
2
4
6
Τ
8
FIG. 1. 共Color online兲 MSD vs time lag for ␭ = 1 / 2, ␥␭ = 1, and
␻0 = 1.4. Solid line corresponds to the exact solution 共37兲, dotted
line to the approximate solution 共44兲, and dashed line to the
asymptotic behavior 共48兲.
冕
⬁
e−stE␣共− ␥t␣兲dt =
0
s␣−1
,
s␣ + ␥
共47兲
one obtains expressions 共44兲 and 共45兲.
Finally, if ␶␭ Ⰷ ␥␭ / ␻20, the behaviors of the MSD and
VACF can be obtained using again the approximation 共43兲.
In this case, we get
␳共␶兲 ⬇
2kBT
m␻20
再
C V共 ␶ 兲 ⬇ −
1−
2
10
冎
1
␥␭
␶−␭ ,
2
␻0 ⌫共1 − ␭兲
共48兲
␥␭ ␭共␭ + 1兲 −共␭+2兲
␶
,
␻40 ⌫共1 − ␭兲
共49兲
showing a pure power-law decay.
In Figs. 1 and 2, we have plotted the MSD vs time lag
using the exact solution 共37兲 and the approximations 共44兲
and 共48兲. Note that the exact solution exhibits a nonmonotonic approach to ␳共⬁兲, while the approximations always
present a monotonic behavior.
More pronounced differences occur in the behavior of
CV共␶兲 as is evidenced in Figs. 3 and 4. Note that the approximate solutions exhibit a negative velocity autocorrelation for
6
Τ
8
10
FIG. 3. 共Color online兲 CV vs time lag for ␭ = 1 / 2, ␥␭ = 1, and
␻0 = 1.4. Solid line corresponds to the exact solution 共38兲, dotted
line to the approximate solution 共45兲, and dashed line to the
asymptotic behavior 共49兲.
all times, whereas the exact solutions alternate between positive and negative values. This behavior is related to the socalled whip-back effect 关27,37兴.
These behaviors can be understood taking into account
that the approximations 共44兲 and 共45兲 only depend on the
one-parameter Mittag-Leffler function E␭共−␻20␶␭ / ␥␭兲. It is
known that the function E␭共−t␭兲 is a completely monotone
function and tends to zero from above as t tends to infinity
for 0 ⬍ ␭ ⬍ 1 关40兴. Then, the approximate solutions are always monotonic for every value of ␻0 and 0 ⬍ ␭ ⬍ 1. However, the exact solutions 共37兲 and 共38兲 are expressed as infi共k兲
2−␭
兲 functions. In this case, the
nite sums of E2−␭,
␤共−␥␭␶
solutions can exhibit a nonmonotonic behavior as is displayed in the previous figures. Then, the overdamped approximation must be used with care in the analysis of the
short- and intermediate-time dynamics where the MSD and
VACF exhibit a relaxation plus an oscillatory behavior.
Interestingly, Burov and Barkai 关41兴 recently arrived to a
similar conclusion using a fractional Langevin equation.
Solving the corresponding fractional differential equation in
terms of roots of regular polynomials, they analyze the position correlation 具X共t兲X共0兲典. As in our case, they found that
the exact solution can exhibit a nonmonotonic decay, while
the overdamped approximation gives a monotonic decay.
1
1
0.8
0.8
0.6
0.4
0.6
CV Τ
ΡΤΡ
4
0.4
0.2
0
0.2
0.2
0.4
2
4
6
Τ
8
10
12
14
FIG. 2. 共Color online兲 Idem Fig. 1 for ␻0 = 0.8.
2
4
6
Τ
8
10
12
FIG. 4. 共Color online兲 Idem Fig. 3 for ␻0 = 0.8.
021111-5
14
PHYSICAL REVIEW E 80, 021111 共2009兲
M. A. DESPÓSITO AND A. D. VIÑALES
IV. SUMMARY
In this work, we have obtained the mean-square displacement and the velocity autocorrelation function for a trapped
particle and immersed in a complex or viscoelastic media.
For this purpose, and starting from a suitable generalized
Langevin equation, we have been able to derive analytic expressions for the two-time dynamics of the processes, valid
for all absolute times and times lags. We have showed that
the MSD and VACF can be expressed as a simple expression
when the memory kernel goes to zero for large times. Furthermore, we have presented a generalization of the Mason
formula valid when a trapping potential is present.
In particular, we have examined the subdiffusive case,
which is a representative example of passive transport in
viscoelastic media. Using a power-law memory kernel, exact
expressions and valid for all time lags have been obtained in
terms of Mittag-Leffler functions and their derivatives. The
behavior for short-, intermediate-, and long-time lags are
given in terms of the involved parameters. Finally, we have
showed that the usual approximations cannot reproduce the
nonmonotonic dynamics present in the exact solutions.
In summary, we have presented a theoretical method to
account for the effects of the trapping potential in the anomalous behavior of the mean-square displacement and the normalized velocity autocorrelation function of a particle embedded in a complex or viscoelastic environment. We believe
that the presented results will be useful to analyze the obtained data from microrheology experiments in viscoelastic
media using trapping potentials.
31980/05 from Agencia Nacional de Promoción Científica y
Tecnológica and Grant No. X099 from Universidad de Buenos Aires, Argentina.
APPENDIX
To calculate the last term of Eqs. 共17兲 and 共18兲, we make
use of a relation given in Ref. 关33兴. Given any stationary
correlation function of the form
具⌿共t兲⌿共t⬘兲典 = Af共兩t − t⬘兩兲,
the corresponding double Laplace transform writes
具⌿̂共s兲⌿̂共s⬘兲典 = A
f̂共s兲 + f̂共s⬘兲
.
s + s⬘
共A2兲
Then, the Laplace domain version of the fluctuationdissipation relation 共6兲 reads 关33兴
具F̂共s兲F̂共s⬘兲典 = kBTm
␥ˆ 共s兲 + ␥ˆ 共s⬘兲
.
s + s⬘
共A3兲
After some algebra and using the relations between the
kernels I共t兲G共t兲 and g共t兲, one can find that
Ĝ共s兲Ĝ共s⬘兲具F̂共s兲F̂共s⬘兲典 = kBTm
冋
Î共s兲 Î共s⬘兲 Î共s兲 + Î共s⬘兲
−
+
s⬘
s
s + s⬘
册
− kBTm关Ĝ共s兲Ĝ共s⬘兲 + ␻20Î共s兲Î共s⬘兲兴,
共A4兲
ĝ共s兲ĝ共s⬘兲具F̂共s兲F̂共s⬘兲典
ACKNOWLEDGMENTS
We would like to thank the anonymous referee for his
suggestions, which were very useful to the final form of the
paper. This work was performed under Grant No. PICT
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