THE JOURNAL OF CHEMICAL PHYSICS 132, 034104 共2010兲
Searching for partially reactive sites: Analytical results for spherical targets
Denis S. Grebenkova兲
Laboratoire de Physique de la Matière Condensée, CNRS, Ecole Polytechnique, 91128 Palaiseau, France
共Received 11 November 2009; accepted 30 December 2009; published online 19 January 2010兲
How do single or multiple 共sub兲diffusing particles search for a target with a partially reactive
boundary? A finite reaction rate which is typical for many chemical or biochemical reactions is
introduced as the possibility for a particle to find a target but not to “recognize” it. The search is not
finished until the target is found and recognized. For a single searching particle, the short- and
long-time regimes are investigated, with a special focus on comparison between perfectly and
partially reactive boundaries. For multiple searching particles, explicit formulas for the probability
density of the search time are given for subdiffusion in one and three dimensions. The dependence
of the mean search time on the density of particles and the reaction rate is analyzed. Unexpectedly,
in the high density limit, the particles undergoing slower subdiffusive motion find a target faster.
© 2010 American Institute of Physics. 关doi:10.1063/1.3294882兴
I. INTRODUCTION
Searching for a reactive site is a common process in
physics, chemistry, and biology. Typical examples are diffusing molecules searching for catalytic grains in a chemical
reactor1–5 or proteins searching for specific DNA target sites
in a living cell.6–8 In both cases, crowding and caging, geometrical traps and energetic barriers, and other mechanisms
often lead to anomalous diffusion with a sublinear growth of
the mean-square displacement: 具关x共t兲 − x共0兲兴2典 ⬀ t␣, with an
exponent ␣ between 0 and 1.9–17 The distribution of search
times in such reactive media characterizes their overall functioning.
Many theoretical, numerical, and experimental studies
are concerned with search times for normal diffusion.18–25
Recently, several groups investigated this question for subdiffusion. Lua and Grosberg26 studied the first-passage times
for a particle, executing one-dimensional diffusive and subdiffusive motions in an asymmetric sawtooth potential, to
exit one of the boundaries. Yuste and Lindenberg27 computed
the asymptotic survival probability of a spherical target in
the presence of a single subdiffusive trap or surrounded by a
sea of subdiffusive traps. Condamin and co-workers28,29 derived a relationship between the moments of the first-passage
time for normal diffusion and the first-passage time density
for subdiffusive processes. All these studies are concerned
with the first-passage time when a subdiffusive process is
stopped 共i.e., a target is found兲 after the first encounter with
the target.
When a target is found, the particle may or may not
interact with the target via chemical transformation, relaxation, adsorption on or transfer through its boundary,
etc.3–5,30–39 For biological applications, the interaction can
also mean a “recognition” of the target. If no interaction
occurs 共the target is not recognized兲, the particle is released
to continue its motion in the bulk until finding a target again.
The subdiffusive process is stopped 共i.e., the search is over兲
a兲
Electronic mail: [email protected].
0021-9606/2010/132共3兲/034104/8/$30.00
when the target is found and recognized. The probability of
interaction 共or recognition兲 at each encounter with the target
is related to permeability, reactivity, or relaxivity W of its
boundary. Among various examples, we mention macromolecules diffusing in the extracellular space and searching for
cells to permeate through the cellular membrane 共as well as
viruses searching for a nucleus inside a living cell and permeating through the boundary of the nucleus兲;40–42 species
diffusing in a chemical reactor and reacting on catalytic
grains with a finite reaction rate,1–5,30,43,44 spin-bearing molecules moving in a medium filled with relaxing sites.45
We aim at answering the following two questions: “how
does the surface reactivity influence the distribution of
search times?” and “what is the role of the exponent ␣ characterizing the subdiffusive process?” The former question
was studied in depth for normal diffusion 共␣ = 1兲 by Barzykin
and Tachiya,5 while the latter question was addressed
by Yuste and Lindenberg27 for perfectly reactive boundary
共W = ⬁兲. We present an extension of these works by considering a general situation of subdiffusive searching 共␣ ⱕ 1兲 for
partially reactive sites 共W ⱕ ⬁兲.
In Sec. II, we calculate the Laplace transform of the
survival probability of a single particle subdiffusing in a
space outside a spherical target. We also consider searching
by many particles and derive formulas for the corresponding
survival probability. For subdiffusion in one and three dimensions, the latter survival probability will be written in an
exact explicit form. We analyze its short-time and long-time
asymptotic behaviors, with a special emphasis on comparison between perfectly and partially reactive targets. In
Sec. III, we compute the mean multiple-particles search time
共MPST兲 and study its dependence on surface relaxivity, density of particles, and exponent ␣.
II. SURVIVAL PROBABILITY
Anomalous diffusions may result from various physical
mechanisms: trapping with a long-tailed distribution of waiting times 关continuous-time random walks 共CTRWs兲兴; strong
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© 2010 American Institute of Physics
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J. Chem. Phys. 132, 034104 共2010兲
Denis S. Grebenkov
corrections between microscopic displacements 共e.g., fractional Brownian motion兲; and diffusion on fractal sets.17 Although all these mechanisms lead to a sublinear growth of
the mean-square displacement, the underlying physics and
mathematics are very different. In this paper, we focus on the
CTRW which can be equivalently described by fractional
Fokker–Planck equations.17 In the CTRW framework, a subdiffusive process is a sequence of random independent microscopic displacements in space with random waiting times
between each two displacements. The probability density of
the waiting times decays at long times as t−1−␣ 共with 0 ⬍ ␣
⬍ 1兲 so that the mean waiting time is infinite, while the
mean-square displacement is sublinear. Since the spatial displacements of a CTRW are governed by the same probability
laws as those for normal diffusion, this subdiffusive process
and normal diffusion differ only by “time clocks.” As a consequence, many results for subdiffusion follow straightforwardly from those for normal diffusion by an appropriate
change of “time clocks,” which is known as
subordination.46–48 In particular, a number of formulas established by Barzykin and Tachiya5 will be directly extended for
subdiffusion.
The classical diffusion equation for the survival
probability3 is replaced by a fractional Fokker–Planck equation for subdiffusion
冋
册
⳵
␣
− D␣ 0D1−
t ⌬ Q␣共r,t兲 = 0
⳵t
⌳
⳵
Q␣共r,t兲 + Q␣共r,t兲 = 0
⳵n
q␣共r,t兲 = −
⳵
Q␣共r,t兲.
⳵t
共1兲
The combined survival probability SN共t兲 for N independent subdiffusing particles is simply the probability that none
of the N particles found and recognized the target up to time
t. If the particles are uniformly distributed in a volume V,
then
SN共t兲 =
冋冕
1
V
⍀
drQ␣共r,t兲
册
N
.
In the thermodynamic limit with N → ⬁ and V → ⬁ holding
the density ␳ = N / V, the survival probability becomes4
S⬁共t兲 = exp关− ␳Rd f共t兲兴,
共2兲
where
f d共t兲 =
1
Rd
冕
⍀
dr关1 − Q␣共r,t兲兴.
共3兲
Note that the derivative of f d共t兲 can be interpreted as the bulk
reaction rate.4,5
共r 苸 ⳵ ⍀兲,
共4兲
共5兲
lim Q␣共r,t兲 = 1,
共6兲
Q␣共r,t = 0兲 = 1,
共7兲
兩r兩→⬁
where ⌬ ⬅ ⳵2 / ⳵21 + . . . +⳵2 / ⳵2d is the d-dimensional Laplace operator, ⳵ / ⳵n is the normal derivative at the target boundary
⳵⍀ pointing toward the exterior of the domain ⍀, ⌳ is a
positive parameter homogeneous to a length, and D␣ is a
generalized diffusion coefficient 共in units m2 / s␣兲. Memory
effects of a subdiffusive process are introduced through the
␣
Riemann-Liouville fractional differential operator 0D1−
t ,
A. Fractional diffusion equation
We consider particles which 共sub兲diffuse in the space ⍀
outside a target. The central quantity of our analysis is the
survival probability Q␣共r , t兲 that is a particle started at
r 苸 ⍀ at time 0 does not find 共and recognize兲 the target up to
a later time t. The time derivative of Q␣共r , t兲 gives the probability density for the single-particle search time 共SPST兲
共r 苸 ⍀兲,
1−␣
0Dt g共t兲 =
1 ⳵
⌫共␣兲 ⳵ t
冕
t
0
dt⬘
g共t⬘兲
,
共t − t⬘兲1−␣
共8兲
which is defined for any sufficiently well-behaved function
g共t兲, and ⌫共z兲 is the Gamma function.49,50
A probabilistic interpretation of Eqs. 共4兲–共7兲 is straightforward. The fractional diffusion Eq. 共4兲 states that the time
evolution of the survival probability is caused by local migration of subdiffusing particles 共described by the Laplace
operator兲. The Robin boundary condition 关Eq. 共5兲兴 includes a
possibility of reflection on the target boundary ⳵⍀ 共the normal derivative兲 when the target is not recognized. The two
other relations state that the survival probability is 1 at the
beginning of the experiment 关Eq. 共7兲兴 and for a particle
which was initially located infinitely far from the target
关Eq. 共6兲兴.
The Robin boundary condition 共also known as radiation,
Fourier, third, etc.兲 is a linear combination of Dirichlet and
Neumann boundary conditions.30–33 This is a form of a mass
conservation law when the diffusive flux toward the boundary is equal to the flux across the boundary. Collins and
Kimball introduced the Robin boundary condition in order to
describe partially diffusion-controlled reactions,30 while Seki
et al.51 and later Eaves and Reichman52 justified this condition for subdiffusion-controlled reactions. They showed that
a positive parameter ⌳ which is homogeneous to a length, is
the ratio between the bulk and surface transport coefficients:
⌳ = D␣ / W␣, W␣ being the 共generalized兲 surface permeability,
relaxivity or reactivity 共although ⌳ may depend on ␣, we
keep writing ⌳ instead of ⌳␣兲. The parameter ⌳ characterizes the recognition phase, i.e., interactions at the encounter
with the target. In the limiting case ⌳ = 0 共infinite reactivity兲,
one retrieves the description by Yuste and Lindenberg27 for a
perfectly reactive target, without recognition phase 共the
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034104-3
J. Chem. Phys. 132, 034104 共2010兲
Searching for partially reactive sites
search is over when a target is reached for the first time兲. The
opposite case of ⌳ = ⬁ 共zero reactivity兲 corresponds to
Neumann boundary condition for a “blind particle” which
never recognizes the target. Although more sophisticated
boundary conditions may sometimes be required in order to
describe surface exchange processes,42,53,54 our focus is on
the Robin boundary condition 关Eq. 共5兲兴.
The set of time-dependent Eqs. 共4兲–共7兲 can be equivalently represented in Laplace domain. We consider a set of
equations for the Laplace transform L of the probability density q␣共r , t兲
q̃␣共r,s兲 ⬅ L关q␣共r,t兲兴共s兲 ⬅
冕
⬁
dte−tsq␣共r,t兲
for computing passage times for normal diffusion in complex
geometries21,56–60 can be adapted for subdiffusion.
B. Spherical target
For a spherical target of radius R, one has
⍀ = 兵r 苸 Rd : 兩r兩 ⬎ R其, and Eq. 共9兲 is reduced to a modified
Bessel equation
g⬙共z兲 +
where g共z兲 ⬅ q̃␣共r , s兲 and z ⬅ 兩r兩冑s␣ / D␣. A regular solution of
this equation satisfying the Robin boundary condition
关Eq. 共10兲兴 is
0
q̃␣共r,s兲 =
共Laplace transformed quantities are denoted by tilde throughout the text兲. Since the fractional derivative in Eq. 共8兲
has a form of a convolution, its Laplace transform is simply
the product of two Laplace transforms, namely,
␣
1−␣
L关g共t兲兴共s兲. The fractional diffusion Eq.
L关 0D1−
t g共t兲兴共s兲 = s
共4兲 for Q␣共r , t兲 is therefore reduced to a Helmholtz equation
for Q̃␣共r , s兲 or q̃␣共r , s兲 which are related as q̃␣共r , s兲 = 1
− sQ̃␣共r , s兲 according to Eq. 共1兲. One gets
s␣q̃␣共r,s兲 − D␣⌬q̃␣共r,s兲 = 0
⌳
⳵
q̃␣共r,s兲 + q̃␣共r,s兲 = 1
⳵n
共r 苸 ⍀兲,
共9兲
共r 苸 ⳵ ⍀兲,
共11兲
Note that the factor s␣ appears as a fixed parameter in the
Helmholtz equation. As a consequence, its solution for subdiffusion 共0 ⬍ ␣ ⬍ 1兲 can be directly derived from the solution for normal diffusion 共␣ = 1兲 by substituting s / D1 by
s␣ / D␣. This is an example of “subordination” meaning that
diffusion and subdiffusion differ by time clocks, the difference being expressed through s␣ / D␣ in Laplace space.46–48
The analysis of the search time for a subdiffusive process is
therefore reduced to “rescaling” of the same characteristics
for normal diffusion. On one hand, one can readily use numerous analytical results for normal diffusion.55 On the other
hand, fast random walk algorithms which were implemented
冉 冊
兩r兩
R
共1−d兲/2
t−1
冑␲共2 − ␣兲
冋
exp − 共2 − ␣兲
冉
共R − 兩r兩兲2␣␣
4D␣t␣
兩r兩
R
1−d/2
Kd/2−1共兩r兩冑s␣/D␣兲
,
Wd共s兲
共12兲
Wd共s兲 = Kd/2−1共R冑s␣/D␣兲 + ⌳冑s␣/D␣Kd/2共R冑s␣/D␣兲 ,
共13兲
where K␯共z兲 is the modified Bessel function of the second
kind 共note that a similar equation was given by Barzykin and
Tachiya5 for normal diffusion with ␣ = 1兲.
In order to calculate the survival probability S⬁共t兲, we
find the Laplace transform of the function f d共t兲 from Eq. 共3兲
共10兲
lim q̃␣共r,s兲 = 0.
冉 冊
with
f̃ d共s兲 =
兩r兩→⬁
q␣共r,t兲 ⯝
d−1
g⬘共z兲 − g共z兲 = 0,
z
1
Rd
冕
dr
兩r兩⬎R
Kd/2共R冑s␣/D␣兲
q̃␣共r,s兲
= Sd
,
s
sWd共s兲R冑s␣/D␣
共14兲
where Sd = 2␲d/2 / ⌫共d / 2兲 is the surface area of the
d-dimensional unit sphere, and we used the identity
冕
⬁
dx xd/2Kd/2−1共xa兲 =
1
Kd/2共a兲
.
a
C. Short-time behavior
The short-time behavior of time-dependent quantities
corresponds to the limit s → ⬁ for their Laplace transforms.
In this limit, we get
q̃␣共r,s兲 ⯝
冉 冊
R
兩r兩
共d−1兲/2
e共R−兩r兩兲
冑s␣/D␣
1 + ⌳冑s␣/D␣
共15兲
.
When 兩r兩 ⬎ R, the short-time asymptotic behavior of the
right-hand side of Eq. 共15兲 was shown to be61
冊 册
1/共2−␣兲
⫻
冦
␣1/2
冉
共R − 兩r兩兲2␣␣
4D␣t␣
␣共1−␣兲/2
冉
冊
1/关2共2−␣兲兴
共R − 兩r兩兲2␣␣
4D␣t␣
冊
共1−␣兲/关2共2−␣兲兴
共⌳ = 0兲,
冑D␣t␣
⌳
共⌳ ⬎ 0兲.
冧
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J. Chem. Phys. 132, 034104 共2010兲
Denis S. Grebenkov
The asymptotic behavior is dominated by the exponential
factor that leads to a rapid decrease of the probability density
when t goes to 0. In the short-time regime, a particle started
far from the target does not have enough time to find it.
We also find the asymptotic behavior of the function
f̃ d共s兲 which determines the survival probability S⬁共t兲
Sd
f̃ d共s兲 ⯝
sR冑s␣/D␣共1 + ⌳冑s␣/D␣兲
f d共t兲 ⯝
冦
⌫共1 +
␣
2
冑D␣t␣
兲
R
S d D ␣t ␣
⌫共1 + ␣兲 ⌳R
共⌳ = 0兲,
共⌳ ⬎ 0兲.
冧
共16兲
Given that
K−1/2共z兲 = K1/2共z兲 =
冑
f̃ 1共s兲 =
⌳
关1 − E␣/2共− 冑D␣t␣/⌳兲兴 ,
R
共17兲
zn
.
⌫共␣n + 1兲
Equation 共17兲 is an exact result which is applicable for any
time t. In the limit ⌳ → 0, the second term in Eq. 共17兲 vanishes, and one retrieves the result by Yuste and Lindenberg.27
In the long-time limit, the Mittag–Leffler function decays as a power law, so that the second term is small, and
f 1共t兲 ⯝
冑4D␣t␣/R
⌫共1 +
␣
2
兲
共18兲
.
Reactivity properties of the target 共the value of ⌳兲 become
irrelevant in this limit.
2. Two dimensions
In two dimensions 共d = 2兲, the long-time asymptotic behavior of f 2共t兲 can be deduced by considering its Laplace
transform f̃ 2共s兲 in the limit of s going to 0. Since
共z → 0兲,
where ␥ ⯝ 0.577 216 is the Euler–Mascheroni constant, one
has
W2共s兲 ⯝ − ln共R冑s␣/D␣/2兲 − ␥ + ⌳/R,
from which
f̃ 2共s兲 ⯝
4␲D␣/R2
,
s ␣ ln共s0/s兲
1+␣
where
冋
2
共␥ − ⌳/R兲
␣
册冉 冊
D␣
R2
1/␣
.
Using the Tauberian theorem,
␲e
,
2 冑z
−z
we get the following exact results:
冑
−R s␣/D␣
␲ e
共1 + ⌳冑s␣/D␣兲,
2 共R冑s␣/D␣兲1/2
冑s␣/D␣
1 + ⌳冑s␣/D␣
兲
−2
where E␣共z兲 is the Mittag–Leffler function,
s0 = 41/␣ exp −
1. One dimension
q̃␣共r,s兲 =
␣
2
K0共z兲 ⯝ − ln共z/2兲 − ␥ + O共z2兲
The analysis of the long-time behavior 共t → ⬁兲 relies on
series expansions of functions q̃␣共r , s兲 and f̃ d共s兲 as s going to
0. The asymptotic properties of the modified Bessel functions suggest to consider the cases d ⬍ 2, d = 2, and d ⬎ 2
separately. Although the computation can be done for any
real dimensionality5,27 d, we focus on three practically important values: d = 1 , 2 , 3. In what follows, we derive exact
explicit formulas for the function f d共t兲 in one and three dimensions. The long-time behavior is also discussed.
e共R−兩r兩兲
⌫共1 +
n=0
D. Long-time behavior
冑
冑4D␣t␣/R
E␣共z兲 = 兺
Since the particles which started near the boundary of a target reach it fast, there is no dominating exponential factor in
Eq. 共16兲, and the distinction between two cases ⌳ = 0 and
⌳ ⬎ 0 is now more explicit. According to Eq. 共2兲, the survival probability S⬁共t兲 decreases slower with time for ⌳ ⬎ 0
because longer time is needed to find and recognize a partially reactive target. The short-time behavior of the survival
probability allows one to determine the length ⌳ and to characterize the surface reactivity.
W1共s兲 =
f 1共t兲 =
⬁
共s → ⬁兲,
from which
Sd
explicit form, except for ␣ = 1.5 In turn, the inverse Laplace
transform of f̃ 1共s兲 is
,
2冑D␣/R
2⌳/R
.
1+␣/2 −
s
s共1 + ⌳冑s␣/D␣兲
Although the probability density q␣共r , t兲 is formally given
through the inverse Laplace transform of q̃␣共r , s兲, there is no
L
冋 册
t␤−1
⌫共␤兲
,
共s兲 ⯝ ␤
ln共ts0兲
s ln共s0/s兲
we get the following approximate result:
f 2共t兲 ⯝
4␲D␣t␣/R2
.
⌫共1 + ␣兲关ln共4D␣t␣/R2兲 − 2␥ + 2⌳/R兴
共19兲
Interestingly, the time derivative of this function, which is
equal to the bulk reaction rate,5 decays much slower 共logarithmically兲 for normal diffusion 共␣ = 1兲 than for subdiffusion
共␣ ⬍ 1兲.
3. Three dimensions
Using explicit formulas for K1/2共z兲 and K3/2共z兲, we find
the following exact results:
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034104-5
J. Chem. Phys. 132, 034104 共2010兲
Searching for partially reactive sites
III. MULTIPLE-PARTICLES SEARCH TIME „MPST…
According to the explicit Eq. 共12兲 for q̃␣共r , s兲, the mean
search time 具␶s典 for a single 共sub兲diffusing particle is infinite
2
具 ␶ s典 =
3
f (t)
10
0
10
冕
⬁
dt t q␣共r,t兲 = −
0
−2
10 −5
10
−4
10
−3
10
−2
10
−1
10
0
10
1
10
2
3
10
FIG. 1. The function f 3共t兲 for d = 3, ␣ = 2 / 3, D␣ = 1, R = 1, and three values of
⌳ / R: 0 共top curve in blue兲, 0.1 共middle curve in green兲, and 1 共bottom curve
in red兲. The time scale is t0 = 共R2 / D␣兲1/␣. Dashed lines show the short-time
asymptotic behavior according to Eq. 共16兲.
W3共s兲 =
冑
q̃␣共r,s兲 =
−R冑s␣/D␣
␲ e
共1 + ⌳/R + ⌳冑s␣/D␣兲 ,
2 共R冑s␣/D␣兲1/2
p␣共t兲 ⯝
冉
冊
As in one dimension, there is no explicit formula for the
probability density q␣共r , t兲 except for ␣ = 1.5 In turn, the inverse Laplace transform of the last relation is
f 3共t兲 = 4␲
−
冉
冑D␣t␣/R
D␣t␣/R2
+
共1 + ⌳/R兲⌫共1 + ␣兲 共1 + ⌳/R兲2⌫共1 +
␣
2
兲
冊
⌳/R
关1 − E␣/2共− 冑D␣t␣共1/⌳ + 1/R兲兲兴 .
共1 + ⌳/R兲3
共20兲
This exact relation which is applicable for any time t, is one
of our main results. Figure 1 shows the behavior of this
function for three values of ⌳ / R and ␣ = 2 / 3 as an arbitrary
example. In the short-time limit, one can clearly notice different slopes for ⌳ = 0 and ⌳ ⬎ 0. In the long-time limit, all
the curves follow a power law with the exponent ␣. The
prefactor 共1 + ⌳ / R兲−1 in front of the first term of Eq. 共20兲 is
responsible for a shift of each curve along the vertical axis
on the logarithmic scale.
4. Higher dimensions
Although explicit formulas are not always available for
higher dimensions 共d ⬎ 3兲, the long-time asymptotic behavior is similar to that of the first term of Eq. 共20兲
f d共t兲 ⯝
冦
Sd
⌫共 ␣2 兲
␳Rd−1冑D␣t␣/2−1 共⌳ = 0兲,
Sd ␳Rd−1D␣ ␣−1
t
⌫共␣兲
⌳
共⌳ ⬎ 0兲,
具␶m
d典=
冕
⬁
dt t p␣共t兲 =
0
冕
⬁
共d − 2兲Sd D␣t /R
.
1 + 共d − 2兲⌳/R ⌫共1 + ␣兲
共23兲
冧
共24兲
dt exp关− ␳Rd f d共t兲兴.
2
共21兲
For ⌳ = 0, one retrieves the result by Yuste and Lindenberg,27
while the case ␣ = 1 was considered by Barzykin and
Tachiya.5
共25兲
0
In what follows, we analyze the mean search time for subdiffusion in one and three dimensions.
A. One dimension
The explicit Eq. 共17兲 for f 1共t兲 allows us to write the
mean search time as
具␶m
1典=
关⌫共1 + ␣2 兲兴2/␣⌫共1 + ␣2 兲
共4D␣␳2兲1/␣
T1,␣共2␳⌳兲,
共26兲
where
T1,␣共x兲 =
1
⌫共 ␣2 兲
冕
⬁
0
dz z2/␣−1
冉 冋
冉
⫻exp − z + x 1 − E␣/2 −
␣
共22兲
while the long-time asymptotic behavior is essentially
stretched-exponential, with the function f d共t兲 given by Eqs.
共18兲, 共19兲, and 共21兲. The stretched-exponential decrease of
p␣共t兲 ensures the existence of all the moments of the MPST.
In particular, the mean search time is
4␲D␣
1
1−
.
⌳Rs1+␣
1 + ⌳/R + ⌳冑s␣/D␣
f̃ 3共s兲 =
= ⬁.
s=0
The short-time asymptotic behavior of p␣共t兲 follows directly
from Eq. 共16兲:
共R−兩r兩兲冑s␣/D␣
e
R
,
兩r兩 1 + ⌳/R + ⌳冑s␣/D␣
冊
⳵
S⬁共t兲 = ␳Rd f d⬘共t兲exp关− ␳Rd f d共t兲兴.
⳵t
p␣共t兲 = −
0
⳵
q̃␣共r,s兲
⳵s
In contrast, many subdiffusing particles find a target with
a finite mean time. We define the probability density p␣共t兲 for
the MPST as
10
t/t
冉
z⌫共1 +
x
␣
2
兲
冊册冊
.
共27兲
Note that the mean search time is independent of the size R
of the target.
For small x, the function T1,␣共x兲 approaches a constant:
T1,␣共0兲 = 1. The prefactor in Eq. 共26兲 is therefore the mean
search time for a perfectly reactive target 共⌳ = 0兲. For normal
2
diffusion 共␣ = 1兲, one gets 具␶m
1 典 = ␲ / 共8D1␳ 兲 at ⌳ = 0.
For large x, one has
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J. Chem. Phys. 132, 034104 共2010兲
Denis S. Grebenkov
T1,␣共x兲 ⯝
⌫共 ␣1 兲
2⌫共 ␣2 兲
冉关 共
⌫共1 + ␣兲
⌫ 1+
␣
2
兲兴2
冊
1/␣
0
10
x1/␣ .
Since corrections are of the order of x1/␣−1/2, they may be
significant for practical use.
B. Three dimensions
−1
T3,α(0,y)
034104-6
10
−2
10
In three dimensions, the explicit formula 共20兲 yields
具␶m
3典=
⌫共 ␣1 兲关⌫共1 + ␣兲兴1/␣
␣共4␲␳RD␣兲1/␣
−3
T3,␣共⌳/R, ␳R3兲,
共28兲
共4␲y兲1/␣
T3,␣共x,y兲 = 1
⌫共 ␣ 兲关⌫共1 + ␣兲兴1/␣ 共1 + x兲2/␣
冕
冉
dz z2/␣−1 exp −
0
+
冋
册冊
4␲ y
z
3
共1 + x兲 ⌫共1 +
z2
− x关1 − E␣/2共− z/x兲兴
⌫共1 + ␣兲
␣
2
兲
For a perfectly reactive target 共⌳ = 0 or x = 0兲, Eq. 共29兲
becomes
2共4␲y兲1/␣
⌫共 ␣1 兲关⌫共1 + ␣兲兴1/␣
冉 冋共
⫻exp − 4␲y
冕
⬁
dz z2/␣−1
0
z
⌫ 1+
␣
2
兲
+
z2
⌫共1 + ␣兲
册冊
. 共30兲
For the low density ␳ of searching particles 共small y兲, the
main contribution to the integral in Eq. 共30兲 comes from
large values of z so that the linear term in z can be neglected
in comparison to the quadratic one, yielding
T3,␣共0,y兲 ⯝ 1 + O共冑y兲
共y → 0兲.
2⌫共 ␣2 兲关⌫共1 +
⌫共
1
␣
1
2
10
10
␣
2
兲兴2/␣
兲关⌫共1 + ␣兲兴1/␣
共4␲y兲−1/␣ ,
冉 冋
T3,␣共0,y兲 ⯝ 1 + 共4␲y兲1/␣
⌫共 ␣1 兲关⌫共1 + ␣兲兴1/␣
2⌫共 ␣2 兲关⌫共1 +
␣
2
兲兴2/␣
册冊
␤ −1/␤
,
共33兲
where ␤ is an adjustable parameter. For an empirical value
␤ = ␣ / 1.5, the Padé approximation is accurate within 10%.
Figure 2 presents T3,␣共0 , y兲 as a function of y = ␳R3 for ␣
= 1 共normal diffusion兲 and ␣ = 2 / 3 共subdiffusion兲. The
asymptotic formulas 共31兲 and 共32兲 and the Padé approximation 关Eq. 共33兲兴 are shown for comparison. It is worth stressing that the Padé formula 共33兲 is an empirical approximation.
At the same time, this approximation correctly describes,
both qualitatively and quantitatively, the behavior of the
mean MPST. For practical purposes, it helps to avoid a numerical computation of the integral in Eq. 共30兲.
Note that for normal diffusion 共␣ = 1兲, the Mittag–Leffler
function E1/2共z兲 can be written in terms of the complemen2
tary error function erfc共z兲 as E1/2共z兲 = ez erfc共−z兲, from
which one gets an explicit form
T3,1共0,y兲 = 1 − 冑4␲ye4y erfc共2冑y兲.
共31兲
According to Eq. 共28兲, the mean search time 具␶m
3 典 diverges as
共␳RD␣兲−1/␣. As expected, a particle undergoing slower subdiffusive motion 共smaller ␣兲 needs longer time to find a target.
For the high density ␳ of searching particles 共large y兲,
the main contribution to the integral in Eq. 共30兲 comes from
small values of z so that the quadratic term z2 can be neglected. In this case, one has
T3,␣共0,y兲 ⯝
0
10
3
共29兲
.
1. Perfectly reactive target
T3,␣共0,y兲 =
−1
10
FIG. 2. The function T3,␣共0 , y兲 for ␣ = 1 共top curve in blue兲 and ␣ = 2 / 3
共bottom curve in red兲, its asymptotic behavior 关Eq. 共32兲兴 at large y 共dashed
lines兲, and Padé approximation 关Eq. 共33兲兴 with ␤ = ␣ / 1.5 共circles兲.
2
⬁
−2
10
y=ρR
where
⫻
10 −3
10
共32兲
2冑
−2/␣
.
and the mean search time 具␶m
3 典 decreases as 共␳R D␣兲
Interestingly, smaller values of ␣ yield a faster decrease of
the mean search time. In other words, slower subdiffusive
motion allows for a faster search of a target. We return to this
seemingly paradoxal result in Sec. III C.
A transition region between two asymptotic limits 关Eqs.
共31兲 and 共32兲兴 can be modeled by a Padé approximation
2. Partially reactive target
For a partially reactive target 共⌳ ⬎ 0 or x ⬎ 0兲, four
asymptotic regimes with small/large values of x and y can be
distinguished. Since T3,␣共x , y兲 has a finite limit as x → 0, the
behavior for small x is similar to that for x = 0 from the previous subsection. For large x, one obtains
T3,␣共x,y兲 ⯝ x1/␣ .
共34兲
This asymptotic result is also valid for any x ⬎ 0 in the limit
y → ⬁. Figure 3 shows T3,␣共x , y兲 for ␣ = 1 and ␣ = 2 / 3 and its
asymptotic behavior at large x.
A Padé approximation between small and large values of
x reads as
冉 冋
T3,␣共x,y兲 ⯝ T3,␣共0,y兲 1 +
x1/␣
T3,␣共0,y兲
册冊
␤ 1/␤
.
共35兲
The adjustable parameter ␤ may depend on ␣ and the second
variable y. Numerical computations show that ␤ is close to ␣
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034104-7
J. Chem. Phys. 132, 034104 共2010兲
Searching for partially reactive sites
2
A qualitative explanation of the “paradox” which was
mentioned in Sec. III B relies in the short-time asymptotic
behavior 关Eq. 共16兲兴. If one introduces a time scale t0 as
10
t0 =
T
3,α
(x,y)
1
10
0
10
−1
(a)
10 −2
10
−1
0
10
1
10
10
x = Λ/R
2
10
2
10
1
(x,y)
10
T
3,α
0
−1
10
−2
10 −2
10
−1
10
0
x = Λ/R
1
10
10
FIG. 3. The function T3,␣共x , y兲 for ␣ = 1 共top plot兲 and ␣ = 2 / 3 共bottom plot兲
with y = 0.01 共top curve in blue兲, y = 0.1 共middle curve in green兲, and y = 1
共bottom curve in red兲. The asymptotic behavior 关Eq. 共34兲兴 at large x is
shown by dashed line, and the Padé approximation 关Eq. 共35兲兴 with ␤ = ␣ is
shown by circles.
for small y and then ␤ slowly decreases with an increase of
y. We stress again that the Padé formula 共35兲 is merely a
convenient empirical approximation. For ␤ = ␣ = 1, the function T3,1共x , y兲 is simply approximated by the sum of
T3,1共0 , y兲 and x = ⌳ / R which can be interpreted as contributions to the mean search time from two search stages,
namely, detection and recognition of the target. A similar
result was reported for the exit time from a spherical
domain.61
C. Other dimensions
The asymptotic behavior of the mean MPST in the limit
of large density ␳ can be derived for any d. In this limit, the
contribution to the integral in Eq. 共25兲 comes from small
times t so that the asymptotic formula 共16兲 for f d共t兲 can be
used to obtain
具␶m
d典
冦冉
兲
⌳R ⌫共1 + ␣兲
Sd
D␣
冊
冊
2/␣
1/␣
共⌳ = 0兲,
共⌳ ⬎ 0兲,
冧
then f d共t兲 ⯝ 共t / t0兲␣/2 for ⌳ = 0 and f d共t兲 ⯝ 共t / t0兲␣ for ⌳ ⬎ 0.
For any fixed time t in the region t ⬍ t0, f d共t兲 as a function of
d
␣ is larger for smaller ␣, while the function e−␳R f d共t兲 is
smaller for smaller ␣. The integral of the latter function from
0 to t0 which provides the major contribution to the mean
search time in the high density limit, is smaller for smaller ␣.
IV. CONCLUSION
10
(b)
冉
␣
2
R ⌫共1 +
冑D ␣ S d
⯝
冦
⌫共1 +
2
␣
兲关⌫共
共Sd冑D␣␳R
⌫共1 +
1
␣
兲兴
1 + ␣2 2/␣
d−1 2/␣
兲
兲关⌫共1 + ␣兲兴1/␣
共SdD␣␳Rd−1/⌳兲1/␣
共⌳ = 0兲,
共⌳ ⬎ 0兲.
冧
A recognition phase was introduced to the classical target search problem through Robin boundary condition on the
target boundary. Each time when a target is found, a subdiffusing particle may either “recognize” it and stop the searching process, or leave the target unrecognized and continue to
search. The recognition phase may account for a finite reaction rate in many chemical or biochemical reactions. Although the problem is formulated for arbitrary target shape,
we focused on spherical targets. The rotation invariance allowed us to derive explicit formulas for the Laplace transform of the probability density q␣共r , t兲 of the single-particle
search time. The short and long-time asymptotic behaviors of
this density were analyzed. The mean SPST is known to be
infinite.
The situation is different when many independent subdiffusing particles search simultaneously for the same target.
We obtained explicit formulas for the probability density of
the MPST for subdiffusion in one and three dimensions. We
computed the mean multiple-particles search time and studied its dependence on the reaction rate 共the length ⌳兲, the
density of particles ␳, and the exponent ␣ characterizing subdiffusion. As expected, lower density or smaller reaction rate
共larger ⌳兲 lead to a longer search. In turn, in the limit of high
density ␳, the particles undergoing slower subdiffusive motion 共smaller ␣兲 find a target faster. This seemingly paradoxal
result can be explained by the short-time asymptotic behavior of the probability density of the MPST.
ACKNOWLEDGMENTS
The work has been partly supported by the ANR program “DYOPTRI.”
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