PHYSICAL REVIEW E 91, 052134 (2015)
Optimization and universality of Brownian search in a basic model of quenched
heterogeneous media
Alja
ˇ
z Godec
1,2,*
and Ralf Metzler
1,3,†
1
Institute of Physics & Astronomy, University of Potsdam, 14776 Potsdam-Golm, Germany
2
National Institute of Chemistry, 1000 Ljubljana, Slovenia
3
Department of Physics, Tampere University of Technology, 33101 Tampere, Finland
(Received 2 March 2015; published 21 May 2015)
The kinetics of a variety of transport-controlled processes can be reduced to the problem of determining the
mean time needed to arrive at a given location for the first time, the so-called mean first-passage time (MFPT)
problem. The occurrence of occasional large jumps or intermittent patterns combining various types of motion
are known to outperform the standard random walk with respect to the MFPT, by reducing oversampling of
space. Here we show that a regular but spatially heterogeneous random walk can significantly and universally
enhance the search in any spatial dimension. In a generic minimal model we consider a spherically symmetric
system comprising two concentric regions with piecewise constant diffusivity. The MFPT is analyzed under
the constraint of conserved average dynamics, that is, the spatially averaged diffusivity is kept constant. Our
analytical calculations and extensive numerical simulations demonstrate the existence of an optimal heterogeneity
minimizing the M FPT to the target. We prove that the MFPT for a random walk is completely dominated by what
we term direct trajectories towards the target and reveal a remarkable universality of the spatially heterogeneous
search with respect to target size and system dimensionality. In contrast to intermittent strategies, which are most
profitable in low spatial dimensions, the spatially inhomogeneous search performs best in higher dimensions.
Discussing our results alongside recent experiments on single-particle tracking in living cells, we argue that the
observed spatial heterogeneity may be beneficial for cellular signaling processes.
DOI: 10.1103/PhysRevE.91.052134 PACS number(s): 05.40.−a, 89.75.−k, 82.70.Gg, 83.10.Rs
I. INTRODUCTION
Random search processes are ubiquitous in nature [1–17],
ranging from the diffusive motion of regulatory molecules
searching for their targets in living biological cells [5,8,18–25]
and bacteria and animals searching for food by active motion
[2,8] all the way to the spreading of epidemics and pandemics
[3,4] and computer algorithms in high-dimensional optimiza-
tion problems [26]. The fact that the search strategy in these
processes is to a large extent random reflects the incapability
of the searcher to keep track of past explorations at least over
more than a certain period [5]. Over the years several different
search strategies have been studied in the literature, including
Brownian motion [8,18,27,28], spatiotemporally decoupled
L
´
evy flights (LFs) [29–34], and coupled L
´
evy walks (LWs)
[35–46] in which the searcher undergoes large relocations
with a heavy-tailed length distribution either instantaneously
(LFs) or with constant speed (LWs), as well as intermittent
search patterns, in which the searcher combines different
types of motion [5], for instance, three-dimensional and one-
dimensional diffusion [18–25,47,48], three-dimensional and
two-dimensional diffusion [49,50], or diffusive and ballistic
motion [ 5,8,51–57].
The efficiency of the search strategy is conventionally
quantified via the mean first-passage time (MFPT) defined
as the average time a random searcher needs to arrive at the
target for the first time [5,27,28,58,59]. The physical principle
underlying an improved search efficiency is an optimized
balance between the sampling of space on a scale much
*
agodec@uni-potsdam.de
†
rmetzler@uni-potsdam.de
larger than the target and on a scale similar to or smaller
than the target [5]. More specifically, periods of less-compact
exploration, for instance, diffusion in higher dimensions, L
´
evy
flights, or ballistic motion, aid towards bringing the searcher
faster into the vicinity of the target. Concurrently, a searcher
in such a less-compact search mode may thereby easily
overshoot the target [32,33]. In contrast, compact exploration
of space (for instance, diffusion in one dimension) is superior
when it comes to hitting the target from close proximity, but
performs worse when it comes to the motion on larger scales
taking the searcher from its starting position into the target’s
vicinity: Typically frequent returns occur to the same location,
a phenomenon referred to as oversampling. Mathematically,
this is connected to the recurrent nature of such compact
random processes. The idea behind search optimization is to
find an optimal balance of both more- and less-compact search
modes in the given physical setting [5]. For instance, in the
so-called facilitated diffusion model for the target search of
regulatory proteins on DNA [19–23], the average duration of
noncompact three-dimensional free diffusion is balanced by
an optimal compact one-dimensional sliding r egime along the
DNA molecule. In an intermittent search [5,8,51] persistent
ballistic excursions are balanced by compact Brownian phases.
This optimization principle intuitively works better in lower
dimensions, where a searcher performing a standard random
walk oversamples the space. Hence, the typically considered
optimized strategies have the largest gain in low dimensions.
In a variety of experimental situations the motion of a
searcher is characterized by the same search strategy but is
not translationally invariant. A typical example is a system
in which the searcher performs a standard random walk but
with a spatially varying rate of making its steps. This type
of motion is actually abundant in biological cells, where
1539-3755/2015/91(5)/052134(17) 052134-1 ©2015 American Physical Society
ALJA
ˇ
Z GODEC AND RALF METZLER PHYSICAL REVIEW E 91, 052134 (2015)
experiments revealed a distinct spatial heterogeneity of the
protein diffusivity [60–62]. Several aspects of such diffusion in
heterogeneous media have already been addressed [27,63–67],
but the generic first-passage time properties remain elusive, in
particular, for quenched environments.
Here was ask the question whether spatial heterogeneity is
generically detrimental for the efficiency of a random search
process or whether it could even be beneficial. Could it
even be true that proteins find their targets on the genome
in the nucleus faster because their diffusivity landscape in
the cell is heterogeneous? On the basis of exact results for
the MFPT in one, two, and three dimensions in a closed
domain under various settings we show here that a spatially
heterogeneous search can indeed significantly enhance the rate
of arrival at the target. We explain the physical basis of this
acceleration compared to a homogeneous search process and
quantify an optimal heterogeneity, which minimizes the MFPT
to the target. Furthermore, we show that heterogeneity can
be generically beneficial in a random system and is thus a
robust means of enhancing the search kinetics. The optimal
heterogeneous search rests on the remarkable observation that
the MFPT is completely dominated by those direct trajectories
heading directly towards the target. We prove that the MFPT
for the heterogeneous system can be exactly described with the
results of a standard random walk. We compare our theoretical
findings to recent experiments on single-particle tracking in
living cells, which are indeed in line with the requirements for
enhanced search.
The paper is organized as follows. Section II introduces our
minimal model for heterogeneous search processes. In Sec. III
we briefly summarize our main general results, which hold
irrespective of the dimension (d = 1, 2, or 3). Section IV is
devoted to the analysis of the most general situation with a
specific starting point and position of the interface. In Sec. V
we focus on the global MFPT, that is, the MFPT averaged
over the initial position. In Sec. VI we analyze a system
with a random position of the interface and optimize the
MFPT averaged over the interface position. In Sec. VII we
address the global MFPT in systems with a random position
of the interface. Throughout we discuss our results in a
biophysical context motivated by recent experimental findings.
We conclude in Sec. VIII by discussing the implications of our
results for more general spatially heterogeneous systems.
II. MINIMAL MODEL FOR SPATIALLY
HETEROGENEOUS RANDOM SEARCH
We focus on the simplest scenario of a spatially hetero-
geneous system. Even for this minimal model the analysis
turns out to be challenging and our exact results reveal a rich
behavior with several apriorisurprising features. We consider
a spherically symmetric system in dimensions d = 1, 2, and
3 with a perfectly absorbing target of radius a located in the
center (Fig. 1). The outer boundary at radius R is taken to be
perfectly reflecting. The system consists of two domains with
uniform diffusivities denoted by D
1
and D
2
in the interior
and exterior domains, respectively. The interface between
these domains is located at r
I
. The microscopic picture we
are considering corresponds to the kinetic interpretation of
the Langevin or corresponding Fokker-Planck equations. In
(a)
(b)
FIG. 1. (Color online) Schematic of the model system: (a) A
spherical target with radius a is placed in the center of a spherical
domain of radius R. The free space between the radii a and R is
divided into two regions denoted by subscripts. The inner region is
bounded by a shell at radius r
I
. The outer region ranges from r
I
to the
reflective boundary at R. Initially, the particle’s starting position is
uniformly distributed over the surface of the sphere with radius r
0
.(b)
Microscopic picture of the problem starting from a discrete random
walk between spherical shells. The hopping rates are assumed to obey
detailed balance and the interface position is chosen to be placed
symmetrically between two concentric spherical surfaces.
particular we assume that the dynamics obey the fluctuation-
dissipation relation and in the steady state agree with the results
of equilibrium statistical mechanics [68].
In the biological context we consider that the system is in
contact with a heat bath at constant and uniform temperature
T . The signaling proteins diffuse in a medium comprising
water and numerous other particles, such as other biomacro-
molecules or cellular organelles. The remaining particles,
which we briefly call crowders, are not uniformly distributed
across the cell: Their identity and relative concentrations differ
within the nucleus and the cytoplasm and can also show
variations across the cytoplasm. The proteins hence experience
a spatially varying friction (r), which originates from spatial
variations in the long-range hydrodynamic coupling to the
motion of the crowders, which is in turn mediated by the
solvent [69,70]. The proteins thus move under the influence of
a position-dependent diffusion coefficient D(r) = 2k
B
T(r)
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OPTIMIZATION AND UNIVERSALITY OF BROWNIAN . . . PHYSICAL REVIEW E 91, 052134 (2015)
and a fluctuation-induced thermal drift F(r) ∼ k
B
T ∇(r )
(see [71] for details). The vital role of such hydrodynamic
interactions in the cell cytoplasm was demonstrated in [72].
Because of the spherical symmetry of the problem we can
reduce the analysis to the radial coordinate alone. That is, we
only trace the projection of the motion of the searcher onto
the radial coordinate and therefore start with a discrete space-
time nearest-neighbor random walk in between thin concentric
spherical shells of equal width R = R
i+1
− R
i
as depicted
in Fig. 1. The shell i denotes the region between surfaces with
radii R
i−1
and R
i
. We assume that the hopping rates between
shells (i → j ) obey detailed balance
p(i)(i → i + 1) = p(i + 1)(i + 1 → i). (1a)
Here p(i) denotes the probability distribution of finding the
particle in shell i. The hopping rates are given as the product
of the intrinsic rate 2D(i)/R
2
and q(i), the probability to
jump from shell i to shell i + 1 [and 1 − q(i) for jumps in the
other direction],
(i → i + 1) =
2D(i)
R
2
q(i). (1b)
Here D(i) is the arithmetic mean of the diffusivity in shell
i.Therateq(i) can be derived as follows. A random walker
located in shell i at time t can either move to shell i + 1
with probability q(i)ortoshelli − 1 with probability 1 −
q(i). Due to the isotropic motion of the random walker, these
probabilities are proportional to the respective surface areas of
the bounding d-dimensional spherical surfaces at R
i−1
and R
i
,
that is, q(i) ∼ R
d−1
i
and 1 − q(i) ∼ R
d−1
i−1
. The proportionality
constant is readily obtained from the normalization condition
leading to
q(i) =
R
d−1
i
R
d−1
i
+ R
d−1
i−1
(1c)
and thus 1 − q(i) = R
d−1
i−1
/(R
d−1
i
+ R
d−1
i−1
). Therefore, while
the random walker moves in all directions (radial, azimuthal, or
polar) we can project its motion on the radial coordinate only.
We assume that the interface is located between two concentric
shells leading to a continuous steady-state probability density
profile. The searcher starts at t = 0 uniformly distributed over
the surface of a d sphere with radius r
0
, as sketched in Fig. 1(a).
In our analytical calculations we model the system in terms
of the probability density function p(r,t |r
0
) to find the particle
at radius r at time t after starting from radius r
0
at t = 0. Here
p(r,t |r
0
) obeys the radial diffusion equation
∂p(r,t |r
0
)
∂t
=
1
r
d−1
∂
∂r
D(r)r
d−1
∂
∂r
p(r,t |r
0
) (2a)
with a position-dependent diffusivity D(r). Here we consider
the s implest version of a piecewise constant diffusivity
D(r) =
D
1
,a<r r
I
D
2
,r
I
<r R.
(2b)
We assume that the target surface at radius a is perfectly
absorbing,
p(a,t|r
0
) = 0, (2c)
to determine the first-passage behavior. At the outer radius R
we use the reflecting boundary condition
∂p(r,t |r
0
)
∂r
r=R
= 0. (2d)
These boundary conditions are complemented by joining
conditions at r
I
by requiring the continuity of the probability
density and the flux, which follow from our microscopic
picture.
To quantify our model system we introduce the ratio
ϕ =
D
1
D
2
(3a)
of the inner and outer diffusivities. Moreover, we demand that
the spatially averaged diffusivity
D =
d
R
d
− a
d
R
a
r
d−1
D(r)dr (3b)
remains constant for varying D
1
and D
2
. Without such
a constraint the problem of finding an optimal ϕ, which
minimizes the MFPT, is ill posed and has a trivial solution
ϕ =∞. More importantly, we want to compare the search
efficiency as a function of the degree of heterogeneity, where
the overall intensity of the dynamics is conserved. Returning
to our microscopic picture of a signaling protein searching
for its target in the nucleus, the heterogeneous diffusivity
is due to spatial variations in the long-range hydrodynamic
coupling to the motion of the crowders. Their identity and
relative concentration in the cell vary in space, but the effect
is mediated by thermal fluctuations in the solvent at a constant
temperature. The constraint in Eq. (3b) then corresponds to
a redistribution of the crowders at constant temperature, cell
volume, and numbers of the various crowders, which would
not affect the spatially averaged diffusivity.
Under the constraint (3b) of constant spatially averaged
diffusivity we obtain for any given ϕ and r
I
that
D
1
=
ϕ
D
(ϕ − 1)χ(r
I
) + 1
,D
2
=
D
(ϕ − 1)χ(r
I
) + 1
, (4a)
where we introduced the hypervolume ratio
χ(r
I
) =
r
d
I
− a
d
R
d
− a
d
(4b)
of the inner versus the entire domain excluding the target
volume. To solve Eq. (2a) we take a Laplace transform in
time and the obtained Bessel-type equation is s olved exactly
as shown in Sec. IV. The MFPT T(r
0
) for the particle to
reach the target surface at r = a is obtained from the Laplace
transformed flux into the target
j(r
0
,s) =
d
D
1
a
d−1
∂
P (r,r
0
,s)
∂r
r=a
(5a)
via the relation
T(r
0
) =−
∂
˜
j
a
(r
0
,s)
∂s
s=0
. (5b)
The angular prefactor
d
= 1ford = 1,
d
= 2π for
d = 2, and
d
= 4π for d = 3. We treat the degree of
heterogeneity ϕ as an adjustable parameter at a fixed value
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ALJA
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Z GODEC AND RALF METZLER PHYSICAL REVIEW E 91, 052134 (2015)
of r
I
and seek an optimal value minimizing the MFPT. The
optimal heterogeneity, which we denote by an asterisk, is thus
obtained by extremizing T
a
(r
0
) with respect to ϕ.
III. SUMMARY OF THE MAIN RESULTS
Since R in combination with the diffusivity D sets the
absolute time scale, we can express, without loss of generality,
time in units of R
2
/D,setD = 1, and focus on the problem in
a unit sphere. We introduce dimensionless spatial units x
a
=
a/R, x
I
= r
I
/R, and x
0
= r
0
/R. For the sake of completeness,
we retain the explicit R and
D dependence in the prefactors.
Our first main result represents the fact that the MFPT to
the target in the inhomogeneous system in dimensions d = 1,
2, and 3 can be expressed exactly in terms of the corresponding
MFPT T
0
(x
0
; D
i
)inahomogeneous system with diffusivity
D
i
, with i = 1 or 2. Remarkably, the MFPT is thus exactly
equal to
T(x
0
) =
T
0
(x
0
; D
1
),x
0
x
I
T
0
(x
I
; D
1
) + T
0
x
I
(x
0
; D
2
),x
0
>x
I
.
(6)
In the second line the argument x
I
stands for the release of
the particle at the interface and the index x
I
of the last term
is used to indicate that this term measures the first passage
to the interface at x
I
. That is, in this case when the particle
starts in the inner region with diffusivity D
1
,Eq.(6) reveals
that the MFPT of the heterogeneous system is equal to that
of a homogeneous system with diffusivity D
1
everywhere and
is independent of the position of the interface. Conversely, if
the searcher starts in the exterior region with diffusivity D
2
the MFPT contains two contributions: (i) the MFPT from r
0
to r
I
in a homogeneous system with diffusivity D
2
and (ii) the
MFPT from r
I
to a in a homogeneous system with diffusivity
D
1
, as shown schematically in Fig. 2. Equation (6) is exact
and independent of the choice for D
1,2
and thus holds for an
arbitrary set of diffusivities and even if D
1
= D
2
. In other
words, it is not a consequence of a conserved
D. Note that
if one starts from different mathematical assumptions for the
inclusion of inhomogeneous step frequencies, a result different
from Eq. (6) emerges [73].
The additivity principle of the individual MFPTs in Eq. (6)
is only possible if the excursions of the searcher in the
directions away from the target are statistically insignificant.
We would expect that some trajectories starting in the inner
region will carry the searcher into the outer region with
diffusivity D
2
before the searcher eventually crosses the
interface and reaches the target by moving through the inner
region with diffusivity D
1
. Such trajectories will obviously
be different in the heterogeneous system in comparison to a
homogeneous system with diffusivity D
1
everywhere. This
(a)
(b)
FIG. 2. (Color online) Schematic of the equivalence of MFPTs
in inhomogeneous and homogeneous systems in the case of (a) a
searcher starting in the inner region and (b) a searcher starting in the
outer region. The radius x
1
of the initial particle position is shown by
the thin dashed circle. We show direct trajectories as solid lines. Each
panel also contains an indirect trajectory (dashed line) that leads the
particle into the outer region of the system. As our analysis shows
direct trajectories dominate the MFPT.
appears to contradict the complete independence of D
2
of
the MFPT in Eq. (6) for trajectories with x
0
x
I
.This
observation can be explained by the dominance of direct
trajectories, whose occupation fraction outside the starting
radius is statistically insignificant: The MFPT for a standard
random walk in dimensions d = 1, 2, and 3 is completely
dominated by direct trajectories. As such, the MFPT is really
a measure for the efficiency of the direct trajectories. We
note that if the walker starts in the outer region (x
0
>x
I
),
as intuitively expected, the MFPT diverges with vanishing D
2
.
Our s econd main result demonstrates that for x
0
>x
I
a
finite optimal heterogeneity exists at a given interface position
and is given by
ϕ
∗
(x
0
) =
1 − χ(x
I
)
χ(x
I
)
T
0
(x
I
;1)
T
0
x
I
(x
0
;1)
. (7)
For this value the MFPT T(x
0
) attains a minimum. Hence, the
optimal heterogeneity is completely determined by the volume
fractions and the MFPT properties and hence strictly by the
direct trajectories. As above, the index x
I
in the MFPT T
0
(x
I
)
indicates the first passage to the interface, while without this
index the MFPT T
0
quantifies the first passage to the target at
x
a
. The explicit results for ϕ
∗
(x
0
)areshowninFig.4 and are
discussed i n detail in Sec. IV.
Often one is interested in the MFPT averaged over an
ensemble of starting positions, the global MFPT
T. As before,
it can be shown that an optimal heterogeneity exists for any
interface position and is universally given by
ϕ
∗
=
⎛
⎜
⎜
⎜
⎝
1 − χ(x
I
)
χ(x
I
)
T
0
(x
I
;1)−
x
I
x
a
x
d
0
(d/dx
0
)T
0
(x
0
;1)
dx
0
T
0
x
I
(1; 1) −
1
x
I
x
d
0
(d/dx
0
)T
0
x
I
(x
0
;1)
dx
0
⎞
⎟
⎟
⎟
⎠
1/2
. (8)
052134-4
OPTIMIZATION AND UNIVERSALITY OF BROWNIAN . . . PHYSICAL REVIEW E 91, 052134 (2015)
Similar to the general case, ϕ
∗
is again proportional to [χ(x
I
)
−1
− 1]
1/2
but here the corresponding MFPTs in the second factor
are reduced by a spatially averaged change of the MFPT with respect to the starting position. The optimal heterogeneity for the
global MFPT is shown in Fig. 5 and discussed in detail in Sec. V.
In a setting when the interface position is random and uniformly distributed we are interested in the MFPT from a given
starting position averaged over the interface position. A measurable quantity for this scenario for an ensemble of random-interface
systems is the MFPT {T
a
(x
0
)}, where the curly brackets denote an average over the interface positions x
I
. Explicit results for
dimensions d = 1, 2, and 3 are given in Sec. VI. Solving for the optimal heterogeneity, we obtain
{ϕ}
∗
=
⎛
⎜
⎜
⎝
T
0
(x
0
) − (1 + 1/d)
x
0
x
a
x[1 − x
d
/(d + 1)][(d/dx
I
)T
0
(x
I
)]dx
I
x
d+1
a
T
0
(x
0
) − (1 + 1/d)
x
0
x
a
x
x
d
/(d + 1) − x
d
a
[(d/dx
I
)T
0
x
I
(x
0
)]dx
I
⎞
⎟
⎟
⎠
1/2
. (9)
The optimal heterogeneity for the MFPT averaged over the random interface position is shown in Fig. 6.
Finally, we compute the global MFPT averaged over the position of t he interface {
T} whose explicit results are given in
Sec. VI. Also here an optimal strategy can be identified as
{
ϕ}
∗
=
⎛
⎜
⎜
⎜
⎝
T
0
−
1
x
a
(1 − x
I
)x
d
I
1 + d − x
d
I
1 − x
d
a
(d/dx
I
)T
0
(x
I
;1)dx
I
x
d+1
a
T
0
−
1
x
a
x
d
I
x
d
I
− (1 + d)x
d
a
/
1 − x
d
a
(d/dx
I
)T
0
x
I
dx
I
⎞
⎟
⎟
⎟
⎠
1/2
. (10)
The optimal heterogeneity for the global MFPT averaged over
the r andom interface position is shown in Fig. 7.
Equation (6) represents a rigorous proof that direct trajec-
tories dominate the MFPT for Brownian motion. In addition,
the heterogeneity does not affect the fraction of direct versus
indirect trajectories but only their respective durations. Due to
the fact that indirect trajectories are statistically insignificant,
we can in principle make them arbitrarily slow if we start in
the inner region. However, as we let the inner diffusivity go to
infinity (and hence the outer one to zero) we are simultaneously
slowing down the arrivals to the interface if starting from the
outer region. The following is the physical principle underlying
the acceleration of search kinetics: The optimal heterogeneity
corresponds to an improved balance between the MFPT to
reach the interface and the one to reach the target from the
interface.
IV. MEAN FIRST-PASSAGE TIME FOR FIXED INITIAL
AND INTERFACE POSITIONS
Here we present the mathematical derivation and the
explicit results for the situation with a specific initial condition
r
0
and interface position r
I
. Equation (2a) is solved by Laplace
transformation and the resulting Green’s function with the
boundary conditions (2c) and (2d) reads
P (r,s|r
0
) =
(rr
0
)
ν
d
D
1
C
ν
(S
1
r,S
1
a)
D
ν
(S
1
a,S
1
r
I
)
C
ν−1
(S
2
r
I
,S
2
R)
+
1
√
ϕ
C
ν
(S
1
a,S
1
r
I
)
D
ν
(S
2
r
I
,S
2
R)
×
⎧
⎪
⎪
⎨
⎪
⎪
⎩
D
ν
(S
1
r
0
,S
1
r
I
)
C
ν−1
(S
2
r
I
,S
2
R)
+
1
√
ϕ
C
ν
(S
1
r
0
,S
1
r
I
)
D
ν
(S
2
r
I
,S
2
R)
,a<r
0
r
I
D
ν
(S
2
r
0
,S
2
R)
r
I
S
1
D
ν
(S
2
r
I
,S
2
R)C
ν−1
(S
2
r
I
,S
2
R)
,r
I
<r
0
R,
(11a)
where we introduced the abbreviation S
1,2
=
s/D
1,2
and the
auxiliary functions
D
ν
(z
1
,z
2
) = I
ν
(z
1
)K
ν−1
(z
2
) + K
ν
(z
1
)I
ν−1
(z
2
),
C
ν
(z
1
,z
2
) = I
ν
(z
1
)K
ν
(z
2
) − I
ν
(z
2
)K
ν
(z
1
).
(11b)
Here the I
ν
(z) and K
ν
(z) denote t he modified Bessel functions
of order ν = 1 − d/2 of the first and second kinds, respec-
tively. The Laplace transformed first-passage time density is
obtained from the flux (5a) into the target and the MFPT then
follows from relation (5b). In the case of d = 1 the target
size only enters the problem by determining the width of the
interval. Using D
ν
(z,z) = 1/z, it can be shown that Eq. (11a)
reduces to the ordinary expression for regular diffusion given
in Ref. [27]forr
I
= R and ϕ = 1.
The MFPT can be written exactly in terms of the expres-
sions for a homogeneous diffusion process in Eq. (6) (compare
Refs. [27,58]) and we obtain
1
T
0
(x
0
; D) =
R
2
2D
2(x
0
− x
a
) + x
2
a
− x
2
0
, (12a)
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