Narrow escape times in microdomains with a particle-surface affinity and

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Narrow escape times in microdomains with a
particle-surface affinity and overlap of Brownian
trajectories.
Mikhail Tamm, Physics Department, Moscow State
University
In collaboration with Gleb Oshanin, Oleg
Vasilyev, Satya Majumdar and Alena Ilyina.
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General outline.
I. Narrow escape problem (NEP) for
surface-mediated diffusion.

1. The definition of the problem.
2. Qualitative discussion and initial guesses. Why
   to expect some optimization here?
3. Mean-field theory of the problem. Why there is no
   optimization in the mean field.
4. Going beyond mean-field (O. Bénichou, D.
   Grebenkov et al.)

II. Overlap of Brownian trajectories.

1. Definition of the problem and its connection to
   the NEP one.
2. General theory of how to calculate the mean
   overlap.
3. Many walkers starting from a single point
   overview of the results and a phase diagram.
4. Two walks starting at separate points numerical
   results for scaling functions.

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NEP problem definitions.
You have a particle in a spherical domain which
is capable of being reversibly adsorbed on its
inner surface. It can diffuse both in the bulk
and on the surface with diffusion constants D2
and D3, respectively. There is a hole (or
target) on the wall which you want the particle
to find. You are interested in the mean time it
takes to find the target depending on the
diffusion constants, the size of the domain R,
the size of the particle a and the size of the
target b and most importantly of the mean
length of a surface excursion t. The case
will be considered.
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NEP problem what to expect?
On the one hand there is much less space on the
surface than in the bulk, so the larger t the
better?
On the other hand diffusion in 2D is marginally
recursive, therefore bulk diffusion may provide
us with useful shortcuts which help to avoid
oversampling the same already visited points on
the surface.
Also, generally speaking, t and D2 are not
independent.
Thus, there rises a possibility that an optimal t
exists!
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NEP problem naïve guess.
The search process consists of recurring steps,
each of them consisting of a search try on the
surface and a bulk excursion. Therefore, one may
expect for the escape time
where is the mean length of a bulk
excursion and N is the average number of attempts
it takes to find a target. Also, if separate
search tries are independent, one expects
where is the mean area visited in one
try.
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Reflecting boundaries infinitely small t.
In turn, if we assume that by touching the
surface once a particle explores the area equal
to its cross-section , we may rewrite S(t)
in the form
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Reflecting boundaries infinitely small t.
and in turn, P(t) can be easily calculated by
solving a diffusion equation with
adsorbing boundary conditions. The answer is
This probability is essentially zero for
, decays as
and has an exponential cut-off at
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Reflecting boundaries infinitely small t.
Thus, we get
which gives us the desired regularization.
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Boundary diffusion calculation of A(t).
To calculate the area sampled in one search try
we recall the solution of a first passage problem
on a sphere. It is known that
where S(t) now is the survival probability of a
particle diffusing on a sphere (with no bulk
excursion, i.e. ), the first
multiplier on the r.h.s. corresponds to the fact
that we may find a target immediately with no
need for diffusion at all, and the coefficients
are given by
Most importantly,
which allows us to get in the large t limit
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NEP the mean-field result.
Now, combining all the ingredients, and assuming
that the different search attempts are
independent one gets for the survival probability
in the Laplace space
Keeping just the first term in these series and
expanding at small l one gets finally for the
survival probability at large t
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NEP problem going beyond mean-field.
This means there is NO optimization, at least
within the simple approximation presented.

• What have gone wrong?
• Tries are actually not independent!
• The interference of the tries depends on the
  length of a bulk excursion between them!
• The small-l expansion kills nonlinearity in
  A(t)!!

However, it turned out to be possible to go
beyond mean-field within a somewhat different
approach.
O.Bénichou, D. Grebenkov, et al. J. Stat.
Physics, 142, 657 (2011).
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Overlap of the Brownian trajectories.
How many common cites will these two trajectories
visit on average?
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Overlap of the Brownian trajectories.
More precisely, assume you have n random walkers
starting at points at time
zero, and making random walks of lengths
, respectively, starting from
these points. The question is what is the
expectation value
of the number of the cites visited by all these
walkers?

• Note, that
• if n 1 the corresponding value is just the
  volume of a Wiener sausage
• the case n 2 and x1 x2 is special. Indeed,
  then

iii) the expectation value of the overlap can be
written in the form
where is the
probability that cite x is visited by all
workers, and
is the probability that it is visited by one
walker, and we have allowed for the fact that
walkers are independent.
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Overlap of the Brownian trajectories.
where
are discrete Laplace transforms, and g(r,m) is a
probability for the end-points of a random walk
of length m to be separated by a distance r.
This allows us to write
Brownian motion is recurring in D lt 2 and
non-recurring in D gt 2, thus
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Many trajectories from one source.
It turns out that asymptotically for large
trajectories three distinct asymptotic regimes
are possible.
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Many trajectories from one source.
1. D lt 2 and any n recurring walks, finite
probability of an overlap at each step,
2. D gt 2 and
the walks intersect only near
the origin 3.
intermediate D and n
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Many trajectories from one source.
Numerical data is supporting the predicted phase
diagram. Here the results for D 1, 2, 3 are
presented, the numbers on the right are the
numbers of walkers corresponding to each curve,
red lines are scaling laws predicted by the
theory, cases D 2 and D n 3 are marginal,
the logarithmic corrections to power laws are
predicted in these cases.
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Two trajectories starting at different points.
can be rewritten in the form
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Two walkers starting at different points.
Thus, for two walks starting at a given distance
there exists an optimal length when they overlap
most!
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Two walkers starting at different points.
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Conclusions and take-home messages.
Concerning narrow-escape problem

1. It is possible to optimize the escape time by
   playing with adsorption-desorption.
2. In 3D with 2D boundary the effect is small.
3. Do not try mean-field here correlations between
   excursions are important.

Concerning the Brownian trajectories overlap
1. There exist three different regimes depending
on the number of particles and dimensionality of
the space. 2. In the intermediate regime there is
an optimal trajectory length, at which the
overlap fraction is maximal. 3. If normalized
by the overlap of trajectories starting at the
same point, the overlap function converges to a
scaling function, which is still to be calculated
explicitly.