Transcript Document

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.
General outline.
I. Narrow escape problem (NEP) for surface-mediated diffusion.
A. The definition of the problem.
B. Qualitative discussion and initial guesses. Why to expect some optimization
here?
C. Mean-field theory of the problem. Why there is no optimization in the mean
field.
D. Going beyond mean-field (O. Bénichou, D. Grebenkov et al.)
II. Overlap of Brownian trajectories.
A. Definition of the problem and its connection to the NEP one.
B. General theory of how to calculate the mean overlap.
C. Many walkers starting from a single point: overview of the results and a phase
diagram.
D. Two walks starting at separate points: numerical results for scaling functions.
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 a  R; b  0 will be
considered.
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!
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 t NE
t NE  t  t bulk N
where t bulk 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


N  At  4 R 2

1
where At  is the mean area visited in
one try.
Reflecting boundaries: infinitely small t.
We start with determining the length of the single bulk excursion t bulk .
To define it, one needs to address a somewhat subtle question: what does it actually
mean that a particle desorbs after time t?
It is not enough to assume that it just changes from “adsorbed” state to “bulk” state, it
should also get significantly far away from the surface (say, by some distance s = ga),
otherwise it will immediately adsorb back.
To define what g (and, therefore, what t bulk ) one should actually choose, we revisit
the known case of searching for a target in a domain with perfectly reflecting
boundaries. The survival probability in this case is known to behave as
S t  ~ exp  t t 3 D  with t 3 D  V 4aD3 
In turn, if we assume that by touching the surface once a particle explores the area
equal to its cross-section  a 2, we may rewrite S(t) in the form

a2 
S t    Pt n 1  2 
n 0
 4R 

n
where n is the number of independent tries in search of a boundary the particle made up
to time t, and Pt n  is the probability that it have made exactly n such tries.
Reflecting boundaries: infinitely small t.
Now, the desired function Pt n  can be expressed in terms of the probability
distribution of a single excursion P(t), i.e. the probability that a particle diffusing from
the point located at a distance s from the wall will survive exactly up to time t.
Indeed, if

   Pt exp  t dt;
0

P n    Pt n exp  t dt
0
are the Laplace transforms of P(t) and Pt n , respectively, then one can show that
P n  1n 1   ,
and in turn, P(t) can be easily calculated by solving a diffusion equation with adsorbing
boundary conditions. The answer is
   n 2

2D3 
  ks 

Pt  
k
sin
exp

D
t



 3 


RR  s  k 1
 R 
  R 

2
3 / 2
if s 2 D3  t  R2 D3
This probability is essentially zero for t  s D3 , decays as t
and has an exponential cut-off at t ~ R 2 D3 .
Reflecting boundaries: infinitely small t.
This allows us to get the following expressions for the distribution Pt n  , the mean
length of the excursion t bulk and the survival probability S t  in the small t limit in
terms of s:
2
 5s 


5s
3
D
t
3

Pt n  
exp 
 N 
 
 4R N 
2  Rn
s R  

t bulk   t Pt  dt  Rs 3D3
 3D3a 2 
S t  ~ exp  
t  at large t.
3 
 4s R 
Thus, we get


t3D  4sR3 3a 2 D3  V 3aD3   s   a 4 ,
which gives us the desired regularization.
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
At  
a2  
D2t 





S t   1 

1

a
exp




1
,

k k
2
2  k
2 

4 R  4 R  k 1
R 

where S(t) now is the survival probability of a particle diffusing on a sphere (with no
bulk excursion, i.e. t   ), 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 ak and νk are given by
Pνk  x0   0;
1  1
ak  1  x0 
 x P k  x dx 
 0

2
 1 P 2  x dx  ,
 x0  k

2
2
where P  x  are Legendre functions, and x0  1  a 4R .
Most importantly,  1  12 ln 2 R a  which allows us to get in the large t limit


S t  ~ exp  t t 2 D  with t 2 D  2R2 D2 ln 2R a .
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

S   S t  exp  t dt 
0
1  


 F0   F 
 1   F 
The first term here corresponds to not finding a surface at all up to time t and is (in
real space) rapidly decaying, in the second one F is a Laplace transform of the
survival probability distribution for one search try, it reads



Dt


F  1  a 4 R  ak 1   k  k  1 22  t 
R


k 1
2
2
1
Keeping just the first term in these series and expanding at small  one gets finally for
the survival probability at large t
S t  ~ exp  t t NE ;
t NE
12 D3 R 2t   aR3
t  t bulk


3D3 a 2  41 1  1D2t
a2 4R2  t t 2 D

 

NEP problem: going beyond mean-field.
t NE 
t  t bulk
a2 4R2  t t 2D


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- 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).
Overlap of the Brownian trajectories.
How many common cites will these two trajectories visit on average?
Overlap of the Brownian trajectories.
More precisely, assume you have n random walkers starting at points xi   x1,.., xn 
at time zero, and making random walks of lengths mi   m1 ,.., mn  , respectively,
starting from these points. The question is what is the expectation value
wn xi , mi 
of the number of the cites visited by all these walkers?
Note, that
i) if n = 1 the corresponding value is just the volume of a Wiener sausage;
ii) the case n = 2 and x1 = x2 is special. Indeed, then
w2 m1, m2   w1 m1   w1 m2   w1 m1  m2 
iii) the expectation value of the overlap can be written in the form
n
wn xi , mi    Pn  x, xi , mi  s  x  xi , mi ,
x
x i 1
where Pn  x, xi , mi  is the probability that cite x is visited by all workers, and
s  x  xi , mi  is the probability that it is visited by one walker, and we have allowed
for the fact that walkers are independent.
Overlap of the Brownian trajectories.
Now, the probability s  x  xi , mi  is easy to express in terms of the propagator of our
random walk. Indeed, in the Laplace space one has
r , s  
1 G r , s 
1  s G 0, s 
where
r,s   s r, ms m ; Gr, s    g r, ms m
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
n
1
G x  xi , si 
Wn xi , si   
 G0, s  ,
1

s
i x i 1
i
i 1
n
Brownian motion is recurring in D < 2 and non-recurring in D > 2, thus
n

1
 1  si  G  x  xi , si , D  2
 x i 1
Wn xi , si   
.
n
d / 2


1  si 
G  x  xi , si , D  2

 x i 1
Many trajectories from one source.
Consider the case of x1  ...  xn  0 , i.e. all walkers starting from origin:
It turns out that asymptotically for large trajectories three distinct asymptotic regimes are
possible.
Many trajectories from one source.
1. D < 2 and any n: recurring walks, finite probability of an overlap at each step,
wn m ~ w1m ~ md / 2 ;
2. D > 2 and n  D D  2 the walks intersect only near the origin wn m   const ;
3. intermediate D and n w m ~ m D  n  D  2  / 2
n
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.
Two trajectories starting at different points.
n
1
G x  xi , si 
Wn xi , si   
,

G0, si 
i 1 1  si x i 1
n
W2 R, s1 , s2  
can be rewritten in the form
s1G R, s1   s2G R, s2 
, R  x1  x2 .
1  s1 1  s2 s1  s2 G0, s1 G0, s2 
Two walkers starting at different points.
These are numerical results on a square lattice for D = 1 (left) and D = 2 (right).
Overlaps for R > 0 are converging to those for R = 0. Note that this gives rise to
The appearance of a maximum on the w2 m, R  w1 m  curve in 2D. Clearly, this
maximum exists for any D belonging to the “intermediate phase” introduced above,
i.e. for 2  D  4 in case of two walkers.
Thus, for two walks starting at a given distance there exists an optimal length when they
overlap most!
Two walkers starting at different points.
Drawing the overlap functions in the properly rescaled variables, namely w2 m, R  w2 m,0
versus R m2, one gets a nice scaling collapse. Finding analytical expressions for these
functions sounds like a nice problem to solve.
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.