brownian flights [email protected] funded by: anr mipomodim, a.batakis (1), d.grebenkov...
TRANSCRIPT
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Brownian flights
Funded by: ANR Mipomodim,
A.Batakis (1), D.Grebenkov (2),K.Kolwankar, P.Levitz (2), B.Sapoval, M.Zinsmeister (1) (2)
(1) (2)
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1)The physics of the problem.
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The polymers that possess electrical charges (polyelectrolytes) are soluble in the water because of hydrogen bondings.
The water molecules exhibit a dynamics made of adsorptions (due to hydrogen bondings) on the polymers followed by Brownian diffusions in the liquid.
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Simulation of an hydrogen bond:
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The dynamics of the water molecules can be seen as an intermittent one: flights around the surface (« quick motion ») followed by a « slow » motion on the surface itself.
In this talk we will ignore the adsorption steps.
We are interested in the statistics of the flights , that is their duration and their length and wish to connect them to the geometry of the surface.
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Brownian flights over a fractal nest:
P.L et al; P.R.L. (May 2006)
)(tFirst passage statistics for and )(r
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Polymers and colloids in suspension exhibit rich and fractal geometry:
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Main point: These statistics can be measured in experiences by using relaxometry methods in NMR (nuclear magnetic resonance).
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I(t+)
ttL /1)(
I(t)B0
L
A
A
L
time
I(t)
L
A
NMR EXPERIMENTATION
NMR Relaxation
R1slow(f) FTt(<I(0).I(t)>)
R1Slow(f)
f
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Mathematical simulation of a flight:
We consider a surface or a curve which is fractal up to a certain scale, typically a piecewise affine approximation of a self-affine curve or surface.
We then choose at random an affine piece and consider a point in the complement of the surface nearby this piece.
We then start a Brownian motion from this point stopped when it hits back the surface and consider the length and duration of this flight.
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We have to precise the term « random »:
If we consider only one flight there are two natural models:
Uniform law: all points at a fixed vicinity of the surface have the same probability of being chosen as the starting point
The point is chosen as the first hitting point on the surface of a Brownian motion started from a distinguished point :=the harmonic measure
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In practise it is impossible to decide if it is the first flight.
If m is a distribution on the boundary, the law of the hitting point at the end of the flight is a new distribution T(m)
Is there an invariant distribution?
If one starts the process with one of the 2 above mentionned distributions, does the law of the nth hitting point converge to an invariant distribution?
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If we want to perform experiments one problem immediately arises:
The flights we consider are not first flights.
2) Experimenal results:
One solution: to use molecules that are non-fractal and homogeneous
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First Test: Probing a Flat Surface
An negatively charged particle AFM Observations
P.L et al Langmuir (2003)
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EXPERIMENTS VERSUS ANALYTICAL MODEL FOR FLAT INTERFACES
0
2
4
6
8
10
12
10-2 10-1 100 101
R1(s
-1)
Frequency (MHz)
-dR1/df
f (MHz)
10-3
10-2
10-1
100
101
102
103
10-2
10-1
100
101
Laponite Glass at C= 4% w/w
2
2
0 2
)2/1(
A
L
))/(2/1)/()//((1)(~ 2/3
002/1
0 L
tt /1)(
P.L. et al Europhysics letters, (2005), P.L. J. Phys: Condensed matter (2005)
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Dilute suspension of Imogolite colloids
Almost )ln()(1 baR
Water NMR relaxation
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Magnetic Relaxation Dispersion of Lithium Ion in Solution of DNA
DNA from Calf Thymus(From B. Bryant et al, 2003)
R1(s-1)
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3) Simulations
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Brownian Exponants exponents: Influence of the surface roughness:
Self similarity/ Self affinity (D. Grebenkov, K.M. Kolwankar, B. Sapoval, P. Levitz)
2D
3D
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2
4 edd 3 edd
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12
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POSSIBLE EXTENSION TO LOW MINKOWSKI DIMENSIONS: (1<dsurface<2 with dambiant=3)
dsurface=1.25
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4) A simple 2D model.
We consider a simple 2D model for which we can rigorously derive the statistics of flights.
In this model the topological structure of the level lines of the distance-to-the-curve function is trivial.
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Case of the second flight:
There is an invariant measure equivalent with harmonic measure
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5) THE 3D CASE.
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u
vU
General case: we want to compare the probability P(u,v,U) that a Brownian path started at u touches the red circle of center v and radius half the distance from v to the boundary of U before the boundary of U with its analogue P(v,u,U)
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This last result is not true in d=2 without some extra condition. But we are going to assume this condition anyhow to hold in any dimension since we will need it for other purposes.
In dimension d it is well-known that sets of co-dimension greater or equal to 2 are not seen by Brownian motion. For the problem to make sense it is thus necessary to assume that the boundary is uniformly « thick ». This condition is usually defined in terms of capacity. It is equivalent to the following condition:
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Every open subset in the d-dimensional space can be partitionned (modulo boundaries) as a union of dyadic cubes such that:
(Whitney decomposition)
For all integers j we define Wj =the number of Whitney cubes of order j.
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We now wish to relate the numbers Wj to numbers related to Minkowski dimension.
For a compact set E and k>0 we define Nk as the number of (closed) dyadic cubes of order k that meet E.
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This gives a rigourous justification of the results of the simulations in the case of the complement of a closed set of zero Lebesgue measure.
For the complement of a curve in 2D in particular, it gives the result if we allow to start from both sides.
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A pair (U,E) where U is a domain and E its boundary is said to be porous if for every x in E and r>0 (r<diam(E)) B(x,r) contains a ball of radius >cr also included in U.
U
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If the pair (U,E) is porous it is obvious that the numbers Wj and Nj are essentially the same.
Problem: the domain left to a SAW is not porous.
So if the domain is porous, have a « thick » boundary and a Minkowski dimension, then the power-law satisfied by the statistics of the flights is the expected one.
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6) Self-avoiding Walks
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P.LEVITZ
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3/52
4 edd
Self-Avoiding Walk, 2ed
3 edd 3/7
S.A.W.
d=4/3
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Theorem (Beffara, Rohde-Schramm): The dimension of the SLEk curve is 1+k/8.
Theorem (Rohde-Schramm): The conformal mapping from UHP onto Ut is Holder continuous (we say that Ut is a Holder domain).
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A Holder domain is weakly porous in the following sense:
If B(x,2-j) is a ball centered at the boundary then the intersection of this ball with the domain contains a Whitney square of order less than j+Cln(j).
As a corollary we can compare the Minkowski and Whitney numbers in the case of Holder domains:
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Relation time-length:
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Perspectives:
Taking into account the adsorption steps
Intermittent dynamics (predator-prey)
Strategy replaced by geometry
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Link with the spectrum of the Laplacian and Weyl-Berry problem.
Link with reflected Brownian Motion and Benyamini-Chen-Rohde work.