(1). (2). Brownian flights. A.Batakis (1), P.Levitz (2), M.Zinsmeister (1) (2). [email protected] Collaboration : D. GREBENKOV, B. SAPOVAL Funded by : ANR Mipomodim,. 1)The physics of the problem.
A.Batakis (1), P.Levitz (2), M.Zinsmeister (1) (2)
D. GREBENKOV, B. SAPOVAL
Funded by: ANR Mipomodim,
1)The physics of the problem.
The polymers that possess electrical charges (polyelectrolytes) are soluble in the water because of the 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.
General theme of this work: rely the statistics of the Brownian flights to the geometry of the polymers.
Simulation of an hydrogen bond:
Polymers and colloids in suspension exhibit rich and fractal geometry:
Mathematical modelling of the first 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 with uniform law 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.
First passage statistics for
Brownian flights over a fractal nest:
P.L et al; P.R.L. (May 2006)
Brownian Exponants exponents: Influence of the surface roughness:
Self similarity/ Self affinity(D. Grebenkov, K.M. Kolwankar, B. Sapoval, P. Levitz)
POSSIBLE EXTENSION TO LOW MINKOWSKI DIMENSIONS:
(1<dsurface<2 with dambiant=3)
Relaxation methods in NMR allow to measure the statistics of flights:
If we want to make experiments one problem immediately arises:
How to be sure that it is the first flight?
One solution: to use molecules that are non-fractal
First Test: Probing a Flat Surface
An negatively charged particle
P.L et al Langmuir (2003)
Laponite Glass at C= 4% w/w
P.L. et al Europhysics letters, (2005), P.L. J. Phys: Condensed matter (2005)
Water NMR relaxation
Dilute suspension of Imogolite colloids
Magnetic Relaxation Dispersion of Lithium Ion in Solution of DNADNA from Calf Thymus(From B. Bryant et al, 2003)
2) 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.
Case of the second flight:
It is likely that there is an invariant measure equivalent with harmonic measure
3) THE 3D CASE.
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)
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:
Every open subset in the d-dimensional space can be partitionned (modulo boundaries) as a union of dyadic cubes such that:
For all integers j we define Wj =the number of Whitney cubes of order j.
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.
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.
4) Self-avoiding Walks
Definition of SAW
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.
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.
It is believed that the SAW is the same as SLE8/3. Definition of SLE:
Let Ut be the set of points in the upper half-plane for which the life-time is >t. gt is a holomorphic bijection between Ut and the UHP.
Rohde and Schramm have shown that for kappa less or equal to 4, UHP\ Ut is a simple curve.
Theorem (Beffara, Rohde-Schramm): The dimension of the SLEkappa curve is 1+kappa/8.
Theorem (Rohde-Schramm): The conformal mapping from UHP onto Ut is Holder continuous (we say that Ut is a Holder domain).
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: