1. Boundary effects for diffusion of particles in finite arrays of traps:
Instutute of Chemical Physics RAS, Moscow
Sergey Traytak
Boundary effects for diffusion of particles
in finite arrays of traps:
Does the classical mean field theory
really work?
St. Petersburg, 18 September 2017
TMCSLS

3. The main goal: Highlight the pitfalls of the theory and
propose a new solution to the problem
Objective
Are you sure you see it entirely?

4. Outline Introduction Diffusive interaction inside arrays of sinks
Edge effects in spherical arrays of sinks
Time-dependent coarse-grained concentration

5. Introduction

6. Diffusion in a cheese like domain (periphractic)
Statement of the problem
cheese
holes (sinks)
absorbing (reflecting)
boundaries
Diffusion in a cheese like domain (periphractic)

7. Total flux into the i-th sink
Diffusion equation
local concentration of particles B
Initial and boundary C’s
Total flux into the i-th sink

8. - sink radius
Diffusion of B
- sink concentration
(1) Coarse-grained concentration inside the cloud of sinks
(2) Find the penetration length

9. Indeed
The problem is solved

10. Previous theories

11. Wrong does not cease to be wrong because the majority share in it
L. Tolstoy

15. Classics - a book which people
praise and don’t read
M. Twain
This is classics!

16. I
Diffusive interaction inside
arrays of sinks

17. One grain does not form a heap. If we
add one more grain a heap still does not
appear. Which grain makes a heap?
Eubulides IV B.C.

18. «discrete system of sinks» «effective sink»
«Phase» transition:
«discrete system of sinks» «effective sink»
critical point
Analog of pair correlation function
dimensionless flux
Parameter of order

19. A cubic arrays of sinks lattice spacing radius of a sink
length of the side

20. Kadanov procedure (MOA - sinks )
lattice spacing
Renormalization Group method
Critical exponents

21. Criteria for a ”heap”
concenration of sinks
sink array size

22. II
Edge effects in spherical
arrays of sinks

24. Concentration inside a cloud of sinks
Normalization

25. RG generator

27. DI near the boundary
The strongest DI

28. Edge effects in the cloud

30. in in
Screening length
Penetration length

31. Zel’dovich’s paradox
However, the physical meaning of this condition is
not quite clear: if there are no particles on the sink
surface how they can penetrate inside?

32. Dimensional analysis Case 1 Case 2 strong DI flux to the whole system
active surface of the system
flux to an isolated sink
active surface of a sink

33. III
Time-dependent coarse-grained
concentration

34. Solution of the problem

35. Coarse-grained concentration inside
Diffusion of B
Coarse-grained concentration inside
Renormalization Group method
Symmetry of the concentration field
radial
Translational
radial
radial

36. Conclusions New approach Classical approach Diffusion equation
Symmetry
Classical approach
Renormgroup
Diffusion equation
Total flux
Concentration field
Local flux
Local flux
Concentration field
Total flux
Diffusion equation