1. Collective diffusion of the interacting surface gas Magdalena Załuska-Kotur Institute of Physics, Polish Academy of Sciences

2. Random walk Diffusion coefficient D

3. + mass conservation Collective diffusion local density

4. The model – noninteracting lattice gas Equilibrium distribution c – microstate Local density

5. single particle result Noninteracting system

6. D o =Wa 2 for small k Single particle diffusion – noninteracting gas.

7. Interacting particles

8. Interacting particles – 2D system with repulsive interactions J’=3/4J Square lattice

9. Questions How diffusion depends on interactions? How minima of the density- diffusion plot are related to the phase diagram? Where are phase transition points? Are there some other characteristic points?

10. Example - hexagonal lattice - repulsion kT=0.25J kT=0.5J kT=J

11. Attraction J<0 T=0.89T c T c =1.8|J|/k J’=2J J’=J J’=0 J=0 J’=J J’=2J J>0 Repulsion

13. Experimental results - Pb/Cu(100 )

14. Simulation methods Harmonic density perturbation Step profile decay

16. kT=0.25J kT=0.5J kT=J

17. Profile evolution Boltzmann –Matano method

18. Definition of transition rates

19. The model Detailed balance condition Equilibrium distribution c – microstate

20. Possible approaches Hierarchy of equations - QCA

21. X Analysis of microscopic equations. Local density L - lattice sites + periodic boundary conditions X

22. Fourier transformation of master equation. when reference particle jumps =1 otherwise For N=2

23. Eigenvalue of matrix M Approximation: EigenvalueLimit

24. Approximate eigenvector for interacting gas one interaction constant J x - number of bonds

25. Definition of transition rates in 1D system Possible transitions ( )

26. Diffusion coefficient of 1D system Grand canonical regime Low temperature approximation

27. Diffusion coefficient - repulsive interactions p=2,10,100

28. Diffusion coefficient - repulsive - QCA p=2,10,100

29. Activation energy –repulsive interactions

30. Diffusion coefficient - attractive interactions p=0.5,0.3,0.1

31. Diffusion coefficient - attractive QCA p=0.5,0.3,0.1

32. Activation energy – attractive interactions

33. Eigenvector for random state Initial configuration

34. Repulsive far from equilibrium case θ θ ν p=100

35. 2x2 ordering –definition of transition rates J J’ M. A. Załuska-Kotur Z.W.Gortel – to be published

36. Equilibrium probability strong repulsion Diagonal matrix

37. Components of eigenvector * * Primary configurations: Secondary configurations (average of neighbouring primary ones):

38. Result Upper line:Lower line:

39. J’=3/4J Ordered phase

40. Other parameters – kT/J=0.3

41. Other parameters – kT/J’=0.4

42. Other parameters – J’=0

43. New approach to the collective diffusion problem, based on many-body function description – analytic theory. Exact solution for noninteracting system. Collective diffusion in 1D system with nearest neighbor attractive and repulsive interactions. Diffusion coefficient in 2D lattice gas of 2X2 ordered phase with repulsive forces. Agrement with numerical results Numerical approaches: step density profile evolution and harmonic density perturbation decay methods Summary

44. Possible applications Analysis of Far from equlibrium systems. More complex interactions – long range Surfaces with steps Phase transitions

45. J=0 J’=2J J=J’ J’=2J ‘

47. Jak dyfuzja zależy od oddziaływań? x i j Gaz cząstek na dwuwymiarowej sieci E init, (i) - lokalna energia jednocząstkowa E bar (ij) - energia cząstki w punkcie siodłowym Szybkość przeskoków jednocząstkowych

48. Analysis of microscopic equations. Local density

49. 1D -- z=2 D o =Wa 2 for small k

50. Calculation = n 1 –n 2 for s clusters Y: Łukasz Badowski, M. A. Załuska-Kotur – to be published

51. D o =Wa 2 Site blocking – noninteracting lattice gas Eigenvalue - For N=2