Collective diffusion of the interacting surface gas
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Transcript of Collective diffusion of the interacting surface gas
Collective diffusion of the interacting surface gas
Magdalena Załuska-Kotur
Institute of Physics,
Polish Academy of Sciences
The model – noninteracting lattice gas
''' )],(),'([),(
ccccc tcPWtcPWtcP
t
Equilibrium distribution
c – microstate
Local density
k
knVcH )(
1)( cPeq
c
xi tcPnn ),(
1,0kn
)()()(2)( keeWkWkt
ikaika
)()( xPekx
ixka
Do=Wa2for small k
20 Dk
Single particle diffusion – noninteracting gas.
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?
The model
),(),'( '' tcPWtcPW eqcceqcc
k
kkl
lk nVnnJcH )(
''' )],(),'([),(
ccccc tcPWtcPWtcP
t
Detailed balance condition
)()( cHeq ecP Equilibrium
distribution
c – microstate
Possible approaches
0 11
2)()0(
)(2
1 N
ii
N
iieq tvvdt
ND
ijijij
jjijii nnWnnWn
ti
)1()1(
Hierarchy of equations
),(
kjijiji nnnnnfnnt
- QCA
X
Analysis of microscopic equations.
c
xx tcPn ),(Local density
],...,,[],[ 11 NmmXXc m
]5,3,1,[X
''' )],(),'([),(
ccccc tcPWtcPWtcP
t
L - lattice sites + periodic boundary conditions
X),( tXPm
Fourier transformation of master equation.
'
'' ),()()(m
mmmm tkPkMkP
),(),(1
tXPetkPL
x
ikxmm
ikacc ek )('F when reference particle jumps
=1 otherwise
)1()()( ''''
''' mmmmmm ccccc
cc WkWkM F
2)1(000
)1(400
0
004)1(
00)1(2
)(ˆ
ika
ika
ika
ika
eW
eWW
W
We
eW
kM
For N=2
Approximate eigenvector for interacting gas
)(cHeq eP one interaction constant J
x - number of bonds xJxeq peP
Activation energy – attractive interactions
AEeD )()(
VeWaD 2)0(
||
)(||
2
)()0()( J
VE
Tk
J A
BeWaD
D
Components of eigenvector
* *
Primary configurations:
Secondary configurations (average of neighbouring primary ones):
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
Possible applications
Analysis of
Far from equlibrium systems.
More complex interactions – long range
Surfaces with steps
Phase transitions
Jak dyfuzja zależy od oddziaływań?
x
i
j
)]},()([exp{),( 0 jiEiEjiW barinit
i
iijiij
iji nVnnJnH })({
Gaz cząstek na dwuwymiarowej sieci
Einit,(i) - lokalna energia jednocząstkowa
Ebar (ij) - energia cząstki w punkcie siodłowym
Szybkość przeskoków jednocząstkowych
)/(1 TkB
Calculation
ssNxeq CpppP
= n1 –n2
for s clusters
Y:
Łukasz Badowski, M. A. Załuska-Kotur – to be published