establishment of stochastic discrete models for continuum langevin equation of surface growths yup...
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Establishment of stochastic discrete models for continuum Langevin equation
of surface growths
Yup Kim and Sooyeon YoonKyung-Hee Univ.Dept. of Physics
Based on the relations among Langevin equation, Fokker- Planck equation, and Master equation for the surface growth phenomena. It can be shown that the deposition (evaporation) rate of one particle to(from) the surface is proportional to . Here , and are from the Langevin . From these rates, we can construct easily the discrete stochastic models of the corresponding continuum equation, which can directly be used to analyze the continuum equation. It is shown that this analysis is successfully applied to the quenched Edward-Wilkinson(EW) equa-tion and quenched Kardar-Parisi-Zhang(KPZ) equation as well as the thermal EW and KPZ equations.
)()( eedd WWWWW
DhxWd 2/)],([
),(),(),,(
242 txhxhhhth
DhxWe 2/)],([
AbstractAbstract
),(),(),( )1( txtxhK
t
txh
Continuum Langevin Equation :
Discretized version : )()( )1( tHK
t
thii
i
NiihH 1
,0)( ti )'(2)'()'()( )2( ttDttKtt ijijijji
Master Equation :
''
),()',(),'(),'(),(
HH
tHPHHWtHPHHWt
tHP
Fokker-Planck Equation :
),(2
1),(
),( )2(2
,
)1( tHPKhh
tHPKht
tHPij
jijiii
i
'
')1( )',()()(H
iii HHWhhHK
)',()()()( '
'
')2( HHWhhhhHK jjH
iiij
HHW ,' is the transition rate from H’ to H.
White noise :
A stochastic analysis of continnum Langevin equation for surface growthsA stochastic analysis of continnum Langevin equation for surface growths
),(),(
),()(),()()()1(
ahhaWahhaW
ahhWhahahhWhahHK
iiii
iiiiiiiii
If we consider the deposition(evaporation) of only one particle at the unit evolution step.
:)()(
)()(
,2
,2
ahhWhah
ahhWhah
iiii
iiii
ji
:0 ji
( a is the lattice constant. )
ahi
ahi
(deposition)
(evaporation)'ih
)(),,,()(242)1( hhhhHK ii
Including quenched disorder in the medium :
)'(2)'()( hhhh ijji ,0)( hi
ijij DK 2)2(
D
HKD
HKW ii
id 2
)(
2
)( )1()1(
D
HKD
HKW ii
ie 2
)(
2
)( )1()1(
Since W (transition rate) > 0 ,
,),( ahhWW iiid ),( ahhWW iiie
2
)1(
2
)(
a
D
a
HKW i
id 2
)1(
2
)(
a
D
a
HKW i
ie
)1( a
• Probability for the unit Monte-Carlo time
)1( idid WP
)1( ieie WP
Calculation RuleCalculation Rule
1. For a given time the transition probability
2. The interface configuration is updated for i site :
)1( idid WP
)1( ieie WP
otherwise
RPthRPthth ieiidi
i ,0
)if,1)((if,1)()1(
1)()1( thth ii
,)1(1If ieid PP
,)1(1if else ieid PP compare with new random value R.
is evaluated for i site.
)1)()1(( thth ii
For the Edward-Wilkinson equation ,
2.0,0.25,0.50 z
iiiii hhhhK 2][ 1122
2)1(
Simulation ResultsSimulation Results
Growth without quenched noise
zL
tfLtLW ),(
z
For the Kardar-Parisi-Zhang equation,
)(2][ 1121
11222
2)1(
iiiiiii hhhhhhhK
,),(
zL
tfLtLW
z
5.1,0.34,0.51 z
Growth with quenched noises
• pinned phase : F < Fc
• critical moving phase : F Fc
• moving phase : F > Fc
• Near but close to the transition threshold Fc, the important physical parameter in the regime is the reduced force f
c
c
F
FFf
• average growth velocityfv
dt
hd~
Question? Is the evaporation process accepted, when the rate Wie>0 ? ( Driving force F makes the interface move forward. )
(cf) Interface depinning in a disordered medium numerical results ( Leschhorn, Physica A 195, 324 (1993))
1. A square lattice where each cell (i , h) is assigned a random pin- ning force i, h which takes the value 1 with probability p and -1 with probability q = 1-p.
3. The interface configuration is updated simultaneously for for all i :
hiiiii thththv ,11 )(2)()(
is determined for all i .
2. For a given time t the value
otherwiseth
vifthth
i
iii ,)(
0,1)()1(
Our results for the quenched Edward-Wilkinson equation
FhhhhFhhK iiiiii )(2])([ 1122
2)1(
on)distributi(uniform]1,1[)( ih
fvdt
hd~
24.0,608.0cF
fvdt
hd~
original Leschhorn’s modeloriginal Leschhorn’s model with evaporation allowed
Comparison with Leschhorn’s results
22.0,8004.0cp
,608.0At cF
Near the threshold Fc
Our results for the quenched Edward-Wilkinson equation
,),(
zL
tfLtLW
z
001.0883.0 ,02.025.1
Near the threshold pc
,8004.0At cp
Comparison Leschhorn’s results
,),(
zL
tfLtLW
z
02.088.0 ,01.025.1
For the quenched Kardar-Parisi-Zhang equation,
Fhhhhhh
FhhhK
iiiiii
ii
)()(2
])([
1121
112
222
)1(
on)distributi(uniform]1,1[)( ih
fvdt
hd~
L = 1024, 2 = 0.1 , = 0.1
6529.0,162.0cF
,),(
zL
tfLtLW
z
Near the threshold Fc
0 2 4 6 8-1
0
1
2
3
4 L = 4096, 2 = 0.1 , = 0.1
= 0.6347(1)
ln W
ln t
,162.0At cF
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.00.5
1.0
1.5
2.0
2.5
3.0
L = 32, 64, 128, 256, 512, 1024
= 0.61(1)
ln W
s
ln L
,635.0,61.0 96.0z
Conclusion and DiscussionsConclusion and Discussions
1. We construct the discrete stochastic models for the given continuum equation. We confirm that the analysis is successfully applied to the quenched Edward-Wilkinson(EW) equation and quenched Kardar- Parisi-Zhang(KPZ) equation as well as the thermal EW and KPZ equations.
2. We expect the analysis also can be applied to
• Linear growth equation , • Kuramoto-Sivashinsky equation , • Conserved volume problem , etc.
3. To verify more accurate application of this analysis, we need
• Finite size scaling analysis for the quenched EW, KPZ equations , • 2-dimensional analysis (phase transition?) .