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Diffusion

D ="#DJ

DJ=1

2d( )t1

N"

r

Ri t( )i=1

N

#$

%

&&

'

(

))

2

" =#

kBT

$

%&

'

()

#ln x=

N

N2 * N

2Thermodynamic

factor

Self Diffusion

Coefficient

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"

r

Ri t( )

"

r

Rj t( )

DJ =1

2d( )t1

N"

r

Ri t( )i=1

N

#$

%

&&

'

(

))

2

Diffusion

D* =1

2d( )t1

N"

r

Ri t( )2

i=1

N

#

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Standard Monte Carlo to study

diffusion

Pick an atom at random

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Standard Monte Carlo to study

diffusion

Pick an atom at random

Pick a hop direction

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Standard Monte Carlo to study

diffusion

Pick an atom at random

Pick a hop direction

Calculate

exp "#Eb kBT( )

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Kinetic Monte Carlo

Consider all hops simultaneously

A. B. Bortz, M. H. Kalos, J. L. Lebowitz, J. Comput Phys, 17, 10 (1975).

F. M. Bulnes, V. D. Pereyra, J. L. Riccardo, Phys. Rev. E, 58, 86 (1998).

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Wi = "* exp #$EikBT

%

&'

(

)*

For each potential hop i,

calculate the hop rate

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Wi = "* exp #$EikBT

%

&'

(

)*

For each potential hop i,

calculate the hop rate

Then randomly choose a hop k, with probability

Wk

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Wi = "* exp #$EikBT

%

&'

(

)*

For each potential hop i,

calculate the hop rate

Then randomly choose a hop k, with probability

Wk

= random number

"1

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Wi = "* exp #$EikBT

%

&'

(

)*

For each potential hop i,

calculate the hop rate

Then randomly choose a hop k, with probability

Wk

= random number

"1

Wi

i=1

k"1

#

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Then randomly choose a hop k, with probability

Wk

= random number

"1

Wi

i=1

k"1

#

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Time

After hop kwe need to update the time

= random number

"2

"t=# 1

Wlog$2

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Kinetic Monte Carlo

Hop every time

Consider all possible hops simultaneously

Pick hop according its relative probability Update the time such that on average

equals the time that we would have waited

in standard Monte Carlo

"t

A. B. Bortz, M. H. Kalos, J. L. Lebowitz, J. Comput Phys, 17, 10 (1975).

F. M. Bulnes, V. D. Pereyra, J. L. Riccardo, Phys. Rev. E, 58, 86 (1998).

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Triangular 2-d lattice, 2NN

pair interactions

Er

"( ) = Vo +Vpo int"l +1

2VNNpair"l "i

i=

NNpairs

# + 12VNNNpair"l "j

j=

NNNpairs

#$

%

&&&

'

(

)))

l=1

Nlattice*sites

#

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Activation barrier

"Ekra =Eactivated#state #1

2

E1 +E2( )

"Ebarrier = "Ekra +1

2Efinal #Einitial( )

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Thermodynamics

" =

#

kBT

$

%&

'

()

#ln x=

N

N2

* N

2

D ="#DJ

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Kinetics

D ="#DJ

DJ =1

2d( )t1

N"

r

Ri t( )i=1

N

#$

%

&&

'

(

))

2