Transcript Kinetic Monte Carlo

Kinetic Monte Carlo
Triangular lattice
Diffusion
D    DJ
Thermodynamic
factor

Self Diffusion

Coefficient
  
 

N
k B T 


 ln x
N2  N
1
DJ 
2dt
 N
2
1 
R i t

N 
i1


2
Diffusion
R j t
R i t
1
DJ 
2dt
 N

1 
R i t

N 
i1


2

D* 
1
2dt
1
N
N

i1
R i t
2
Standard Monte Carlo to study
diffusion
• Pick an atom at random
Standard Monte Carlo to study
diffusion
• Pick an atom at random
• Pick a hop direction
Standard Monte Carlo to study
diffusion
• Pick an atom at random
• Pick a hop direction
• Calculate exp E b k B T 

Standard Monte Carlo to study
diffusion
•
•
•
•
Pick an atom at random
Pick a hop direction
Calculate exp E b k B T 
If ( exp E b k B T  >
random number) do the hop

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).
For each potential hop i,
calculate the hop rate
E 
i
Wi   * exp 

 k B T 

For each potential hop i,
calculate the hop rate
E 
i
Wi   * exp 

 k B T 
Then randomly choose a hop k, with probability


Wk
For each potential hop i,
calculate the hop rate
E 
i
Wi   * exp 

 k B T 
Then randomly choose a hop k, with probability

1 = random number


Wk
For each potential hop i,
calculate the hop rate
E 
i
Wi   * exp 

 k B T 
Then randomly choose a hop k, with probability

1 = random number
k1
i
i1

Nhops
k
W   W  W
1
i
i 0
Wk
W 
W
i 0
i
Then randomly choose a hop k, with probability
1
k1
= random number
Nhops
k
W   W  W
i
i1


Wk
1
W 
i
i 0
W
i 0

i
Time
After hop k we need to update the time
2


= random number
1
t   log  2
W
Two independent stochastic
variables:
the hop k and the waiting time t
k1
k


i1
i 0
Wi  1 W 

Wi

Nhops
W

1
t   log  2
W
E 
i
Wi   * exp 

 k B T 

W
i 0
i
Kinetic Monte Carlo
•
•
•
•
Hop every time
Consider all possible hops simultaneously
Pick hop according its relative probability
Update the time such that t on average
equals the time that we would have waited
in standard Monte Carlo

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).
Triangular 2-d lattice, 2NN
pair interactions
E 
N latticesites 

l1

1
1


V

V


V



V


o
point l
i
j 

2 NNpair l
2 NNNpair l


i
NNpairs
j
NNNpairs




Activation barrier
E kra

1
 E activatedstate  E1  E 2 
2
E barrier  E kra


1
 E final  E initial
2
Thermodynamics
D    DJ
  
 

N
k B T 


 ln x
N2  N


2
Kinetics
D    DJ
DJ 


1
2dt
 N
2
1 
R i t

N 
i1

