Kinetic%20Monte%20Carlo

1
Kinetic Monte Carlo

• Triangular lattice

2
Diffusion
Thermodynamic factor
Self Diffusion Coefficient
3
Diffusion
4
Standard Monte Carlo to study diffusion

• Pick an atom at random

5
Standard Monte Carlo to study diffusion

• Pick an atom at random
• Pick a hop direction

6
Standard Monte Carlo to study diffusion

• Pick an atom at random
• Pick a hop direction
• Calculate

7
Standard Monte Carlo to study diffusion

• Pick an atom at random
• Pick a hop direction
• Calculate
• If ( gt random number)
  do the hop

8
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).
9
(No Transcript)
10
(No Transcript)
11
(No Transcript)
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
(No Transcript)
16
(No Transcript)
17
For each potential hop i, calculate the hop rate
18
For each potential hop i, calculate the hop rate
Then randomly choose a hop k, with probability
19
For each potential hop i, calculate the hop rate
Then randomly choose a hop k, with probability
random number
20
For each potential hop i, calculate the hop rate
Then randomly choose a hop k, with probability
random number
21
Then randomly choose a hop k, with probability
random number
22
Time

• After hop k we need to update the time

random number
23
Two independent stochastic variables the hop k
and the waiting time
24
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

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).
25
Triangular 2-d lattice, 2NN pair interactions
26
Activation barrier
27
Thermodynamics
28
Kinetics