Kinetic Monte Carlo
description
Transcript of Kinetic Monte Carlo
Kinetic Monte Carlo
Triangular lattice
Diffusion
€
D = Θ ⋅DJ
€
DJ =1
2d( )t1N
Δr R i t( )
i=1
N
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
€
Θ=∂
μkBT
⎛
⎝ ⎜
⎞
⎠ ⎟
∂ ln x=
N
N 2 − N 2Thermodynamic
factor
Self DiffusionCoefficient
€
Δr R i t( )
€
Δr R j t( )
€
DJ =1
2d( )t1N
Δr R i t( )
i=1
N
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
Diffusion
€
D* =1
2d( )t1N
Δr R i t( )2
i=1
N
∑
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 −ΔEb kBT( )
Standard Monte Carlo to study diffusion
• Pick an atom at random• Pick a hop direction• Calculate • If ( >
random number) do the hop€
exp −ΔEb kBT( )
€
exp −ΔEb kBT( )
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).
€
Wi = ν * exp−ΔEikBT
⎛
⎝ ⎜
⎞
⎠ ⎟
For each potential hop i, calculate the hop rate
€
Wi = ν * exp−ΔEikBT
⎛
⎝ ⎜
⎞
⎠ ⎟
For each potential hop i, calculate the hop rate
Then randomly choose a hop k, with probability
€
Wk
€
Wi = ν * exp−ΔEikBT
⎛
⎝ ⎜
⎞
⎠ ⎟
For each potential hop i, calculate the hop rate
Then randomly choose a hop k, with probability
€
Wk= random number
€
ξ1
€
Wi = ν * exp−ΔEikBT
⎛
⎝ ⎜
⎞
⎠ ⎟
For each potential hop i, calculate the hop rate
Then randomly choose a hop k, with probability
€
Wk= random number
€
ξ1
€
Wii=1
k−1
∑ <ξ1 ⋅W ≤ Wii= 0
k
∑
€
W = Wii= 0
Nhops
∑
Then randomly choose a hop k, with probability
€
Wk= random number
€
ξ1
€
Wii=1
k−1
∑ <ξ1 ⋅W ≤ Wii= 0
k
∑
€
W = Wii= 0
Nhops
∑
Time
After hop k we need to update the time
= random number
€
ξ 2
€
Δt = −1W
logξ 2
Two independent stochastic variables:
the hop k and the waiting time
€
Wii=1
k−1
∑ <ξ1 ⋅W ≤ Wii= 0
k
∑
€
Δt = −1W
logξ 2 €
W = Wii= 0
Nhops
∑€
Wi = ν * exp−ΔEikBT
⎛
⎝ ⎜
⎞
⎠ ⎟
€
Δt
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
€
Δt
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
€
Er
σ ( ) = Vo +Vpo intσ l +12
VNNpairσ l σ ii= NNpairs∑ +
12
VNNNpairσ l σ jj= NNNpairs
∑ ⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
l=1
N lattice−sites
∑
Activation barrier
€
ΔEkra = Eactivated −state −12
E1 + E2( )
€
ΔEbarrier = ΔEkra +12
E final − Einitial( )
Thermodynamics
€
Θ=∂
μkBT
⎛
⎝ ⎜
⎞
⎠ ⎟
∂ ln x=
N
N 2 − N 2
€
D = Θ ⋅DJ
Kinetics
€
D = Θ ⋅DJ
€
DJ =1
2d( )t1N
Δr R i t( )
i=1
N
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2