kinetic monte carlo - illinoiskinetic monte carlo •hop every time •consider all possible hops...
TRANSCRIPT
![Page 1: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/1.jpg)
Kinetic Monte Carlo
Triangular lattice
![Page 2: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/2.jpg)
Diffusion
€
D =Θ⋅DJ
€
DJ =12d( )t
1N
Δr R i t( )
i=1
N
∑
2
€
Θ =
∂µkBT
∂ ln x=
N
N 2 − N 2Thermodynamic
factor
Self DiffusionCoefficient
![Page 3: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/3.jpg)
€
Δr R i t( )
€
Δr R j t( )
€
DJ =12d( )t
1N
Δr R i t( )
i=1
N
∑
2
Diffusion
€
D* =12d( )t
1N
Δr R i t( )2
i=1
N
∑
![Page 4: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/4.jpg)
Standard Monte Carlo to studydiffusion
• Pick an atom at random
![Page 5: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/5.jpg)
Standard Monte Carlo to studydiffusion
• Pick an atom at random• Pick a hop direction
![Page 6: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/6.jpg)
Standard Monte Carlo to studydiffusion
• Pick an atom at random• Pick a hop direction• Calculate
€
exp −ΔEb kBT( )
![Page 7: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/7.jpg)
Standard Monte Carlo to studydiffusion
• Pick an atom at random• Pick a hop direction• Calculate• If ( >
random number) do the hop€
exp −ΔEb kBT( )
€
exp −ΔEb kBT( )
![Page 8: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/8.jpg)
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).
![Page 9: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/9.jpg)
![Page 10: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/10.jpg)
![Page 11: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/11.jpg)
![Page 12: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/12.jpg)
![Page 13: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/13.jpg)
![Page 14: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/14.jpg)
![Page 15: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/15.jpg)
![Page 16: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/16.jpg)
![Page 17: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/17.jpg)
€
Wi = ν * exp −ΔEikBT
For each potential hop i, calculate the hop rate
![Page 18: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/18.jpg)
€
Wi = ν * exp −ΔEikBT
For each potential hop i, calculate the hop rate
Then randomly choose a hop k, with probability
€
Wk
![Page 19: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/19.jpg)
€
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
![Page 20: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/20.jpg)
€
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
∑
![Page 21: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/21.jpg)
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
∑
![Page 22: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/22.jpg)
Time
After hop k we need to update the time
= random number
€
ξ 2
€
Δt = −1Wlogξ 2
![Page 23: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/23.jpg)
Two independent stochasticvariables:
the hop k and the waiting time
€
Wii=1
k−1
∑ <ξ1 ⋅W ≤ Wii= 0
k
∑
€
Δt = −1Wlogξ 2 €
W = Wii= 0
Nhops
∑€
Wi = ν * exp −ΔEikBT
€
Δt
![Page 24: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/24.jpg)
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 waitedin 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).
![Page 25: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/25.jpg)
Triangular 2-d lattice, 2NNpair interactions
€
E r σ ( ) = Vo +Vpo intσ l +
12VNNpairσ l σ i
i=NNpairs∑ +
12VNNNpairσ l σ j
j=NNNpairs∑
l=1
Nlattice−sites
∑
![Page 26: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/26.jpg)
Activation barrier
€
ΔEkra = Eactivated−state −12E1 +E2( )
€
ΔEbarrier = ΔEkra +12E final −Einitial( )
![Page 27: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/27.jpg)
Thermodynamics
€
Θ =
∂µkBT
∂ ln x=
N
N 2 − N 2
€
D =Θ⋅DJ
![Page 28: Kinetic Monte Carlo - IllinoisKinetic Monte Carlo •Hop every time •Consider all possible hops simultaneously •Pick hop according its relative probability •Update the time such](https://reader034.vdocuments.net/reader034/viewer/2022052616/60a1f4e72922d87b23067a93/html5/thumbnails/28.jpg)
Kinetics
€
D =Θ⋅DJ
€
DJ =12d( )t
1N
Δr R i t( )
i=1
N
∑
2