Kinetic Monte Carlo

Triangular lattice

Diffusion

€

D = Θ ⋅DJ

€

DJ =1

2d( )t

1

NΔ

r R i t( )

i=1

N

∑ ⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

2

€

Θ=∂ μ

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

∂ ln x=

N

N 2 − N2

Thermodynamic factor

Self DiffusionCoefficient

€

Δ r

R i t( )

€

Δ r

R j t( )

€

DJ =1

2d( )t

1

NΔ

r R i t( )

i=1

N

∑ ⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

2

Diffusion

€

D* =1

2d( )t

1

NΔ

r 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−ΔEi

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

For each potential hop i, calculate the hop rate

€

Wi = ν * exp−ΔEi

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

For each potential hop i, calculate the hop rate

Then randomly choose a hop k, with probability

€

Wk

€

Wi = ν * exp−ΔEi

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

For each potential hop i, calculate the hop rate

Then randomly choose a hop k, with probability

€

Wk

= random number

€

ξ1

€

Wi = ν * exp−ΔEi

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

For each potential hop i, calculate the hop rate

Then randomly choose a hop k, with probability

€

Wk

= random number

€

ξ1

€

Wi

i=1

k−1

∑ <ξ1 ⋅W ≤ Wi

i= 0

k

∑

€

W = Wi

i= 0

Nhops

∑

Then randomly choose a hop k, with probability

€

Wk

= random number

€

ξ1

€

Wi

i=1

k−1

∑ <ξ1 ⋅W ≤ Wi

i= 0

k

∑

€

W = Wi

i= 0

Nhops

∑

Time

After hop k we need to update the time

= random number

€

ξ 2

€

Δt = −1

Wlogξ 2

Two independent stochastic variables:

the hop k and the waiting time

€

Wi

i=1

k−1

∑ <ξ1 ⋅W ≤ Wi

i= 0

k

∑

€

Δt = −1

Wlogξ 2 €

W = Wi

i= 0

Nhops

∑€

Wi = ν * exp−ΔEi

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

€

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 +1

2VNNpairσ l σ i

i= NNpairs

∑ +1

2VNNNpairσ l σ j

j= NNNpairs

∑ ⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

l=1

N lattice−sites

∑

Activation barrier

€

ΔEkra = Eactivated −state −1

2E1 + E2( )

€

ΔEbarrier = ΔEkra +1

2E final − Einitial( )

Thermodynamics

€

Θ=∂ μ

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

∂ ln x=

N

N 2 − N2

€

D = Θ ⋅DJ

Kinetics

€

D = Θ ⋅DJ

€

DJ =1

2d( )t

1

NΔ

r R i t( )

i=1

N

∑ ⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

2