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Kinetic Monte Carlo. Triangular lattice. Diffusion. Thermodynamic factor. Self Diffusion Coefficient. Diffusion. 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.

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Kinetic monte carlo

Kinetic Monte Carlo

Triangular lattice


Diffusion
Diffusion

Thermodynamic

factor

Self Diffusion

Coefficient




Standard monte carlo to study diffusion1
Standard Monte Carlo to study diffusion

  • Pick an atom at random

  • Pick a hop direction


Standard monte carlo to study diffusion2
Standard Monte Carlo to study diffusion

  • Pick an atom at random

  • Pick a hop direction

  • Calculate


Standard monte carlo to study diffusion3
Standard Monte Carlo to study diffusion

  • Pick an atom at random

  • Pick a hop direction

  • Calculate

  • If ( > random number) do the hop


Kinetic monte carlo1
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


For each potential hop i,

calculate the hop rate

Then randomly choose a hop k, with probability


For each potential hop i,

calculate the hop rate

Then randomly choose a hop k, with probability

= random number


For each potential hop i,

calculate the hop rate

Then randomly choose a hop k, with probability

= random number


Then randomly choose a hop k, with probability

= random number


Time

After hop k we need to update the time

= random number


Two independent stochastic variables the hop k and the waiting time
Two independent stochastic variables: the hop k and the waiting time


Kinetic monte carlo2
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).






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