Rare Event Simulations

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Rare Event Simulations

• Theory 16.1
• Transition state theory 16.1-16.2
• Bennett-Chandler Approach 16.2
• Diffusive Barrier crossings 16.3
• Transition path ensemble 16.4

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Diffusion in porous material
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Theory macroscopic phenomenological
Chemical reaction
Total number of molecules
Make a small perturbation
Equilibrium
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Theory microscopic linear response theory
Microscopic description of the reaction
Reaction coordinate
Reaction coordinate
Reactant A
Product B
Lowers the potential energy in A
Increases the concentration of A
Perturbation
Probability to be in state A
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Theory microscopic linear response theory
Microscopic description of the reaction
Reaction coordinate
Reactant A
Product B
Lowers the potential energy in A
Increases the concentration of A
Perturbation
Probability to be in state A
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Reaction coordinate
Reaction coordinate
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Very small perturbation linear response theory
Linear response theory static
Outside the barrier gA 0 or 1 gA (x) gA (x) gA
(x)
Switch of the perturbation dynamic linear
response
Holds for sufficiently long times!
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Very small perturbation linear response theory
Outside the barrier gA 0 or 1 gA (x) gA (x) gA
(x)
Switch of the perturbation dynamic linear
response
Holds for sufficiently long times!
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Linear response theory static
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? has disappeared because of derivative
Derivative
Stationary
For sufficiently short t
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Stationary
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Eyrings transition state theory
Only products contribute to the average
At t0 particles are at the top of the barrier
Let us consider the limit t ?0
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Transition state theory

• One has to know the free energy accurately
• Gives an upper bound to the reaction rate
• Assumptions underlying transition theory should
  hold no recrossings

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Bennett-Chandler approach
Probability to find q on top of the barrier
Computational scheme

1. Determine the probability from the free energy
2. Compute the conditional average from a MD
   simulation

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Reaction coordinate
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Ideal gas particle and a hill
q is the true transition state
q1 is the estimated transition state
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Bennett-Chandler approach

• Results are independent of the precise location
  of the estimate of the transition state, but the
  accuracy does.
• If the transmission coefficient is very low
• Poor estimate of the reaction coordinate
• Diffuse barrier crossing

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Transition path sampling
xt is fully determined by the initial condition
Path that starts at A and is in time t in B
importance sampling in these paths
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Walking in the Ensemble
Shooting
Shifting
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