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Rare Event Simulations

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. Diffusion in porous material. Chemical reaction. Theory: macroscopic phenomenological. Total number of molecules.

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Rare Event Simulations

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  1. 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

  2. Diffusion in porous material

  3. Chemical reaction Theory: macroscopic phenomenological Total number of molecules Make a small perturbation Equilibrium:

  4. Heaviside θ-function q q* βF(q) q Microscopic description of the reaction Theory: microscopic linear response theory 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

  5. 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!

  6. Δ has disappeared because of derivative Derivative Stationary For sufficiently short t

  7. Eyring’s transition state theory Only products contribute to the average At t=0 particles are at the top of the barrier Let us consider the limit: t →0+

  8. 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

  9. Conditional average: given that we start on top of the barrier Bennett-Chandler approach Probability to find q on top of the barrier Computational scheme: • Determine the probability from the free energy • Compute the conditional average from a MD simulation

  10. cage window cage cage window cage q* q* βF(q) βF(q) q q Reaction coordinate

  11. Ideal gas particle and a hill q* is the true transition state q1 is the estimated transition state

  12. 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

  13. 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

  14. dp pt pt o n rT r0 rT rT rT r0 rt o o n n o o r0 r0 n n B A B A Walking in the Ensemble Shooting Shifting

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