Individual Hopping Is Random in Nature Residence time show broad distributions

Individual hopping is random in nature residence time

This preview shows page 5 - 9 out of 15 pages.

Individual Hopping Is Random in Nature Residence time show broad distributions staying in a certain configuration when hopping happens
Image of page 5

Subscribe to view the full document.

Previous model is over-simplified, each timestep ∆? is non-uniform in reality Individual Hopping Is Random in Nature 0 ~ exp[ ] b B E k k k T Transition State Theory (Arrhenius Law) Frequently trying with a success rate of k , the system can jump out the current basin at any time 0< t <∞ Define the survival probability at time t : P s (t) ( there must be P s (t=0)=1, P s (t= )=0 ) 𝑃 ? ? + ∆? = 𝑃 ? ? ∙ (1 − ? ∙ ∆?) 𝑑𝑃 𝑠 ? 𝑑? = 𝑃 𝑠 ?+∆? −𝑃 𝑠 ? ∆? = −? ∙ 𝑃 ? ? ֜ 𝑃 ? ? = ? −𝑘∙? probability density function ( PDF ) of the first jump happening at time t is therefore: ? ? = − ?𝑃 ? ? ?? = ? ∙ ? −𝑘∙? (??? ?𝑎? ?𝑎???? ?ℎ???: න 0 ? ? ∙ ?? = 1) the average residence time is therefore: ? = න 0 ? ? ∙ ? ∙ ?? = 1/? therefore the earlier model is only a first order approximation
Image of page 6
7 How to Sample Arbitrary p(x) by Monte Carlo Define the cumulative distribution function ( CDF ) of p(x) : 𝐶 ? ≡ න ?𝑖? 𝑥 ? ? ∙ ?? 𝐶 ? 0 1 ξ 2 generate a uniformly distributed random number ξ 2 ~(0,1) the corresponding x would then be selected so that C(x)= ξ 2 . ? ? ? ? the high probability positions in p(x) correspond to larger cross-sections in C(x)
Image of page 7

Subscribe to view the full document.