Individual Hopping Is Random in Nature•Residence time show broad distributionsstaying in a certain configurationwhen hopping happens
Subscribe to view the full document.
•Previous model is over-simplified, each timestep ∆?is non-uniform in reality Individual Hopping Is Random in Nature0~exp−bBEkkk TTransition 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 : Ps(t)( there must be Ps(t=0)=1, Ps(t=∞)=0 )𝑃?? + ∆?= 𝑃??∙ (1 − ? ∙ ∆?)𝑑𝑃𝑠?𝑑?=𝑃𝑠?+∆? −𝑃𝑠?∆?= −? ∙ 𝑃??֜ 𝑃??= ?−𝑘∙?probability density function (PDF) of the first jump happening at time tis therefore:??= −?𝑃????= ? ∙?−𝑘∙?(??? ?𝑎? ?𝑎???? ?ℎ???: න0∞??∙ ?? = 1)the average residence time is therefore:?= න0∞??∙ ? ∙ ?? = 1/?therefore the earlier model is only a first order approximation
7How to Sample Arbitrary p(x) byMonte Carlo•Define the cumulative distribution function (CDF) of p(x):𝐶?≡ න?𝑖?𝑥??∙ ??𝐶?01ξ2generate a uniformly distributed random number ξ2~(0,1)the corresponding xwould then be selected so that C(x)=ξ2 .????the high probability positions in p(x)correspond to larger cross-sections in C(x)