Central limit theorem Brownian motion

We now explicitly consider the waiting time distribution. First we reiterate that the Einstein theory of the Brownian motion relies on the central limit theorem that a sum of independent identically distributed random variables (the sum of the elementary displacements of the Brownian particle)... 

Equation (1.20) has many applications in this book, such as the diffusion motion of a Brownian particle (Appendix 3.D) and the probability distribution of the end-to-end vector of a long polymer chain. The latter case will be studied in this chapter. Essentially, (x ) / jjj gq (1.20) can be regarded as the mean projection of an independent segment (bond) vector in one of the three coordinate directions (i.e. x, y or z all three directions are equivalent). As long as the considered polymer chain is very long, we can always apply the central limit theorem, regardless of the local chemical structure. 

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