Download Random Walk - Thermodynamics and Statistical Mechanics - Lecture Slides and more Slides Thermodynamics in PDF only on Docsity! Thermodynamics and Statistical Mechanics Random Walk Docsity.com Random Walk •Often called “drunkard’s walk”. Steps in random directions, but on average, how far does he move, and what is the standard deviation? Do for one dimension. Consider each step is of length s0, but it can be either forward or backward. •Probability of going forward is p. Docsity.com Random Walk Standard deviation. ( ) ( ) ( )[ ] ( ) step)per (s.d. 4 where 2 4 44 22 22 0 22 0 2 0 2 2 0 22 0 2 2 00 22 spqs sN NpqspqsN Npqsnns sNnsNnSS ∆= ∆= == =−= −−−=−= σ σσ σ σ Docsity.com Random Walk As before, NS Np pq spN Npqs S 1 1 )12( 2 )12( 2 0 0 ∝ − = − = σ σ Docsity.com 3D Random Walk Assume the direction of each step is random. 0sin)cos( 4 1cos 00 === ∫∫ φθθθπθ ddsssz Average distance moved per step is zero. Docsity.com Gaussian Distribution •When dealing with very large numbers of particles, it is often convenient to deal with a continuous function to describe the probability distribution, rather than the binomial distribution. The Gaussian distribution is the function that approximates the binomial distribution for very large numbers. Docsity.com Binomial Distribution •Let us develop a differential equation for P in terms of n, and treat n as continuous. Then we can solve the equation for P. nNnqp nNn NnP − − = )!(! !)( Docsity.com Binomial Distribution • If n increased by one, then the change in P is )!()!( ! ))(1()!()!( ! )!()!( ! )!1()!1( ! )()1( 1 1 11 nNn qpN nNn pq nNn qpNP nNn qpN nNn qpNP nPnPP nNnnNn nNnnNn − − −+− =∆ − − −−+ =∆ −+=∆ − − −− −−−+ Docsity.com Binomial Distribution dnnn P dP nnnP dn dP nnnP Npq NpnnPP −−= −−= −−= −−=∆ 2 2 2 )( )()( σ σ σ Docsity.com Binomial to Gaussian Distribution 2 2 2 )( 2 2 2 1 2 const.)(ln so , σ σ σ nn CeP nnP dnnn P dP − − = +−−= −−= Docsity.com Gaussian Distribution • What is C? ∫==∫ = ==− ∞ ∞− −∞ ∞− − dxeCdxxP CexP dxdnxnn x x 2 2 2 2 2 2 1)( Then, )( so , and ,Let σ σ Docsity.com Gaussian Distribution 2 2 2 2 2 )( 2 2 1)( 2 1)( σ σ σπ σπ nn x enP exP − − − = = Docsity.com Properties of Gaussian Distribution 997.0)( 954.0)( 683.0)( 0)( 3 3 2 2 =∫ =∫ =∫ ==∫= − − − ∞ ∞− σ σ σ σ σ σ dxxP dxxP dxxP nndxxxPx Docsity.com Properties of Gaussian Distribution 22 2 2 )( at 0)( )(or 0at 0)( σ σ =∫ == ∂ ∂ === ∂ ∂ ∞ ∞− dxxPx x x xP nnx x xP Docsity.com
