PHY 341641 Thermodynamics and Statistical Physics Lecture 10

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PHY 341/641 Thermodynamics and Statistical Physics Lecture 10 Probability concepts (Chapter 3 in STP)

PHY 341/641 Thermodynamics and Statistical Physics Lecture 10 Probability concepts (Chapter 3 in STP) A. Binomial distribution (continued) B. Central Limit theorem C. Poisson distribution and others 10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 1

Comment on HW problem: 10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 2

Comment on HW problem: 10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 2

Binomial distribution (review) – Assume each elemental event has 2 possible outcomes (for example

Binomial distribution (review) – Assume each elemental event has 2 possible outcomes (for example H with probability p and T with probability q=1 -p). PN(n) gives the probability distribution for N elemental events having n instances of H. 10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 3

10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 4

10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 4

More general importance of Gaussian probability distribution – “central limit theorem” One-dimensional random walk

More general importance of Gaussian probability distribution – “central limit theorem” One-dimensional random walk • It is possible to describe a version of a random walk process as a binomial distribution. If each step has a fixed length s, then an elementary event is a step to the right with probability p or a step to the left with probability q. The probability of n steps to the right is then the binomial distribution. 10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 5

Example of random walk: stp_Random. Walk 2 D. jar 10/19/2021 PHY 341/641 Spring 2012

Example of random walk: stp_Random. Walk 2 D. jar 10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 6

One-dimensional fixed step walk, continued For N total steps and n steps to the

One-dimensional fixed step walk, continued For N total steps and n steps to the right with probability PN(n), displacement is x=ns-(N-n)s=(2 n-N)s <x>=(2<n>-N)s=(2 p-1)Ns <x 2>-<x>2 =4 pq. Ns 2 Generalization – one-dimensional variable step (si) walk (derivations following Fundamentals of statistical and thermal physics by F. Reif (1965)) 10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 7

10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 8

10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 8

10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 9

10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 9

10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 10

10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 10

Poisson probability distribution – Approximation to binomial distribution for small p 10/19/2021 PHY 341/641

Poisson probability distribution – Approximation to binomial distribution for small p 10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 11

Other probability distributions: 10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 12

Other probability distributions: 10/19/2021 PHY 341/641 Spring 2012 -- Lecture 10 12

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