Thermo & Stat Mech - Spring 2006 Class 27 1 Thermodynamics and Statistical Mechanics Random Walk

Thermo & Stat Mech - Spring 2006 Class 272 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 s 0, but it can be either forward or backward. Probability of going forward is p.

Thermo & Stat Mech - Spring 2006 Class 273 Random Walk After N steps, if n are forward, the distance traveled is, S = [n – (N – n)]s 0 = (2n – N) s 0 The probability of this occurring is,

Thermo & Stat Mech - Spring 2006 Class 274 Random Walk The average distance covered after N steps is,

Thermo & Stat Mech - Spring 2006 Class 275 Random Walk Standard deviation.

Thermo & Stat Mech - Spring 2006 Class 276 Random Walk As before,

Thermo & Stat Mech - Spring 2006 Class 277 3D Random Walk Assume the direction of each step is random. Average distance moved per step is zero.

Thermo & Stat Mech - Spring 2006 Class 278 Standard Deviation since.

Thermo & Stat Mech - Spring 2006 Class 279 Standard Deviation Let cos  = x, and sin  d  = – dx. Then,

Thermo & Stat Mech - Spring 2006 Class 2710 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.

Thermo & Stat Mech - Spring 2006 Class 2711 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.

Thermo & Stat Mech - Spring 2006 Class 2712 Binomial Distribution If n increased by one, then the change in P is

Thermo & Stat Mech - Spring 2006 Class 2713 Binomial Distribution

Thermo & Stat Mech - Spring 2006 Class 2714 Binomial Distribution

Thermo & Stat Mech - Spring 2006 Class 2715 Binomial Distribution

Thermo & Stat Mech - Spring 2006 Class 2716 Binomial to Gaussian Distribution

Thermo & Stat Mech - Spring 2006 Class 2717 Gaussian Distribution What is C?

Thermo & Stat Mech - Spring 2006 Class 2718 Gaussian Distribution

Thermo & Stat Mech - Spring 2006 Class 2719 Gaussian Distribution

Thermo & Stat Mech - Spring 2006 Class 2720 Gaussian Distribution

Thermo & Stat Mech - Spring 2006 Class 2721 Properties of Gaussian Distribution

Thermo & Stat Mech - Spring 2006 Class 2722 Properties of Gaussian Distribution

Thermo & Stat Mech - Spring 2006 Class 2723 Problem A bottle of ammonia is opened briefly. The molecules move s 0 = m in any direction before a collision. There are 10 7 collisions per second. How long until 32% of the molecules are 6 m or more from the bottle?

Thermo & Stat Mech - Spring 2006 Class 2724 Solution  = 6 m, where N = (10 7 s -1 )t

Thermo & Stat Mech - Spring 2006 Class 2725 Solution t = 1.08 ×10 5 s = 30 hr = 1.25 days