Thermodynamics and Statistical Mechanics Random Walk Thermo Stat

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

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 27 2

Random Walk After N steps, if n are forward, the distance traveled is, S = [n – (N – n)]s 0 = (2 n – N) s 0 The probability of this occurring is, Thermo & Stat Mech - Spring 2006 Class 27 3

Random Walk The average distance covered after N steps is, Thermo & Stat Mech - Spring 2006 Class 27 4

Random Walk Standard deviation. Thermo & Stat Mech - Spring 2006 Class 27 5

Random Walk As before, Thermo & Stat Mech - Spring 2006 Class 27 6

3 D Random Walk Assume the direction of each step is random. Average distance moved per step is zero. Thermo & Stat Mech - Spring 2006 Class 27 7

Standard Deviation since. Thermo & Stat Mech - Spring 2006 Class 27 8

Standard Deviation Let cos q = x, and sin q dq = – dx. Then, Thermo & Stat Mech - Spring 2006 Class 27 9

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 27 10

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 27 11

Binomial Distribution If n increased by one, then the change in P is Thermo & Stat Mech - Spring 2006 Class 27 12

Binomial Distribution Thermo & Stat Mech - Spring 2006 Class 27 13

Binomial Distribution Thermo & Stat Mech - Spring 2006 Class 27 14

Binomial Distribution Thermo & Stat Mech - Spring 2006 Class 27 15

Binomial to Gaussian Distribution Thermo & Stat Mech - Spring 2006 Class 27 16

Gaussian Distribution What is C? Thermo & Stat Mech - Spring 2006 Class 27 17

Gaussian Distribution Thermo & Stat Mech - Spring 2006 Class 27 18

Gaussian Distribution Thermo & Stat Mech - Spring 2006 Class 27 19

Gaussian Distribution Thermo & Stat Mech - Spring 2006 Class 27 20

Properties of Gaussian Distribution Thermo & Stat Mech - Spring 2006 Class 27 21

Properties of Gaussian Distribution Thermo & Stat Mech - Spring 2006 Class 27 22

Problem A bottle of ammonia is opened briefly. The molecules move s 0 = 10 -5 m in any direction before a collision. There are 107 collisions per second. How long until 32% of the molecules are 6 m or more from the bottle? Thermo & Stat Mech - Spring 2006 Class 27 23

Solution s =6 m , where N = (107 s-1)t Thermo & Stat Mech - Spring 2006 Class 27 24

Solution t = 1. 08 × 105 s = 30 hr = 1. 25 days Thermo & Stat Mech - Spring 2006 Class 27 25