Lect05 - Le cture5: Statistical Processes Random Walk and...

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Lecture 5, p 1 Random Walk and Particle Diffusion Counting and Probability Microstates and Macrostates The meaning of equilibrium Lecture 5: Statistical Processes 0 20 40 60 80 100 0.00 0.02 0.04 0.06 0.08 0.10 Probability (N 1 , N 2 ) N 1
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Lecture 5, p 2 The Random Walk Problem (1) We’ll spend a lot of time in P213 studying random processes. As an example, consider a gas. The molecules bounce around randomly, colliding with other molecules and the walls. How far on average does a single molecule go in time? The motion that results from a random walk is called diffusion. http://intro.chem.okstate.edu/1314F00/Laboratory/GLP.htm This picture can also apply to: impurity atoms in an electronic device defects in a crystal sound waves carrying heat in solid!
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Lecture 5, p 3 We’ll analyze a simplified model of diffusion. A particle travels a distance in a straight line, then scatters off another particle and travels in a new, random direction. Assume the particles have average speed v. As we saw before, there will be a distribution of speeds. We are interested in averages. Each step takes an average time Note: is also an average, called the “ mean free path ”. We’d like to know how far the particle gets after time t. First, answer a simpler question: How many steps, M, will the particle have taken? The Random Walk Problem (2) t M = τ v τ =
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Lecture 5, p 4 Random walk with constant step size ( always the same): Random walk with random step size ( varies, but has the same average): Random Walk Simulation
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Lecture 5, p 5 Simplify the problem by considering 1-D motion and constant step size. At each step, the particle moves After M steps, the displacement is Repeat it many times and take the average: The average (mean) displacement is: The average squared displacement is: The average distance is the square root (the “root-mean-square” displacement ) The average distance moved is proportional to t. M i i 1 x s = = M i i 1 x s 0 = = = M M M 2 2 2 i j i i j i 1 j 1 i 1 i j x s s s ss M = = = = = + = ∑ ∑ x - + The Random Walk Problem (3) i s = ± 2 1/ 2 t rms x x M τ = = =   Cross terms cancel, due to randomness. Note: This is the same square root we obtained last lecture when we looked at thermal conduction. It’s generic to diffusion problems.
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Lecture 5, p 6 Look at the distribution of displacements after 10 steps. n L = # steps left. x = 0 when n L = 5 Consider what happens to this distribution as the number of The Random Walk Problem (4) +/- s.d . 0 2 4 6 8 10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Probability (n L steps left, out of N total) n L ± x rms x = 0 ( 29 L 10 n L 10 C P n 2 =
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Lecture 5, p 7 Here are the probability distributions for various number of steps (N). The
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This note was uploaded on 02/21/2011 for the course PHYS 213 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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Lect05 - Le cture5: Statistical Processes Random Walk and...

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