Lecture 10: Random Walks in One Dimension

Transcription

1 Physical Principles in Biology Biology 3550 Fall 2018 Lecture 10: Random Walks in One Dimension Wednesday, 12 September 2018 c David P. Goldenberg University of Utah goldenberg@biology.utah.edu

2 Announcements Quiz 2: Wednesday, 19 Sept. Problem Set 2 is now posted on course web page. Due Monday, 24 Sept. Start early!

3 A Random Walk in One Dimension 1 Start at position x = 0. 2 Flip coin. Heads, take step of length δ to the right. Tails - left Heads - right Tails, take step of length δ to the left. 3 Repeat 2 another (n 1) times. Final position after n steps is x n. Generally expect a distribution of x n if the random walk is repeated a large number, N, of times.

4 Like a Plinko, with variable x x represents the position of the bucket, relative to the central bucket.

5 What Do We Know About x n, the End Point? The maximum value of x n is δn. The minimum value of x n is δn. If we repeat the random walk many times, the distribution of x n will be binomial. But, if n is very large, calculating the binomial distribution will be difficult!

6 Clicker Question #1 What is the largest value of n for which your calculator can calculate n!? A) 1 25 B) C) D) E) > 100

7 Calculate The Average Final Position (The Expected Value of x n ) For a single random walk, the final position will be: x n = n i=1 δ i where i is the step number, and δ i is either +δ or δ, with probabilities p +δ and p δ. For each step, δ i is a random variable, with an expected value, E(δ i ): E(δ i ) = δp +δ δp δ = δp +δ δ(1 p +δ ) = δp +δ δ + δp +δ = 2δp +δ δ = δ ( 2p +δ 1 )

8 Clicker Question #2 If the random-walk step size is 0.5 m, and the probability of a forward step, p +δ, is 0.3, what is the expected value for the displacement in a single step, E(δ i )? A) -0.5 m B) -0.2 m C) 0 m D) 0.2 m E) 0.5 m E(δ i ) = δp +δ δp δ = 0.5 m m 0.7 = 0.5 m( ) = 0.2 m

9 Calculating The Expected Value of x n An important theorem: If x and y are two independent random variables, then the expected value of the sum is calculated as: E(x + y) = E(x) + E(y) Since: x n = n i=1 δ i The expected value of x n is calculated as: E ( ) n x n = E(δ i ) = ne(δ i ) i=1 = nδ ( 2p +δ 1 )

10 Expected Value of x n for a One-dimensional Random Walk

11 Some Different Kinds of Average For N random walks of n steps each: The mean: x n = 1 N x N n,j, for large N Angle brackets,, indicate average over a large sample. The mean-square: N x 2 n = 1 N x 2 n,j The root-mean-square (RMS): RMS(x n ) = x 2 n = 1 N N x 2 n,j

12 An Application of Mean-square and Root-mean-square Averages: Household Power (US) Voltage versus time

13 An Application of Mean-square and Root-mean-square Averages: Household Power (US) Voltage squared versus time

14 An Application of Mean-square and Root-mean-square Averages: Household Power (US) V 2 versus time

15 Clicker Question #3 For the numbers: 4, 2, 3, 1, 5, Calculate the root-mean-square average A) 0.2 B) 1.5 C) 2.9 D) 3.3 E) 4.8 RMS = = = 55 5 = 11

16 Calculating the Mean-Square Displacement for a 1-d Random Walk For a single random walk, the final position will be: x n = n i=1 δ i where i is the step number, and δ i is either δ or +δ, with equal probability (if the coin is fair), for each step. We can also express x n in terms of the position after the next-to-last step, x n 1 : x n = x n 1 + δ n Something tricky is coming up!

17 Calculating The Mean-Square Displacement If we do a large number, N, of random walks, the mean-square displacement, x, will be: ( xn 2 = 1 N x 2 = 1 N n ) 2 δ n,j N N j,i where j is the random walk number, and δ j,i is the displacement for step i of walk j. i=1 We can also write the mean-square average as: xn 2 = 1 N ) 2 (x N (n 1),j + δ j,n = 1 N N ( ) x 2 + 2x (n 1),j (n 1),jδ j,n + δ 2 j,n δ n,j is the change in position for the last step in random walk j.

18 Calculating The Mean-Square Displacement Following from the previous slide: x 2 n = 1 N N ( ) x 2 + 2x (n 1),j (n 1),jδ j,n + δ 2 j,n N N N = 1 N x 2 (n 1),j + 1 N (2x (n 1),j δ j,n + 1 N δ 2 j,n = x 2 n 1 + 2x n 1δ n + δ 2 n For each walk, δ n is either +δ or δ, with equal probability (for an unbiased walk), and is independent of the position before the last step, x n 1. The average of 2x n 1 δ n over N walks, 2x n 1 δ n, is expected to be 0.

19 Calculating The Mean-Square Displacement From the previous slide: xn 2 = x 2 n 1 + δ2 n Following the same logic: x 2 n 1 = x 2 n 2 + δ2 n 1 and xn 2 = x 2 n 2 + δ2 n 1 + δ2 n = x 2 n δ2

20 Calculating The Mean-Square Displacement Continuing in the same way: x 2 n = x 2 n δ2 = x 2 n 3 + δ2 n δ2 = x 2 n δ2 x 2 n = x 2 n δ2 and so on, until we have: x 2 n = x (n 1) δ2 = x n δ2 = n δ 2 A derivation based on recursion!