Random Numbers Random Walk

Transcription

1 Random Numbers Random Walk Computational Physics Random Numbers Random Walk

2 Outline Random Systems Random Numbers Monte Carlo Integration Example Random Walk Exercise 7 Introduction

3 Random Systems Deterministic Systems Describe with equations Exact solution Random or Stochastic Systems Models with random processes Describe behavior with statistics

4 Consider 1cm 3 box ~10 19 particles motion and collisions Not interested in detailed trajectories Model behavior as result action random processes Statistical Description Results: e.g. probability finding particle at particular location Example Particles in a Box

5 Generation Random Numbers Most computing systems and computer languages have a means to generate random numbers between 0 and 1. Sequence generated from recursive relationship: x n+1 (a x n + b) mod m need a "seed" to start the process same sequence generated by each seed "pseudorandom" in real systems, sequences may repeat eventually. Caveat Emptor!

6 Python Generation Random Numbers with object object is used to generate streams random numbers. Important Methods: rand() generates a sequence uniformly distributed random numbers. randn() generates a sequence normally distributed random numbers. seed(arg) seeds the random number stream with a fixed value arg. rand randn

7 About... is a class in NumPy. We use an instance this class to manage random number generation. Random numbers are generated by methods in the class (e.g. the rand or randn methods). Each instance comes with its own specific random number stream. The random number stream is initialized ( seeded ) when you create a instance. You can reset the seed using the seed() method. We can use the objects to have different random streams or to reset a stream.

8 Examples with from from numpy.random numpy.random import import # an an instance instance the the class class # used used to to make make a a stream stream random random numbers numbers t t () () 'generate 'generate array array 5 5 random random numbers numbers - - uniform uniform dist.' dist.' t.rand(5) t.rand(5) # if if we we seed seed the the with with an an integer integer # we we always always get get the the same same stream stream t2 t2 (12345) (12345) # a a random random stream stream t3 t3 (12345) (12345) # another another one one - - same same seed seed # these these give give the the same same results!! results!! 'check 'check two two streams streams with with same same seed seed - - normal normal dist.' dist.' 'first: 'first: ',t2.randn(5) ',t2.randn(5) 'second: 'second: ',t3.randn(5) ',t3.randn(5) Output: generate array 5 random numbers - uniform dist. [ ] check two streams with same seed - normal dist. first: [ ] second: [ ]

9 Monte Carlo Integration Algorithm: Select random number pair (x,y) from uniformly distributed sample 1 f(x) Check whether (x,y) is above or below f(x) curve. y (x,y) Repeat Fraction points below curve is fraction area below curve x

10 import import numpy numpy as as np np from from numpy.random numpy.random import import def def f(x): f(x): # define define function function to to be be integrated integrated return return x**2 x**2 # create create an an instance instance t t () () # create create random random x x and and y y arrays arrays n n input('number input('number monte monte carlo carlo trials: trials: ') ') x x t.rand(n) t.rand(n) y y t.rand(n) t.rand(n) sum sum for for i i in in range(n): range(n): # compare compare y y to to f(x) f(x) if( if( y[i] y[i] < < f(x[i]) f(x[i]) ): ): sum sum sum sum '{0:d} '{0:d} Monte Monte Carlo Carlo trials'.format(n) trials'.format(n) 'Monte 'Monte Carlo Carlo Answer: Answer: {0:10.7f}'.format(sum/n) {0:10.7f}'.format(sum/n) 'Exact 'Exact Answer: Answer: {0:10.7f}'.format(1./3.) {0:10.7f}'.format(1./3.) Monte Carlo Integration Example: f(x) x 2 creates 1D array with n uniformly distributed random numbers initialize "sum" to 0 to accumulate number points below curve compare each point to function and add to sum points below curve number points below curve out total number is fraction area

11 Random Walk

12 1D Random Walk import import numpy numpy as as np np import import matplotlib.pyplot matplotlib.pyplot as as pl pl from from numpy.random numpy.random import import n n # number number steps steps r r () () p p np.zeros(n) np.zeros(n) p[0] p[0] for for i i in in range(n-1): range(n-1): if if (r.rand() (r.rand() > > 0.5): 0.5): p[i+1] p[i+1] p[i] p[i] else: else: p[i+1] p[i+1] p[i] p[i] initialize array for number steps start at position 0 loop through n-1 steps rand is uniformly distributed: 0->1 take forward step if > 0.5 take backward step if < 0.5

13 7 Random Walks 100 steps

14 Random Walk Results Random Walk Properties: 1. Zero Mean 2. Dispersion which increases with number steps

15 Random Walk with 10,000,000 Walkers demonstrate that: Mean Square Distance Number Random Steps

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19 Mean Square Distance Number Random Steps

20 Understanding this result...

21 Exercise 2D Random Walk y Θ is random number: 0-2π S Θ x s is step size x step s cosθ y step s sinθ

22 Things we want to demonstrate For an ensemble particles: Mean Square Displacement is proportional to number steps. Constant proportionality depends on the step size. For track a single particle: Mean Square Displacement after N steps for a single particle track is the same as Mean Square displacement for an ensemble particles.