M etodos Matem aticos e de Computa c ao I

Transcription

1 Métodos Matemáticos e de Computação I

2 Complex Systems 01/16 General structure Microscopic scale Individual behavior Description of the constituents Model Macroscopic scale Collective behavior Emergence of regularities with size

3 Computer simulations 02/16 Python (2., 3. ) Installation of Anaconda

4 Computer simulations 03/16 Python Simulation of the agents Iterative processes Random numbers Complexity Randomness

5 Python 04/16 Installation - Anaconda Open a terminal Linux (Ubuntu): Ctrl + Alt + t Mac: Applications / Utilities Windows: Remark: Windows: dir Directory1\Directory2\ icon.png Linux/Mac: cd Directory1/Directory2/

6 Python 05/16 Paths for the execution of a program 1) Write / change a code 2) Save the code 3) Execute (at the terminal) python programa.py Remark: The file program.py should be at the same directory where the third command is executed

7 Python - Print command 06/16 Command: print print 'Hello, world!' Hello, world! # The symbol '#' indicates # comment, which is not an # effective part of the code

8 Python - Print command 06/16 Command: print print 'Hello, world!' Hello, world! 1+2 i print '1+2' i = 0 print 'i' # The symbol '#' indicates # comment, which is not an # effective part of the code 3 0 print 1+2 i = 0 print i

9 Coin flip 07/16 Single coin flip Commands: if and else Idea Code (Python) Generate a random number r [0.0, 1.0] If r > 0.5: Write 'Head' Otherwise: Write 'Tail'

10 Coin flip 07/16 Single coin flip Commands: if and else Idea Code (Python) import random Generate a random number r [0.0, 1.0] If r > 0.5: Write 'Head' Otherwise: Write 'Tail' r = random.uniform(0.0, 1.0) if r > 0.5: print 'Head' else: print 'Tail'

11 Coin flip 08/16 Ten coin flips Command: for import random for i in range(10): r = random.uniform(0.0, 1.0) if r > 0.5: print 'Head' else: print 'Tail'

12 Coin flip 09/16 Ten coin flips Command: while import random i = 0 # Store 0 at i while i<10: r = random.uniform(0.0, 1.0) if r > 0.5: print 'Head' else: print 'Tail' i = i+1 # Store i+1 at i

13 Two urns with N balls L balls in the left urn Initial condition R balls in the right urn Example: Dynamics (1) Choose one of the N balls randomly (2) The chosen ball is moved to the opposite urn (3) Return to (1) 10/16 L + R = N L = 7 R = 3 N = 10

14 11/16 File ehrenfest.py L: Bleft R: Bright Ttot iterations (total time) t list: List for time left list: List for L (...) for t in range (1, Ttot+1): # Time: 1 a Ttot r = random.uniform(0.0, 1.0) if r < Bleft / float(bleft + Bright): Bleft -= 1 # Same as Bleft = Bleft - 1 Bright += 1 # Same as Bright = Bright + 1 else: Bleft += 1 Bright -= 1 t list.append(t) # Attaching t to t list left list.append(bleft) # Attaching Bleft to left list (...)

15 12/16 File ehrenfest.py Initial condition: L = 10 and R = 0 Ttot = 100 Ttot = 5000

16 13/16 Ttot = 100 Ttot = 5000

17 13/16 Ttot = 100 Ttot = 5000 Extremal situations Intuition : Memory loss & concentration around L+R 2

18 14/16 Increase of number of balls & decrease of relative fluctuations

19 14/16 Increase of number of balls & decrease of relative fluctuations balls and Ttot =

20 14/16 Increase of number of balls & decrease of relative fluctuations balls and Ttot = balls and Ttot =

21 14/16 Increase of number of balls & decrease of relative fluctuations balls and Ttot = balls and Ttot =

22 15/16 File: ehrenfest multirun.py Increase of the number of runs (...) for n in range (run): # Repetitions: run for t in range (1, Ttot+1): # Time: 1 a Ttot r = random.uniform(0.0, 1.0) if r < Bleft / float(bleft + Bright): Bleft -= 1 # Same as Bleft = Bleft - 1 Bright += 1 # Same as Bright = Bright + 1 else: Bleft += 1 Bright -= 1 (...)

23 16/16 File: ehrenfest multirun.py Increase of the number of runs 1 iteration

24 16/16 File: ehrenfest multirun.py Increase of the number of runs 1 iteration 10 iterations

25 16/16 File: ehrenfest multirun.py Increase of the number of runs 1 iteration 10 iterations 1000 iterations Estimation of uncertainties