Diffusion CFTP.py

From Werner KRAUTH

Revision as of 15:10, 6 June 2024 Werner (Talk | contribs) ← Previous diff		Revision as of 15:03, 7 June 2024 Werner (Talk | contribs) Next diff →
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	==Context==		==Context==
	This page is part of my [[BegRohu_Lectures_2024|2024 Beg Rohu Lectures]] on "The second Markov chain revolution" at the [https://www.ipht.fr/Meetings/BegRohu2024/index.html Summer School] "Concepts and Methods of Statistical Physics" (3 - 15 June 2024).		This page is part of my [[BegRohu_Lectures_2024|2024 Beg Rohu Lectures]] on "The second Markov chain revolution" at the [https://www.ipht.fr/Meetings/BegRohu2024/index.html Summer School] "Concepts and Methods of Statistical Physics" (3 - 15 June 2024).
		+
		+	In the example of a particle diffusing on a path graph with five sites, with moves from configuration i to [i-1, i, i] being proposed, we consider the formulation of a Markov chain in terms of random maps, but run from time t=-infinity up to time t=0.
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		+	[[Image:One d cftp.png|left|600px|border|Coupling-from-the-past approach to sampling.]]
		+	<br clear="all" />
	==Python program==		==Python program==
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	plt.savefig('backward_position_t0.png')		plt.savefig('backward_position_t0.png')
	plt.show()		plt.show()
		+
		+	==Output==

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Revision as of 15:03, 7 June 2024

Context

This page is part of my 2024 Beg Rohu Lectures on "The second Markov chain revolution" at the Summer School "Concepts and Methods of Statistical Physics" (3 - 15 June 2024).

In the example of a particle diffusing on a path graph with five sites, with moves from configuration i to [i-1, i, i] being proposed, we consider the formulation of a Markov chain in terms of random maps, but run from time t=-infinity up to time t=0.

Python program

import random
import matplotlib.pyplot as plt

N = 5
pos = []
for stat in range(100000):
   all_arrows = {}
   time_tot = 0
   while True:
      time_tot -= 1
      arrows = [random.choice([-1, 0, 1]) for i in range(N)]
      if arrows[0] == -1: arrows[0] = 0
      if arrows[N - 1] == 1: arrows[N - 1] = 0
      all_arrows[time_tot] = arrows
      positions=set(range(0, N))
      for t in range(time_tot, 0):
         positions = set([b + all_arrows[t][b] for b in positions])
      if len(positions) == 1: break
   a = positions.pop()
   pos.append(a)
plt.title('Backward coupling: 1-d with walls: position at t=0')
plt.hist(pos, bins=N, range=(-0.5, N - 0.5), density=True)
plt.savefig('backward_position_t0.png')
plt.show()

Output