Diffusion CFTP.py
From Werner KRAUTH
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| ==Further information== | ==Further information== | ||
| + | Rather than to run the simulation until t=0, one might be tempted to stop it as soon as the configurations have coupled at some time t < 0. This is tested in the following program, but it is wrong for obvious reasons: The goal was to infer the position at t=0 of a simulation that has run since time t=-infinity. | ||
| ==References== | ==References== | ||
Revision as of 15:42, 7 June 2024
Contents |
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.
At time t=-infinity, as shown, the pebble starts on configuration 1, so that, by time t=0, it must sample the stationary distribution (the uniform distribution on the five sites). The below program constructs just as many set of arrows as needed to conclude on where it finishes it infinitely long journey.
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
Here is output of the above Python program. The histogram is absolutely flat, without any corrections. But this is normal, given that the simulation has run, for each of the realizations of the random map, an infinite number of iterations.
Further information
Rather than to run the simulation until t=0, one might be tempted to stop it as soon as the configurations have coupled at some time t < 0. This is tested in the following program, but it is wrong for obvious reasons: The goal was to infer the position at t=0 of a simulation that has run since time t=-infinity.
References
This program illustrates the coupling-from-the-past approach to Markov-chain sampling introduced by Propp and Wilson, in 1997.
The reference is:
