We want to study a physical system which evolves towards equilibrium, from som given initial conditions. Recall the simple example of particles in a box. At an initial time t_0 all particles are in the left half of the box. Thereafter they are allowed to diffuse into the two halves of the box.
xxxxxxxxxx#!/usr/bin/env pythonfrom matplotlib import pyplot as pltfrom math import expimport numpy as npimport random# initial number of particlesN0 = 1000MaxTime = 10*N0values = np.zeros(MaxTime)time = np.zeros(MaxTime)random.seed()# initial number of particles in left halfnleft = N0for t in range (0, MaxTime, 1): if N0*random.random() <= nleft: nleft -= 1 else: nleft += 1 time[t] = t values[t] = nleft# Finally we plot the resultsplt.plot(time, values,'b-')plt.axis([0,MaxTime, N0/4, N0])plt.xlabel('$t$')plt.ylabel('$N$')plt.title('Number of particles in left half')plt.savefig('box.pdf')plt.show()