We give here an example of how a system evolves towards a well defined equilibrium state.

Consider a box divided into two equal halves separated by a wall. At the beginning, time \( t=0 \), there are \( N \) particles on the left side. A small hole in the wall is then opened and one particle can pass through the hole per unit time.

After some time the system reaches its equilibrium state with equally many particles in both halves, \( N/2 \). Instead of determining complicated initial conditions for a system of \( N \) particles, we model the system by a simple statistical model. In order to simulate this system, which may consist of \( N \gg 1 \) particles, we assume that all particles in the left half have equal probabilities of going to the right half. We introduce the label \( n_l \) to denote the number of particles at every time on the left side, and \( n_r=N-n_l \) for those on the right side.