Generally speaking, it is difficult to get the distribution, so it is usually to find the state distribution when the system reaches equilibrium, which is recorded as

In order to obtain a smooth distribution, consider any state n that the system may be in. Suppose that the number of times the system enters and leaves the state n within a period of time is recorded. Because "enter" and "leave" appear alternately, these two numbers are either equal or different by 1. But as far as the average incidence of these two events is concerned, it can be considered as equal. That is, when the system runs for a long time and reaches an equilibrium state, for any state n, the average number of times to enter the state per unit time and the average number of times to leave the state per unit time should be equal, which is the principle of "inflow = outflow" of the system under statistical equilibrium. According to this principle, the equilibrium equation in any state can be obtained as follows:

Solvable:

Please remember:

The distribution of the equilibrium state is:

By the requirements of probability distribution:

There are:

So: