Suppose a system with energy $E_r$ is in contact with a thermal reservoir of energy $E'$.

A thermal reservoir is understood as a system whose temperature remains constant. One way to achieve this is by having a large reservoir (like a water bath).

If both systems are isolated from the surroundings, the sum of their energies will remain constant, which can be expressed with list as

This version provides clarity while retaining the essential information.

The probability of finding the system in a state where it has an energy $E_r$, while the reservoir has an energy $E' = E_0 - E_r$, is defined as

$P_r = C \cdot \Omega_r(E_r) \cdot \Omega'(E')$

where $C$ is a constant that is adjusted to ensure the probability is normalized.

Since $P_r$ represents the probability of finding the system in a particular state $r$, the number of states in state $r$ is equal to one. In other words, this implies that

$\Omega_r(E_r) = 1$

Therefore, the probability can be expressed with respect to list as

If we sum the probabilities of each state $r$, the result should be one. This signifies that it is normalized with the list:

This is equivalent to stating that the system must necessarily be in one of the possible states.

As the energy $E_r$ is much smaller than the total energy $E_0$, the logarithm of the number of states can be expanded around the energy $E_r$ as follows:

$\ln\Omega'(E_0-E_r)=\ln\Omega'(E_0)-\left.\displaystyle\frac{\partial\Omega'}{\partial E'}\right\vert_0E_r\ldots$

Since the derivative of the logarithm of the number of states is equal to the beta function:

$\beta=\left.\displaystyle\frac{\partial\Omega'}{\partial E'}\right\vert_0$

It follows that, in a first approximation with list,

If we substitute the expression

equation=3523

with list=3523

into the equation for probability with list=3521,

,

we obtain with list the probability

where $C$ is a constant to be determined using the normalization condition.

The expression $e^{-\beta E}$ is referred to as the Boltzmann factor, and the distribution it describes is known as the canonical distribution.

Under the normalization condition with list=3522,

,

it is obtained that the normalization constant $C$ is equal to list: