A small system in thermal contact with a much larger reservoir can exchange energy while the combined system remains isolated. The probability that the small system occupies a microstate of energy $E_i$ is proportional to the reservoir's number of compatible states.

Expanding the reservoir entropy around its equilibrium energy gives $$P_i\propto e^{-E_i/(k_BT)}.$$ The factor $$e^{-\beta E_i},\qquad \beta=\frac{1}{k_BT},$$ is the Boltzmann factor.

After normalization, $$P_i=\frac{e^{-\beta E_i}}{Z},$$ where $$Z=\sum_i e^{-\beta E_i}$$ is the partition function. This probability distribution is the canonical ensemble, appropriate to fixed temperature $T$, volume $V$ and particle number $N$.

For a two-level system with energies $0$ and $\Delta$, the probability of the excited state is $$P_1=\frac{e^{-\beta\Delta}}{1+e^{-\beta\Delta}}.$$ At low temperature, $\beta\Delta\gg1$, so excitation is strongly suppressed. At high temperature, $\beta\Delta\ll1$, the two levels approach equal probability.

The exponential form reflects a balance between energy and the enormous number of reservoir microstates. It explains why high-energy states are possible but exponentially less probable at finite temperature. The canonical ensemble is the central bridge from microscopic energy levels to measurable thermal averages.