Statistical Mechanics/Boltzmann and Gibbs factors and Partition functions/Boltzmann Factors

The first 'method of simplification' involves considering a thermal reservoir, basically a temperature bath that will keep our system of consideration at a constant temperature T.

Then by the fundamental assumption, given two energy states:

${\begin{aligned}{\frac {P(\varepsilon _{1})}{P(\varepsilon _{2})}}&{}={\frac {g_{R}(U_{0}-\varepsilon _{1})}{g_{R}(U_{0}-\varepsilon _{2})}}\\&{}={\frac {e^{S_{R}(U_{0}-\varepsilon _{1})}}{e^{S_{R}(U_{0}-\varepsilon _{2})}}}.\end{aligned}}$

Now, because of the Taylor Series, and in the presence of an infinitely large reservoir the higher-order terms vanish:

${\begin{aligned}S_{R}(U_{0}-\varepsilon )&{}=S_{R}(U_{0})-\varepsilon {\frac {\partial S_{R}}{\partial U}}{\big |}_{V,N}\\&{}=S_{R}(U_{0})-{\frac {\varepsilon }{T}}.\end{aligned}}$

Using this simplification we can write the previous exponential form of the ratio of probabilities:

${\frac {P(\varepsilon _{1})}{P(\varepsilon _{2})}}={\frac {e^{-\varepsilon _{1}/T}}{e^{-\varepsilon _{2}/T}}},$

where $e^{-\varepsilon /T}$ is known as a Boltzmann factor. We will expand on its usefulness in the next section.