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:

$\begin{array}{cc}\frac{P({\epsilon }_1)}{P({\epsilon }_2)}& =\frac{g_R(U_0-{\epsilon }_1)}{g_R(U_0-{\epsilon }_2)}\\ & =\frac{e^{S_R(U_0-{\epsilon }_1)}}{e^{S_R(U_0-{\epsilon }_2)}}.\end{array}$

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

$\begin{array}{cc}{S}_R(U_0-\epsilon )& =S_R(U_0)-\epsilon \frac{\partial S_R}{\partial U}{|}_{V,N}\\ & =S_R(U_0)-\frac{\epsilon }{T}.\end{array}$

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

$\frac{P({\epsilon }_1)}{P({\epsilon }_2)}=\frac{e^{-{\epsilon }_1/T}}{e^{-{\epsilon }_2/T}},$

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