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

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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:

\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)}}.

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

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}.

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.