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:

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

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

where is known as a Boltzmann factor. We will expand on its usefulness in the next section.