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: 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.