2: The Boltzmann Factor and Partition Functions
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 199151





 2.1: The Boltzmann Factor
 The proportionality constant \(k\) (or \(k_B\)) is named after him: the Boltzmann constant. It plays a central role in all statistical thermodynamics. The Boltzmann factor is used to approximate the fraction of particles in a large system. The Boltzmann factor is given by: \[ \exp(\beta E_i)\]
 2.2: The Thermal Boltzman Distribution
 The Boltzmann distribution represents a thermally equilibrated most probable distribution over all energy levels. There is always a higher population in a state of lower energy than in one of higher energy.
 2.3: The Average Ensemble Energy
 The probability of finding a molecule with energy \(E_i\) is equal to the fraction of the molecules with energy \(E_i\). The average energy is obtaining by multiplying \(E_i\) with its probability and summing over all \(i\): \[ \langle E \rangle = \sum_i E_i P_i \]
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