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2: The Boltzmann Factor and Partition Functions

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    • 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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