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Boltzmann Factor & Distribution

November 20, 2008

The Boltzmann factor gives the unnormalised relative probability for a state of energy \epsilon_i of being occupied at temperature \displaystyle T.

The partition function is the proportional factor between the probability (for a state of energy \epsilon_i of being occupied at temperature T) and the Boltzmann factor:

\displaystyle P(\epsilon_i)=\frac{\exp\{-\epsilon_i/k_BT\}}{Z(T)}

equivalently:

\displaystyle \frac{N(\epsilon_i)}{N}=\frac{\exp\{-\epsilon_i/k_BT\}}{Z(T)}

If the system can exchange particles, it is necessary to introduce the chemical potential \mu by which it is possible to define the Boltzmann distribution function that represents the classical limit* (high energies) of the Fermi-Dirac and the Bose-Einsten distributions:

\displaystyle f(\epsilon_i)=\frac{1}{\exp\{(\epsilon_i-\mu)/k_BT\}}

*in despite of the term “classical” the Boltzmann distribution is valid for quantum particles.

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