Boltzmann Average

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The Boltzmann average (sometimes known as the thermal average) for a given quantity or observable, let us say $A$ , is given by

\[ \langle A \rangle= \frac{\sum_{i}Ae^{-E_i/k_BT}}{\sum_{i}e^{^{-E_{i}/k_BT}}} \nonumber\]

where $k B$ is the Boltzmann constant, and T is the temperature. This provides the expected value of the property in question at a given temperature. This equation assumes non-degenerate states.

Boltzmann Distribution
The Maxwell-Boltzmann distribution function is a function f(E) which gives the probability that a system in contact with a thermal bath at temperature T has energy E. This distribution is classical and is used to describe systems with identical but distinguishable particles.
Fluctuations
The methods developed allow us to calculate thermodynamic averages. The deviation of a mechanical variable from its mean value is called a fluctuation. The theory of fluctuations is useful for understanding how the different ensembles (NVT, NPT etc.). Fluctuations are also important in theories of light scattering and in the study of transport processes.
Ideal Gas Partition Function
Proof that β = 1/kT
The Boltzmann constant
The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. It is named after the Austrian physicist Ludwig Eduard Boltzmann.