Maxwell–Boltzmann statistics

In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the particle density is low enough to render quantum effects negligible.

The expected number of particles with energy ${\displaystyle \varepsilon _{i}}$ for Maxwell–Boltzmann statistics is

${\displaystyle \langle N_{i}\rangle ={\frac {g_{i}}{e^{(\varepsilon _{i}-\mu )/kT}}}={\frac {N}{Z}}\,g_{i}e^{-\varepsilon _{i}/kT},}$

where:

• ${\displaystyle \varepsilon _{i}}$ is the energy of the i-th energy level,
• ${\displaystyle \langle N_{i}\rangle }$ is the average number of particles in the set of states with energy ${\displaystyle \varepsilon _{i}}$,
• ${\displaystyle g_{i}}$ is the degeneracy of energy level i, that is, the number of states with energy ${\displaystyle \varepsilon _{i}}$ which may nevertheless be distinguished from each other by some other means,
• μ is the chemical potential,
• k is the Boltzmann constant,
• T is absolute temperature,
• N is the total number of particles:
  $${\displaystyle N=\sum _{i}N_{i},}$$
• Z is the partition function:
  $${\displaystyle Z=\sum _{i}g_{i}e^{-\varepsilon _{i}/kT},}$$
• e is Euler's number

Equivalently, the number of particles is sometimes expressed as

${\displaystyle \langle N_{i}\rangle ={\frac {1}{e^{(\varepsilon _{i}-\mu )/kT}}}={\frac {N}{Z}}\,e^{-\varepsilon _{i}/kT},}$

where the index i now specifies a particular state rather than the set of all states with energy ${\displaystyle \varepsilon _{i}}$, and ${\textstyle Z=\sum _{i}e^{-\varepsilon _{i}/kT}}$.