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Concept
Maxwell–Boltzmann statistics
Summary
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 for Maxwell–Boltzmann statistics is
where:
is the energy of the i-th energy level,
is the average number of particles in the set of states with energy ,
is the degeneracy of energy level i, that is, the number of states with energy 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:
Z is the partition function:
e is Euler's number
Equivalently, the number of particles is sometimes expressed as
where the index i now specifies a particular state rather than the set of all states with energy , and .
Maxwell–Boltzmann statistics grew out of the Maxwell–Boltzmann distribution, most likely as a distillation of the underlying technique. The distribution was first derived by Maxwell in 1860 on heuristic grounds. Boltzmann later, in the 1870s, carried out significant investigations into the physical origins of this distribution. The distribution can be derived on the ground that it maximizes the entropy of the system.
Maxwell–Boltzmann statistics is used to derive the Maxwell–Boltzmann distribution of an ideal gas. However, it can also be used to extend that distribution to particles with a different energy–momentum relation, such as relativistic particles (resulting in Maxwell–Jüttner distribution), and to other than three-dimensional spaces.
Maxwell–Boltzmann statistics is often described as the statistics of "distinguishable" classical particles. In other words, the configuration of particle A in state 1 and particle B in state 2 is different from the case in which particle B is in state 1 and particle A is in state 2.
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Partition function (statistical mechanics)
In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless.