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The Maxwell–Boltzmann distribution is a result of the kinetic theory of gases, which provides a simplified explanation of many fundamental gaseous properties, including pressure and diffusion. [3] The Maxwell–Boltzmann distribution applies fundamentally to particle velocities in three dimensions, but turns out to depend only on the speed ...
The Boltzmann equation can be used to derive the fluid dynamic conservation laws for mass, charge, momentum, and energy. [ 8 ] : 163 For a fluid consisting of only one kind of particle, the number density n is given by n = ∫ f d 3 p . {\displaystyle n=\int f\,d^{3}\mathbf {p} .}
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.
Thermal velocity or thermal speed is a typical velocity of the thermal motion of particles that make up a gas, liquid, etc. Thus, indirectly, thermal velocity is a measure of temperature. Technically speaking, it is a measure of the width of the peak in the Maxwell–Boltzmann particle velocity distribution.
As an example: the partition function for the isothermal-isobaric ensemble, the generalized Boltzmann distribution, divides up probabilities based on particle number, pressure, and temperature. The energy is replaced by the characteristic potential of that ensemble, the Gibbs Free Energy.
In a plasma, the Boltzmann relation describes the number density of an isothermal charged particle fluid when the thermal and the electrostatic forces acting on the fluid have reached equilibrium. In many situations, the electron density of a plasma is assumed to behave according to the Boltzmann relation, due to their small mass and high mobility.
Although Boltzmann first linked entropy and probability in 1877, the relation was never expressed with a specific constant until Max Planck first introduced k, and gave a more precise value for it (1.346 × 10 −23 J/K, about 2.5% lower than today's figure), in his derivation of the law of black-body radiation in 1900–1901. [11]
where is the Boltzmann constant and is the absolute temperature defined by the ideal gas law, to obtain =, which leads to a simplified expression of the average translational kinetic energy per molecule, [33] =.