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The Maxwell-Boltzmann distribution can be derived using statistical mechanics (see the derivation of the partition function) or by using the principle of extreme physical information. It corresponds to the most probable energy distribution, in a collisionally-dominated system consisting of a large number of non-interacting particles. Since interactions between the molecules in a gas are generally quite small, the Maxwell-Boltzmann distribution provides a very good approximation of the conditions in a gas.
In many other cases, however, the condition of elastic collisions dominating all other processes is not even approximately fulfilled. That is true, for instance, for the physics of the ionosphere and space plasmas where recombination and collisional excitation (i.e. radiative processes) are of far greater importance: in particular for the electrons. Not only would the assumption of a Maxwell distribution yield quantitatively wrong results, but even prevent a correct qualitative understanding of the physics involved.
The Maxwell-Boltzmann distribution can be expressed as:
where Ni is the number of molecules at equilibrium temperature T, having energy level Ei, N is the total number of molecules in the system and k is the Boltzmann constant. Essentially Equation 1 provides a means for calculating the fraction of molecules (Ni/N) that have energy Ei at a given temperature, T. Because velocity and speed are related to energy, Equation 1 can be used to derive relationships between temperature and the speeds of molecules in a gas. The denominator in this equation is known as the partition function.
What follows is a derivation wildly different from the derivatin described by James Clerk Maxwell and later described with fewer assumptions by Ludwig Boltzmann.
For the case of an "ideal gas" consisting of non-interacting atoms in the ground state, all energy is in the form of kinetic energy. From the particle in a box problem in quantum mechanics we know that the energy levels for a gas in a rectangular box with sides of lengths ax, ay, az are given by:
where, nx, ny, and nz are the quantum numbers for x, y, and z motion, respectively. However, for a macroscopic sized box, the energy levels are very closely spaced, so the energy levels can be considered continuous and we can replace the sum with an integral. Furthermore, we can recognize that (h2ni2/4ai2) corresponds to the square of the ith component of momentum, pi2 giving:
where q corresponds to the denominator in Equation 1. Here m is the molecular mass of the gas, T is the thermodynamic temperature and k is the Boltzmann constant. This distribution of Ni/N is proportional to the probability distribution function fp for finding a molecule with these values of momentum components, so:
The constant of proportionality, c, can be determined by recognizing that the probability of a molecule having any momentum must be 1. Therefore the integral of equation 4 over all px, py, and pz must be 1.
It can be shown that:
so in order for the integral of equation 4 to be 1,
Substituting Equation 6 into Equation 4 and using pi = mvi for each component of momentum gives:
Finally recognizing that the velocity probability distribution, fv is proportional to the momentum probability distribution function as
we get:
Which is the Maxwell-Boltzmann velocity distribution.