Maxwell–Boltzmann distribution

In physics, the Maxwell–Boltzmann distribution, or Maxwell distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.
It was first defined and used for describing particle speeds in idealized gases, where the particles move freely inside a stationary container without interacting with one another, except for very brief collisions in which they exchange energy and momentum with each other or with their thermal environment. The term "particle" in this context refers to gaseous particles only, and the system of particles is assumed to have reached thermodynamic equilibrium. The energies of such particles follow what is known as Maxwell–Boltzmann statistics, and the statistical distribution of speeds is derived by equating particle energies with kinetic energy.
Mathematically, the Maxwell–Boltzmann distribution is the chi distribution with three degrees of freedom, with a scale parameter measuring speeds in units proportional to the square root of .
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. The Maxwell–Boltzmann distribution applies fundamentally to particle velocities in three dimensions, but turns out to depend only on the speed of the particles. A particle speed probability distribution indicates which speeds are more likely: a randomly chosen particle will have a speed selected randomly from the distribution, and is more likely to be within one range of speeds than another. The kinetic theory of gases applies to the classical ideal gas, which is an idealization of real gases. In real gases, there are various effects that can make their speed distribution different from the Maxwell–Boltzmann form. However, rarefied gases at ordinary temperatures behave very nearly like an ideal gas and the Maxwell speed distribution is an excellent approximation for such gases. This is also true for ideal plasmas, which are ionized gases of sufficiently low density.
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. A list of derivations are:

Maximum entropy probability distribution in the phase space, with the constraint of conservation of average energy
Canonical ensemble.
Distribution function

For a system containing a large number of identical non-interacting, non-relativistic classical particles in thermodynamic equilibrium, the fraction of the particles within an infinitesimal element of the three-dimensional velocity space, centered on a velocity vector of magnitude, is given by
where:

is the particle mass;
is the Boltzmann constant;
is thermodynamic temperature;
is a probability distribution function, properly normalized so that over all velocities is unity.

Image:MaxwellBoltzmann-en.svg|right|thumb|340px|The speed probability density functions of the speeds of a few noble gases at a temperature of 298.15 K. The y-axis is in s/m so that the area under any section of the curve is dimensionless.
One can write the element of velocity space as, for velocities in a standard Cartesian coordinate system, or as in a standard spherical coordinate system, where is an element of solid angle and.
Alternately, the distribution function can also be written in momentum space aswhere is the momentum vector.
The Maxwellian distribution function for particles moving in only one direction, if this direction is, is a normal distribution with a standard deviation of :
which can be obtained by integrating the three-dimensional form given above over and.
Recognizing the symmetry of, one can integrate over solid angle and write a probability distribution of speeds as the function
This probability density function gives the probability, per unit speed, of finding the particle with a speed near. This equation is simply the Maxwell–Boltzmann distribution with distribution parameter
The Maxwell–Boltzmann distribution is equivalent to the chi distribution with three degrees of freedom and scale parameter
The simplest ordinary differential equation satisfied by the distribution is:
or in unitless presentation:
With the Darwin–Fowler method of mean values, the Maxwell–Boltzmann distribution is obtained as an exact result.

Relaxation to the 2D Maxwell–Boltzmann distribution

For particles confined to move in a plane, the speed distribution is given by
This distribution is used for describing systems in equilibrium. However, most systems do not start out in their equilibrium state. The evolution of a system towards its equilibrium state is governed by the Boltzmann equation. The equation predicts that for short range interactions, the equilibrium velocity distribution will follow a Maxwell–Boltzmann distribution. To the right is a molecular dynamics simulation in which 900 hard sphere particles are constrained to move in a rectangle. They interact via perfectly elastic collisions. The system is initialized out of equilibrium, but the velocity distribution quickly converges to the 2D Maxwell–Boltzmann distribution.

Typical speeds

The mean speed, most probable speed , and root-mean-square speed can be obtained from properties of the Maxwell distribution.
This works well for nearly ideal, monatomic gases like helium, but also for molecular gases like diatomic oxygen. This is because despite the larger heat capacity due to their larger number of degrees of freedom, their translational kinetic energy is unchanged.
In summary, the typical speeds are related as follows:
The root mean square speed is directly related to the speed of sound in the gas, by
where is the adiabatic index, is the number of degrees of freedom of the individual gas molecule. For the example above, diatomic nitrogen at, and
the true value for air can be approximated by using the average molar weight of air, yielding at .
The average relative velocity
where the three-dimensional velocity distribution is
The integral can easily be done by changing to coordinates and

Limitations

The Maxwell–Boltzmann distribution assumes that the velocities of individual particles are much less than the speed of light, i.e. that. For electrons, the temperature of electrons must be. For distribution of speeds of relativistic particles, see Maxwell–Jüttner distribution.

Derivation and related distributions

Maxwell–Boltzmann statistics

The original derivation in 1860 by James Clerk Maxwell was an argument based on molecular collisions of the Kinetic theory of gases as well as certain symmetries in the speed distribution function; Maxwell also gave an early argument that these molecular collisions entail a tendency towards equilibrium. After Maxwell, Ludwig Boltzmann in 1872 also derived the distribution on mechanical grounds and argued that gases should over time tend toward this distribution, due to collisions. He later derived the distribution again under the framework of statistical thermodynamics. The derivations in this section are along the lines of Boltzmann's 1877 derivation, starting with result known as Maxwell–Boltzmann statistics. Maxwell–Boltzmann statistics gives the average number of particles found in a given single-particle microstate. Under certain assumptions, the logarithm of the fraction of particles in a given microstate is linear in the ratio of the energy of that state to the temperature of the system: there are constants and such that, for all,
The assumptions of this equation are that the particles do not interact, and that they are classical; this means that each particle's state can be considered independently from the other particles' states. Additionally, the particles are assumed to be in thermal equilibrium.
This relation can be written as an equation by introducing a normalizing factor:
where:

is the expected number of particles in the single-particle microstate,
is the total number of particles in the system,
is the energy of microstate,
the sum over index takes into account all microstates,
is the equilibrium temperature of the system,
is the Boltzmann constant.

The denominator in is a normalizing factor so that the ratios add up to unity — in other words it is a kind of partition function.
Because velocity and speed are related to energy, Equation can be used to derive relationships between temperature and the speeds of gas particles. All that is needed is to discover the density of microstates in energy, which is determined by dividing up momentum space into equal sized regions.

Distribution for the momentum vector

The potential energy is taken to be zero, so that all energy is in the form of kinetic energy.
The relationship between kinetic energy and momentum for massive non-relativistic particles is
where is the square of the momentum vector. We may therefore rewrite Equation as:
where:

is the partition function, corresponding to the denominator in ;
is the molecular mass of the gas;
is the thermodynamic temperature;
is the Boltzmann constant.

This distribution of is proportional to the probability density function for finding a molecule with these values of momentum components, so:
The normalizing constant can be determined by recognizing that the probability of a molecule having some momentum must be 1.
Integrating the exponential in over all,, and yields a factor of
So that the normalized distribution function is:
The distribution is seen to be the product of three independent normally distributed variables,, and, with variance. Additionally, it can be seen that the magnitude of momentum will be distributed as a Maxwell–Boltzmann distribution, with. The Maxwell–Boltzmann distribution for the momentum can be obtained more fundamentally using the H-theorem at equilibrium within the Kinetic theory of gases framework.