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Probability density function
Cumulative distribution function
|CDF||where erf is the error function|
In statistics the Maxwell–Boltzmann distribution is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and used in physics (in particular in statistical mechanics) 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. Particle in this context refers to gaseous particles (atoms or molecules), and the system of particles is assumed to have reached thermodynamic equilibrium. While the distribution was first derived by Maxwell in 1860 on heuristic grounds, Boltzmann later carried out significant investigations into the physical origins of this distribution.
A particle speed probability distribution indicates which speeds are more likely: a particle will have a speed selected randomly from the distribution, and is more likely to be within one range of speeds than another. The distribution depends on the temperature of the system and the mass of the particle. The Maxwell–Boltzmann distribution applies to the classical ideal gas, which is an idealization of real gases. In real gases, there are various effects (e.g., van der Waals interactions, vortical flow, relativistic speed limits, and quantum exchange interactions) 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. Thus, it forms the basis of the Kinetic theory of gases, which provides a simplified explanation of many fundamental gaseous properties, including pressure and diffusion.
The Maxwell–Boltzmann distribution is the function
The simplest ordinary differential equation satisfied by the distribution is:
Typical speedsThe mean speed, most probable speed (mode), and root-mean-square can be obtained from properties of the Maxwell distribution.
- The most probable speed, vp, is the speed most likely to be possessed by any molecule (of the same mass m) in the system and corresponds to the maximum value or mode of f(v). To find it, we calculate the derivative df/dv, set it to zero and solve for v:
For diatomic nitrogen (N2, the primary component of air) at room temperature (300 K), this gives m/s
- The mean speed is the expected value of the speed distribution
- The root mean square speed is the second-order moment of speed:
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 (see H-theorem). He later (1877) 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 (from statistical thermodynamics). Maxwell–Boltzmann statistics gives the average number of particles found in a given single-particle microstate, under certain assumptions:
- i and j are indices (or labels) of the single-particle micro states,
- Ni is the average number of particles in the single-particle microstate i,
- N is the total number of particles in the system,
- Ei is the energy of microstate i,
- T is the equilibrium temperature of the system,
- k is the Boltzmann constant.
Because velocity and speed are related to energy, Equation (1) 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 vectorThe 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
It can be shown that:
Distribution for the energyThe energy distribution is found imposing
By the equipartition theorem, this energy is evenly distributed among all three degrees of freedom, so that the energy per degree of freedom is distributed as a chi-squared distribution with one degree of freedom:
The Maxwell–Boltzmann distribution can also be obtained by considering the gas to be a type of quantum gas for which the approximation ε >> k T may be made.
Distribution for the velocity vectorRecognizing that the velocity probability density fv is proportional to the momentum probability density function by
The Maxwell–Boltzmann distribution for the speed follows immediately from the distribution of the velocity vector, above. Note that the speed is