In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
(in particular in
statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular
probability distribution named after
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and li ...
and
Ludwig Boltzmann
Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of ther ...
.
It was first defined and used for describing particle
speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (ma ...
s in
idealized gases, where the particles move freely inside a stationary container without interacting with one another, except for very brief
collision
In physics, a collision is any event in which two or more bodies exert forces on each other in a relatively short time. Although the most common use of the word ''collision'' refers to incidents in which two or more objects collide with great fo ...
s 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 (
atoms or
molecules), and the system of particles is assumed to have reached
thermodynamic equilibrium
Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In the ...
.
[''Statistical Physics'' (2nd Edition), F. Mandl, Manchester Physics, John Wiley & Sons, 2008, ] The energies of such particles follow what is known as
Maxwell–Boltzmann statistics
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 ...
, and the statistical distribution of speeds is derived by equating particle energies with
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
.
Mathematically, the Maxwell–Boltzmann distribution is the
chi distribution
In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the positive square root of the sum of squares of a set of independent random variables each following a standard norm ...
with three
degrees of freedom (the components of the
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
vector in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
), with a
scale parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution.
Definition
If a family o ...
measuring speeds in units proportional to the square root of
(the ratio of temperature and particle mass).
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
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
and
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemica ...
. The Maxwell–Boltzmann distribution applies fundamentally to particle velocities in three dimensions, but turns out to depend only on the speed (the
magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
of the velocity) 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
An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
, which is an idealization of real gases. In real gases, there are various effects (e.g.,
van der Waals interaction
In molecular physics, the van der Waals force is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical electronic bond; they are comparatively weak and the ...
s,
vortical flow,
relativistic speed limits, and quantum
exchange interaction
In chemistry and physics, the exchange interaction (with an exchange energy and exchange term) is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an exchange force in an analogy to classic ...
s) 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.
[See:
* Maxwell, J.C. (1860 A): ''Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science'', 4th Series, vol.19, pp.19-32]
* Maxwell, J.C. (1860 B): ''Illustrations of the dynamical theory of gases. Part II. On the process of diffusion of two or more kinds of moving particles among one another. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science'', 4th Ser., vol.20, pp.21-37
/ref> 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#Distributions with measured constants, Maximum entropy probability distribution in the phase space, with the constraint of conservation of average energy ;
# Canonical ensemble
In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat ...
.
Distribution function
For a system containing a large number of identical non-interacting, non-relativistic classical particles in themodynamic 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
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
, and is thermodynamic temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic ...
.
is a probability distribution function, properly normalized so that over all velocities is unity.
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.
The Maxwellian distribution function for particles moving in only one direction, if this direction is , is
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
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
gives the probability, per unit speed, of finding the particle with a speed near . This equation is simply the Maxwell–Boltzmann distribution (given in the infobox) with distribution parameter .
The Maxwell–Boltzmann distribution is equivalent to the chi distribution
In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the positive square root of the sum of squares of a set of independent random variables each following a standard norm ...
with three degrees of freedom and scale parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution.
Definition
If a family o ...
.
The simplest ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
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.
Relation 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 Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
. 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
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of t ...
(MD) 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 (in blue) quickly converges to the 2D Maxwell–Boltzmann distribution (in orange).
Typical speeds
The mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
speed , most probable speed (mode
Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to:
Arts and entertainment
* '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine
* ''Mode'' magazine, a fictional fashion magazine which is ...
) , and root-mean-square speed can be obtained from properties of the Maxwell distribution.
This works well for nearly ideal, monatomic
In physics and chemistry, "monatomic" is a combination of the words "mono" and "atomic", and means "single atom". It is usually applied to gases: a monatomic gas is a gas in which atoms are not bound to each other. Examples at standard conditions ...
gases like helium
Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic table. ...
, but also for molecular gases like diatomic oxygen
Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as ...
. This is because despite the larger heat capacity
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity ...
(larger internal energy at the same temperature) due to their larger number of degrees of freedom, their translational kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
(and thus their speed) is unchanged.
* The most probable speed, , is the speed most likely to be possessed by any molecule (of the same mass ) in the system and corresponds to the maximum value or the mode
Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to:
Arts and entertainment
* '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine
* ''Mode'' magazine, a fictional fashion magazine which is ...
of . To find it, we calculate the derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
, set it to zero and solve for : with the solution: is the gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
and is molar mass of the substance, and thus may be calculated as a product of particle mass, , and Avogadro constant
The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining c ...
, : For diatomic nitrogen (N2, the primary component of air
The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing f ...
) at room temperature (), this gives
* The mean speed is the expected value of the speed distribution, setting :
* The mean square speed is the second-order raw moment of the speed distribution. The "root mean square speed" is the square root of the mean square speed, corresponding to the speed of a particle with median kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
, setting :
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
In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant vol ...
, is the number of degrees of freedom of the individual gas molecule. For the example above, diatomic nitrogen (approximating air
The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing f ...
) at , and
the true value for air can be approximated by using the average molar weight of air
The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing f ...
(), yielding at (corrections for variable humidity
Humidity is the concentration of water vapor present in the air. Water vapor, the gaseous state of water, is generally invisible to the human eye. Humidity indicates the likelihood for precipitation, dew, or fog to be present.
Humidity dep ...
are of the order of 0.1% to 0.6%).
The average relative velocity
where the three-dimensional velocity distribution is
The integral can easily be done by changing to coordinates and
Derivation and related distributions
Maxwell–Boltzmann statistics
The original derivation in 1860 by James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and li ...
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
Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of ther ...
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
In classical statistical mechanics, the ''H''-theorem, introduced by Ludwig Boltzmann in 1872, describes the tendency to decrease in the quantity ''H'' (defined below) in a nearly-ideal gas of molecules.
L. Boltzmann,Weitere Studien über das Wä ...
). He later (1877) derived the distribution again under the framework of statistical thermodynamics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
. The derivations in this section are along the lines of Boltzmann's 1877 derivation, starting with result known as Maxwell–Boltzmann statistics
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 ...
(from statistical thermodynamics). 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 proportional to the ratio of the energy of that state to the temperature of the system:
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 Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
.
The denominator in Equation () is a normalizing factor so that the ratios add up to unity — in other words it is a kind of partition function (for the single-particle system, not the usual partition function of the entire system).
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 ''p''2 is the square of the momentum vector . We may therefore rewrite Equation () as:
where ''Z'' is the partition function, corresponding to the denominator in Equation (). Here ''m'' is the molecular mass of the gas, ''T'' is the thermodynamic temperature and ''k'' is the Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
. This distribution of is proportional to the probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
''f''p for finding a molecule with these values of momentum components, so:
The normalizing constant
The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. The normalizing constant is used to reduce any probability function to a probability density function with total probability of one.
...
can be determined by recognizing that the probability of a molecule having ''some'' momentum must be 1.
Integrating the exponential in () over all ''p''''x'', ''p''''y'', and ''p''''z'' 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 (or equally for the velocities) can be obtained more fundamentally using the H-theorem
In classical statistical mechanics, the ''H''-theorem, introduced by Ludwig Boltzmann in 1872, describes the tendency to decrease in the quantity ''H'' (defined below) in a nearly-ideal gas of molecules.
L. Boltzmann,Weitere Studien über das Wä ...
at equilibrium within the Kinetic theory of gases framework.
Distribution for the energy
The energy distribution is found imposing
where is the infinitesimal phase-space volume of momenta corresponding to the energy interval .
Making use of the spherical symmetry of the energy-momentum dispersion relation , this can be expressed in terms of as
Using then () in (), and expressing everything in terms of the energy , we get
and finally
Since the energy is proportional to the sum of the squares of the three normally distributed momentum components, this energy distribution can be written equivalently as a gamma distribution, using a shape parameter, and a scale parameter, .
Using the equipartition theorem
In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. T ...
, given that the energy is evenly distributed among all three degrees of freedom in equilibrium, we can also split into a set of chi-squared distribution
In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squar ...
s, where the energy per degree of freedom, , is distributed as a chi-squared distribution with one degree of freedom,
Appendix N, page 434
At equilibrium, this distribution will hold true for any number of degrees of freedom. For example, if the particles are rigid mass dipoles of fixed dipole moment, they will have three translational degrees of freedom and two additional rotational degrees of freedom. The energy in each degree of freedom will be described according to the above chi-squared distribution with one degree of freedom, and the total energy will be distributed according to a chi-squared distribution with five degrees of freedom. This has implications in the theory of the specific heat
In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
of a gas.
Distribution for the velocity vector
Recognizing that the velocity probability density ''f''v is proportional to the momentum probability density function by
and using p = ''m''v we get
which is the Maxwell–Boltzmann velocity distribution. The probability of finding a particle with velocity in the infinitesimal element about velocity is
Like the momentum, this distribution is seen to be the product of three independent normally distributed variables , , and , but with variance .
It can also be seen that the Maxwell–Boltzmann velocity distribution for the vector velocity
is the product of the distributions for each of the three directions:
where the distribution for a single direction is
Each component of the velocity vector has a normal distribution with mean and standard deviation , so the vector has a 3-dimensional normal distribution, a particular kind of multivariate normal distribution
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One ...
, with mean and covariance , where is the identity matrix.
Distribution for the speed
The Maxwell–Boltzmann distribution for the speed follows immediately from the distribution of the velocity vector, above. Note that the speed is
and the volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form
:dV ...
in spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
where and are the spherical coordinate
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
angles of the velocity vector. Integration of the probability density function of the velocity over the solid angles yields an additional factor of .
The speed distribution with substitution of the speed for the sum of the squares of the vector components:
In ''n''-dimensional space
In ''n''-dimensional space, Maxwell–Boltzmann distribution becomes:
Speed distribution becomes:
The following integral result is useful:
where is the Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
. This result can be used to calculate the moments of speed distribution function:
which is the mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
speed itself .
which gives root-mean-square speed .
The derivative of speed distribution function:
This yields the most probable speed (mode
Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to:
Arts and entertainment
* '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine
* ''Mode'' magazine, a fictional fashion magazine which is ...
) .
See also
* Quantum Boltzmann equation
* Maxwell–Boltzmann statistics
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 ...
* Maxwell–Jüttner distribution
* Boltzmann distribution
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability th ...
* Rayleigh distribution
In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distribut ...
* Kinetic theory of gases
References
Further reading
* Physics for Scientists and Engineers – with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008,
* Thermodynamics, From Concepts to Applications (2nd Edition), A. Shavit, C. Gutfinger, CRC Press (Taylor and Francis Group, USA), 2009,
* Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971,
* Elements of Statistical Thermodynamics (2nd Edition), L.K. Nash, Principles of Chemistry, Addison-Wesley, 1974,
* Ward, CA & Fang, G 1999, 'Expression for predicting liquid evaporation flux: Statistical rate theory approach', Physical Review E, vol. 59, no. 1, pp. 429–40.
* Rahimi, P & Ward, CA 2005, 'Kinetics of Evaporation: Statistical Rate Theory Approach', International Journal of Thermodynamics, vol. 8, no. 9, pp. 1–14.
External links
"The Maxwell Speed Distribution"
from The Wolfram Demonstrations Project at Mathworld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Di ...
{{DEFAULTSORT:Maxwell-Boltzmann Distribution
Continuous distributions
Gases
Ludwig Boltzmann
James Clerk Maxwell
Normal distribution
Particle distributions