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Encyclopedia > Boltzmann distribution
Boltzmann distribution
Probability mass function
Cumulative distribution function
Parameters
Support
Probability mass function (pmf)
Cumulative distribution function (cdf)
Mean
Median
Mode
Variance
Skewness
Excess Kurtosis
Entropy
mgf
Char. func.

In physics, the Boltzmann distribution predicts the distribution function for the fractional number of particles Ni / N occupying a set of states i which each has energy Ei: In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In probability theory, a probability mass function (abbreviated pmf) gives the probability that a discrete random variable is exactly equal to some value. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability theory and statistics, the variance of a random variable (or equivalently, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... Example of the experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ... Entropy of a Bernoulli trial as a function of success probability, often called the binary entropy function. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space and time. ... In physics, a particles distribution function is a function of seven variables, , which gives the number of particles per unit volume in phase space. ...

{{N_i}over{N}} = {{g_i e^{-E_i/k_BT}}over{Z(T)}}

where kB is the Boltzmann constant, T is temperature (assumed to be a sharply well-defined quantity), gi is the degeneracy, or number of states having energy Ei, N is the total number of particles: Ludwig Boltzmann The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...

N=sum_i N_i,

and Z(T) is called the partition function, which can be seen to be equal to In number theory, see Partition function (number theory) In statistical mechanics, see Partition function (statistical mechanics) In quantum field theory, see Partition function (quantum field theory) In game theory, see Partition function (game theory) This is a disambiguation page — a navigational aid which lists other pages that might otherwise...

Z(T)=sum_i g_i e^{-E_i/k_BT}

Alternatively, for a single system at a well-defined temperature, it gives the probability that the system is in the specified state. The Boltzmann distribution applies only to particles at a high enough temperature and low enough density that quantum effects can be ignored, and the particles are obeying Maxwell-Boltzmann statistics. (See that article for a derivation of the Boltzmann distribution.) It has been suggested that the section Physical applications of the Maxwell-Boltzmann distribution from the article Maxwell-Boltzmann distribution be merged into this article or section. ...


The Boltzmann distribution is often expressed in terms of β=1/kT where β is referred to as thermodynamic beta. The term exp(-βEi) or exp(-Ei/kT), which gives the (unnormalised) relative probability of a state, is called the Boltzmann factor and appears often in the study of physics and chemistry. In statistical mechanics, thermodynamic beta is defined as the inverse of the product of the temperature of a macroscopic system and the Boltzmann constant: Thermodynamic beta serves as a theoretical connection between the microscopic picture of a physical system, and the macroscopic picture of the system. ... In physics, the Boltzmann factor is a weighting factor determining the relative probability of a system in thermodynamic equilibrium at a temperature T being in a state with energy E: (kB is Boltzmanns constant. ...


When the energy is simply the kinetic energy of the particle

E_i = {begin{matrix} frac{1}{2} end{matrix}} mv^{2},

then the distribution correctly gives the Maxwell-Boltzmann distribution of gas molecule speeds, previously predicted by Maxwell in 1859. The Boltzmann distribution is, however, much more general. For example, it also predicts the variation of the particle density in a gravitational field with height, if E_i = {begin{matrix} frac{1}{2} end{matrix}} mv^{2} + mgh. In fact the distribution applies whenever quantum considerations can be ignored. The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematical physicist, born in Edinburgh. ... 1859 (MDCCCLIX) is a common year starting on Saturday of the Gregorian calendar (or a common year starting on Monday of the Julian calendar). ...


In some cases, a continuum approximation can be used. If there are g(E)dE states with energy E to E+dE, then the Boltzmann distribution predicts a probability distribution for the energy:

p(E)dE = {g(E) exp({-beta E})over {int g(E') exp {(-beta E')}}dE'} dE

g(E) is then called the density of states if the energy spectrum is continuous. Density of states (DOS) is a property in statistical and condensed matter physics that quantifies how closely packed energy levels are in some physical system. ...


Classical particles with this energy distribution are said to obey Maxwell-Boltzmann statistics. It has been suggested that the section Physical applications of the Maxwell-Boltzmann distribution from the article Maxwell-Boltzmann distribution be merged into this article or section. ...


In the classical limit, i.e. at large values of E/kT or at small density of states - when wave functions of particles practically do not overlap, both the Bose-Einstein or Fermi-Dirac distribution become the Boltzmann distribution. Density of states (DOS) is a property in statistical and condensed matter physics that quantifies how closely packed energy levels are in some physical system. ... For other topics related to Einstein see Einstein (disambig) In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ... In statistical mechanics, Fermi-Dirac statistics determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. ...


Derivation

See Maxwell-Boltzmann statistics. It has been suggested that the section Physical applications of the Maxwell-Boltzmann distribution from the article Maxwell-Boltzmann distribution be merged into this article or section. ...

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Univariate Multivariate
Discrete: BernoullibinomialBoltzmanncompound Poissondegenerate • Gauss-Kuzmin • geometrichypergeometriclogarithmicnegative binomialparabolic fractalPoissonRademacherSkellamuniform • Yule-Simon • zetaZipf • Zipf-Mandelbrot Ewensmultinomial
Continuous: BetaBeta primeCauchychi-squareDirac delta functionErlangexponentialexponential powerFfading • Fisher's z • Fisher-Tippett • Gammageneralized extreme valuegeneralized hyperbolicgeneralized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square • inverse gaussianinverse gammaKumaraswamyLandauLaplaceLévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speednormal (Gaussian)ParetoPearsonpolarraised cosineRayleigh • relativistic Breit-Wigner • Rice • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniformVoigtvon MisesWeibullWigner semicircle DirichletKentmatrix normalmultivariate normal • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous: Cantorconditionalexponential family • infinitely divisible • location-scale family • marginalmaximum entropy • phase-type • posteriorprior • quasi • samplingsingular

  Results from FactBites:
 
Ludwig Boltzmann (765 words)
Boltzmann was awarded a doctorate from the University of Vienna in 1866 for a thesis on the kinetic theory of gases supervised by Josef Stefan.
Boltzmann, at least half jokingly, used to say that the reason he moved around so much was that he was born during the dying hours of a Mardi Gras ball.
Boltzmann obtained the Maxwell-Boltzmann distribution in 1871, namely the average energy of motion of a molecule is the same for each direction.
Distribution functions for identical particles (249 words)
The distribution function f(E) is the probability that a particle is in energy state E. The distribution function is a generalization of the ideas of discrete probability to the case where energy can be treated as a continuous variable.
The term A in the denominator of each distribution is a normalization term which may change with temperature.
The Maxwell-Boltzmann distribution is the classical distribution function for distribution of an amount of energy between identical but distinguishable particles.
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