In physics, the Boltzmann distribution predicts the distribution function for the fractional number of particles N_{i} / N occupying a set of states i which each has energy E_{i}: In mathematics, the support of a realvalued 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 realvalued random variable, X. For every real number x, the cdf is given by where the righthand 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 nonzero 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 realvalued random variable. ...
In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a realvalued 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 momentgenerating 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, (phÃºsis), nature and (phusikÃ©), 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. ...
where k_{B} is the Boltzmann constant, T is temperature (assumed to be a sharply welldefined quantity), g_{i} is the degeneracy, or number of states having energy E_{i}, N is the total number of particles: Ludwig Boltzmann The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...
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...
Alternatively, for a single system at a welldefined 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 MaxwellBoltzmann statistics. (See that article for a derivation of the Boltzmann distribution.) It has been suggested that the section Physical applications of the MaxwellBoltzmann distribution from the article MaxwellBoltzmann 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(βE_{i}) or exp(E_{i}/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  ,
then the distribution correctly gives the MaxwellBoltzmann 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 . 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 â€“ 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: 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 MaxwellBoltzmann statistics. It has been suggested that the section Physical applications of the MaxwellBoltzmann distribution from the article MaxwellBoltzmann 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 BoseEinstein or FermiDirac 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, BoseEinstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ...
In statistical mechanics, FermiDirac statistics determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. ...
Derivation
See MaxwellBoltzmann statistics. It has been suggested that the section Physical applications of the MaxwellBoltzmann distribution from the article MaxwellBoltzmann distribution be merged into this article or section. ...
