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Boltzmann's distribution is an exponential distribution. Boltzmann factor (vertical axis) as a function of temperature T for several energy differences ε i − ε j.. In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution [1]) is a probability distribution or probability measure that gives the probability that a system will be in a certain ...
The typical problem begins with a system for which the Hamiltonian is known, it is at a given temperature and it follows the Boltzmann statistics. To obtain the mean value of some macroscopic variable, say A, the general approach is to compute, over all the phase space, PS for simplicity, the mean value of A using the Boltzmann distribution:
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 (MD) simulation in which 900 hard sphere particles are constrained to move in a rectangle.
As an example: the partition function for the isothermal-isobaric ensemble, the generalized Boltzmann distribution, divides up probabilities based on particle number, pressure, and temperature. The energy is replaced by the characteristic potential of that ensemble, the Gibbs Free Energy .
Using the results from either Maxwell–Boltzmann statistics, Bose–Einstein statistics or Fermi–Dirac statistics we use the Thomas–Fermi approximation (gas in a box) and go to the limit of a very large trap, and express the degeneracy of the energy states as a differential, and summations over states as integrals.
Maxwell–Boltzmann statistics is used to derive the Maxwell–Boltzmann distribution of an ideal gas. However, it can also be used to extend that distribution to particles with a different energy–momentum relation, such as relativistic particles (resulting in Maxwell–Jüttner distribution), and to other than three-dimensional spaces.
The beta negative binomial distribution; The Boltzmann distribution, a discrete distribution important in statistical physics which describes the probabilities of the various discrete energy levels of a system in thermal equilibrium. It has a continuous analogue. Special cases include: The Gibbs distribution; The Maxwell–Boltzmann distribution
This link is provided by Boltzmann's fundamental assumption written as S = k B ln Ω , {\displaystyle S=k_{\rm {B}}\ln \Omega ,} where k B is the Boltzmann constant , S is the classical thermodynamic entropy, and Ω is the number of microstates.