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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, probability measure, or frequency distribution of particles in a system over various possible states. The distribution is expressed in the form where is state energy (which varies from state to state), and (a constant of the distribution) is the product of Boltzmann's constant and thermodynamic temperature. In statistical mechanics, the Boltzmann distribution is a probability distribution that gives the probability that a system will be in a certain state as a function of that state’s energy and the temperature of the system. It is given as 〔McQuarrie, A. (2000) Statistical Mechanics, University Science Books, California〕 where ''pi'' is the probability of state i, ''εi'' the energy of state i, ''k'' the Boltzmann constant, ''T'' the temperature of the system and ''M'' is the number of states accessible to the system.〔Atkins, P. W. (2010) Quanta, W. H. Freeman and Company, New York〕 The sum is over all states accessible to the system of interest. The term system here has a very wide meaning; it can range from a single atom to a macroscopic system such as a natural gas storage tank. Because of this Boltzmann distribution can be used to solve a very wide variety of problems. The distribution shows that states with lower energy will always have a higher probability of being occupied than the states with higher energy.〔 The ''ratio'' of a Boltzmann distribution computed for two states is known as the Boltzmann factor and characteristically only depends on the states' energy difference. The Boltzmann distribution is named after Ludwig Boltzmann who first formulated it in 1868 during his studies of the statistical mechanics of gases in thermal equilibrium. The distribution was later investigated extensively, in its modern generic form, by Josiah Willard Gibbs in 1902. The Boltzmann distribution should not be confused with the Maxwell-Boltzmann distribution. The former gives the probability that a system will be in a certain state as a function of that state's energy. When applied to particles such as atoms or molecules, it gives the distribution of particles over energy states.〔 Maxwell-Boltzmann distribution is used to describe particle speeds in idealized gases. ==The distribution== The Boltzmann distribution is a probability distribution that gives the probability of a certain state as a function of that state’s energy and temperature of the system to which the distribution is applied.〔 It is given as where ''pi'' is the probability of state i, ''εi'' the energy of state i, ''k'' the Boltzmann constant, ''T'' the temperature of the system and ''M'' is the number of all states accessible to the system.〔〔 The sum is over all states accessible to the system of interest. The right hand side denominator of the equation above is also known as the canonical partition function, commonly denoted by Q (or by some authors by Z). Therefore the Boltzmann distribution can also be written as The partition function can be calculated if we know the energies of the levels accessible to the system of interest. For atoms the partition function values can be found at the NIST Atomic Spectra Database.〔http://physics.nist.gov/PhysRefData/ASD/levels_form.html〕 The distribution shows that states with lower energy will always have a higher probability of being occupied than the states with higher energy. It can also give us the quantitative relationship between the probabilities of the two states being occupied. The ratio of probabilities for states i and j is given as where ''pi'' is the probability of state i, ''pj'' the probability of state j, and ''εi'' and ''εi'' are the energies of states i and j, respectively. The Boltzmann distribution is often used to describe the distribution of particles, such as atoms or molecules, over energy states accessible to them. If we have a system consisted of many particles, the probability of a particle being in state i is practically the probability that, if we pick a random particle from that system and check what state it is in, we will find it is in state i. This probability is equal to the number of particles in state i divided by the total number of particles in the system, that is the fraction of particles that occupy state i. where ''Ni'' is the number of particles in state i and ''N'' is the total number of particles in the system. We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. So the equation that gives the fraction of particles in state i as a function of the energy of that state is 〔 This equation is of great importance to spectroscopy. In spectroscopy we observe a spectral line if atoms or molecules that we are interested in go from one state to another.〔〔Atkins, P. W.; de Paula J. (2009) Physical Chemistry, 9th edition, Oxford University Press, Oxford, UK〕 In order for this to be possible, there must be some particles in the first state to undergo the transition. We may find that this condition is fulfilled by finding the fraction of particles in the first state. If it is negligible, the transition is very likely not be observed at the temperature for which the calculation was done. In general, a larger fraction of molecules in the first state means a higher number of transitions to the second state.〔Skoog, D. A.; Holler, F. J.; Crouch, S. R. (2006) Principles of Instrumental Analysis, Brooks/Cole, Boston, MA〕 This gives a stronger spectral line. However, there are other factors that influence the intensity of a spectral line, such as whether it is caused by an allowed or a forbidden transition. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Boltzmann distribution」の詳細全文を読む スポンサード リンク
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