Bose-Einstein statistics In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. Bose-Einstein (or B-E) statistics are closely related to Maxwell-Boltzmann statistics (M-B) and Fermi-Dirac statistics (F-D). While F-D statistics holds for fermions, M-B statistics holds for classical particles, i.e. identical but distinguishable particles, and represents the classical or high-temperature limit of both F-D and B-E statistics. (M-B, B-E, and F-D statistics are all derived from the Boltzmann factor probability weight applied to the problem of classical particles and discrete energy quanta with boson/fermion behavior, respectively.) Bosons, unlike fermions, are not subject to the Pauli exclusion principle: an unlimited number of particles may occupy the same state at the same time. This explain why, at low temperatures, bosons can behave very differently than fermions; all the particles will tend to congregate together at the same lowest-energy state, forming what is known as a Bose-Einstein condensate. B-E statistics was introduced for photons in 1920 by Bose and generalized to atoms by Einstein in 1924. The expected number of particles in an energy state i  for B-E statistics is: $n\_i\; =\; \backslash frac\; )\; -\; 1\}$ where: ni  is the number of particles in state i gi  is the degeneracy of state i εi  is the energy of the i-th state μ is the chemical potential k is Boltzmann's constant T is absolute temperature exp is the exponential function Derivation of the Bose-Einstein distribution Suppose we have a number of energy levels, labelled by index i , each level having energy εi  and containing a total of ni  particles. Suppose each level contains gi  distinct sublevels, all of which have the same energy, and which are distinguishable. For example, two particles may have different momenta, in which case they are distinguishable from each other, yet they can still have the same energy. The value of gi  associated with level i is called the "degeneracy" of that energy level. Any number of bosons can occupy the same sublevel. Let w(n,g) be the number of ways of distributing n particles among the g sublevels of an energy level. There is only one way of distributing n particles with one sublevel, therefore w(n,1)=1. Its easy to see that there are n+1 ways of distributing n particles in two sublevels which we will write as: $w(n,2)=\backslash frac$ With a little thought it can be seen that the number of ways of distributing n particles in 3 sublevels is w(n,3)=w(n,2)+w(n-1,2)+...+w(0,2) so that $w(n,3)=\backslash sum\_^n\; w(n-k,2)\; =\; \backslash sum\_^n\backslash frac=\backslash frac$ where we have used the following theorem involving binomial coefficients: $$ \sum_^n\frac=\frac Continuing this process, we can see that w(n,g) is just a binomial coefficient $w(n,g)=\backslash frac$ The number of ways that a set of occupation numbers ni  can be realized is the product of the ways that each individual energy level can be populated: $W\; =\; \backslash prod\_i\; w(n\_i,g\_i)\; =\; \backslash prod\_i\; \backslash frac$ Following the same procedure used in deriving the Maxwell-Boltzmann distribution, we wish to find the set of ni  for which W  is maximised, subject to the constraint that there be a fixed number of particles, and a fixed energy. We constrain our solution using Lagrange multipliers forming the function: $f(n\_i)=\backslash ln(W)-\backslash alpha(N-\backslash sum\; n\_i)+\backslash beta(E-\backslash sum\; n\_i\; \backslash epsilon\_i)$ Taking the derivative with respect to ni  and setting the result to zero and solving for ni  yields the Bose-Einstein population numbers. (Note: we have assumed that gi >>1) $n\_i\; =\; \backslash frac-1\}$ It can be shown thermodynamically that β=1/kT  where k  is Boltzmann's constant and T  is the temperature. The term containing α is variously written: $\backslash left.\backslash right.\; e^\backslash alpha\; =\; e^\; =\; 1/z$ where μ is the chemical potential and z  is the fugacity. See Also Planck's_law_of_black_body_radiation Maxwell Boltzmann statistics (derivation) Fermi-Dirac statistics parastatistics. FAVORITE ARTICLES: History of the World    History of the United States    History of Europe    Ancient History    History    Military History    World War I    Bombing of Dresden    California    US Constitution    Frank Sinatra    Boston    Marbury v. Madison This article is licensed under the GNU Free Documentation License at http://www.gnu.org/copyleft/fdl.html You may copy and modify it as long as the entire work (including additions) remains under this license. You must provide a link to http://www.gnu.org/copyleft/fdl.html To view or edit this article at Wikipedia go to http://www.wikipedia.org/wiki/Bose-Einstein_statistics