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Bose–Einstein statistics
In statistical mechanics, Bose-Einstein statistics (or more colloquially B-E statistics) determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. Fermi-Dirac and Bose-Einstein statistics apply when quantum effects have to be taken into account and the particles are considered "indistinguishable". The quantum effects appear if the concentration of particles (N/V) ≥ n_{q} (where n_{q} is the quantum concentration). The quantum concentration is when the interparticle distance is equal to the thermal de Broglie wavelength i.e. when the wavefunctions of the particles are touching but not overlapping. As the quantum concentration depends on temperature; high temperatures will put most systems in the classical limit unless they have a very high density e.g. a White dwarf. Fermi-Dirac statistics apply to fermions (particles that obey the Pauli exclusion principle), Bose-Einstein statistics apply to bosons. Both Fermi-Dirac and Bose-Einstein become Maxwell-Boltzmann statistics at high temperatures or low concentrations. Maxwell-Boltzmann statistics are often described as the statistics of "distinguishable" classical particles. In other words the configuration of particle A in state 1 and particle B in state 2 is different from the case where particle B is in state 1 and particle A is in state 2. When this idea is carried out fully, it yields the proper (Boltzmann) distribution of particles in the energy states, but yields non-physical results for the entropy, as embodied in Gibbs paradox. These problems disappear when it is realized that all particles are in fact indistinguishable. Both of these distributions approach the Maxwell-Boltzmann distribution in the limit of high temperature and low density, without the need for any ad hoc assumptions. Maxwell-Boltzmann statistics are particularly useful for studying gases. Fermi-Dirac statistics are most often used for the study of electrons in solids. As such, they form the basis of semiconductor device theory and electronics. 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 explains why, at low temperatures, bosons can behave very differently from 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: with and where:
This reduces to M-B statistics for energies ( ε_{i} − μ ) >> kT. Additional recommended knowledge
HistoryIn the early 1920s Satyendra Nath Bose, a professor of University of Dhaka was intrigued by Einstein's theory of light waves being made of particles called photons. Bose was interested in deriving Planck's radiation formula, which Planck obtained largely by guessing. In 1900 Max Planck had derived his formula by manipulating the math to fit the empirical evidence. Using the particle picture of Einstein, Bose was able to derive the radiation formula by systematically developing a statistics of massless particles without the constraint of particle number conservation. Bose derived Planck's Law of Radiation by proposing different states for the photon. Instead of statistical independence of particles, Bose put particles into cells and described statistical independence of cells of phase space. Such systems allow two polarization states, and exhibit totally symmetric wavefunctions. He developed a statistical law governing the behaviour pattern of photons quite successfully. However, he was not able to publish his work; no journals in Europe would accept his paper, being unable to understand it. Bose sent his paper to Einstein, who saw the significance of it and used his influence to get it published. A derivation of the Bose–Einstein distributionSuppose we have a number of energy levels, labelled by index , each level having energy and containing a total of particles. Suppose each level contains 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 associated with level is called the "degeneracy" of that energy level. Any number of bosons can occupy the same sublevel. Let be the number of ways of distributing particles among the sublevels of an energy level. There is only one way of distributing particles with one sublevel, therefore . It is easy to see that there are ways of distributing particles in two sublevels which we will write as: With a little thought (See Notes below) it can be seen that the number of ways of distributing particles in three sublevels is so that where we have used the following theorem involving binomial coefficients: Continuing this process, we can see that is just a binomial coefficient (See Notes below) The number of ways that a set of occupation numbers can be realized is the product of the ways that each individual energy level can be populated: where the approximation assumes that . Following the same procedure used in deriving the Maxwell–Boltzmann statistics, we wish to find the set of for which is maximised, subject to the constraint that there be a fixed number of particles, and a fixed energy. The maxima of and occur at the value of and, since it is easier to accomplish mathematically, we will maximise the latter function instead. We constrain our solution using Lagrange multipliers forming the function: Using the approximation and using Stirling's approximation for the factorials gives Taking the derivative with respect to , and setting the result to zero and solving for , yields the Bose–Einstein population numbers: It can be shown thermodynamically that , where is Boltzmann's constant and is the temperature. It can also be shown that , where is the chemical potential, so that finally: Note that the above formula is sometimes written: where is the absolute activity. NotesThe purpose of these notes is to clarify some aspects of the derivation of the Bose-Einstein (B-E) distribution for beginners. The enumeration of cases (or ways) in the B-E distribution can be recast as follows. Consider a game of dice throwing in which there are dice, with each dice taking values in the set , for . The constraints of the game is that the value of a dice , denoted by , has to be greater or equal to the value of dice , denoted by , in the previous throw, i.e., . Thus a valid sequence of dice throws can be described by an -tuple , such that . Let denote the set of these valid -tuples:
Then the quantity (defined above as the number of ways to distribute particles among the sublevels of an energy level) is the cardinality of , i.e., the number of elements (or valid -tuples) in . Thus the problem of finding and expression for becomes the problem of counting the elements in . Example n=4, g=3:
Subset is obtained by fixing all indices to , except for the last index, , which is incremented from to . Subset is obtained by fixing , and increment from to ; due to the constraint on the indices in , the index must automatically take values in . The construction of subsets and follows in the same manner. Each element of can be thought of as a multiset of cardinality ; the elements of such multiset are taken from the set of cardinality , and the number of such multisets is the multiset coefficient More generally, each element of is a multiset of cardinality (number of dice) with elements taken from the set of cardinality (number of possible values of each dice), and the number of such multisets, i.e., is the multiset coefficient
which is exactly the same as the formula for , as derived above with the aid of a theorem involving binomial coefficients, namely
To understand the decomposition
or for example, and To this end, let's rearrange the elements of as follows Clearly, the subset of is the same as the set By deleting the index (shown in red with double underline) in the subset of , one obtain the set In other words, there is a one-to-one correspondence between the subset of and the set . We write Similarly, it is easy to see that
Thus we can write or more generally,
and since the sets are non-intersecting, we thus have
with the convention that
Continue the process, we arrive at the following formula Using the convention (7)_{2} above, we obtain the formula
keeping in mind that for and being constants, we have
It can then be verified that (8) and (2) give the same result for , , , etc. ReferencesAnnett, James F., "Superconductivity, Superfluids and Condensates", Oxford University Press, 2004, New York. Carter, Ashley H., "Classical ans Statistical Thermodynamics", Prentice-Hall, Inc., 2001, New Jersey. Griffiths, David J., "Introduction to Quantum Mechanics", 2nd ed. Pearson Education, Inc., 2005. See also
Categories: Statistical mechanics | Particle statistics |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Bose–Einstein_statistics". A list of authors is available in Wikipedia. |