Bose–Einstein statistics

In quantum statistics, Bose–Einstein statistics describes one of two possible ways in which a collection of non-interacting identical particles may occupy a set of available discrete energy states at thermodynamic equilibrium. The aggregation of particles in the same state, which is a characteristic of particles obeying Bose–Einstein statistics, accounts for the cohesive streaming of laser light and the frictionless creeping of superfluid helium. The theory of this behaviour was developed by Satyendra Nath Bose, who recognized that a collection of identical and indistinguishable particles could be distributed in this way. The idea was later adopted and extended by Albert Einstein in collaboration with Bose.
Bose–Einstein statistics apply only to particles that do not follow the Pauli exclusion principle restrictions. Particles that follow Bose-Einstein statistics are called bosons, which have integer values of spin. In contrast, particles that follow Fermi-Dirac statistics are called fermions and have half-integer spins.

Bose–Einstein distribution

At low temperatures, bosons behave differently from fermions in a way that an unlimited number of them can "condense" into the same energy state. This apparently unusual property also gives rise to the special state of matter - the Bose–Einstein condensate. Fermi–Dirac and Bose–Einstein statistics apply when quantum effects are important and the particles are "indistinguishable". Quantum effects appear if the concentration of particles satisfies
where is the number of particles, is the volume, and is the quantum concentration, for which the interparticle distance is equal to the thermal de Broglie wavelength, so that the wavefunctions of the particles are barely overlapping.
Fermi–Dirac statistics applies to fermions, and Bose–Einstein statistics applies to bosons. As the quantum concentration depends on temperature, most systems at high temperatures obey the classical limit, unless they also have a very high density, as for a white dwarf. Both Fermi–Dirac and Bose–Einstein become Maxwell–Boltzmann statistics at high temperature or at low concentration.
Bose–Einstein statistics was introduced for photons in 1924 by Bose and generalized to atoms by Einstein in 1924–25.
The expected number of particles in an energy state for Bose–Einstein statistics is:
with and where is the occupation number in state, is the degeneracy of energy level, is the energy of the th state, μ is the chemical potential, is the Boltzmann constant, and is the absolute temperature.
The variance of this distribution is calculated directly from the expression above for the average number.
For comparison, the average number of fermions with energy given by Fermi–Dirac particle-energy distribution has a similar form:
As mentioned above, both the Bose–Einstein distribution and the Fermi–Dirac distribution approaches the Maxwell–Boltzmann distribution in the limit of high temperature and low particle density, without the need for any ad hoc assumptions:

In the limit of low particle density,, therefore or equivalently. In that case,, which is the result from Maxwell–Boltzmann statistics.
In the limit of high temperature, the particles are distributed over a large range of energy values, therefore the occupancy on each state is again very small,. This again reduces to Maxwell–Boltzmann statistics.

In addition to reducing to the Maxwell–Boltzmann distribution in the limit of high and low density, Bose–Einstein statistics also reduces to Rayleigh–Jeans law distribution for low energy states with, namely

History

In 1900, Max Planck derived the Planck law to explain blackbody radiation. For this purpose, he introduced the concept of quanta of energy.
Władysław Natanson in 1911 concluded that Planck's law requires indistinguishability of "units of energy", although he did not frame this in terms of Einstein's light quanta.
While presenting a lecture at the University of Dhaka on the theory of radiation and the ultraviolet catastrophe, Satyendra Nath Bose intended to show his students that the contemporary theory was inadequate, because it predicted results not in accordance with experimental results. During this lecture, Bose committed an error in applying the theory, which unexpectedly gave a prediction that agreed with the experiment. The error was a simple mistake – similar to arguing that flipping two fair coins will produce two heads one-third of the time – that would appear obviously wrong to anyone with a basic understanding of statistics. However, the results it predicted agreed with experiment, and Bose realized it might not be a mistake after all. For the first time, he took the position that the Maxwell–Boltzmann distribution would not be true for all microscopic particles at all scales. Thus, he studied the probability of finding particles in various states in phase space, where each state is a little patch having phase volume of h³, and the position and momentum of the particles are not kept particularly separate but are considered as one variable.
Bose adapted this lecture into a short article called "Planck's law and the hypothesis of light quanta" and submitted it to the Philosophical Magazine. However, the referee's report was negative, and the paper was rejected. Undaunted, he sent the manuscript to Albert Einstein requesting publication in the italic=yes. Einstein immediately agreed, personally translated the article from English into German, and saw to it that it was published. Bose's theory achieved respect when Einstein sent his own paper in support of Bose's to italic=yes, asking that they be published together. The paper came out in 1924.
The reason Bose produced accurate results was that since photons are indistinguishable from each other, one cannot treat any two photons having equal quantum numbers as being two distinct identifiable photons. Bose originally had a factor of 2 for the possible spin states, but Einstein changed it to polarization. By analogy, if in an alternate universe coins were to behave like photons and other bosons, the probability of producing two heads would indeed be one-third, and so is the probability of getting a head and a tail which equals one-half for the conventional coins. Bose's "error" leads to what is now called Bose–Einstein statistics.
Bose and Einstein extended the idea to atoms and this led to the prediction of the existence of phenomena which became known as Bose–Einstein condensate, a dense collection of bosons, which was demonstrated to exist by experiment in 1995.

Derivation

Derivation from the microcanonical ensemble

In the microcanonical ensemble, one considers a system with fixed energy, volume, and number of particles. We take a system composed of identical bosons, of which have energy and are distributed over levels or states with the same energy, i.e. is the degeneracy associated with energy. The total energy of the system is. Calculation of the number of arrangements of particles distributed among states is a problem of combinatorics. Since particles are indistinguishable in the quantum mechanical context here, the number of ways for arranging particles in boxes, where each box is capable of containing an infinite number of bosons, would be :
where is the k-combination of a set with m elements. The total number of arrangements in an ensemble of bosons is simply the product of the binomial coefficients above over all the energy levels, i.e.
which for very large and can be simplified using Stirling's approximation to
The entropy of the system can then be expressed as
The three constraints we can impose on the system can be expressed as
,
, and
.
This final constraint can be expanded to be in terms of :
Now we can write
for which to be true, it must be the case that for any i
By solving for and simplifying we obtain
which for sufficiently large reduces to
which is the form of the Bose-Einstein distribution. Note that this form holds even for a system of interacting bosons.

Derivation from the grand canonical ensemble

The Bose–Einstein distribution, which applies only to a quantum system of non-interacting bosons, is naturally derived from the grand canonical ensemble without any approximations. In this ensemble, the system is able to exchange energy and exchange particles with a reservoir.
Due to the non-interacting quality, each available single-particle level forms a separate thermodynamic system in contact with the reservoir. That is, the number of particles within the overall system that occupy a given single particle state form a sub-ensemble that is also grand canonical ensemble; hence, it may be analysed through the construction of a grand partition function.
Every single-particle state is of a fixed energy,. As the sub-ensemble associated with a single-particle state varies by the number of particles only, it is clear that the total energy of the sub-ensemble is also directly proportional to the number of particles in the single-particle state; where is the number of particles, the total energy of the sub-ensemble will then be. Beginning with the standard expression for a grand partition function and replacing with, the grand partition function takes the form
This formula applies to fermionic systems as well as bosonic systems. Fermi–Dirac statistics arises when considering the effect of the Pauli exclusion principle: whilst the number of fermions occupying the same single-particle state can only be either 1 or 0, the number of bosons occupying a single particle state may be any integer. Thus, the grand partition function for bosons can be considered a geometric series and may be evaluated as such:
Note that the geometric series is convergent only if, including the case where. This implies that the chemical potential for the Bose gas must be negative, i.e.,, whereas the Fermi gas is allowed to take both positive and negative values for the chemical potential.
The average particle number for that single-particle substate is given by
This result applies for each single-particle level and thus forms the Bose–Einstein distribution for the entire state of the system.
The variance in particle number,, is:
As a result, for highly occupied states the standard deviation of the particle number of an energy level is very large, slightly larger than the particle number itself:. This large uncertainty is due to the fact that the probability distribution for the number of bosons in a given energy level is a geometric distribution; somewhat counterintuitively, the most probable value for N is always 0.