Bose–Einstein statistics
The theory of this behaviour was developed (1924–25) by Satyendra Nath Bose, who recognized that a collection of identical and indistinguishable particles can be distributed in this way.
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.
At low temperatures, bosons behave differently from fermions (which obey the Fermi–Dirac statistics) 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.
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.
[2][3] While presenting a lecture at the University of Dhaka (in what was then British India and is now Bangladesh) 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 (remarkably, this error resembled the famous blunder by d'Alembert known from his Croix ou Pile article[4][5]).
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"[6][7] and submitted it to the Philosophical Magazine.
Undaunted, he sent the manuscript to Albert Einstein requesting publication in the Zeitschrift für Physik.
Einstein immediately agreed, personally translated the article from English into German (Bose had earlier translated Einstein's article on the general theory of relativity from German to English), and saw to it that it was published.
Bose originally had a factor of 2 for the possible spin states, but Einstein changed it to polarization.
[9] 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 (classical, distinguishable) coins.
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 are particles with integer spin, named after Bose), which was demonstrated to exist by experiment in 1995.
In the microcanonical ensemble, one considers a system with fixed energy, volume, and number of particles.
Since particles are indistinguishable in the quantum mechanical context here, the number of ways for arranging
The total number of arrangements in an ensemble of bosons is simply the product of the binomial coefficients
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.
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.
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
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:
, whereas the Fermi gas is allowed to take both positive and negative values for the chemical potential.
This result applies for each single-particle level and thus forms the Bose–Einstein distribution for the entire state of the system.
These derivations are lengthy and only yield the above results in the asymptotic limit of a large number of particles.
[10] The fluctuations of the ground state in the condensed region are however markedly different in the canonical and grand-canonical ensembles.
A much simpler way to think of Bose–Einstein distribution function is to consider that n particles are denoted by identical balls and g shells are marked by g-1 line partitions.
