In
quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting, indistinguishable
particles
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from s ...
may occupy a set of available discrete
energy states at
thermodynamic equilibrium
Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In the ...
. 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
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The firs ...
and the frictionless creeping of
superfluid helium. The theory of this behaviour was developed (1924–25) by
Satyendra Nath Bose
Satyendra Nath Bose (; 1 January 1894 – 4 February 1974) was a Bengali mathematician and physicist specializing in theoretical physics. He is best known for his work on quantum mechanics in the early 1920s, in developing the foundation for ...
, who recognized that a collection of identical and indistinguishable particles can be distributed in this way. The idea was later adopted and extended by
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
in collaboration with Bose.
The Bose–Einstein statistics applies only to the particles not limited to single occupancy of the same state – that is, particles that do not obey the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
restrictions. Such particles have integer values of
spin and are named
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s. Particles with half-integer spins are called
fermions
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
and obey
Fermi-Dirac statistics.
Bose–Einstein distribution
At low temperatures, bosons behave differently from
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s (which obey the
Fermi–Dirac statistics
Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac di ...
) 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
In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67&n ...
. Fermi–Dirac and Bose–Einstein statistics apply when
quantum effects
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
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
In physics, the thermal de Broglie wavelength (\lambda_, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. We can take the average interparticle spacing in ...
, so that the
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
s of the particles are barely overlapping.
Fermi–Dirac statistics applies to fermions (particles that obey the
Pauli exclusion principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
), and Bose–Einstein statistics applies to
bosons
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer ...
. As the quantum concentration depends on temperature, most systems at high temperatures obey the classical (Maxwell–Boltzmann) limit, unless they also have a very high density, as for a
white dwarf
A white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's faint luminosity comes fro ...
. Both Fermi–Dirac and Bose–Einstein become
Maxwell–Boltzmann statistics
In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of Classical physics, classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the ...
at high temperature or at low concentration.
B–E statistics was introduced for
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
s in 1924 by
Bose
Bose may refer to:
* Bose (crater), a lunar crater
* ''Bose'' (film), a 2004 Indian Tamil film starring Srikanth and Sneha
* Bose (surname), a surname (and list of people with the name)
* Bose, Italy, a ''frazioni'' in Magnano, Province of Biella ...
and generalized to atoms by
Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
in 1924–25.
The expected number of particles in an energy state for B–E statistics is:
with and where is the occupation number (the number of particles) in state ,
is the
degeneracy of energy level , is the
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
of the -th state, ''μ'' is the
chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
, is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
, and is
absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
.
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 physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.
It was first defined and use ...
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 (especially the high energy ones with
) is again very small,
. This again reduces to Maxwell–Boltzmann statistics.
In addition to reducing to the
Maxwell–Boltzmann distribution
In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.
It was first defined and use ...
in the limit of high
and low density, B–E statistics also reduces to
Rayleigh–Jeans law
In physics, the Rayleigh–Jeans law is an approximation to the spectral radiance of electromagnetic radiation as a function of wavelength from a black body at a given temperature through classical arguments. For wavelength λ, it is:
B_ (T) = \ ...
distribution for low energy states with
, namely
History
Władysław Natanson
Władysław Natanson (1864–1937) was a Polish physicist.
Life
Natanson was head of Theoretical Physics at Kraków University from 1899 to 1935. 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
The University of Dhaka (also known as Dhaka University, or DU) is a public research university located in Dhaka, Bangladesh. It is the oldest university in Bangladesh. The university opened its doors to students on July 1st 1921. Currently i ...
(in what was then
British India
The provinces of India, earlier presidencies of British India and still earlier, presidency towns, were the administrative divisions of British governance on the Indian subcontinent. Collectively, they have been called British India. In one ...
and is now
Bangladesh
Bangladesh (}, ), officially the People's Republic of Bangladesh, is a country in South Asia. It is the eighth-most populous country in the world, with a population exceeding 165 million people in an area of . Bangladesh is among the mos ...
) on the theory of radiation and the
ultraviolet catastrophe
The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of energy ...
,
Satyendra Nath Bose
Satyendra Nath Bose (; 1 January 1894 – 4 February 1974) was a Bengali mathematician and physicist specializing in theoretical physics. He is best known for his work on quantum mechanics in the early 1920s, in developing the foundation for ...
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
Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the '' Encyclopé ...
known from his ''Croix ou Pile'' article). 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 ''h''
3, 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
The ''Philosophical Magazine'' is one of the oldest scientific journals published in English. It was established by Alexander Tilloch in 1798;John Burnett"Tilloch, Alexander (1759–1825)" Oxford Dictionary of National Biography, Oxford Univer ...
''. However, the referee's report was negative, and the paper was rejected. Undaunted, he sent the manuscript to Albert Einstein requesting publication in the . 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's theory achieved respect when Einstein sent his own paper in support of Bose's to , 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 (e.g., polarization and momentum vector) as being two distinct identifiable photons. 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'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 are particles with integer spin, named after Bose), which was demonstrated to exist by experiment in 1995.
Derivation
Derivation from the microcanonical ensemble
In the
microcanonical ensemble
In statistical mechanics, the microcanonical ensemble is a statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assumed to be isolated in the sense that it canno ...
, 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
of total energy
. Calculation of the number of arrangements of
particles distributed among
states is a problem of
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
. Since particles are indistinguishable in the quantum mechanical context here, the number of ways for arranging
particles in
boxes (for the
th energy level) would be (see image).
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.
The maximum number of arrangements determining the corresponding occupation number
is obtained by maximizing the
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
, or equivalently, setting
and taking the subsidiary conditions
into account (as
Lagrange multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied e ...
s).
The result for
,
,
is the Bose–Einstein distribution.
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
In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium ...
without any approximations.
In this ensemble, the system is able to exchange energy and exchange particles with a reservoir (temperature ''T'' and chemical potential ''µ'' fixed by the reservoir).
Due to the non-interacting quality, each available single-particle level (with energy level ''ϵ'') 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
In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregat ...
.
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
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
: 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
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each suc ...
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 probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
in particle number,
, is:
As a result, for highly occupied states the
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
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
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
for the number of bosons in a given energy level is a
geometric distribution
In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions:
* The probability distribution of the number ''X'' of Bernoulli trials needed to get one success, supported on the set \;
* ...
; somewhat counterintuitively, the most probable value for ''N'' is always 0. (In contrast,
classical particles have instead a
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
in particle number for a given state, with a much smaller uncertainty of
, and with the most-probable ''N'' value being near
.)
Derivation in the canonical approach
It is also possible to derive approximate Bose–Einstein statistics in the
canonical ensemble
In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat b ...
. These derivations are lengthy and only yield the above results in the asymptotic limit of a large number of particles. The reason is that the total number of bosons is fixed in the canonical ensemble. The Bose–Einstein distribution in this case can be derived as in most texts by maximization, but the mathematically best derivation is by the
Darwin–Fowler method
In statistical mechanics, the Darwin–Fowler method is used for deriving the distribution functions with mean probability. It was developed by Charles Galton Darwin and Ralph H. Fowler in 1922–1923.
Distribution functions are used in statist ...
of mean values as emphasized by Dingle. See also Müller-Kirsten.
[H. J. W. Müller-Kirsten, ''Basics of Statistical Physics'', 2nd ed., World Scientific (2013), .] The fluctuations of the ground state in the condensed region are however markedly different in the canonical and grand-canonical ensembles.
[Ziff R. M.; Kac, M.; Uhlenbeck, G. E. (1977)]
"The ideal Bose–Einstein gas, revisited"
''Physics Reports
''Physics Reports'' is a peer-reviewed scientific journal, a review section of '' Physics Letters'' that has been published by Elsevier since 1971. The journal publishes long and deep reviews on all aspects of physics. In average, the length of the ...
'' 32: 169–248.
Suppose we have a number of energy levels, labeled 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
Note, notes, or NOTE may refer to:
Music and entertainment
* Musical note, a pitched sound (or a symbol for a sound) in music
* Notes (album), ''Notes'' (album), a 1987 album by Paul Bley and Paul Motian
* ''Notes'', a common (yet unofficial) sho ...
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 coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s:
Continuing this process, we can see that
is just a binomial coefficient
(See
Notes
Note, notes, or NOTE may refer to:
Music and entertainment
* Musical note, a pitched sound (or a symbol for a sound) in music
* Notes (album), ''Notes'' (album), a 1987 album by Paul Bley and Paul Motian
* ''Notes'', a common (yet unofficial) sho ...
below)
For example, the population numbers for two particles in three sublevels are 200, 110, 101, 020, 011, or 002 for a total of six which equals 4!/(2!2!). 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
In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of Classical physics, classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the ...
, we wish to find the set of
for which ''W'' is maximised, subject to the constraint that there be a fixed total number of particles, and a fixed total energy. The maxima of
and
occur at the same value of
and, since it is easier to accomplish mathematically, we will maximise the latter function instead. We constrain our solution using
Lagrange multipliers
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied e ...
forming the function:
Using the
approximation and using
Stirling's approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less p ...
for the factorials
gives
Where ''K'' is the sum of a number of terms which are not functions of the
. Taking the derivative with respect to
, and setting the result to zero and solving for
, yields the Bose–Einstein population numbers:
By a process similar to that outlined in the
Maxwell–Boltzmann statistics
In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of Classical physics, classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the ...
article, it can be seen that:
which, using Boltzmann's famous relationship
becomes a statement of the
second law of thermodynamics
The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
at constant volume, and it follows that
and
where ''S'' is the
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
,
is the
chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
, ''k''
B is
Boltzmann's constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant ...
and ''T'' is the
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
, so that finally:
Note that the above formula is sometimes written:
where
is the absolute
activity, as noted by McQuarrie.
[See McQuarrie in citations]
Also note that when the particle numbers are not conserved, removing the conservation of particle numbers constraint is equivalent to setting
and therefore the chemical potential
to zero. This will be the case for photons and massive particles in mutual equilibrium and the resulting distribution will be the
Planck distribution.
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. It is clear that the
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
s of these n balls and g − 1 partitions will give different ways of arranging bosons in different energy levels. Say, for 3 (= ''n'') particles and 3 (= ''g'') shells, therefore , the arrangement might be , ●●, ●, or , , ●●●, or , ●, ●● , etc. Hence the number of distinct permutations of objects which have ''n'' identical items and (''g'' − 1) identical items will be:
See the image for a visual representation of one such distribution of ''n'' particles in ''g'' boxes that can be represented as partitions.
![BosonicParticleStatistics](https://upload.wikimedia.org/wikipedia/commons/5/54/BosonicParticleStatistics.png)
OR
The 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 die taking values in the set
, for
. The constraints of the game are that the value of a die
, denoted by
, has to be ''greater than or equal to'' the value of die
, denoted by
, in the previous throw, i.e.,
. Thus a valid sequence of die throws can be described by an ''n''-tuple
, such that
. Let
denote the set of these valid ''n''-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 ''n''-tuples) in
. Thus the problem of finding an expression for
becomes the problem of counting the elements in
.
Example ''n'' = 4, ''g'' = 3:
(there are
elements in
)
Subset
is obtained by fixing all indices
to
, except for the last index,
, which is incremented from
to
. Subset
is obtained by fixing
, and incrementing
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
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that e ...
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
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that e ...
of cardinality
(number of dice) with elements taken from the set
of cardinality
(number of possible values of each die), and the number of such multisets, i.e.,
is the
multiset coefficient
which is exactly the same as the
formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
for
, as derived above with the aid of a
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
involving binomial coefficients, namely
To understand the decomposition
or for example,
and
let us 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 obtains 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
Continuing 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.
Interdisciplinary applications
Viewed as a pure
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
, the Bose–Einstein distribution has found application in other fields:
* In recent years, Bose–Einstein statistics has also been used as a method for term weighting in
information retrieval
Information retrieval (IR) in computing and information science is the process of obtaining information system resources that are relevant to an information need from a collection of those resources. Searches can be based on full-text or other co ...
. The method is one of a collection of DFR ("Divergence From Randomness") models,
[Amati, G.; C. J. Van Rijsbergen (2002).]
Probabilistic models of information retrieval based on measuring the divergence from randomness
'' ACM TOIS'' 20(4):357–389. the basic notion being that Bose–Einstein statistics may be a useful indicator in cases where a particular term and a particular document have a significant relationship that would not have occurred purely by chance. Source code for implementing this model is available from th
Terrier projectat the University of Glasgow.
* The evolution of many complex systems, including the
World Wide Web
The World Wide Web (WWW), commonly known as the Web, is an information system enabling documents and other web resources to be accessed over the Internet.
Documents and downloadable media are made available to the network through web se ...
, business, and citation networks, is encoded in the dynamic web describing the interactions between the system's constituents. Despite their irreversible and nonequilibrium nature these networks follow Bose statistics and can undergo Bose–Einstein condensation. Addressing the dynamical properties of these nonequilibrium systems within the framework of equilibrium quantum gases predicts that the "first-mover-advantage", "fit-get-rich" (FGR) and "winner-takes-all" phenomena observed in competitive systems are thermodynamically distinct phases of the underlying evolving networks.
[ Bianconi, G.; Barabási, A.-L. (2001).]
Bose–Einstein Condensation in Complex Networks
. ''Physical Review Letters
''Physical Review Letters'' (''PRL''), established in 1958, is a peer-reviewed, scientific journal that is published 52 times per year by the American Physical Society. As also confirmed by various measurement standards, which include the ''Journa ...
'' 86: 5632–5635.
See also
*
Bose–Einstein correlations In physics, Bose–Einstein correlations are correlations between identical bosons. They have important applications in astronomy, optics, particle and nuclear physics.
From intensity interferometry to Bose–Einstein correlations
The interf ...
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Bose–Einstein condensate
In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67&n ...
*
Bose gas
An ideal Bose gas is a quantum-mechanical phase of matter, analogous to a classical ideal gas. It is composed of bosons, which have an integer value of spin, and abide by Bose–Einstein statistics. The statistical mechanics of bosons were deve ...
*
Einstein solid
The Einstein solid is a model of a crystalline solid that contains a large number of independent three-dimensional quantum harmonic oscillators of the same frequency. The independence assumption is relaxed in the Debye model.
While the model provi ...
*
Higgs boson
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Parastatistics
In quantum mechanics and statistical mechanics, parastatistics is one of several alternatives to the better known particle statistics models (Bose–Einstein statistics, Fermi–Dirac statistics and Maxwell–Boltzmann statistics). Other alt ...
*
Planck's law of black body radiation
In physics, Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature , when there is no net flow of matter or energy between the body and its environment.
At ...
*
Superconductivity
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
*
Fermi–Dirac statistics
Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac di ...
*
Maxwell–Boltzmann statistics
In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of Classical physics, classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the ...
Notes
References
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{{DEFAULTSORT:Bose-Einstein statistics
Concepts in physics
Quantum field theory
Albert Einstein
Statistical mechanics
Satyendra Nath Bose