Bose–Einstein statistics

In quantum statistics, Bose–Einstein statistics (or more colloquially B–E statistics) is one of two possible ways in which a collection of non-interacting indistinguishable 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 (1924–25) by Satyendra Nath Bose, 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 in collaboration with Bose.

The Bose–Einstein statistics apply only to those particles not limited to single occupancy of the same state—that is, particles that do not obey the Pauli exclusion principle restrictions. Such particles have integer values of spin and are named bosons, after the statistics that correctly describe their behaviour. There must also be no significant interaction between the particles.

Concept

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 – 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,

$\frac{N}{V} \ge n_q$

where N is the number of particles and V is the volume and n_q 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 apply to fermions (particles that obey the Pauli exclusion principle), and Bose–Einstein statistics apply to bosons. As the quantum concentration depends on temperature, most systems at high temperatures obey the classical (Maxwell–Boltzmann) limit unless they 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.

B–E 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 i  for B–E statistics is

$n_i(\varepsilon_i) = \frac{g_i}{e^{(\varepsilon_i-\mu)/kT}-1}$

with ε_i > μ and where n_i  is the number of particles in state i, g_i  is the degeneracy of state i, ε_i  is the energy of the ith state, μ is the chemical potential, k is the Boltzmann constant, and T is absolute temperature. For comparison, the average number of fermions with energy $\epsilon_i$ given by Fermi–Dirac particle-energy distribution has a similar form,

$\bar{n}_i(\epsilon_i) = \frac{g_i}{e^{(\epsilon_i-\mu) / k T} + 1}$

B–E statistics reduces to the Rayleigh–Jeans Law distribution for $\varepsilon_i-\mu \gg kT$, namely $n_i = \frac{g_i kT}{\varepsilon_i-\mu}$.

History

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 (remarkably, this error resembled the famous blunder by d'Alembert 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 volume 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 Zeitschrift für Physik. Einstein immediately agreed, personally translated the article into German (Bose had earlier translated Einstein's article on the theory of General 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 Zeitschrift für Physik, asking that they be published together. This was done 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 energy 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.

Two derivations of 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 easily derived from the grand canonical ensemble. 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. In other words, each single-particle level is a separate, tiny grand canonical ensemble. With bosons there is no limit on the number of particles N in the level, but due to indistinguishability each possible N corresponds to only one microstate (with energy Nϵ). The resulting partition function for that single-particle level therefore forms a geometric series:

$\begin{align}\mathcal Z & = \sum_{N=0}^{\infty} \exp(N(\mu - \epsilon)/k_B T) = \sum_{N=0}^{\infty} [\exp((\mu - \epsilon)/k_B T)]^N \\ & = \frac{1}{1 - \exp((\mu - \epsilon)/k_B T)}\end{align}$

and the average particle number for that single-particle substate is given by

$\langle N\rangle = k_B T \frac{1}{\mathcal Z} \left(\frac{\partial \mathcal Z}{\partial \mu}\right)_{V,T} = \frac{1}{\exp((\epsilon-\mu)/k_B T)-1}$

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 (due to thermal fluctuations) may also be derived:

$\langle (\Delta N)^2 \rangle = k_B T \left(\frac{d\langle N\rangle}{d\mu}\right)_{V,T} = \langle N^2\rangle - \langle N\rangle^2$

This level of fluctuation is much larger than for distinguishable particles, which would instead show Poisson statistics ($\langle (\Delta N)^2 \rangle = \langle N\rangle$). This is because the probability distribution for the number of bosons in a given energy level is a geometric distribution, not a Poisson distribution.

Derivation in the canonical approach

It is also possible to derive approximate Bose–Einstein statistics in the canonical ensemble. 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. That contradicts the implication in Bose–Einstein statistics that each energy level is filled independently from the others (which would require the number of particles to be flexible).

Derivation

Suppose we have a number of energy levels, labeled by index $\displaystyle i$, each level having energy $\displaystyle \varepsilon_i$ and containing a total of $\displaystyle n_i$ particles. Suppose each level contains $\displaystyle g_i$ 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 $\displaystyle g_i$ associated with level $\displaystyle i$ is called the "degeneracy" of that energy level. Any number of bosons can occupy the same sublevel.

Let $\displaystyle w(n,g)$ be the number of ways of distributing $\displaystyle n$ particles among the $\displaystyle g$ sublevels of an energy level. There is only one way of distributing $\displaystyle n$ particles with one sublevel, therefore $\displaystyle w(n,1)=1$. It is easy to see that there are $\displaystyle (n+1)$ ways of distributing $\displaystyle n$ particles in two sublevels which we will write as:

$w(n,2)=\frac{(n+1)!}{n!1!}.$

With a little thought (see Notes below) it can be seen that the number of ways of distributing $\displaystyle n$ particles in three sublevels is

$w(n,3) = w(n,2) + w(n-1,2) + \cdots + w(1,2) + w(0,2)$

so that

$w(n,3)=\sum_{k=0}^n w(n-k,2) = \sum_{k=0}^n\frac{(n-k+1)!}{(n-k)!1!}=\frac{(n+2)!}{n!2!}$

where we have used the following theorem involving binomial coefficients:

$\sum_{k=0}^n\frac{(k+a)!}{k!a!}=\frac{(n+a+1)!}{n!(a+1)!}.$

Continuing this process, we can see that $\displaystyle w(n,g)$ is just a binomial coefficient (See Notes below)

$w(n,g)=\frac{(n+g-1)!}{n!(g-1)!}.$

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 $\displaystyle n_i$ can be realized is the product of the ways that each individual energy level can be populated:

$W = \prod_i w(n_i,g_i) = \prod_i \frac{(n_i+g_i-1)!}{n_i!(g_i-1)!} \approx\prod_i \frac{(n_i+g_i)!}{n_i!(g_i-1)!}$

where the approximation assumes that $n_i \gg 1$.

Following the same procedure used in deriving the Maxwell–Boltzmann statistics, we wish to find the set of $\displaystyle n_i$ 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 $\displaystyle W$ and $\displaystyle \ln(W)$ occur at the same value of $\displaystyle n_i$ 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:

$f(n_i)=\ln(W)+\alpha(N-\sum n_i)+\beta(E-\sum n_i \varepsilon_i)$

Using the $n_i \gg 1$ approximation and using Stirling's approximation for the factorials $\left(x!\approx x^x\,e^{-x}\,\sqrt{2\pi x}\right)$ gives

$f(n_i)=\sum_i (n_i + g_i) \ln(n_i + g_i) - n_i \ln(n_i) +\alpha\left(N-\sum n_i\right)+\beta\left(E-\sum n_i \varepsilon_i\right)+K.$

Where K is the sum of a number of terms which are not functions of the $n_i$. Taking the derivative with respect to $\displaystyle n_i$, and setting the result to zero and solving for $\displaystyle n_i$, yields the Bose–Einstein population numbers:

$n_i = \frac{g_i}{e^{\alpha+\beta \varepsilon_i}-1}.$

By a process similar to that outlined in the Maxwell–Boltzmann statistics article, it can be seen that:

$d\ln W=\alpha\,dN+\beta\,dE$

which, using Boltzmann's famous relationship $S=k\,\ln W$ becomes a statement of the second law of thermodynamics at constant volume, and it follows that $\beta = \frac{1}{kT}$ and $\alpha = - \frac{\mu}{kT}$ where S is the entropy, $\mu$ is the chemical potential, k is Boltzmann's constant and T is the temperature, so that finally:

$n_i = \frac{g_i}{e^{(\varepsilon_i-\mu)/kT}-1}.$

Note that the above formula is sometimes written:

$n_i = \frac{g_i}{e^{\varepsilon_i/kT}/z-1},$

where $\displaystyle z=\exp(\mu/kT)$ is the absolute activity, as noted by McQuarrie.

Also note that when the particle numbers are not conserved, removing the conservation of particle numbers constraint is equivalent to setting $\alpha$ and therefore the chemical potential $\mu$ to zero. This will be the case for photons and massive particles in mutual equilibrium and the resulting distribution will be the Planck distribution.

Notes

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 permutations 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 (g-1)=2, the arrangement might be |●●|●, or ||●●●, or |●|●● , etc. Hence the number of distinct permutations of n + (g-1) objects which have n identical items and (g-1) identical items will be:

$\frac{(g-1 + n)!} {(g-1)! n!}$

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 $\displaystyle n$ dice, with each die taking values in the set $\displaystyle \left\{ 1, \dots, g \right\}$, for $g \ge 1$. The constraints of the game are that the value of a die $\displaystyle i$, denoted by $\displaystyle m_i$, has to be greater than or equal to the value of die $\displaystyle (i-1)$, denoted by $\displaystyle m_{i-1}$, in the previous throw, i.e., $m_i \ge m_{i-1}$. Thus a valid sequence of die throws can be described by an n-tuple $\displaystyle \left( m_1 , m_2 , \dots , m_n \right)$, such that $m_i \ge m_{i-1}$. Let $\displaystyle S(n,g)$ denote the set of these valid n-tuples:

$S(n,g) = \Big\{ \left( m_1 , m_2 , \dots , m_n \right) \Big| \Big. m_i \ge m_{i-1} , m_i \in \left\{ 1, \dots, g \right\} , \forall i = 1, \dots , n \Big\}.$

(1)

Then the quantity $\displaystyle w(n,g)$ (defined above as the number of ways to distribute $\displaystyle n$ particles among the $\displaystyle g$ sublevels of an energy level) is the cardinality of $\displaystyle S(n,g)$, i.e., the number of elements (or valid n-tuples) in $\displaystyle S(n,g)$. Thus the problem of finding an expression for $\displaystyle w(n,g)$ becomes the problem of counting the elements in $\displaystyle S(n,g)$.

Example n = 4, g = 3:

$S(4,3) = \left\{ \underbrace{(1111), (1112), (1113)}_{(a)}, \underbrace{(1122), (1123), (1133)}_{(b)}, \underbrace{(1222), (1223), (1233), (1333)}_{(c)}, \right.$

$\left. \underbrace{(2222), (2223), (2233), (2333), (3333)}_{(d)} \right\}$

$\displaystyle w(4,3) = 15$ (there are $\displaystyle 15$ elements in $\displaystyle S(4,3)$)

Subset $\displaystyle (a)$ is obtained by fixing all indices $\displaystyle m_i$ to $\displaystyle 1$, except for the last index, $\displaystyle m_n$, which is incremented from $\displaystyle 1$ to $\displaystyle g=3$. Subset $\displaystyle (b)$ is obtained by fixing $\displaystyle m_1 = m_2 = 1$, and incrementing $\displaystyle m_3$ from $\displaystyle 2$ to $\displaystyle g=3$. Due to the constraint $\displaystyle m_i \ge m_{i-1}$ on the indices in $\displaystyle S(n,g)$, the index $\displaystyle m_4$ must automatically take values in $\displaystyle \left\{ 2, 3 \right\}$. The construction of subsets $\displaystyle (c)$ and $\displaystyle (d)$ follows in the same manner.

Each element of $\displaystyle S(4,3)$ can be thought of as a multiset of cardinality $\displaystyle n=4$; the elements of such multiset are taken from the set $\displaystyle \left\{ 1, 2, 3 \right\}$ of cardinality $\displaystyle g=3$, and the number of such multisets is the multiset coefficient

$\displaystyle \left\langle \begin{matrix} 3 \\ 4 \end{matrix} \right\rangle = {3 + 4 - 1 \choose 3-1} = {3 + 4 - 1 \choose 4} = \frac {6!} {4! 2!} = 15$

More generally, each element of $\displaystyle S(n,g)$ is a multiset of cardinality $\displaystyle n$ (number of dice) with elements taken from the set $\displaystyle \left\{ 1, \dots, g \right\}$ of cardinality $\displaystyle g$ (number of possible values of each die), and the number of such multisets, i.e., $\displaystyle w(n,g)$ is the multiset coefficient

$\displaystyle w(n,g) = \left\langle \begin{matrix} g \\ n \end{matrix} \right\rangle = {g + n - 1 \choose g-1} = {g + n - 1 \choose n} = \frac{(g + n - 1)!} {n! (g-1)!}$

(2)

which is exactly the same as the formula for $\displaystyle w(n,g)$, as derived above with the aid of a theorem involving binomial coefficients, namely

$\sum_{k=0}^n\frac{(k+a)!}{k!a!}=\frac{(n+a+1)!}{n!(a+1)!}.$

(3)

To understand the decomposition

$\displaystyle w(n,g) = \sum_{k=0}^{n} w(n-k, g-1) = w(n, g-1) + w(n-1, g-1) + \cdots + w(1, g-1) + w(0, g-1)$

(4)

or for example, $\displaystyle n=4$ and $\displaystyle g=3$

$\displaystyle w(4,3) = w(4,2) + w(3,2) + w(2,2) + w(1,2) + w(0,2),$

let us rearrange the elements of $\displaystyle S(4,3)$ as follows

$S(4,3) = \left\{ \underbrace{ (1111), (1112), (1122), (1222), (2222) }_{(\alpha)}, \underbrace{ (111{\color{Red}{\underset{=}{3}}}), (112{\color{Red}{\underset{=}{3}}}), (122{\color{Red}{\underset{=}{3}}}), (222{\color{Red}{\underset{=}{3}}}) }_{(\beta)}, \right.$

$\left. \underbrace{ (11{\color{Red}{\underset{==}{33}}}), (12{\color{Red}{\underset{==}{33}}}), (22{\color{Red}{\underset{==}{33}}}) }_{(\gamma)}, \underbrace{ (1{\color{Red}{\underset{===}{333}}}), (2{\color{Red}{\underset{===}{333}}}) }_{(\delta)} \underbrace{ ({\color{Red}{\underset{====}{3333}}}) }_{(\omega)} \right\}.$

Clearly, the subset $\displaystyle (\alpha)$ of $\displaystyle S(4,3)$ is the same as the set

$\displaystyle S(4,2) = \left\{ (1111), (1112), (1122), (1222), (2222) \right\}$.

By deleting the index $\displaystyle m_4=3$ (shown in red with double underline) in the subset $\displaystyle (\beta)$ of $\displaystyle S(4,3)$, one obtains the set

$\displaystyle S(3,2) = \left\{ (111), (112), (122), (222) \right\}$.

In other words, there is a one-to-one correspondence between the subset $\displaystyle (\beta)$ of $\displaystyle S(4,3)$ and the set $\displaystyle S(3,2)$. We write

$\displaystyle (\beta) \longleftrightarrow S(3,2)$.

Similarly, it is easy to see that

$\displaystyle (\gamma) \longleftrightarrow S(2,2) = \left\{ (11), (12), (22) \right\}$

$\displaystyle (\delta) \longleftrightarrow S(1,2) = \left\{ (1), (2) \right\}$

$\displaystyle (\omega) \longleftrightarrow S(0,2) = \varnothing$ (empty set).

Thus we can write

$\displaystyle S(4,3) = \bigcup_{k=0}^{4} S(4-k,2)$

or more generally,

$\displaystyle S(n,g) = \bigcup_{k=0}^{n} S(n-k,g-1)$;

(5)

and since the sets

$\displaystyle S(i,g-1) \ , \ {\rm for} \ i = 0, \dots , n$

are non-intersecting, we thus have

$\displaystyle w(n,g) = \sum_{k=0}^{n} w(n-k,g-1)$,

(6)

with the convention that

$\displaystyle w(0,g) = 1 \ , \forall g \ , {\rm and} \ w(n,0) = 1 \ , \forall n$.

(7)

Continuing the process, we arrive at the following formula

$\displaystyle w(n,g) = \sum_{k_1=0}^{n} \sum_{k_2=0}^{n-k_1} w(n - k_1 - k_2, g-2) = \sum_{k_1=0}^{n} \sum_{k_2=0}^{n-k_1} \cdots \sum_{k_g=0}^{n-\sum_{j=1}^{g-1} k_j} w(n - \sum_{i=1}^{g} k_i, 0).$

Using the convention (7)₂ above, we obtain the formula

$\displaystyle w(n,g) = \sum_{k_1=0}^{n} \sum_{k_2=0}^{n-k_1} \cdots \sum_{k_g=0}^{n-\sum_{j=1}^{g-1} k_j} 1,$

(8)

keeping in mind that for $\displaystyle q$ and $\displaystyle p$ being constants, we have

$\displaystyle \sum_{k=0}^{q} p = q p$.

(9)

It can then be verified that (8) and (2) give the same result for $\displaystyle w(4,3)$, $\displaystyle w(3,3)$, $\displaystyle w(3,2)$, etc.

Interdisciplinary applications

Viewed as a pure probability distribution, the Bose–Einstein distribution has found application in other fields:

• In recent years, Bose Einstein statistics have also been used as a method for term weighting in information retrieval. The method is one of a collection of DFR ("Divergence From Randomness") models, 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 the Terrier project at the University of Glasgow.

• Main article: Bose–Einstein condensation (network theory)
  The evolution of many complex systems, including the World Wide Web, 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.

See also

• Bose–Einstein correlations
• Einstein solid
• Higgs boson
• Parastatistics
• Planck's law of black body radiation
• Superconductivity
• Fermi-Dirac Statistics
• Maxwell-Boltzmann Statistics

References

• Bose (1924). "Plancks Gesetz und Lichtquantenhypothese", Zeitschrift für Physik 26:178–181. doi:10.1007/BF01327326 (Einstein's translation into German of Bose's paper on Planck's law).