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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bose–Einstein statistics | 2/4 | https://en.wikipedia.org/wiki/Bose–Einstein_statistics | reference | science, encyclopedia | 2026-05-05T13:41:28.703444+00:00 | kb-cron |
== 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 (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 Jean le Rond 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 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 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'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. 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. 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 (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, one considers a system with fixed energy, volume, and number of particles. We take a system composed of
N
=
∑
i
n
i
{\textstyle N=\sum _{i}n_{i}}
identical bosons,
n
i
{\displaystyle n_{i}}
of which have energy
ε
i
{\displaystyle \varepsilon _{i}}
and are distributed over
g
i
{\displaystyle g_{i}}
levels or states with the same energy
ε
i
{\displaystyle \varepsilon _{i}}
, i.e.
g
i
{\displaystyle g_{i}}
is the degeneracy associated with energy
ε
i
{\displaystyle \varepsilon _{i}}
. The total energy of the system is
E
=
∑
i
n
i
ε
i
{\textstyle E=\sum _{i}n_{i}\varepsilon _{i}}
. Calculation of the number of arrangements of
n
i
{\displaystyle n_{i}}
particles distributed among
g
i
{\displaystyle g_{i}}
states is a problem of combinatorics. Since particles are indistinguishable in the quantum mechanical context here, the number of ways for arranging
n
i
{\displaystyle n_{i}}
particles in
g
i
{\displaystyle g_{i}}
boxes (for the
i
{\displaystyle i}
th energy level), where each box is capable of containing an infinite number of bosons (because for bosons the Pauli exclusion principle does not apply), would be (see image):
w
i
,
BE
=
(
n
i
+
g
i
−
1
)
!
n
i
!
(
g
i
−
1
)
!
=
C
n
i
n
i
+
g
i
−
1
,
{\displaystyle w_{i,{\text{BE}}}={\frac {(n_{i}+g_{i}-1)!}{n_{i}!(g_{i}-1)!}}=C_{n_{i}}^{n_{i}+g_{i}-1},}
where
C
k
m
{\displaystyle C_{k}^{m}}
is the k-combination of a set with m elements (Note also that
w
i
,
BE
{\displaystyle w_{i,{\text{BE}}}}
represents the absolute non-normalized probability of an energy state with
n
i
{\displaystyle n_{i}}
bosons and a degeneracy of
g
i
{\displaystyle g_{i}}
, it is not the same as the
w
i
{\displaystyle w_{i}}
associated with the Gibbs formulation of entropy). The total number of arrangements in an ensemble of bosons is simply the product of the binomial coefficients
C
n
i
n
i
+
g
i
−
1
{\displaystyle C_{n_{i}}^{n_{i}+g_{i}-1}}
above over all the energy levels, i.e.
W
BE
=
∏
i
w
i
,
BE
=
∏
i
(
n
i
+
g
i
−
1
)
!
(
g
i
−
1
)
!
n
i
!
,
{\displaystyle W_{\text{BE}}=\prod _{i}w_{i,{\text{BE}}}=\prod _{i}{\frac {(n_{i}+g_{i}-1)!}{(g_{i}-1)!n_{i}!}},}
which for very large
n
i
{\displaystyle n_{i}}
and
g
i
{\displaystyle g_{i}}
can be simplified using Stirling's approximation to
W
BE
=
∏
i
(
n
i
+
g
i
−
1
e
)
n
i
+
g
i
−
1
(
g
i
−
1
e
)
g
i
−
1
(
n
i
e
)
n
i
.
{\displaystyle W_{\text{BE}}=\prod _{i}{\frac {({\frac {n_{i}+g_{i}-1}{e}})^{n_{i}+g_{i}-1}}{({\frac {g_{i}-1}{e}})^{g_{i}-1}({\frac {n_{i}}{e}})^{n_{i}}}}.}
The entropy of the system can then be expressed as