15 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bose–Einstein statistics | 1/4 | https://en.wikipedia.org/wiki/Bose–Einstein_statistics | reference | science, encyclopedia | 2026-05-05T13:41:28.703444+00:00 | kb-cron |
In quantum statistics, Bose–Einstein statistics (B–E 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 (1924–25) 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 (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. 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
N
V
≥
n
q
,
{\displaystyle {\frac {N}{V}}\geq n_{\text{q}},}
where N is the number of particles, V is the volume, and nq 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 (particles that obey the Pauli exclusion principle), and Bose–Einstein statistics applies to bosons. 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. 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 i for Bose–Einstein statistics is:
with εi > μ and where ni is the occupation number (the number of particles) in state i,
g
i
{\displaystyle g_{i}}
is the degeneracy of energy level i, εi is the energy of the ith state, μ is the chemical potential (zero for a photon gas), kB is the Boltzmann constant, and T is the absolute temperature. The variance of this distribution
V
(
n
)
{\displaystyle V(n)}
is calculated directly from the expression above for the average number.
V
(
n
)
=
k
T
∂
∂
μ
n
¯
i
=
⟨
n
⟩
(
1
+
⟨
n
⟩
)
=
n
¯
+
n
¯
2
{\displaystyle V(n)=kT{\frac {\partial }{\partial \mu }}{\bar {n}}_{i}=\langle n\rangle (1+\langle n\rangle )={\bar {n}}+{\bar {n}}^{2}}
For comparison, the average number of fermions with energy
ε
i
{\displaystyle \varepsilon _{i}}
given by Fermi–Dirac particle-energy distribution has a similar form:
n
¯
i
(
ε
i
)
=
g
i
e
(
ε
i
−
μ
)
/
k
B
T
+
1
.
{\displaystyle {\bar {n}}_{i}(\varepsilon _{i})={\frac {g_{i}}{e^{(\varepsilon _{i}-\mu )/k_{\text{B}}T}+1}}.}
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,
n
¯
i
=
g
i
e
(
ε
i
−
μ
)
/
k
B
T
±
1
≪
1
{\displaystyle {\bar {n}}_{i}={\frac {g_{i}}{e^{(\varepsilon _{i}-\mu )/k_{\text{B}}T}\pm 1}}\ll 1}
, therefore
e
(
ε
i
−
μ
)
/
k
B
T
±
1
≫
1
{\displaystyle e^{(\varepsilon _{i}-\mu )/k_{\text{B}}T}\pm 1\gg 1}
or equivalently
e
(
ε
i
−
μ
)
/
k
B
T
≫
1
{\displaystyle e^{(\varepsilon _{i}-\mu )/k_{\text{B}}T}\gg 1}
. In that case,
n
¯
i
≈
g
i
e
(
ε
i
−
μ
)
/
k
B
T
=
1
Z
e
−
(
ε
i
−
μ
)
/
k
B
T
{\displaystyle {\bar {n}}_{i}\approx {\frac {g_{i}}{e^{(\varepsilon _{i}-\mu )/k_{\text{B}}T}}}={\frac {1}{Z}}e^{-(\varepsilon _{i}-\mu )/k_{\text{B}}T}}
, 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
ε
i
−
μ
≫
k
B
T
{\displaystyle \varepsilon _{i}-\mu \gg k_{\text{B}}T}
) is again very small,
n
¯
i
=
g
i
e
(
ε
i
−
μ
)
/
k
B
T
±
1
≪
1
{\displaystyle {\bar {n}}_{i}={\frac {g_{i}}{e^{(\varepsilon _{i}-\mu )/k_{\text{B}}T}\pm 1}}\ll 1}
. This again reduces to Maxwell–Boltzmann statistics. In addition to reducing to the Maxwell–Boltzmann distribution in the limit of high
T
{\displaystyle T}
and low density, Bose–Einstein statistics also reduces to Rayleigh–Jeans law distribution for low energy states with
ε
i
−
μ
≪
k
B
T
{\displaystyle \varepsilon _{i}-\mu \ll k_{\text{B}}T}
, namely
n
¯
i
=
g
i
e
(
ε
i
−
μ
)
/
k
B
T
−
1
≈
g
i
(
ε
i
−
μ
)
/
k
B
T
=
g
i
k
B
T
ε
i
−
μ
.
{\displaystyle {\begin{aligned}{\bar {n}}_{i}&={\frac {g_{i}}{e^{(\varepsilon _{i}-\mu )/k_{\text{B}}T}-1}}\\&\approx {\frac {g_{i}}{(\varepsilon _{i}-\mu )/k_{\text{B}}T}}={\frac {g_{i}k_{\text{B}}T}{\varepsilon _{i}-\mu }}.\end{aligned}}}