16 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bose–Einstein statistics | 3/4 | https://en.wikipedia.org/wiki/Bose–Einstein_statistics | reference | science, encyclopedia | 2026-05-05T13:41:28.703444+00:00 | kb-cron |
S
BE
=
k
B
ln
W
BE
=
k
B
∑
i
[
(
n
i
+
g
i
−
1
)
(
ln
(
n
i
+
g
i
−
1
)
−
1
)
−
(
g
i
−
1
)
(
ln
(
g
i
−
1
)
−
1
)
−
n
i
(
ln
n
i
−
1
)
]
.
{\displaystyle S_{\text{BE}}=k_{B}{\text{ln}}W_{\text{BE}}=k_{B}\sum _{i}[(n_{i}+g_{i}-1)({\text{ln}}(n_{i}+g_{i}-1)-1)-(g_{i}-1)({\text{ln}}(g_{i}-1)-1)-n_{i}({\text{ln}}n_{i}-1)].}
The three constraints we can impose on the system can be expressed as
∑
i
δ
n
i
=
0
{\displaystyle \sum _{i}\delta n_{i}=0}
(conservation of N),
∑
i
ϵ
i
δ
n
i
=
0
{\displaystyle \sum _{i}\epsilon _{i}\delta n_{i}=0}
(conservation of E), and
δ
S
BE
=
0
{\displaystyle \delta S_{\text{BE}}=0}
(second law of thermodynamics for a system at equilibrium). This final constraint can be expanded to be in terms of
n
i
{\displaystyle n_{i}}
:
δ
S
BE
=
∂
∂
n
i
S
BE
δ
n
i
=
k
B
∑
i
[
ln
(
n
i
+
g
i
−
1
)
−
ln
n
i
]
δ
n
i
=
0.
{\displaystyle \delta S_{\text{BE}}={\frac {\partial }{\partial n_{i}}}S_{\text{BE}}\delta n_{i}=k_{B}\sum _{i}[{\text{ln}}(n_{i}+g_{i}-1)-{\text{ln}}n_{i}]\delta n_{i}=0.}
Now we can write
∑
i
[
ln
(
n
i
+
g
i
−
1
)
−
ln
n
i
]
δ
n
i
+
C
∑
i
δ
n
i
−
β
∑
i
ϵ
i
δ
n
i
=
0
,
{\displaystyle \sum _{i}[{\text{ln}}(n_{i}+g_{i}-1)-{\text{ln}}n_{i}]\delta n_{i}+C\sum _{i}\delta n_{i}-\beta \sum _{i}\epsilon _{i}\delta n_{i}=0,}
for which to be true, it must be the case that for any i
ln
(
n
i
+
g
i
−
1
)
−
ln
n
i
+
C
−
β
ϵ
i
=
0.
{\displaystyle {\text{ln}}(n_{i}+g_{i}-1)-{\text{ln}}n_{i}+C-\beta \epsilon _{i}=0.}
By solving for
n
i
{\displaystyle n_{i}}
and simplifying we obtain
n
i
=
g
i
−
1
α
e
β
ϵ
i
−
1
,
{\displaystyle n_{i}={\frac {g_{i}-1}{\alpha e^{\beta \epsilon _{i}}-1}},}
which for sufficiently large
g
i
{\displaystyle g_{i}}
reduces to
n
i
=
g
i
α
e
β
ϵ
i
−
1
,
{\displaystyle n_{i}={\frac {g_{i}}{\alpha e^{\beta \epsilon _{i}}-1}},}
which is the form of the Bose-Einstein distribution. Note that this form holds even for a system of interacting bosons.
=== 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 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. Every single-particle state is of a fixed energy,
ε
{\displaystyle \varepsilon }
. 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
N
{\displaystyle N}
is the number of particles, the total energy of the sub-ensemble will then be
N
ε
{\displaystyle N\varepsilon }
. Beginning with the standard expression for a grand partition function and replacing
E
{\displaystyle E}
with
N
ε
{\displaystyle N\varepsilon }
, the grand partition function takes the form
Z
=
∑
N
exp
(
(
N
μ
−
N
ε
)
/
k
B
T
)
=
∑
N
exp
(
N
(
μ
−
ε
)
/
k
B
T
)
{\displaystyle {\mathcal {Z}}=\sum _{N}\exp((N\mu -N\varepsilon )/k_{\text{B}}T)=\sum _{N}\exp(N(\mu -\varepsilon )/k_{\text{B}}T)}
This formula applies to fermionic systems as well as bosonic systems. 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:
Z
=
∑
N
=
0
∞
exp
(
N
(
μ
−
ε
)
/
k
B
T
)
=
∑
N
=
0
∞
[
exp
(
(
μ
−
ε
)
/
k
B
T
)
]
N
=
1
1
−
exp
(
(
μ
−
ε
)
/
k
B
T
)
.
{\displaystyle {\begin{aligned}{\mathcal {Z}}&=\sum _{N=0}^{\infty }\exp(N(\mu -\varepsilon )/k_{\text{B}}T)=\sum _{N=0}^{\infty }[\exp((\mu -\varepsilon )/k_{\text{B}}T)]^{N}\\&={\frac {1}{1-\exp((\mu -\varepsilon )/k_{\text{B}}T)}}.\end{aligned}}}
Note that the geometric series is convergent only if
e
(
μ
−
ε
)
/
k
B
T
<
1
{\displaystyle e^{(\mu -\varepsilon )/k_{\text{B}}T}<1}
, including the case where
ε
=
0
{\displaystyle \varepsilon =0}
. This implies that the chemical potential for the Bose gas must be negative, i.e.,
μ
<
0
{\displaystyle \mu <0}
, 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
⟨
N
⟩
=
k
B
T
1
Z
(
∂
Z
∂
μ
)
V
,
T
=
1
exp
(
(
ε
−
μ
)
/
k
B
T
)
−
1
{\displaystyle \langle N\rangle =k_{\text{B}}T{\frac {1}{\mathcal {Z}}}\left({\frac {\partial {\mathcal {Z}}}{\partial \mu }}\right)_{V,T}={\frac {1}{\exp((\varepsilon -\mu )/k_{\text{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,
σ
N
2
=
⟨
N
2
⟩
−
⟨
N
⟩
2
{\textstyle \sigma _{N}^{2}=\langle N^{2}\rangle -\langle N\rangle ^{2}}
, is: