--- title: "Bose–Einstein statistics" chunk: 3/4 source: "https://en.wikipedia.org/wiki/Bose–Einstein_statistics" category: "reference" tags: "science, encyclopedia" date_saved: "2026-05-05T13:41:28.703444+00:00" instance: "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: