???????? ?????????? ??????? ????????? ??????? (???????? Thermodynamic Equilibrium)?? ??????? ??? ???????? ???????? ???????? ????????? ???? ????? ????? ????? ?????? ??????? ????????? ???? ????? ?????? ??????? ??????? ? ??? ????? ???????? ??? ?????? ????? ???? - ???-???????????? ?????????? (???????? Bose?Einstein Statistics ?? B?E Statistics) ? ???-???????????? ?????????? ???? ?? ?????? ??????? ? ?? ??????? ????? ???????? ??????? ??????????? ????????? ????? ????? ????? ???????? ?????? ? ?????? ??????? ???????? ?????? ??? ?????? ???????? ??? ?  ???????????? ???? ??????? ????-?? ??????? ?????????? ??? ???????????? ???????? ??????? ? ?????????? ???? ???????? ?? ?? ? ?????? ???????? ????????? ?????? ????????? ????????? ? ??? ??????? ?????????? ????? ??????? ? ????????????? ?? ????? ??????? ???? ?????? ??????? ? 

?????? ???????? ????????? (???????? Pauli Exclusion Principle) ??????? ????? ????? ????????? ????????? ??????? ???? ???????? ????? ???????? ?????? ???? ???????? ????? ? ????? ?????? ?????? ???????? ?????? ??? ??????? ?????? ???????? ?????? ???????? ?????? ??????? ???? ???? ???? ? ?????? ???????? ????????? ?????? ???????? ??????????? ????? ?????? ???-???????????? ?????????? ???????? ?????? ?????? ?  ???? ???????????? ?????????? ????????? ??? ?? ????????????? ?????? ? ???????? “??????” ?? "?????" ??????? ?

????????? [ ???????? ]

????? ??????????? ?????? ??????????? ??????? ??????????? ????? ? ???????? ???????? ?????-?????? ??????????? ?????? ?????? ? ????? ??????????? ???? ?????? ?????? ?????? ??? ???? ????? ???????? ?????????? ? ???? ????? ?????????????? ????????????? ???-???????????? ???????? (???????? Bose?Einstein Condensate) ???? ??????? ?  ?????????? ?????? ???????? ????? ?????? ???? ? ????????? ?????????? ?????? ???????, ?????-?????? ? ???-???????????? ??????????? ????? ?????? ? ???????????? ?????? ????? ????? ??????? ?????? ?????? ???? ???????? ?????? ??????? ?

${\displaystyle {\frac {N}{V}}\geq n_{q},}$

????? N ????? ???????????? ??????? ??????, V ????? ???? ? n _q ???????? ?????????? (Quantum Concentration) ? ??????????? ?????? ????? ????? ?? ??????? ???????????? (???????? Thermal de Broglie Wavelength) ???? ???? ???? ???????? ??????? ?????? ???? ?????????? ??????? ????? ? ???????? ?????????? ?????? ?????? ??? ?

?????? ???????? ????????? ?????? ??????? ???????? ???????? ?????-?????? ??????????? ???? ?????? ?????? ? ????-?????????? ?????????? ?????? ??????????? ???? ???????? ? ???????? ?????????? ????????? ???? ????????? ? ???? ???????????? ????? ???? ??????? ?????? ???? ??????????? ????????? ?????????-??????????? ???? ?????? ????????? ? ?????- ?????? ?????????? ? ???-???????????? ?????????? ???? ????????? ?? ??? ???????????? ?????????-??????????? ???????????? ????? ??????? ?

???? ??????? ?????????? ??? ???? ???????????? ???? ??? ?????????? ?????? ??????? ? ????-?? ??????? ?????? ???? ?????? ??? ?????????? ???????????? ???? ???? ??????????? ???? ?????? ??? ? ?? ??????????? ????? ?????? ????? ????-?????????? ??????????? ????? ???? i ?? ????????? ???? ????? ???????? ??????? :

${\displaystyle n_{i}(\varepsilon _{i})={\frac {g_{i}}{e^{(\varepsilon _{i}-\mu )/kT}-1}}}$

??? ???????? ε _i  > μ , i ??????????? ?????? ????? ???????? ????? ??? ????? n _i , g _i ????? i ?????? ??????? ???????? ???? ??????? ?????????, ε _i ???? i ?? ??????? ?????, μ ????? ???????? ????, k ??????????? ????????? ? T ?????????? ???? ?

?????-?????? ????? ????? ?????? ????? ???? ??? ???? ?????? ? ?????? ???????? ${\displaystyle \epsilon _{i}}$ ??????????? ????? ???? ?????????? ?????-?????? ???????????? ??????? ?

${\displaystyle {\bar {n}}_{i}(\epsilon _{i})={\frac {g_{i}}{e^{(\epsilon _{i}-\mu )/kT}+1}}}$

${\displaystyle kT\gg \epsilon _{i}-\mu }$ ???? ???-???????????? ?????????? ????????? ??? ?????????? ????-??????? ???? (Rayleigh?Jeans Law)?? ?????????? ??? ?

${\displaystyle n_{i}={\frac {g_{i}kT}{\varepsilon _{i}-\mu }}}$

?????? [ ???????? ]

???? ??????????????? ?????? ? ????????? ????? ????????? ????????? ??????? ???? ?????????? ??? ??? ???? ???? ??????? ??????????????? ??? ?????? ????????? ????????? ????????? ?????????? ???? ???????? ???????? ??? ??? ??????????? ???????? ???????? ??????? ?????? ??????? ? ?? ????? ???? ?? ?????????? ????????? ???????? ???????? ????? ??? ???? ? ?????? ??????? ?????????? ??? ?????????? ????? ??? ???? ???????? ????? ???? ?? ??? ??? ???????? ? ?????? ?????? ??????? ???? ??? ???? ????? ??????? ?????? ?????? ? ??? ???????? ???????? ?????? ???? ????????? ?????????? ?????? ??????? ?? ?????? ??? ?????? ??? ??????? ??? ????? ? ????????????? ???? ????? ?????? ???? ?? ?????????-??????????? ?????????? ????? ??????? ????? ???? ???? ????? ? ???? ?????? ???????? ? ??????? ????? ???????????? ???? ???? ????? ??? ??????? ???????? ?? ??????? ?????? ???????? ?????????? ??????? ?????? ??? ?

?????????? ???? ??????????? ?? ???? ?? ?????? “????????? ???? ? ???? ???????? ????????? ????” (Planck's Law and the Hypothesis of Light Quanta) ???? ?????? ??????? ????? ??? ? ^{[?] [?]} ?? ??? ?????? ???????????? ??????????? ?????? ?????? ??? ?????? ??????????? ??????? ????????? ????? ? ????? ?????? ? ??? ?????????? ??? ???????? ????? ??????? ????????? (Zeitschrift fur Physik)?? ?????? ????? ???? ?????????? ???????? ??????? ?????????????? ????? ???????? ? ???????????? ????? ???? ????? ?????? ??? ???????? ???????? ??????? ?????? ?????? ??????? ? ??? ??????? ?????????? ???? ?????????????????? ????? ?????? ???????? ???????? ???????? ???????? ?????? ??????? ? ???????????? ??? ????? ?????? ???????? ? ??? ???? ???????? ???? ????? ????? ????? ?????? ??????? ? ??????? ???? ?????? ???????? ? ^[?] ???? ??????? ?????????????? ?????? ??? ????????????? ?????? ? ?????????? ?????? ?????????? ?

????? ???????? ????? ???????? ?? ??? (Toss) ??? ????? ??????? / ???? / ???? (Head) ?????? ???????? ???? ? ????? ??????? ??????? ?????? ???????? ?/? ? ??????? ??????? ??????? ??????? ???????? ???? ?/? ? ???? ????? ??????? ??????? ?????? ???????? ?/? * ?/? = ?/? ? ??? ????????? ????? ???????? ?????? ??? ????? ?????? ???? ?????? ???? ????? ?? ?????? ????? ?? ???????? ; ?????? ????? ?????????? ????? ? ??????? ???? ???? ????? ?????????? ?????? ?? ??? ??? ?????????? ? ??? ????? ?????? ??????? ???????? ???? ????? ???????? ?? ????? ?????? ???? ?????? ???? ???????? ???????? ???? ? ??? ???????? ??????? ??????, ??????????? ??????? ?????? ? ????? ???????? ?????????? ???? ?? ?????? ? ?????? ???????? ???? ????? ???????? ??????? ?????? ???????? ?/? ? ??? ????? ?????? ??????? ????????? ?????????? ????? ??????? ???? ????? ???? ??????? ??????? ???????? ??????? ? ?????? ??? ?????????? ????? ????????? ????? ???????? ???? ??????? ???? ???????? ??? ? ??? ???????? ????????????? ????????????????? ????? ?????? ? ?????? ??? ????????? ?????????? ?????? ???????? ? ???? ????? ?????? ???? ????? ?????? ????? “????? ????? ??????” ???? ??????????? ? ???? ???? ?? ?????? ??????? ?????? ??? ?? ?????? ? ?? ???????????? ????? ?????? ??? ???? ?? ??????, ???? ???????? ????? ??????? ??????? ???????? ?/? ??? ? ????????? ????? ????? ?????? ???????????? ???????? ?/? ??? ? ???? ??? ????????????? ??? ??? ??? ???-???????????? ???????????? ?????? ? ???????? ???? ? ?????????? ? ???????????? ??? ???????? ?????? ???? ?????? ??? ???? ?????? ? ?????? ???????? ? ???? ??????? ????? ??????????? ??????????? ???????? ???-???????????? ???????? ?????? ????????? ??? ?????? ?

???-???????????? ?????? ????? ??? [ ???????? ]

???????? ???????????? ????????? [ ???????? ]

?????? ???? ?????? ????????? ?????????? ???????? ????????? ??? ???????? ???????????? ??????????? ???-???????????? ????? (Bose?Einstein distribution) ??? ????? ??????? ???????? ? ^[?] ????? ????? ?????? ??????? ?????? ??????? ????? ? ??? ?????????? T ????????? ? μ ???????? ????????? ?? ???? ???? ????? ? ????? ?????? ??????? ? ?????? ????? ????????? ?????? ???????? ????? (????? ????????? ?), ???? ???? ?????? ?? ????????? ???????? ??? ?????? ? ????? ??????????? ???????? ????? ???? ?? ??????? ???????? ???????????? ????????? ? ?????????? ???? ??????????? N? ????? ???? ????? ???? ???????? ?????? ??????? ??????????? N? ???? ?????? ?????? ? ????? ????????? ??? ????? ?? ????????? ?????? ??? ????????????:

${\displaystyle {\begin{aligned}{\mathcal {Z}}&=\sum _{N=0}^{\infty }\exp(N(\mu -\epsilon )/k_{B}T)=\sum _{N=0}^{\infty }[\exp((\mu -\epsilon )/k_{B}T)]^{N}\\&={\frac {1}{1-\exp((\mu -\epsilon )/k_{B}T)}}\end{aligned}}}$

???? ?????? ???? ???????? ????? ?????? ???? :

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

??? ??????? ????? ????????? ???????? ? ??? ???-???????????? ????? ????? ????????? ?????? ?????? ? ^{[?] [?]} ??????? ?????????? ????? ????? ??????? ?????????? ?????? ????? ???? ????? ????? ?????? ?????? :

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

????????? ????? ????????? ???????????? ??????????? ?????? ???? ? ???? ???????? ?????? ??????? ????? ?????????? (Poisson statistics) ??????? ?

${\displaystyle \langle (\Delta N)^{2}\rangle =\langle N\rangle ^{2}}$

???????????? ?????? [ ???????? ]

???????????? ??????? ????????? ???-???????????? ??????????? ????? ?????????? ?????? ? ??? ?????? ??????? ??? ????? ???????? ???????????? ?????????? ???? ???? ???? ??? ? ?? ???????????? ??????????? ?????????? ?????? ????? ? ???????-??????, ???????-??????????, ?????? ???????????? ?????????? ??????? ????????? ??????? ??? ???????????? ?????????? ???-???????????? ??????????? ?????? ??????????? ? ??? ?????? ${\displaystyle \displaystyle i}$ ?????? ????????? ?????, ???????? ?????????? ????? ???? ${\displaystyle \displaystyle \varepsilon _{i}}$ ? ?????? ${\displaystyle \displaystyle n_{i}}$?? ????? ????? ? ??? ?????? ???????? ?????? ${\displaystyle \displaystyle g_{i}}$ ???? ????? ????? ?????? ????? ? ??? ????? ?????? ????? ???? ? ?????? ????? ???? ????????? ? ?????? ?????? ???? ????????? ????? ?????? ????? ?????, ???? ?? ??????????? ????????? ??????? ????? ????????? ??? ????? ???? ?????? ? ${\displaystyle \displaystyle i}$ ?????? ${\displaystyle \displaystyle g_{i}}$? ???????? ???? ????? ????????? ??????? ? ?? ????? ???? ????? ?? ??????????? ??????? ????????? ? ????, ??? ?????? ?? ${\displaystyle \displaystyle n}$ ??????? ${\displaystyle \displaystyle g}$ ???????? ${\displaystyle \displaystyle w(n,g)}$ ?????? ?????? ????? ?? ??????? ??? ??????????? ? ${\displaystyle \displaystyle n}$ ???? ??????? ????? ???????? ???? ????? ??????? ??????? ??????????? ; ???? ${\displaystyle \displaystyle w(n,1)=1}$ ? ??? ???????? ${\displaystyle \displaystyle n}$?? ?????? ${\displaystyle \displaystyle (n+1)}$ ??????? ????? ${\displaystyle \displaystyle n}$?? ????? ?????? ???????????? :

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

??? ???????? ${\displaystyle \displaystyle n}$ ?????? ?????? ?????? ????? ????????, ???? ???? :

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

???????

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

????? ?????? ??????? ????????? ????? ?????????? ??? ??????? ????? :

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

??? ????????? ???? ??????? ??? ?????? ?? ${\displaystyle \displaystyle w(n,g)}$ ???? ?? ?????? ??????? ????? ?

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

?????? ?????? ??? ????? ??? ???????? ?????? ??????? ???????? ???? ? ???, ???, ???, ???, ??? ? ??? ? ???? ?!/(?!?!) = ??? ???????? ????? ? ???? ${\displaystyle \displaystyle n_{i}}$ ????????? ???? ??????? ???????? ???????? ?????? ????????????????? ????? ??????? ??? ????? ?????? ????? ??????? ??? ?

${\displaystyle W=\prod _{i}w(n_{i},g_{i})=\prod _{i}{\frac {(n_{i}+g_{i}-1)!}{n_{i}!(g_{i}-1)!}}\approx \prod _{i}{\frac {(n_{i}+g_{i})!}{n_{i}!(g_{i})!}}}$

????? ${\displaystyle n_{i}\gg 1}$?? ???? ??? ??????? ????? ??????? ????? ?

??????????? ? ?????? ?????? ????? ??? ??? ???????????-??????????? ???????????? ??????? ??????? ????? ??? ${\displaystyle \displaystyle n_{i}}$? ?? ???? ?????????? ???????????, ???????? W ? ?????? ???????? ??? ? ${\displaystyle \displaystyle n_{i}}$? ???? ???? ?????? ???? ${\displaystyle \displaystyle W}$ ? ${\displaystyle \displaystyle \ln(W)}$? ??? ???????? ??? ????????? ??? ? ??????????? ????? ??????? ??? ??? ????? ?????? :

${\displaystyle f(n_{i})=\ln(W)+\alpha (N-\sum n_{i})+\beta (E-\sum n_{i}\varepsilon _{i})}$

${\displaystyle n_{i}\gg 1}$ ???? ? ????????? ????????? ??????? ??? ${\displaystyle \left(x!\approx x^{x}\,e^{-x}\,{\sqrt {2\pi x}}\right)}$ ????? ????? ????? ?????

${\displaystyle f(n_{i})=\sum _{i}(n_{i}+g_{i})\ln(n_{i}+g_{i})-n_{i}\ln(n_{i})+\alpha \left(N-\sum n_{i}\right)+\beta \left(E-\sum n_{i}\varepsilon _{i}\right)+K.}$

????? K ????? ${\displaystyle n_{i}}$? ??? ???????? ????????? ?????? ? ${\displaystyle \displaystyle n_{i}}$?? ?????? ??? ?????????? ????? ???, ??????? ??? ????? ??? ???? ? ${\displaystyle \displaystyle n_{i}}$?? ?????? ??? ???-???????????? ??????????? ????????? ??????? ?????? ?

${\displaystyle n_{i}={\frac {g_{i}}{e^{\alpha +\beta \varepsilon _{i}}-1}}}$

???-???????????? ??????????? ?????? [ ???????? ]

?? ????????? ?????????? ????????? ?????? ????????? ???-???????????? ?????????? ???? ????????? ??????? ??? ? - ???? ?????????? ????? ???? ??????? ?????? ??? ??????????? ??????? ??? ? ??????????? ????? (Divergence From Randomness - DFR) ????? ???????? ??? ??? ????? ???? ????? ?????? ? - ????? ????? ???? (WWW) ? ??????? ????????? ???? ???? ?????????? ???-???????????? ??????? ???????? ?????? ? ???-???????????? ?????????? ??????? ??? ?

???? [ ???????? ]