إحصاء بوز-أينشتاين (بالإنجليزية Bose-Einstein Statistics) وإحصاء فرمي-ديراك Fermi-Dirac Statistics هي نظم لتوزيع الجسيمات الأولية في الإحصاء الكمومي. وتنتمي الپوزونات إلى إحصاء بوز-أينشتاين، وتنتمي الفرميونات إلى إحصاء فرمي-ديراك.

ويعطي جميع نظام منها عدد الجسيمات ${\displaystyle ⟨<!--\; ⟨\; -->n(E)⟩<!--\; ⟩\; -->}$

في حالة عدم وجود تآثر بين تلك الجسيمات تعطينا المعادلة الأتية توزيع البوزونات (تتميز البوزونات بعزم مغزلي 0 أوSpin=1):

${\displaystyle \langle n(E)\rangle ={\frac {1 {e^{\beta (E-\mu ) -1$

حيث:

μ الجهد الكيميائي

${\displaystyle \beta$ تساوي عادة ${\displaystyle 1/(k_{B T)$

k_B ثابت بولتزمان

T درجة الحرارة كلفن

ويعتمد الجهد الكيميائي على درجة الحرارة.

تعطينا المعادلة عدد الجسيمات في الحالة الكمومية E. وإذا كانت الحالة E منفطرة (مفصصة طبقا لميكانيكا الكم ) فيجب ضرب درجة الانفطار g_i في المعادلة السابقة.

عند درجة الحرارة الحرجة المنخفضة جدا ${\displaystyle T_{\lambda$ نحصل على الحالة الخاصة في عدم وجود تآثر بين الجسيمات، مع افتراض حتى الجهد الكيميائي μ قريب من مستواه الأدنى، نحصل على تكثف بوز-أينشتاين.

وفي حالة توزيع فرمي-ديراك نحصل على المعادلة السابقة ولكنقد يكون المقام مجموع أجزائه (+) بدلا من الفرق بين جزئيه(-).

أي:

${\displaystyle \langle n(E)\rangle ={\frac {1 {e^{\beta (E-\mu ) +1$

وبالنسبة للفرميونات فهي تتبع إحصاء فرمي-ديراك، وهي تتحول إلى عند الطاقات العالية E إلى توزيع بولتزمان، كما يتحول أيضا توزيع بوز-أينشتاين عند الطاقات العالية إلى توزيع بولتزمان. وكان توزيع بولتزمان أصلا يصف توزيع الذرات أوالجزيئات في نظام غازي في حالة توازن حراري.

تتميز الفرميونات حتى لها عزم مغزلي 1/2.

ملاحظات

Example n = 4, g = 3:

${\displaystyle S(4,3)=\left\{\underbrace {(1111),(1112),(1113) _{(a) ,\underbrace {(1122),(1123),(1133) _{(b) ,\underbrace {(1222),(1223),(1233),(1333) _{(c) ,\right.$

${\displaystyle \left.\underbrace {(2222),(2223),(2233),(2333),(3333) _{(d) \right\$

${\displaystyle \displaystyle w(4,3)=15$ (there are ${\displaystyle \displaystyle 15$ elements in ${\displaystyle \displaystyle S(4,3)$)

Subset ${\displaystyle \displaystyle (a)$ is obtained by fixing all indices ${\displaystyle \displaystyle m_{i$ to ${\displaystyle \displaystyle 1$, except for the last index, ${\displaystyle \displaystyle m_{n$, which is incremented from ${\displaystyle \displaystyle 1$ to ${\displaystyle \displaystyle g=3$. Subset ${\displaystyle \displaystyle (b)$ is obtained by fixing ${\displaystyle \displaystyle m_{1 =m_{2 =1$, and incrementing ${\displaystyle \displaystyle m_{3$ from ${\displaystyle \displaystyle 2$ to ${\displaystyle \displaystyle g=3$. Due to the constraint ${\displaystyle \displaystyle m_{i \geq m_{i-1$ on the indices in ${\displaystyle \displaystyle S(n,g)$, the index ${\displaystyle \displaystyle m_{4$ must automatically take values in ${\displaystyle \displaystyle \left\{2,3\right\$. The construction of subsets ${\displaystyle \displaystyle (c)$ and ${\displaystyle \displaystyle (d)$ follows in the same manner.

Each element of ${\displaystyle \displaystyle S(4,3)$ can be thought of as a multiset of cardinality ${\displaystyle \displaystyle n=4$; the elements of such multiset are taken from the set ${\displaystyle \displaystyle \left\{1,2,3\right\$ of cardinality ${\displaystyle \displaystyle g=3$, and the number of such multisets is the multiset coefficient

${\displaystyle \displaystyle \left\langle {\begin{matrix 3\\4\end{matrix \right\rangle ={3+4-1 \choose 3-1 ={3+4-1 \choose 4 ={\frac {6! {4!2! =15$

More generally, each element of ${\displaystyle \displaystyle S(n,g)$ is a multiset of cardinality ${\displaystyle \displaystyle n$ (number of dice) with elements taken from the set ${\displaystyle \displaystyle \left\{1,\dots ,g\right\$ of cardinality ${\displaystyle \displaystyle g$ (number of possible values of each die), and the number of such multisets, i.e., ${\displaystyle \displaystyle w(n,g)$ is the multiset coefficient

${\displaystyle \displaystyle w(n,g)=\left\langle {\begin{matrix g\\n\end{matrix \right\rangle ={g+n-1 \choose g-1 ={g+n-1 \choose n ={\frac {(g+n-1)! {n!(g-1)!$

(2)

which is exactly the same as the formula for ${\displaystyle \displaystyle w(n,g)$, as derived above with the aid of a theorem involving binomial coefficients, namely

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

(3)

To understand the decomposition

${\displaystyle \displaystyle w(n,g)=\sum _{k=0 ^{n w(n-k,g-1)=w(n,g-1)+w(n-1,g-1)+\cdots +w(1,g-1)+w(0,g-1)$

(4)

or for example, ${\displaystyle \displaystyle n=4$ and ${\displaystyle \displaystyle g=3$

${\displaystyle \displaystyle w(4,3)=w(4,2)+w(3,2)+w(2,2)+w(1,2)+w(0,2),$

let us rearrange the elements of ${\displaystyle \displaystyle S(4,3)$ as follows

${\displaystyle S(4,3)=\left\{\underbrace {(1111),(1112),(1122),(1222),(2222) _{(\alpha ) ,\underbrace {(111{\color {Red {\underset {= {3 ),(112{\color {Red {\underset {= {3 ),(122{\color {Red {\underset {= {3 ),(222{\color {Red {\underset {= {3 ) _{(\beta ) ,\right.$

${\displaystyle \left.\underbrace {(11{\color {Red {\underset {== {33 ),(12{\color {Red {\underset {== {33 ),(22{\color {Red {\underset {== {33 ) _{(\gamma ) ,\underbrace {(1{\color {Red {\underset {=== {333 ),(2{\color {Red {\underset {=== {333 ) _{(\delta ) \underbrace {({\color {Red {\underset {==== {3333 ) _{(\omega ) \right\ .$

Clearly, the subset ${\displaystyle \displaystyle (\alpha )$ of ${\displaystyle \displaystyle S(4,3)$ is the same as the set

${\displaystyle \displaystyle S(4,2)=\left\{(1111),(1112),(1122),(1222),(2222)\right\$.

By deleting the index ${\displaystyle \displaystyle m_{4 =3$ (shown in red with double underline) in the subset ${\displaystyle \displaystyle (\beta )$ of ${\displaystyle \displaystyle S(4,3)$, one obtains the set

${\displaystyle \displaystyle S(3,2)=\left\{(111),(112),(122),(222)\right\$.

In other words, there is a one-to-one correspondence between the subset ${\displaystyle \displaystyle (\beta )$ of ${\displaystyle \displaystyle S(4,3)$ and the set ${\displaystyle \displaystyle S(3,2)$. We write

${\displaystyle \displaystyle (\beta )\longleftrightarrow S(3,2)$.

Similarly, it is easy to see that

${\displaystyle \displaystyle (\gamma )\longleftrightarrow S(2,2)=\left\{(11),(12),(22)\right\$

${\displaystyle \displaystyle (\delta )\longleftrightarrow S(1,2)=\left\{(1),(2)\right\$

${\displaystyle \displaystyle (\omega )\longleftrightarrow S(0,2)=\varnothing$ (empty set).

Thus we can write

${\displaystyle \displaystyle S(4,3)=\bigcup _{k=0 ^{4 S(4-k,2)$

or more generally,

${\displaystyle \displaystyle S(n,g)=\bigcup _{k=0 ^{n S(n-k,g-1)$;

(5)

and since the sets

${\displaystyle \displaystyle S(i,g-1)\ ,\ {\rm {for \ i=0,\dots ,n$

are non-intersecting, we thus have

${\displaystyle \displaystyle w(n,g)=\sum _{k=0 ^{n w(n-k,g-1)$,

(6)

with the convention that

${\displaystyle \displaystyle w(0,g)=1\ ,\forall g\ ,{\rm {and \ w(n,0)=1\ ,\forall n$.

(7)

Continuing the process, we arrive at the following formula

${\displaystyle \displaystyle w(n,g)=\sum _{k_{1 =0 ^{n \sum _{k_{2 =0 ^{n-k_{1 w(n-k_{1 -k_{2 ,g-2)=\sum _{k_{1 =0 ^{n \sum _{k_{2 =0 ^{n-k_{1 \cdots \sum _{k_{g =0 ^{n-\sum _{j=1 ^{g-1 k_{j w(n-\sum _{i=1 ^{g k_{i ,0).$

Using the convention (7)₂ above, we obtain the formula

${\displaystyle \displaystyle w(n,g)=\sum _{k_{1 =0 ^{n \sum _{k_{2 =0 ^{n-k_{1 \cdots \sum _{k_{g =0 ^{n-\sum _{j=1 ^{g-1 k_{j 1,$

(8)

keeping in mind that for ${\displaystyle \displaystyle q$ and ${\displaystyle \displaystyle p$ being constants, we have

${\displaystyle \displaystyle \sum _{k=0 ^{q p=qp$.

(9)

It can then be verified that (8) and (2) give the same result for ${\displaystyle \displaystyle w(4,3)$, ${\displaystyle \displaystyle w(3,3)$, ${\displaystyle \displaystyle w(3,2)$, etc.

التطبيقات متداخلة المجالات

طالع أيضا

• Bose–Einstein correlations
• بوزون
• بوزون هيگز
• Parastatistics
• Planck's law of black body radiation
• جسيم أولي
• إحصاء فرمي-ديراك
• إحصاء ماكسويل-بولتزمان
• توزيع بولتزمان
• ميكانيكا إحصائية

الهامش

المراجع

• Annett, James F. (2004). Superconductivity, Superfluids and Condensates. New York: Oxford University Press. ISBN .
• Carter, Ashley H. (2001). Classical and Statistical Thermodynamics. Upper Saddle River, NJ: Prentice Hall. ISBN .
• Griffiths, David J. (2005). Introduction to Quantum Mechanics (2nd ed.). Upper Saddle River, NJ: Pearson, Prentice Hall. ISBN .