Bose einstein Statistical Distribution

Bose–Einstein (BE) statistics describes the distribution of identical particles with integer spin (bosons) over different energy states when the particles are indistinguishable and there is no restriction on the number of particles occupying the same state. It was developed by: Satyendra Nath Bose Albert Einstein in 1924–1925.

Thermodynamic Probability (W)

Consider a system of bosons distributed among different energy levels.

Total number of particles = \( N \)
Energy levels = \( \epsilon_1, \epsilon_2, \epsilon_3, \dots \)
Degeneracy of level \( i \) = \( g_i \)
Number of particles in level \( i \) = \( n_i \)

\[ \sum_i n_i = N \]

\[ \sum_i n_i \epsilon_i = E \]

Step 2: Nature of Bosons

Particles are indistinguishable
Any number of particles can occupy the same state
No restriction like Pauli exclusion principle

Step 3: Combinatorial Problem

We calculate the number of ways to distribute \( n_i \) indistinguishable particles among \( g_i \) quantum states. The number of possible ways to distribute n distinct particles in g number of cells , so that any number of particles can occupy any cell can be understood by a model called Stars and Bars Method or Partition method. In this method : we can assume that there are \( n+g-1 \) number of particles , where \( g-1 \) is the number of partitions or bars. Now these particles (n+g-1) numbers can be distributed among themselves in \( (n+g-1)! \) ways . But these ways include several ways in which partitions or bars have been exchanged among themselves which are identical in nature and considering such ways si meaning less , hence we have to divide by \( (g-1)! \) . Similarly we had thought the particles to be distinct for simplicity of calculation but in fact these are bososns which are indistinguishables so we have to divide \( n!\) to subtract such no of ways where they have been treated identical . Hence the Possible ways or thermodynamic probability of n bosons in g number of cells is : \[ W= \dfrac{(n+g-1)!}{n! (g-1) !} \]

Mathematically:

\[ x_1 + x_2 + x_3 + \dots + x_{g_i} = n_i \]

where each \( x_k \ge 0 \).

Step 4: Stars and Bars Method

Total symbols:

\[ n_i + g_i - 1 \]

We choose positions of \( g_i - 1 \) separators.

Step 5: Thermodynamic Probability for One Energy Level

\[ W_i = \frac{(n_i + g_i -1)!}{n_i! (g_i -1)!} \]

Step 6: Total Thermodynamic Probability

Since energy levels are independent:

\[ W = \prod_i W_i \]

\[ \boxed{ W = \prod_i \frac{(g_i + n_i -1)!}{n_i! (g_i -1)!} } \]

Step 7: Large Number Approximation

Using Stirling’s approximation:

\[ \ln n! \approx n \ln n - n \]

Taking logarithm:

\[ \ln W = \sum_i \left[ (g_i+n_i)\ln(g_i+n_i) - n_i \ln n_i - g_i \ln g_i \right] \]

Final Result

Maximizing \( \ln W \) under constraints gives the Bose–Einstein distribution:

\[ n_i = \frac{g_i}{e^{(\epsilon_i - \mu)/kT} - 1} \]

Conclusion:
The thermodynamic probability for Bose–Einstein statistics is derived using combinatorics for indistinguishable particles, leading to the factorial expression above.

\[\begin{align} {{W}_{1}}&={{g}_{1}}\times \left( {{g}_{1}}-1 \right)\times \left( {{g}_{1}}-2 \right).....\left( {{g}_{1}}-\left( {{n}_{1}}-1 \right) \right) \\ & ={{g}_{1}}\left( {{g}_{1}}-1 \right)\left( {{g}_{1}}-2 \right).....\left( {{g}_{1}}-{{n}_{1}}+1 \right) \\ & =\frac{{{g}_{1}}\left( {{g}_{1}}-1 \right)\left( {{g}_{1}}-2 \right).....\left( {{g}_{1}}-{{n}_{1}}+1 \right)\left( {{g}_{1}}-{{n}_{1}} \right)\left( {{g}_{1}}-{{n}_{1}}-1 \right)....3\times 2\times 1}{\left( {{g}_{1}}-{{n}_{1}} \right)\left( {{g}_{1}}-{{n}_{1}}-1 \right)....3\times 2\times 1} \\ & =\frac{{{g}_{1}}!}{\left( {{g}_{1}}-{{n}_{1}} \right)!} \\ & \Rightarrow {{W}_{2}}=\frac{{{g}_{2}}!}{\left( {{g}_{2}}-{{n}_{2}} \right)!},{{W}_{3}}=\frac{{{g}_{3}}!}{\left( {{g}_{3}}-{{n}_{3}} \right)!}... \\ & \Rightarrow W={{W}_{1}}{{W}_{2}}{{W}_{3}}.... \\ & =\frac{{{g}_{1}}!}{\left( {{g}_{1}}-{{n}_{1}} \right)!}\frac{{{g}_{2}}!}{\left( {{g}_{2}}-{{n}_{2}} \right)!}\frac{{{g}_{3}}!}{\left( {{g}_{3}}-{{n}_{3}} \right)!}... \\ & =\prod\limits_{i=1}^{i=n}{\frac{{{g}_{i}}!}{\left( {{g}_{i}}-{{n}_{i}} \right)!}} \\ \end{align}\]

\[\begin{align} & \ln W=\ln \prod\limits_{i=1}^{i=n}{\frac{\left( n{{ & }_{i}}+{{g}_{i}}-1 \right)!}{{{n}_{i}}!\left( {{g}_{i}}-1 \right)!}} \\ & =\ln \prod\limits_{i=1}^{i=n}{\frac{\left( n{{ & }_{i}}+{{g}_{i}} \right)!}{{{n}_{i}}!{{g}_{i}}!}} \\ & =\sum{\ln \left( n{{ & }_{i}}+{{g}_{i}} \right)!}-\sum{\ln \left( {{n}_{i}}!{{g}_{i}}! \right)} \\ & =\sum{\left( {{n}_{i}}+{{g}_{i}} \right)\ln \left( {{n}_{i}}+{{g}_{i}} \right)-}\sum{\left( {{n}_{i}}+{{g}_{i}} \right)}-\sum{\ln {{n}_{i}}!}-\sum{\ln {{g}_{i}}!} \\ & =\sum{\left( {{n}_{i}}+{{g}_{i}} \right)\ln \left( {{n}_{i}}+{{g}_{i}} \right)-}N-\sum{{{g}_{i}}-}\sum{\ln {{n}_{i}}!}-\sum{{{g}_{i}}\ln {{g}_{i}}+}\sum{{{g}_{i}}} \\ & \Rightarrow d\left( \ln W \right)=\sum{d{{n}_{i}}\ln \left( {{n}_{i}}+{{g}_{i}} \right)-\sum{\left( {{n}_{i}}+{{g}_{i}} \right)\frac{1}{\left( {{n}_{i}}+{{g}_{i}} \right)}d{{n}_{i}}-}}\sum{d{{n}_{i}}\ln {{n}_{i}}+\sum{{{n}_{i}}\frac{1}{{{n}_{i}}}d{{n}_{i}}}} \\ & =\sum{d{{n}_{i}}\ln \left( {{n}_{i}}+{{g}_{i}} \right)-\sum{d{{n}_{i}}-}}\sum{d{{n}_{i}}\ln {{n}_{i}}+\sum{d{{n}_{i}}}} \\ & =\sum{d{{n}_{i}}\ln \left( {{n}_{i}}+{{g}_{i}} \right)-}\sum{d{{n}_{i}}\ln {{n}_{i}}} \\ & =\sum{\ln \frac{\left( {{n}_{i}}+{{g}_{i}} \right)}{{{n}_{i}}}d{{n}_{i}}} \\ & \sum{\left( \alpha +\beta {{E}_{i}} \right)d{{n}_{i}}}=0 \\ & \Rightarrow \sum{\ln \frac{\left( {{n}_{i}}+{{g}_{i}} \right)}{{{n}_{i}}}d{{n}_{i}}}=\sum{\left( \alpha +\beta {{E}_{i}} \right)d{{n}_{i}}}=0 \\ & \Rightarrow \ln \frac{\left( {{n}_{i}}+{{g}_{i}} \right)}{{{n}_{i}}}=\left( \alpha +\beta {{E}_{i}} \right) \\ & \Rightarrow \frac{\left( {{n}_{i}}+{{g}_{i}} \right)}{{{n}_{i}}}={{e}^{\alpha +\beta {{E}_{i}}}} \\ & \Rightarrow 1+\frac{{{g}_{i}}}{{{n}_{i}}}={{e}^{\alpha +\beta {{E}_{i}}}} \\ & \Rightarrow \frac{{{g}_{i}}}{{{n}_{i}}}={{e}^{\alpha +\beta {{E}_{i}}}}-1 \\ & \Rightarrow {{n}_{i}}=\frac{{{g}_{i}}}{{{e}^{\alpha +\beta {{E}_{i}}}}-1} \\ & \Rightarrow \frac{{{n}_{i}}}{{{g}_{i}}}=\frac{1}{{{e}^{\alpha +\beta {{E}_{i}}}}-1} \\ \end{align}\]