Maxwell Boltzmann Statistics

Maxwell Boltzmann Statistics

It is applied to distinguishable particles.
Particles are distinguishable from each other.
Each cell may contain 0, 1, 2, … n_i particles.
Total number of particles of system remain constant, n = Σn_i = constant
Sum of energies of all the particles in the different groups taken together i.e., total energy of the system remain constant E = Σn_iε_i = constant

Consider a system of n distinguishable particles.
These particles be divided into groups or levels such that
Energy levels ε_1,ε_2,ε_{3, ...}ε_i
Degeneracies g_1,g_2,g_{3, ...}g_i
Occupation number n_1,n_2,n_{3, ...}n_i

Consider a box, divide it into g_i sections, distribute n_iparticles among them.

Number of ways to distribute n₁ particles in first state are

$https://latex.codecogs.com/svg.image?_{}^{n}\textrm{C}_{n_{1}}=\frac{n!}{n_{1}! (n-n_{1}!)}$

Number of ways to put n₂ particles in the second state are

$https://latex.codecogs.com/svg.image?_{}^{(n-n_{1})}\textrm{C}_{n_{2}}=\frac{(n-n_{1})!}{n_{2}! (n-n_{1}-n_{2})!}$

… … … … … …

Total ways to distributions

First particle can be accommodated in any of the g_i group by g_i ways.
Since there is no restriction, so second particle can be accommodated in g_i group by g_i ways.

Thus n_i particle can be accommodated in g_i group by g_iⁿⁱ
Total number of eigen state for the whole system

According to the postulates of a priori probability of eigen state

Sterling approximation log x! = x log x – x

But δn_i is arbitrary

To know more about Maxwell-Boltzmann statistics please visit on https://youtu.be/BZdsg2Um_7w and for bilingual (Hindi/English) https://youtu.be/a35dIjtSGC0

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