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_i particles among them.

• Number of ways to distribute n₁ particles in first state are

                  

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

                  

            …     …     …     …     …     … 

• 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