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

• For maximum probability, δ log ω = 0

• Other condition

           n = Σ n_i = constant

or       δn = Σ δn_i = 0    …(2)

and   E = Σ n_i ε_i = constant

or      δE = Σ ε_iδn_i = 0    …(3)

• Lagrange’s method of undetermined multiplier, (1) + (2) × α + (3) × β

           Σ [{log (n_i / g_i ) + 1)} + α + βε_i ] δn_i = 0

• But δn_i is arbitrary

∴      log (n_i / g_i ) + 1 + α + βε_i = 0

or     log (n_i / g_i ) + α + βε_i = 0

or      log (g_i / n_i ) = α + βε_i

or      (g_i / n_i ) = exp (α + βε_i )

or      n_i = g_i / exp (α + βε_i )