Detailed discussion about Maxwell-Boltzmann statistics with the derivation of the relevant equations .Probable applications and the limitations.  

Maxwell Boltzmann Statistics

    The Maxwell-Boltzmann statistics takes classical principles into consideration and do not take any quantum principle into consideration.   

Basic Features of Maxwell Boltzmann statistics:

    The basic postulates of MB statistics are given by:

1. The particles of the system are spin-less, identifiable and distinguishable.
2. Neither the Heisenberg's uncertainty principle nor the exclusion principle of Pauli applies to the particles.   
3. There is no apriority restriction applies to the particles i.e. any of the particles can be accommodated in any energy state independently.  

Calculation for the Maxwell-Boltzmann Distribution Function:

    Let, N be the total number of distinguishable particles in the system,  be the number of particles with energies  respectively and for the shake of generality   be the number of  quantum  states for the energy level .

    If the system is Isolated the total number of particles is conserved. i.e. 

               [  is the number of particles in the i-th energy state. ]

 ------------------- (1)

   

 Also, if the particles are non-interacting then the total energy of the system U is constant

i.e. 

---------------------- (2)

    The equation (1) and (2) are the two condition equation for the motion.

    

    Now, the number of ways in which the  N particles with groups of   particles could be distributed is given by,

------------------- (3)

Where,   stands for product of particles following it and n is the number of  energy levels.

    Now,  particles are to be arranged in  states, each state having the same a priori probability of being occupied independent of the particles. So, the number of ways in which each particular i-th group with  particles, having energy  can be distributed in the  quantum states is given by .This is true for any value of i . 

    Thus considering all the values of i , the total number ways in which the particles can be distributed in quantum states is given by,

---------------------------(4)

( Thus,  total number of microstates of the system is  given by Ω . )

    Now, taking logarithm of both sides of equation (4) and making use of starling's approximation

we get ,

---------------------------- (5)

    Now, for the equilibrium condition the most probable distribution takes place. For that particular distribution the entropy of the system (S) must be maximum. Then we must have,

Again we know, 

 [  From the Boltzmann relation. Click here for details derivation of the equation ]

therefore, 

---------------------------- (6)

In this condition, putting the value of  ln Ω  from equation (5)  in equation (6) we get,

------------------------------ (7)

    Using the method of  Lagrange's undermined multipliers , multiplying  equation (1) and (2) with -α and -β respectively and adding with equation (7) we get ,

Now, since the  's are in effect independent then the term in the bracket Vanishes for all i .

[ Where , α and β are constants to be determined ]

--------------------------------- (8)

    Now, the function -

------------------------------------ (9)

is called the Maxwell-Boltzmann distribution function. It gives the average number of particles per quantum state.

Thus the total number of particles ,

The summation   plays an important role in statistical theory and is termed as Partition Function(Z) or Sum of States.

---------------------------------- (10)

Thus,

--------------------------------- (11)

Evaluation of β  :

From equation (8) we have, 

putting,

  

We get,

Now, Putting these values in equation (5) we get,

-----------------------------(12)

-------------------------------(13)

We shall now introduce temperature T from thermodynamics and write,

------------------------------------(14)

Now, from equation (13)

------------------------------------(15)

From the definition of U,Z and  , therefore we obtain

------------------------------------(16)

Using this expression in equation (15) the first and the third term of the right hand side  cancel out,

So, finally, the Maxwell-Boltzmann distribution function assumes the following form,

Applications of MB statistics :

    The MB- statistics is applicable to the ensembles of particles forming a dilute gas and also to free electrons in the conduction band of a semiconductor at ordinary temperatures, provided the doping of the semiconductor is not too high.

Limitations of MB statistics:

1. This statistics is applicable only to isolated gas molecular system in equilibrium, when the mean potential energy due to mutual interaction between the molecules is negligible compared to their mean kinetic energy, and the gas is dilute. [ For dilute gases the number of molecules per unit volume is small so that the average separation between the molecules is large. Individual molecules could then be distinguished. ]   
2. The expression for MB-count does not lead to the correct expression for entropy of an ideal gas. It leads to the Gibbs paradox which can be resolved if expression is divided by N! .
3. If we put T=0 in the expression for entropy of an ideal gas, the entropy S becomes negative quantity which is at variance with the Third Law of Thermodynamics S → 0 at T→ 0. 
4.     All the difficulties with MB-statistics have been satisfactorily removed by the quantum statistics which will be discussed in the next parts.

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