1. Chapter-2
Maxwell-Boltzmann Statistics

2. Phase Space
A combination of position and momentum space is known as phase space.
We need 6n co-ordinates to describe the behavior of dimensional space.
So, if a system consists of n particles then the system in the phase space.
The concept of phase space is very useful since it describe the behavior of a dynamic system.

3. The volume available in phase space can be divide into a large no
The volume available in phase space can be divide into a large no. of compartments, each compartment is further divided into a large no. of elementary cells of equal volumes.
Volume of elementary cell in phase space is given by
Let

4. According to uncertainty principle
Total no. of elementary cells in phase space

5. Three kinds of statistics
Quantum Statistics
Classical Statistics
Maxwell-Boltzmann
Statistics
Bose-Einstein
Statistics
Fermi-Dirac
Statistics

6. CLASSICAL OR M-B STATISTICS BOSE-EINSTEIN STATISTICS
PARAMETER
CLASSICAL OR M-B STATISTICS
BOSE-EINSTEIN STATISTICS
FERMI-DIRAC STATISTICS
Particles
Particles are distinguishable
Particles are indistinguishable
Size of phase space cell
The size of phase space cell can be as small as we require
The size of the phase space cell cannot be less than h3
The size cannot be less than h3
Number of cells
If ni be the number of particles and gi the number of cells then gi>>ni so ni/gi<<1. Thus,number of cells can be made as large as possible
The number of cells is less than or comparable to the number of particles gi < ni
ni/gi  1
The number of cells has to be greater or equal to the number of particles so ni /gi  1
Restriction on particles
No restriction
Restriction due to Pauli Exclusion Principle
Two particle distribution in two cells
The particles are distinguishable and can be arranged in four ways
The particles are indistinguishable and can be arranged in three ways
No two particles can occupy the same cell. Hence, only one arrangement

7. Basic Approach in three Statistics
In any dynamic isolated system, the total number of particles (n) and the total energy (U) has to remain constant
When the system is in equilibrium then it exists only in the most probable state. In the most probable state of the system W must be maximum.
For all natural systems W is a very large number. We, therefore, deal with log W.

8. The three equations (1), (2) and (3) must be simultaneously satisfied by the system irrespective of the kind of statistics to be applied. These three conditions can be incorporated into a single eq. by the method of Lagrange’s undetermined multipliers. We multiply eq. (1) with - , eq. (2) by - and add to eq. (3)

9. Maxwell – Boltzmann statistics applied to an ideal gas in Equilibrium
The total energy (U) is divided into 1,2,3…..k intervals (compartments) of magnitude in 1,2,3……k. Let represent the no. of molecules in these energy intervals 1,2,3….k.
The thermodynamic prob. for macrostate

10. Taking logarithm
Using stirling’s formula

11. Differentiating as
For most probable state W= maximum therefore

12. Putting this in
This is Maxwell Boltzmann distribution law

13. In Maxwell Boltzmann distribution

14. Momentum of particle is related with the kinetic energyas
From eqn. (5)

15. where n(p) is number of molecules in the range
Therefore total no. of molecules in the range p and (p+dp)
where g(p)dp is the no. of cells in momentum interval p and p+dp

16. Number of cells in phase space corresponding to momentum interval p and (p+dp)

17. Substituting eq (x) in eq (2)

18. Using standard integral

19. Substituting these in (7)

20. Total energy of the system

21. Using standard integral
Substituting this in (8)

22. Putting in (9)