Chapter-2 Maxwell-Boltzmann Statistics

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.

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

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

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

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

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.

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)

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

Taking logarithm Using stirling’s formula

Differentiating as For most probable state W= maximum therefore

Putting this in This is Maxwell Boltzmann distribution law

In Maxwell Boltzmann distribution

Momentum of particle is related with the kinetic energy as From eqn. (5)

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

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

Substituting eq (x) in eq (2)

Using standard integral

Substituting these in (7)

Total energy of the system

Using standard integral Substituting this in (8)

Putting in (9)