STATISTICAL

MECHANICS

MICROSCOPIC AND MACROSCOPIC SYSTEM

• Macroscopic system: system made up of large no of particles.

• For describing such a system, macroscopically measurable independent parameters.(temperature,pressure,volume)Eg: Gas in container

• For describing microscopic particle, velocity and position of each particles are requred.

But it is impractical.We can predict the behaviour of system in terms of

macroscopic properties only.Relation connecting microscopic and macroscopic

properties gives the idea about microscopic systems.

MICROSCOPIC AND MACROSCOPIC SYSTEM

PHASE SPACE

PHASE SPACE• In classical mechanics the dynamical state

of system can be described by 3 position co ordinates x , y ,z and 3 momentum co-ordinates Px,Py,Pz.

• This combined position and momentum space is called PHASE SPACE.

• For this we have to imagine 6D space with 6 mutually perpendicular axis.

• If there are N no of particles,Total no of co ordinates-6NPosition co ordinates 3N and Momentum co

ordinates 3N.

STATISTICAL

DISTRIBUTION

STATISTICAL DISTRIBUTIONConsider system of N particles in thermal equlibrium at temp T.If E is the total energy, SM deals with the distribution of energy E among N particles.Thus we can establish how many particles having energy E1 ,how many having E2 and so on.The interaction between the particles are between one another and with the walls of the container.Here more than one particle state have same energy and also more than one particle have same Energy state. There are different no of ways W in which particles can be arranged among available states.(Greater W, more probable is the distribution)

N(E)=g(E)f(E) g(E) ---- No of states with energy E F(E)----Probability of occupancy of each state

Identical, DistinguishableOverlapping of wave function negligible extent.Particles having any spinObeys Maxwell- Boltzmannstatistics.Eg : Gas molecules

CLASSICAL PARTICLES

Identical, inistinguishablewave functionOverlaps.Integral spin(0,1,2,3…)Do not Obeys pauli’s exclusion principle.Obeys Bose –Einstein statistics.Eg : photons, particles

BOSONS

Identical, inistinguishablewave functionOverlaps.Half Integral spin(1/2)Obeys pauli’s exclusion principle.Obeys Fermi –Dirac statistics.Eg : electrons, protons,nutrons

FERMIONS

ni=

ni =

Maxwell -Boltzman statistics

BOSE-EINSTIEN statistics

ni =

ni =

-1

-1

BOSE-EINSTIEN statistics

ni =

ni =

+1

+1

Fermi-dirac statistics

COMPARISON BETWEEN

3 STATISTICS

APPLIES TO SYSTEM OF

IDENTICAL,DISTINGUSHABLE

Identical, inistinguishableDo not Obeys pauli’s exclusion principle.

Identical, inistinguishableObeys pauli’s exclusion principle.

CATEGORY

CLASSICAL PARTICLES BOSONS FERMIONS

SPIN AND WAVE FUNCTION

Any spinOverlapping of wave function wave function

Overlaps.

Integral spin(0,1,2,3..)

Odd half Integral spin(1/2,3/2,5/2,…)

wave functionOverlaps.

DISTRIBUTION FUNCTION -1 +1

Examples

Gas molecules

Photons,Phonons,liquid Helium Electrons,protons

,nuetrons

MB BE FD