STATISTICAL MECHANICS

MICROSCOPIC AND MACROSCOPIC SYSTEM

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

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-6N Position co ordinates 3N and Momentum co ordinates 3N.

STATISTICAL DISTRIBUTION

 Consider 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, Distinguishable  Overlapping of wave function negligible extent.  Particles having any spin  Obeys Maxwell- Boltzmann statistics.  Eg : Gas molecules CLASSICAL PARTICLES

 Identical, inistinguishable  wave function Overlaps.  Integral spin(0,1,2,3…)  Do not Obeys pauli’s exclusion principle.  Obeys Bose –Einstein statistics.  Eg : photons, particles BOSONS

 Identical, inistinguishable  wave function Overlaps.  Half Integral spin(1/2)  Obeys pauli’s exclusion principle.  Obeys Fermi –Dirac statistics.  Eg : electrons, protons,nutrons FERMIONS

ni=ni= n i = Maxwell -Boltzman statistics

BOSE-EINSTIEN statistics n i = BOSE-EINSTIEN statistics

n i = Fermi-dirac statistics

COMPARISON BETWEEN 3 STATISTICS

APPLIES TO SYSTEM OF  IDENTICAL, DISTINGUSHABLE  Identical, inistinguishable  Do not Obeys pauli’s exclusion principle.  Identical, inistinguishable  Obeys pauli’s exclusion principle. CATEGORY CLASSICAL PARTICLES BOSONSFERMIONS SPIN AND WAVE FUNCTION  Any spin  Overlapping of wave function  wave function Overlaps.  Integral spin(0,1,2,3..)  Odd half Integral spin(1/2,3/2,5/2,…)  wave function Overlaps. DISTRIBUTION FUNCTION +1 Examples Gas molecules Photons,Phonons,liquid Helium Electrons,protons,nuetrons MBBEFD