Statistical MechanicsThe purpose of Stat Mech is tod Furnish a microscopic basis for theLaws of Thermodynamics

Ei To provide a systematic method forextending the concepts of Thermodynamics

to new situationsClassical MechanicsQuantum Mechanics

We have learnedto predict behaviourof systems with a few

degrees of freedomThis procedure becomes impractical when dealingwith N 10 degrees of freedom Macroscopici e for a Thermodynamic System

Thermodynamic Parameters measure ablemacroscopic quantities associated with system such asP V T magnetic field etc defined

experimentallystart Mech relate these to microscopic

propertiesBeyond purely intellectual satisfaction ofunderstanding ordinary ThermodynamicsThere is new physics associated with interactingmany body systems

e g superconductivitysuper fluiditymagnetismlasers non equilibrium

phase transitionssupernovae of distant stars

Modern Stat Mech provides a predictivecapability for many new and striking cooperative

phenomenaA look at the overall forest

FundamentIPrincinishidMchanicsIf a system in Equilibrium can be in one

of N states then the probab izf that systemin a state n having an energy Enis given by Iz e En1kg 1

Nwhere 2 E e

En1h51 partition functionh I

1 temperaturebB Boltzmann's constant

1.38 X 15 erg degClarification

Equilibrium is a state in which the ThermodynamicParameters do not change with time

est

i Large system system could be small butreservoir is very large

Ci Long measurement time scale i e the systemhas had a large number of interactions with itsenvironment on the time scale of measurementThis is constnasted with Late in QuantumMechanics Coherent evolution of state requiresthat time scale of interaction be very longotherwise there is collapse of wavefunction to a

stationary state

Let II be the state rector of thesystemwith energy Ei and A is a quantum mechanicaloperator for a Physical Observable then theexpected value of the observable is

LA Lz Lil Ali eEilkBT

Fundamental Law ofStat mech

EgeanigeapIIawmaghtegmperatue

µ pplicationThermodynamic to variouslarified new cases

Assumption Equal a Priori Probabities

Consider two distinct states r and s of aThermodynamic system that have the same energy Er Es

Ergodic Small random fluctuations from heat bathHypothesis will cause system initially in a state Er to

sample all other States such as Es

i.e The probabilities that system is in staterand state s are equetprobability that system has energy F

is only a function of EPCE