Statistical'sPurpose i Furnish a microscopic basisfor the Laws of ThermodynamicsIi to provide a systematic method forextending the concepts of thermodynamicsto new situations

Classical MechanicsQuantum Mechanics

learnedtopredict understand

systems with a fewdegrees offreedom

This procedure becomes impracticalwhendealing with Nn 1023 degrees of freedom

Macroscopici e for a Thermodynamic System

Thermodynamic Parameters

measurableFacroscopicquantities associatedwith large system such as P V T magneticfield H

stat Mechanics relates these tomicroscopic properties

Beyond the purely intellectual satisfactionof understanding ordinary Thermodynamicsthere is entirely newphysies associated withinteracting many body systems

e g superconductivitysuper fluiditymagnetismphase transitionssupernovae of distant stateslasers non equilibrium

Modern Statistical Mechanics provides apredictive capability for striking and unexpectedCooperative phenomena

fundameatdprincitahs.catMechoenics

If a system in Equilibrium can be inone of N states then the probability ofthat system being in a state n and havingenergy En is given by

eEn1451

where 2 eElknT partition

functionT temperatureKB Boltzmann's constant

1.38 10 erg deg

ClarificationEquilibrium a state in which Thermodynamic

Parameters do not change with time

IiEnvironment

i Large System system can be small butReservoir is very large

Ci Long Measurement timescalei e the system has had a largenumberofinteractions with it's environment on time scale ofmeasurement

This is contrasted to Pare State inQ M coherent evolution of state requiresthat time scale of interaction is very longotherwise there is collapse of wavefunctionto stationary state

Let Ii be the stale rector ofsystem with energy Ei and let A be a

quantum mechanical operator for a physicalobservable then the expected value of theobservable is

LA Iz Lil Ali e Either

Fundamental Law ofstatistical Mechanics

and i ie is pand thermodynamicsclarified systems

what is the microscopic meaning of theconcept of Temperature

Assumption Equal a Priori Probabilities

Consider two distinct states r and s of thethermodynamic system that have the same

energy Er Es

Ergodic Small random fluctuations fromHypothesis Heat Bath will cause the system

initially in stater with energy En tosample all other states such as Es

i e The Probabilities that the systemis in stater and states are equal

probability that system has an energyE is only a function of E

PCE