Page - 34 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 P(b)= +∞ ∑ N=0 1 N! ( 16πV c3b3 )N = exp ( 16πV c3b3 ) . ThenumberNofphotons in thevesselat thermodynamicequilibriumisastochastic function whichtakes thevaluenwiththeprobability Probability ( [N=n] ) = 1 n! ( 16πV c3b3 )n exp ( −16πV c3b3 ) . Theexpressionof thepartitionfunctionPallowsthecalculationof theinternalenergy, theentropy and all other thermodynamic functions of the system. However, the formula so obtained for the distributionofphotonsofvarious energies at agiven temperaturedoesnot agreewith the law, in verygoodagreementwithexperiments,obtainedbyMaxPlanck(1858–1947) in1900.Anassembly ofphotons in thermodynamicequilibriumevidentlycannotbedescribedasaclassicalHamiltonian system.This factplayedanimportantpart for thedevelopmentofquantummechanics. 6.3.5. SpecificHeatofSolids Themotionofaone-dimensionalharmonicoscillatorcanbedescribedbyaHamiltoniansystem with,asHamiltonian, H(p,q)= p2 2m + μq2 2 . The idea that theheatenergyofasolidcomes fromthesmallvibrations,atamicroscopicscale,of its constitutiveatoms, leadphysicists toattempt tomathematicallydescribeasolidasanassemblyofa largenumberNof three-dimensionalharmonicoscillators. Bydealingseparatelywitheachproper oscillationmode, thesolidcanevenbedescribedasanassemblyof3None-dimensionalharmonic oscillators. Exangesofenergybetweentheseoscillators isallowedbytheexistenceofsmall couplings betweenthem.However, for thedeterminationof the thermodynamicequilibriaof thesolidwewill, as in theprevioussection for idealgases, considerasnegligible theenergyof interactionsbetweenthe oscillators.Wetherefore take forHamiltonianof thesolid H= 3N ∑ i=1 ( pi2 2mi + μiqi2 2 ) . Thevalueof theparititionfunctionP, foranyb>0, is P(b)= ∫ R6N exp [ −b 3N ∑ i=1 ( pi2 2mi + μiqi2 2 )] 3N ∏ i=1 (dpidqi)= 3N ∏ i=1 ( 1 νi ) b−3N , where νi= 1 2π √ μi mi is the frequencyof the i-thharmonicoscillator. The internalenergyof thesolid is E(b)=−dlogP(b) db = 3N b . Weobserve that itonlydependsonthe the temperatureandonthenumberofatomsin thesolid, notonthe frequenciesνiof theharmonicoscillators.Withb= 1 kT this result is inagreementwith the empirical lawfor thespecificheatof solids, ingoodagreementwithexperimentsathightemperature, discovered in1819bytheFrenchscientistsPierreLouisDulong(1785–1838)andAlexisThérèsePetit (1791–1820). 34