On an Improvement of the Planck radiation Energy D

arXiv:physics/0603117v3[physics.gen-ph]18Jul2006ONANIMPROVEMENTOFTHEPLANCKRADIATIONENERGYDISTRIBUTIONDiegoSa´a1Copyrightc2006AbstractTheprobabilitydistributionfunctionforthermodynamicsandecono-physicsisobtainedbysolvinganequilibriumequation.Thisapproachisdifferentfromthecommononeofoptimizingtheentropyofthesys-temorobtainingthestateofmaximumprobability,whichusuallyobtainsasaresulttheBoltzmanndistribution.TheGammadistri-butionisproposedasabetterequationtodescribetheblackbodyradiationinsubstitutionofPlanck’sradiationequation.Also,anewformofentropyisproposed,thatmaintainsthecorrectrelationwiththeClausius’formula.PACS:02.50.-r,05.20.-y,65.50.+mkeywords:statisticalmechanics,thermodynamics,econophysics,thermo-dynamicequilibrium,probabilitydistributions1.INTRODUCTIONKirchhoffintroducedtheconceptofaperfectlyblackbody,beingonewhichabsorbsallincidentradiation.Hewasthefirstinpresentingtheenunci-ationoftheprincipleofthebalancebetweenemissionandabsorptionaround1860.Theprincipleofequilibriumofemissionandabsorptionbetweendiffer-entagentsis,essentially,themethodthatwillbeusedinthepresentpapertodevelopanewequationthatdescribestheradiationdistributionofablackbody.By1900Planckdevisedtheknownequationthatdescribesthedistribu-tionofradiationemittedbyablackbody,basedonhunchandpureguessing.Althoughitisratheraccurateovertheentirerangeoffrequenciesitissug-gestedherethatitisnotappropriatetodescribetheblackbodyradiationdistribution.1EscuelaPolit´ecnicaNacional.Quito–Ecuador.email:[e-mail]1“Althoughtheintroductionofenergyquantaacenturyagohasledtothecurrentlyacceptedexplanationwithinquantumtheory,thereisstillnofirmconclusionastowhetherornotblackbodyradiationcanbeexplainedwithinclassicalphysics.”[7]Oneofthebasicassumptionsinthispaperisthatenergyexistsinpackets,orquanta,whichhavecontinuousvalues.ThisdeservessomeexplanationbecausethisisagainstoneofthemostbasicandacceptedtenetsofPhysics,intheso-calledareaofquantumtheory.Theelementaryparticlescalledfermions,suchastheelectronandproton,havesomeveryspecificamountsofenergyandalsotheorbitsoftheelectronsintheatomsseemtobeatdiscretelevels.Consequently,itcanbesafelyacceptedthatthoseelementaryparticlessatisfytheclassicalquantumtheory.Moreover,itcanbeacceptedthat,whentheelectronsjumpbetweenlevelsintheatomicorbits,theyemitandabsorbphotonswithdiscretelevelsofenergy.However,thisdiscretecharacterisnotlogicallyandnecessarilyextrapo-latedtothephotonsproduced,forexample,intheblackbodyoranantennaradiation.Ithasnotbeenprovedthatsuchphotonshavediscretelevelsofenergy.Onthecontrary,electromagneticradiationseemstohaveacontinu-ousspectruminthewholerangeoffrequenciesorwavelengths.Thepresentauthorhasnotbeenabletofindanyexperimentsupportingtheoppositeposition.Itseemsthatsomephysicistsmighthavesomemisconceptionsaboutthecurrentblackbodyradiationtheory.Forexample,thepeerreviewersofPhysicaA,undertheordersofProf.C.Tsallis,usedtheargumentthat“ItisobviousthattheenergyspectrumgivenbythePlanckdistributioniscontinuous”,torejectaformerversionofthepresentpaper.Onthelightofthepointsofviewsuggestedbythefollowingreference,suchargumentgivestheimpressionofbeingwrong.Letusreferas“A&F”tothebookofAlonso&Finn,“FundamentalUni-versityPhysics,VolumeIII,QuantumandStatisticalMechanics”,Addison-Wesley,1968,whichisreferredtointhepresentpaperas[1].Bydefinition,QuantumMechanicsdealswithenergylevels.Thisissug-gestedinFig.2-10ofA&F.Insection10-3,A&Fcarryouttheproofofthe“Maxwell-Boltzmanndistribution.”Inparticular,inthesecondparagraphofthatsection,theauthorsmention“thenumberofparticlespresentineachoftheenergylevels”.2Fig.10-1,appearinginthesamesection,isalsoagoodillustrationofthisconcept.Thederivationofthe“probabilitydistributionaccordingtotheMaxwell-Boltzmanndistribution”,equation(10.8),isveryclearandshouldbereviewedtoconfirmthattheauthorsA&Falwaysusediscreteenergylevels.Afterusingasimplemathematicaloptimizationtechniquetheyfindthe“partitionofmaximumprobability”andthelawof“Maxwell-Boltzmanndistribution”,equations(10.9)and(10.11),whichprovidetheintegernumberofparticlesineachenergylevel.Itisobviousthattheproductsofintegernumbersbydiscreteenergylevelsproducediscretelevelsfortheenergydistribution.MorerelevanttothepresentpaperisthederivationofthePlanckradia-tionenergydistribution,eq.(13.20),whichiscarriedoutbyA&Finsection13.6,thattheycall“maybethemostimportantapplicationoftheBose-Einsteinstatistics”(rathersimilartotheMaxwell-Boltzmanndistribution,giventheadditionalconsiderationsidentifiedinthecitedbook).Theauthorsmentionthatitisonlyanapproximationthattheenergyspectrumcouldbeconsideredcontinuous,undertheconditionthat“thecavitybegreatrelativetothemeanradiationwavelengthbecausetheenergydifferenceamongtwosuccessiveenergylevelsissmall.”Thislastconditionissatisfiedbecause,ingeneral,wehavea“bigbox”,aspicturedinFig.2-10(b),comparedwiththesmallvaluesobtainedforenergy,whichisjusttheproductofthefrequenciesofthepacketsbythePlanckconstant.ButthisdoesnotmeanthatQuantumMechanicsdoesnotconsiderenergylevelsorthatthoselevelshavesomehowdisappearedduringthederivation.Moreov