entropy and partial differential equations - evans l.c.

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1、EntropyandPartialDifferentialEquationsLawrenceC.EvansDepartmentofMathematics,UCBerkeleyInspiringQuotationsAgoodmanytimesIhavebeenpresentatgatheringsofpeoplewho,bythestandardsoftraditionalculture,arethoughthighlyeducatedandwhohavewithconsiderablegustobeenexpressingtheirincredulity

2、attheilliteracyofscientists.OnceortwiceIhavebeenprovokedandhaveaskedthecompanyhowmanyofthemcoulddescribetheSecondLawofThermodynamics.Theresponsewascold:itwasalsonegative.YetIwasaskingsomethingwhichisaboutthescientificequivalentof:HaveyoureadaworkofShakespeare’s?–C.P.Snow,TheTwoCu

3、lturesandtheScientificRevolution...C.P.Snowrelatesthatheoccasionallybecamesoprovokedatliterarycolleagueswhoscornedtherestrictedreadinghabitsofscientiststhathewouldchallengethemtoexplainthesecondlawofthermodynamics.Theresponsewasinvariablyacoldnegativesilence.Thetestwastoohard.Eve

4、nascientistwouldbehard-pressedtoexplainCarnotenginesandrefrigerators,reversibilityandirreversibility,energydissipationandentropyincrease...allinthespanofacocktailpartyconversation.–E.E.Daub,“Maxwell’sdemon”Hebeganthen,bewilderingly,totalkaboutsomethingcalledentropy...Shedidgathe

5、rthatthereweretwodistinctkindsofthisentropy.Onehavingtodowithheatengines,theotherwithcommunication...“Entropyisafigureofspeechthen”...“ametaphor”.–T.Pynchon,TheCryingofLot491CONTENTSIntroductionA.OverviewB.ThemesI.EntropyandequilibriumA.ThermalsystemsinequilibriumB.Examples1.Simp

6、lefluids2.OtherexamplesC.Physicalinterpretationsofthemodel1.Equilibrium2.Positivityoftemperature3.Extensiveandintensiveparameters4.ConcavityofS5.ConvexityofE6.Entropymaximization,energyminimizationD.Thermodynamicpotentials1.ReviewofLegendretransform2.Definitions3.MaxwellrelationsE

7、.CapacitiesF.Moreexamples1.Idealgas2.VanderWaalsfluidII.EntropyandirreversibilityA.Amodelmaterial1.Definitions2.Energyandentropya.Workingandheatingb.FirstLaw,existenceofEc.Carnotcyclesd.SecondLawe.ExistenceofS3.Efficiencyofcycles4.Addingdissipation,ClausiusinequalityB.Somegeneralthe

8、ories1.Entropyandefficiency1a.Definitionsb.Existen