1、 iiirNffiqHppHqiiii,21 （4）在一般物理问题中，哈密顿函数）在一般物理问题中，哈密顿函数H及其微分都及其微分都是单值函数，决定了是单值函数，决定了dqdpdpdpdqdqdff 11dtpqdtppqqff),();,;,(11Ndtpq ),(dtppdtqq ,dtdtdtpqdttdtppdtqqfff ),(),(111 iiiiitpptqqtdtd iiiiippqqt iiiiiqHppHqt 0 dtd iiiiiqHppHqt 一个重要的推论：一个重要的推论：0 t 。NrhNd!irNiiihNd!dttpqBtB),(1 ssBttB 1 dtpqp
2、qBtB),(),()(1 sst)(sssBttB)()(EEpqHEpqHpqEEpqHEpq ),(),(),(),(),(，常常数数0 1 s EEpqHENrdhN),(!11222 BBB EEpqHEirNidhNii),(!1)()(),()(2211210EEEE lnkS )(021EEE )()(),()()()(102111010EEEEEE 11EE 010 E)(0122221122111 EEEEEEEE)()()()(112 EE2211222111VNVNEEEE,)(ln)(ln 1E2EVNEVEN,),(ln dVTpdUTdS 122112211VNV
3、NUSUS,TUSVN1 ,kT1 lnkSSS 2122112211VNVNEE,lnln 22112211ENENVV,lnln 22112211VEVENN,lnln 010 E)(010 V)(010 N)(ENVVEN,),(ln VENVEN,),(ln 212121 ,能量能量dNdVdEd lndNTdVTpdUTdS 1kT1 kTkTp ,lnkS 212121 ,ppTTVNVkTp lnNVSET,NSVEp,NiimpH3122 EEpqHENNNdpdpdqdqhNE),(!)(313131 EpqHNNNdpdpdqdqhNE),(!)(313131 EmpNNN
4、NiidpdphNV3122313!iixmEp2 2332NNNmEhNVKE!)(!2323311312NdxdxKNNxNii !)(232233NNmEhVENN )()(EEENEEE 23 lnkS EENkNkNmENhVNk lnlnln232534233mmmm ln!ln0 NNNlnlim 35323235243NkSemVNhVSNE),(NkNmENhVNkS2534233 lnVEVEpENkSETSNVN3232 ,NkTpVNkTE 23NkhmkTNVNkS252232 ln NiiixmmpH31222212)()(srsEE 0)(ln)(ln)(ln)(
5、)()(sEErrrsrEEEEEr 000srEE )(ln)(0kTEEErrr10 )(ln sEse sEse sEseZ1 sEseZ lElleZ 1 lElleZ dZehNdpqpqENr),(!),(1 dehNZpqENr),(!1 sssEEU sEsseEZ 1ZeZsEsln 1ZUln sEssssseyEZyEY 1ZyeyZsEsln 111ZyYln 1ZVpln 1dSYdydUT )(1 ZZdYdydUlnln)(kT1 ZZkSlnln ZkTTSUFln sEseZ 1 sEseZ ZZkSlnln sssssEkZkUZk ln)(ln sssEZ
6、k lnssEZ lnln ssskS lnBABAEEE lElleEZ)(BAEEBEEAZZeeBBAA BABAUUZZZU lnlnln BABAFFFSSS ,sssEEEE22)(22222EEEEEEssss )(sEsEssseZeEZE 222211 EEEZZ21 VCkTTEkTEEE222 2222ECkTEEEV )(NEEE122 )(NEEE3222 )(jiijNkkrmpE)(3122 NNENdpdpdqdqehNZ313131!NrNmpNdqdqedpdpehNjiijNkk3131233121)(!AQhNN31!2331222NNkkmpmdpeA
7、k NrNrddedqdqeQjiijjiij 131)()(NrNrddedqdqeQjiijjiij 131)()(QhmNZN 23221 !NNVddQ 1 rdfVNVQN12221 rdfVNVNQ12221lnlnln很小很小假设假设 rdfVN1222 rdfVNVNQ1222lnlnQhmNNZlnln!lnln 2223 。1 )(ijrijef jiNijddfQ 11)(jiNjijiijjiijddfff 11)(上式的积分展开称为上式的积分展开称为。jiNjijiijjiijddfffQ 11)(NjiijNNjiijddfVddfQ 111)(jijiijNjiN
8、jiijNjiijddfVddddfddf 21121221112mmrmrmrrrrC rdfVddfddfC12122112 2112212 ddfNVVQNN rdfVNVN12221 rdfVNVNQ12221lnlnln12122 rdfVN rdfVNVNQ1222lnlnQVZVplnln 11 rdfVNVNp12211 rdfVNVNQ1222lnln VnBNkTpV1这就是实际气体物态方程的近似表达式，式中这就是实际气体物态方程的近似表达式，式中。rdfNBA122 6012002rrrrr )(06000rrrrrrrr )()(rdfNBA122 0020212rkT
9、rrAdrredrrNB)()(0212drreNkTrA)()(kTekT 1 kTrrNBA33230030 kTNabBA mAAVvNrNb0304432 mAAVNrNa003002432 ddrdreNBkTrAsin)()(212VanVnbNkTpV21 NkTVnbVVanp 1221 VnbVnbVnb 111NkTnbVVanp )(22固体的一般模型：固体的一般模型：Nimpi3122 jijijiiii,2002100 i 02 jiija 0312212 jijiijNiampEi,03122221 NiiiiqpE)(NiiiinnE31021021,)(iiii
10、snnsEeeeZ)(210 iiininee)(210 iinniiiiieeeee 1202100)(iiieeeZ 120 NiiieUZU3101 ln NiiU31002 Nii312 NieUU3101 1323 eNNkckctl ,dccVdDtl2332212 )(332212tlccVB dBdD2)(DdeBUdeDUUkT 0300011)(NiiieUU3101 固体的内能固体的内能NdBD302 BND93 )(,DDDDkTkTxkTy xyedyyxxD03313)()(xDNkTUU 30 xyedyykTNkTUU033019D DDdBdD0)(2 xye
11、dyyxxD03313)(NkCNkTUUV330 ,)(xDNkTUU 30 xyedyyxxD03313)(13023 xdyyx xyedyyxxD03313)(03313yedyyx34435153xx 34344054353 DVDTNkCTNkUU ,)(xDNkTUU 30 xyedyyxxD03313)(NiiiinnE31021021,)(,)(2121 nn ,kpkckctl ,pcpctl ,NiiieUU3101 系系统统源源能量能量粒子粒子)()(,00NNNEEErr )()(,00NNEE),()()(srNsEENN 00 SEErrNNrrrrEENNENE
12、ENNrr )()(lnln),(ln),(ln)()()()(000000srrENENEENN ),(ln),(ln)()()()(0000kTEVENVN1 ,),(ln kTNVENVE ,),(lnsENNse 0NsENse sENNse 110 NsNs s 0N dhNedqdppqNrpqENN!),(),(1 0NpqENrNdehNe ),(!NsENNSNSsNeNN 1 11NsENse ln N NsENsNSNSSseEEEU 1 ln 11NsENse ln U NsENsseyEY 1 lnyy 111 NsENsey 11 lnyY 1 lnVp 1 dSN
13、dYdydUT 1 NdYdydU dNdYdydUlnlnln)(NdYdydU NdYdydU kT1 kT kddSlnlnln kSlnlnln21NNNN 222NNNN )(NSENSeNN 221 NSENSNe 1 N1VTyNkTNNN,)(2VTNNkTNNN,)(222TVkTNNN 22)(0000NNNNNNsENATZees),(sNsNNeeATZ100 ),()!(!NNNNeN 000 0000000NNNNNNNNNNNeATZe)!(!),()(001Ne)(kTNlnln )(010eN2322 mkThkTpekT kTehmkTpkTNN023202
14、11 )(0110eNNsENNse 1 NsENse NsNsllaa lllllaEaN,llllsaaNsENee)(laaalallllllee)()(ll lllaale)(NsENlNsNsllseaaa 1 mmmmaalea)(1 mmmamalea)(1 mmmllalmaaleea)()(1 lmaaaalmmmllleea)()(1 lllaallea)(1lllln 1lllleeaal 110)(lel 1lnln11 leall ln)()(lllleeaal 110 lel 1lnln11 leall ln1 leall lllaaa2 llllaaaa 12 mlmlmmllaaaaaaaa NsENmlmlseaaaa 1 kkkkaamleaa 1 mlkaaaamaalkkkmmmllleeaea,)()()(1mlaamaalmlaaeaeammmlll )()(11 0 mmllaaaa
