.MCAD 306000000 Z  docDocument<mcObjectI (d2_graph_format graphData axisFormatLLtrace2D      dim_formatCmasslengthtimecharge temperature luminosity substanceNumericalFormat@dii shpRectEA\mcDocumentObjectStateJ mcPageModel:????mcHeaderFooter99 ComputeEngine=BuiltInsB  SerialAnyvalH@!@H @"@H#@HMbP?$@H units_classA TextState0 TextStyle/@ Arial Normalfont_style_list> font_style?  VariablesTimes New Roman?  ConstantsTimes New Roman? TextArial? Greek VariablesSymbol? User^1Arial? User^2 Courier New? User^3System? User^4Script? User^5Roman? User^6Modern? User^7Times New Roman? SymbolsSymbol? Current Selection FontArial? Undefined Font? քOgHeaderArial? քOgFooterArial? Rotated Math FontTimes New Roman TextRegion docRegion7shpBoxD;H;; CharacterMapRangeMap,BnCOPYRIGHT MIRON KAUFMAN, 1998 Statistical Physics Computer Lab #2b We consider the two-state model. Each atom has an energy of -e or e. There are N atoms and the total energy is U. Since N = Nplus + Nminus and U = e(Nplus - Nminus) we can solve for the numbers: Nplus = 1/2(N + U/e) and Nminus= 1/2(N - U/e). In the microcanonical ensemble the number of microstates corresponding to the total energy U and the number of atoms N is W = N!/Nplus!Nminus! . The entropy is, according to the Boltzmann postulate, S = kBlnW. In what follows we express the entropy in units of kB and the energy in units of e. 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Arial0,0,128CC- C)@ Arial0,0,128CC-C)@ Arial0,0,128CC-C)@ Arial0,0,128CC-C)@ Symbol0,0,128CC-C)@ Arial0,0,128CC ParPropMap*nC-nC ParPropData+EmbedMap"C-LinkMap nC-nCLinkData!@NormalArial CeqRegion3@D8e HlCtree1 pC1 CC1dCNC1C170C3@DkC1 pC1 CC1dCN.plusC1CC1tC0C1CNC3@D+Uh@C1 pC1 CC1@CC1dC\WC1pCC1 CC1dCNC1CN.plusC1CC1@CC1dCNC1CC1@CC1dCN.plusC1CC1p@CC1CC1dCNC1CN.plusC3@D2I@C1 pC1 CC1@CC1dCentropyC1pCC1 CC1dCNC1CN.plusC1CC1dClnC1pCC1CC1dC\WC1pCC1 CC1dCNC1CN.plusC3@DoCpC1 pC1CC1@CC1@CC1@CC1fC 117.835021C1C0C1CC1@CC1CC1 CC1@CC1dCentropyC1pCC1 CC1dCNC1CN.plusC1CC1dCNC1CC1dClnC1pCC1C2C1CC1@CC1@CC1tC170C1C0C1CC1@CC1CC1CN.plusC8 LL     C3@D'9(C1 pC1CC1@CC1@CC1@CC1BCC1dC 9.144842C1CC1dC10C1C49C1C1C1CD1@CD1CD1CD1dD\WD1pDD1 DD1dDND1DN.plusD1CD 1@DD 1@D D 1dD ND 1D 0D 1D D1@D D1dDND1D2D1D D1DN.plusD9 \L     D@D@>Note the huge number of microstates for Nplus = Nminus = N/2 (6)>D-)D)Dn Arial0,0,128D-D)DN Arial0,0,128DD-D)Dn Arial0,0,128DD-D)DN Arial0,0,128DD-D)Dn Arial0,0,128DDD*>D->D +"D!- >D"->D#!@NormalArial D$@D+(@COPYRIGHT @MIRON KAUFMAN(*D%-D&+"D'- D(-D)!@NormalArial D*@DYkh@ENext we define the entropy as function of energy and number of atoms.(ED+-ED,)D*n Arial0,0,128*ED--ED.+"D/- ED0-ED1!@NormalArial D23@Dsi&rD31 pD41 D3D51@D4D61dD5N.plusD71pD5D81 D7D91dD8UD:1D8ND;1D4D<1@D;D=1tD<1D>1D<2D?1pD;D@1D?DA1dD@NDB1D@UDC3@D(ZDD1 pDE1 DDDF1@DEDG1dDFSDH1pDFDI1 DHDJ1dDIUDK1DINDL1DEDM1dDLentropyDN1pDLDO1 DNDP1dDONDQ1DODR1dDQN.plusDS1pDQDT1 DSDU1dDTUDV1DTNDW@D*("$"$@|To compute the entropy, U and N must be either both even integers or both odd integers. Furthermore: -N < U < N. Examples:(|DX-|DY)DW~ 0,0,128*|DZ-|D[+"D\- |D]-|D^!@NormalArial D_3@D0 aD`1 pDa1D`Db1@DaDc1dDbSDd1pDbDe1 DdDf1tDe5Dg1De97Dh1DaDi1+@DhSerial_DisplayNodeFDj1DhDk3@D2 Dl1 pDm1DlDn1@DmDo1dDnSDp1pDnDq1 DpDr1tDq168Ds1Dq170Dt1DmDu1+@Dt@FDv1DtDw3@D`;LpHDx1 pDy1 DxDz1dDyUD{1DyD|1 @D{D}1K@D|D~1D}170D1KD|D1D168D1D{168D3@D;LHD1 pD1 DD1dDND1D170D3@DW<XD1 pD1DD1@DD1@DD1@DD1fD 117.835021D1D0D1DD1@DD1DD1 DD1@DD1dDSD1pDD1 DD1dDUD1DND1DD1dDND1DD1dDlnD1pDD1D2D1DD1@DD1@DD1tD170D1KDD1D170D1DD1@DD1DD1DUD; LL     D@D' @@LNote the maximum entropy is Nln(2) and it occurs for U = 0 or Nplus= Nminus.(K?LD-?D)Dn Arial0,0,128D-D)DN Arial0,0,128DD-D)Dn Arial0,0,128DD-D)DN Arial0,0,128DD-D)Dn Arial0,0,128DDD*LD-LD+"D- LD-LD!@NormalArial D@DYhOOAThe temperature is obtained from 1/T = dS/dU. We are going to obtained the derivative numerically 1/T = DS/DU with the smallest possible DU = 2. The temperature is negative for U > 0 when Nplus > Nminus. The temperature is positive for U <0 when Nplus < Nminus .( 'D-'D)Dn Arial0,0,128D-D)D@ Symbol0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Symbol0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Arial0,0,128DD-(D)D@ Arial0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Symbol0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Symbol0,0,128DD-D)D@ Arial0,0,128DD- D)D@ Arial0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Symbol0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Arial0,0,128DD-D)D~ 0,0,128DD-D)D~ 0,0,128DD-D)D^ 0,0,128DD-D)D~ 0,0,128DD-D)D^ 0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Arial0,0,128DD-D)D@ Arial0,0,128DD- E)D~ 0,0,128DE-E)D@ Arial0,0,128DE-E)D@ Arial0,0,128EE-E)D@ Arial0,0,128EE-E)D@ Arial0,0,128EE -E )D@ Arial0,0,128EE -E )D@ Arial0,0,128E E -E)D@ Arial0,0,128E E-E)D@ Arial0,0,128E E-E)D@ Arial0,0,128EE-E)D@ Arial0,0,128EE-E)D@ Arial0,0,128EE-E)D@ Arial0,0,128EE-E)D@ Arial0,0,128EE-E)D@ Arial0,0,128EE-E)D@ Arial0,0,128EE-E )D@ Arial0,0,128EE!-E")D@ Arial0,0,128EE#-E$)D@ Arial0,0,128E!E%-E&)D@ Arial0,0,128E#E'-E()D@ Arial0,0,128E%E)-E*)D@ Arial0,0,128E'E+-E,)D@ Arial0,0,128E)E--E.)D@ Arial0,0,128E+E/-E0)D@ Arial0,0,128E-E1-E2)D@ Arial0,0,128E/E3-E4)D@ Arial0,0,128E1E5-E6)D@ Arial0,0,128E3E7-E8)D@ Arial0,0,128E5E9-E:)D~ 0,0,128E7E;-E<)D^ 0,0,128E9E=-E>)D~ 0,0,128E;E?-E@)D@ Arial0,0,128E=EA-EB)D~ 0,0,128E?EC-ED)D^ 0,0,128EAEE-EF)D@ Arial0,0,128ECEG-EH)D@ Arial0,0,128EEEGD*EI-EJ+"EK- EL-EM!@NormalArial EN3@DEO1 pEP1 EOEQ1@EPER1dEQTES1pEQET1 ESEU1dETUEV1ETNEW1EPEX1tEW1EY1EWEZ1@EYE[1@EZE\1dE[SE]1pE[E^1 E]E_1@E^E`1dE_UEa1E_2Eb1E^NEc1EZEd1dEcSEe1pEcEf1 EeEg1dEfUEh1EfNEi1EY2Ej@DI[X8.(Ek-El)Ejn Arial0,0,128*Em-En+"Eo- Ep-Eq!@NormalArial Er@DSc`0COPYRIGHT @MIRON KAUFMAN 1998(*Es-Et+"Eu- Ev-Ew!@NormalArial Ex3@DEy1 pEz1 EyE{1dEztE|1EzE}1 @E|E~1K@E}E1E~170E1KE}E1E165E1E|170E3@D5E1 pE1EE1@EE1@EE1@EE1fE168E1KEE1E170E1EE1@EE1EE1 EE1dEUE1EE1K@EE1ENE1EE1dEtanhE1pEE1EE1tE1E1EtE1EE1@EE1@EE1tE170E1KEE1E170E1EE1@EE1EE1 EE1@EE1dETE1pEE1 EE1dEUE1ENE1EtE9 LL     E@D70'='=@The energy for a very large system is U = - N e tanh(e/kBT). (see lecture notes) The heat capacity C = dU/dT is obtained numerically as: DU/DT with the smallest possible DU =2. Note that C vs T exhibits the famous Schottky hump. (+E-+E)E~ 0,0,128E-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ Symbol0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ Symbol0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E^ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)E~ 0,0,128EE-E)En Arial0,0,128EE-E)E@ Symbol0,0,128EE-E)E@ Arial0,0,128EE-F)E@ Arial0,0,128EF-F)E@ Symbol0,0,128EF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF -F )E@ Arial0,0,128FF -F )E@ Arial0,0,128F F -F)E@ Arial0,0,128F F-F)E@ Arial0,0,128F F-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F )E@ Arial0,0,128FF!-F")E@ Arial0,0,128FF#-F$)E@ Arial0,0,128F!F%-F&)E@ Arial0,0,128F#F'-F()E@ Arial0,0,128F%F)-F*)E@ Arial0,0,128F'F+-F,)E@ Arial0,0,128F)F--F.)E@ Arial0,0,128F+F/-F0)E@ Arial0,0,128F-F1-F2)E@ Arial0,0,128F/F3-F4)E@ Arial0,0,128F1F5-F6)E@ Arial0,0,128F3F7-F8)E@ Arial0,0,128F5F9-F:)E@ Arial0,0,128F7F;-F<)E@ Arial0,0,128F9F=-F>)E@ Arial0,0,128F;F?-F@)E@ Symbol0,0,128F=FA-FB)E@ Arial0,0,128F?FC-FD)E@ Arial0,0,128FAFE-FF)E@ Symbol0,0,128FCFG-FH)E@ Arial0,0,128FEFI-FJ)E@ Arial0,0,128FGFK-FL)E@ Arial0,0,128FIFM-FN)E@ Arial0,0,128FKFO-FP)E@ Arial0,0,128FMFQ-FR)E@ Arial0,0,128FOFS-FT)E@ Arial0,0,128FQFU-FV)E@ Arial0,0,128FSFW-FX)E@ Arial0,0,128FUFY-FZ)E@ Arial0,0,128FWF[-F\)E@ Arial0,0,128FYF]-F^)E@ Arial0,0,128F[F_-F`)E@ Arial0,0,128F]Fa-Fb)E@ Arial0,0,128F_Fc-Fd)E@ Arial0,0,128FaFe-Ff)E@ Arial0,0,128FcFg-Fh)E@ Arial0,0,128FeFi-Fj)E@ Arial0,0,128FgFk-Fl)E@ Arial0,0,128FiFm-Fn)E@ Arial0,0,128FkFo-Fp)E@ Arial0,0,128FmFq-Fr)E@ Arial0,0,128FoFs-Ft)E@ Arial0,0,128FqFu-Fv)E@ Arial0,0,128FsFw-Fx)E@ Arial0,0,128FuFy-Fz)E@ Arial0,0,128FwF{-F|)E@ Arial0,0,128FyF}-F~)E@ Arial0,0,128F{F-F)E@ Arial0,0,128F}F-F)E@ Symbol0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FF-F)E@ Arial0,0,128FFE*F-F+"F- F-F!@NormalArial F3@D 0S F1 pF1 FF1@FF1dFCF1pFF1 FF1dFUF1FNF1FF1tF2F1FF1@FF1dFTG1pFG1 GG1@GG1dGUG1G2G1GNG1FG1dGTG1pGG 1 GG 1dG UG 1G NG 3@DP,[(G 1 pG1 G G1dGtG1GG1 @GG1K@GG1G10G1KGG1G9.9G1G10G@DQ0v`0(%(%@aThe heat capacity for a very large system is C = NkB(e/kBT)2/(cosh(e/kBT))2 (see lecture notes).(L3aG-3G)Gn Arial0,0,128G-G)GN Arial0,0,128GG-G)Gn Arial0,0,128GG-G)G@ Symbol0,0,128GG -G!)G@ Arial0,0,128GG"-G#)G@ Arial0,0,128G G$-G%)G@ Arial0,0,128G"G&-G')G@ Arial0,0,128G$G(-G))G@ Arial0,0,128G&G*-G+)G@ Arial0,0,128G(G,-G-)G@ Arial0,0,128G*G.-G/)G@ Arial0,0,128G,G0-G1)G@ Arial0,0,128G.G2-G3)G@ Arial0,0,128G0G4-G5)G@ Arial0,0,128G2G6-G7)G@ Arial0,0,128G4G8-G9)G@ Arial0,0,128G6G:-G;)G@ Symbol0,0,128G8G<-G=)G@ Arial0,0,128G:G>-G?)G@ Arial0,0,128GGB-GC)G@ Arial0,0,128G@GD-GE)G@ Arial0,0,128GBGF-GG)G@ Arial0,0,128GDGH-GI)G@ Arial0,0,128GFGJ-GK)G@ Arial0,0,128GHGL-GM)G@ Arial0,0,128GJGLG*aGN-aGO+"GP- aGQ-aGR!@NormalArial GS3@DMt 6GT1 pGU1GTGV1@GUGW1@GVGX1@GWGY1fGX 75.091344GZ1GX0G[1GWG\1dG[ _n_u_l_l_G]1G[ _n_u_l_l_G^1 GVG_1@G^G`1dG_CGa1pG_Gb1 GaGc1dGbUGd1GbNGe1G^Gf1dGeNGg1GeGh1tGg1Gi1GgGj1@GiGk1dGjtGl1Gj2Gm1GiGn1p@GmGo1GnGp1dGocoshGq1pGoGr1GqGs1tGr1Gt1GrtGu1Gm2Gv1GUGw1@GvGx1@GwGy1tGx10Gz1KGxG{1Gz10G|1GwG}1dG| _n_u_l_l_G~1G| _n_u_l_l_G1 GvG1@GG1dGTG1pGG1 GG1dGUG1GNG1GtG; LL     G@D HCOPYRIGHT MIRON KAUFMAN, 1998(G-G)G0,0,128*G-G+"G- G-G!@NormalArial