.MCAD 306000000 Z  docDocument<mcObjectIÿÿÿÿÀÀ (d2_graph_format graphData axisFormatLLtrace2D      dim_formatCmasslengthtimecharge temperature luminosity substanceNumericalFormat@dii shpRectE—A\mcDocumentObjectStateJð´ÿÿÿÿ mcPageModel:š™™?œ?š™™?š™™?mcHeaderFooter99 ComputeEngine=BuiltInsB  SerialAnyvalH@!@H @"@H#@Hü©ñÒMbP?$@H units_classA TextState0 TextStyle/@ Arial Normalÿÿÿÿÿÿÿÿ€font_style_list> font_style?óÿÿÿöÿÿÿ ÿÿÿÿ VariablesTimes New Roman?óÿÿÿöÿÿÿ ÿÿÿÿ ConstantsTimes New Roman?óÿÿÿöÿÿÿ ÿÿÿÿTextArial?óÿÿÿöÿÿÿ ÿÿÿÿGreek VariablesSymbol?óÿÿÿöÿÿÿ ÿÿÿÿUser^1Arial?óÿÿÿöÿÿÿ ÿÿÿÿUser^2 Courier New?óÿÿÿóÿÿÿ ÿÿÿÿUser^3System?óÿÿÿóÿÿÿ ÿÿÿÿUser^4Script?óÿÿÿóÿÿÿ ÿÿÿÿUser^5Roman?óÿÿÿöÿÿÿ ÿÿÿÿÿUser^6Modern?óÿÿÿöÿÿÿ ÿÿÿÿUser^7Times New Roman?óÿÿÿöÿÿÿ ÿÿÿÿSymbolsSymbol?óÿÿÿóÿÿÿ ÿÿÿÿCurrent Selection FontArial?óÿÿÿóÿÿÿ ÿÿÿÿUndefined Font?óÿÿÿóÿÿÿ Ö„OgHeaderArial?óÿÿÿóÿÿÿ Ö„OgFooterArial?óÿÿÿöÿÿÿ ÿÿÿÿRotated Math FontTimes New Roman TextRegion docRegion7shpBoxD;àH;ß;ß 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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Á€C¬C®1Î@C­C¯1dC®\WC°1Žp€C®C±1 ÀC°C²1dC±NC³1¤C±N.plusC´1û€C­Cµ1žŒ@C´C¶1dCµNC·1Ê€C´C¸1žŒ@C·C¹1dC¸N.plusCº1žŒ€C·C»1Žp@CºC¼1ˆÇ€C»C½1dC¼NC¾1¤C¼N.plusC¿3@D2éI@‘CÀ1 pCÁ1 Á€CÀCÂ1Î@CÁCÃ1dCÂentropyCÄ1Žp€CÂCÅ1 ÀCÄCÆ1dCÅNCÇ1¤CÅN.plusCÈ1΀CÁCÉ1dCÈlnCÊ1Žp€CÈCË1΀CÊCÌ1dCË\WCÍ1Žp€CËCÎ1 ÀCÍCÏ1dCÎNCÐ1¤CÎN.plusCÑ3@DoCp�CÒ1 pCÓ1Á€CÒCÔ1ŸÁ@CÓCÕ1ŸÁ@CÔCÖ1ŸÁ@CÕC×1fCÖ 117.835021CØ1¦CÖ0CÙ1ŸÁ€CÕCÚ1@CÙCÛ1€CÙCÜ1 ÀCÔCÝ1Î@CÜCÞ1dCÝentropyCß1Žp€CÝCà1 ÀCßCá1dCàNCâ1¤CàN.plusCã1Ê€CÜCä1dCãNCå1΀CãCæ1dCålnCç1Žp€CåCè1´Cç2Cé1ŸÁ€CÓCê1ŸÁ@CéCë1ŸÁ@CêCì1tCë170Cí1¦Cë0Cî1ŸÁ€CêCï1@CîCð1€CîCñ1¤CéN.plusCò8 LL     Có3@D'9Ð(ºCô1 pCõ1Á€CôCö1ŸÁ@CõC÷1ŸÁ@CöCø1ŸÁ@C÷Cù1ÊBCøCú1dCù 9.144842Cû1ü€CùCü1dCû10Cý1¤Cû49Cþ1¦Cø1Cÿ1ŸÁ€C÷D1@CÿD1€CÿD1΀CöD1dD\WD1Žp€DD1 ÀDD1dDND1¤DN.plusD1ŸÁ€CõD 1ŸÁ@DD 1ŸÁ@D D 1dD ND 1¦D 0D 1ŸÁ€D D1û@D D1dDND1´D2D1€D D1¤DN.plusD9 \L     D@D³ª@««>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 +ÿÿÿÿÿÿÿÿÿÿÿÿ"D!- >D"->D#!ÿÿÿÿ@NormalArialÿÿÿ D$@DÒ+(”@ÊÊCOPYRIGHT @MIRON KAUFMAN(*D%-D&+ÿÿÿÿÿÿÿÿÿÿÿÿ"D'- D(-D)!ÿÿÿÿ@NormalArialÿÿÿ D*@DYêkh¡ââ@ENext we define the entropy as function of energy and number of atoms.(ED+-ED,)D*n Arial0,0,128*ED--ED.+ÿÿÿÿÿÿÿÿÿÿÿÿ"D/- ED0-ED1!ÿÿÿÿ@NormalArialÿÿÿ D23@DØsi˜&ˆrD31 pD41 Á€D3D51Î@D4D61dD5N.plusD71Žp€D5D81 ÀD7D91dD8UD:1¤D8ND;1Ê€D4D<1û@D;D=1tD<1D>1´D<2D?1Žp€D;D@1‰Ç€D?DA1dD@NDB1¤D@UDC3@D(£ñ¹Z°£DD1 pDE1 Á€DDDF1Î@DEDG1dDFSDH1Žp€DFDI1 ÀDHDJ1dDIUDK1¤DINDL1΀DEDM1dDLentropyDN1Žp€DLDO1 ÀDNDP1dDONDQ1΀DODR1dDQN.plusDS1Žp€DQDT1 ÀDSDU1dDTUDV1¤DTNDW@DÑ*õà¢("$"$@|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[+ÿÿÿÿÿÿÿÿÿÿÿÿ"D\- |D]-|D^!ÿÿÿÿ@NormalArialÿÿÿ D_3@D0û¤ a¤D`1 pDa1ŒÁ€D`Db1Î@DaDc1dDbSDd1Žp€DbDe1 ÀDdDf1tDe5Dg1´De97Dh1–ô€DaDi1+@DhSerial_DisplayNodeFDj1€DhDk3@D¸û2 û¥Dl1 pDm1ŒÁ€DlDn1Î@DmDo1dDnSDp1Žp€DnDq1 ÀDpDr1tDq168Ds1´Dq170Dt1–ô€DmDu1+@Dt@FDv1€DtDw3@D`;ÍLpH˜Dx1 pDy1 Á€DxDz1dDyUD{1€DyD|1 Ã@D{D}1•K@D|D~1´D}170D1•K€D|D€1´D168D�1´D{168D‚3@DØ;LèH™Dƒ1 pD„1 Á€DƒD…1dD„ND†1´D„170D‡3@DW<üXšDˆ1 pD‰1Á€DˆDŠ1ŸÁ@D‰D‹1ŸÁ@DŠDŒ1ŸÁ@D‹D�1fDŒ 117.835021DŽ1¦DŒ0D�1ŸÁ€D‹D�1@D�D‘1€D�D’1 ÀDŠD“1Î@D’D”1dD“SD•1Žp€D“D–1 ÀD•D—1dD–UD˜1¤D–ND™1Ê€D’Dš1dD™ND›1΀D™Dœ1dD›lnD�1Žp€D›Dž1´D�2DŸ1ŸÁ€D‰D 1ŸÁ@DŸD¡1ŸÁ@D D¢1tD¡170D£1•K€D¡D¤1´D£170D¥1ŸÁ€D D¦1@D¥D§1€D¥D¨1¤DŸUD©; LL     Dª@D' ·@@LNote the maximum entropy is Nln(2) and it occurs for U = 0 or Nplus= Nminus.(K?LD«-?D¬)Dªn Arial0,0,128D­-D®)DªN Arial0,0,128D«D¯-D°)Dªn Arial0,0,128D­D±-D²)DªN Arial0,0,128D¯D³-D´)Dªn Arial0,0,128D±D³D«*LDµ-LD¶+ÿÿÿÿÿÿÿÿÿÿÿÿ"D·- LD¸-LD¹!ÿÿÿÿ@NormalArialÿÿÿ Dº@DYý¨h°õõOõOAThe 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»-'D¼)Dºn Arial0,0,128D½-D¾)Dº@ Symbol0,0,128D»D¿-DÀ)Dº@ Arial0,0,128D½DÁ-DÂ)Dº@ Symbol0,0,128D¿DÃ-DÄ)Dº@ Arial0,0,128DÁDÅ-DÆ)Dº@ Arial0,0,128DÃDÇ-DÈ)Dº@ Arial0,0,128DÅDÉ-(DÊ)Dº@ Arial0,0,128DÇDË-DÌ)Dº@ Arial0,0,128DÉDÍ-DÎ)Dº@ Symbol0,0,128DËDÏ-DÐ)Dº@ Arial0,0,128DÍDÑ-DÒ)Dº@ Symbol0,0,128DÏDÓ-DÔ)Dº@ Arial0,0,128DÑDÕ- DÖ)Dº@ Arial0,0,128DÓD×-DØ)Dº@ Arial0,0,128DÕDÙ-DÚ)Dº@ Arial0,0,128D×DÛ-DÜ)Dº@ Arial0,0,128DÙDÝ-DÞ)Dº@ Arial0,0,128DÛDß-Dà)Dº@ Arial0,0,128DÝDá-Dâ)Dº@ Arial0,0,128DßDã-Dä)Dº@ Arial0,0,128DáDå-Dæ)Dº@ Symbol0,0,128DãDç-Dè)Dº@ Arial0,0,128DåDé-Dê)Dº@ Arial0,0,128DçDë-Dì)Dº@ Arial0,0,128DéDí-Dî)Dº~ 0,0,128DëDï-Dð)Dº~ 0,0,128DíDñ-Dò)Dº^ 0,0,128DïDó-Dô)Dº~ 0,0,128DñDõ-Dö)Dº^ 0,0,128DóD÷-Dø)Dº@ Arial0,0,128DõDù-Dú)Dº@ Arial0,0,128D÷Dû-Dü)Dº@ Arial0,0,128DùDý-Dþ)Dº@ Arial0,0,128DûDÿ- E)Dº~ 0,0,128DýE-E)Dº@ Arial0,0,128DÿE-E)Dº@ Arial0,0,128EE-E)Dº@ Arial0,0,128EE-E)Dº@ Arial0,0,128EE -E )Dº@ Arial0,0,128EE -E )Dº@ Arial0,0,128E E -E)Dº@ Arial0,0,128E E-E)Dº@ Arial0,0,128E E-E)Dº@ Arial0,0,128EE-E)Dº@ Arial0,0,128EE-E)Dº@ Arial0,0,128EE-E)Dº@ Arial0,0,128EE-E)Dº@ Arial0,0,128EE-E)Dº@ Arial0,0,128EE-E)Dº@ Arial0,0,128EE-E )Dº@ Arial0,0,128EE!-E")Dº@ Arial0,0,128EE#-E$)Dº@ Arial0,0,128E!E%-E&)Dº@ Arial0,0,128E#E'-E()Dº@ Arial0,0,128E%E)-E*)Dº@ Arial0,0,128E'E+-E,)Dº@ Arial0,0,128E)E--E.)Dº@ Arial0,0,128E+E/-E0)Dº@ Arial0,0,128E-E1-E2)Dº@ Arial0,0,128E/E3-E4)Dº@ Arial0,0,128E1E5-E6)Dº@ Arial0,0,128E3E7-E8)Dº@ Arial0,0,128E5E9-E:)Dº~ 0,0,128E7E;-E<)Dº^ 0,0,128E9E=-E>)Dº~ 0,0,128E;E?-E@)Dº@ Arial0,0,128E=EA-EB)Dº~ 0,0,128E?EC-ED)Dº^ 0,0,128EAEE-EF)Dº@ Arial0,0,128ECEG-EH)Dº@ Arial0,0,128EEEGD»*EI-EJ+ÿÿÿÿÿÿÿÿÿÿÿÿ"EK- EL-EM!ÿÿÿÿ@NormalArialÿÿÿ EN3@DÈË‚ûà¸EO1 pEP1 Á€EOEQ1Î@EPER1dEQTES1Žp€EQET1 ÀESEU1dETUEV1¤ETNEW1û€EPEX1tEW1EY1û€EWEZ1ˆÇ@EYE[1Î@EZE\1dE[SE]1Žp€E[E^1 ÀE]E_1‰Ç@E^E`1dE_UEa1´E_2Eb1¤E^NEc1΀EZEd1dEcSEe1Žp€EcEf1 ÀEeEg1dEfUEh1¤EfNEi1´EY2Ej@DI[X´8.(Ek-El)Ejn Arial0,0,128*Em-En+ÿÿÿÿÿÿÿÿÿÿÿÿ"Eo- Ep-Eq!ÿÿÿÿ@NormalArialÿÿÿ Er@DSc`²0êêCOPYRIGHT @MIRON KAUFMAN 1998(*Es-Et+ÿÿÿÿÿÿÿÿÿÿÿÿ"Eu- Ev-Ew!ÿÿÿÿ@NormalArialÿÿÿ Ex3@D€‹îœ‹˜»Ey1 pEz1 Á€EyE{1dEztE|1€EzE}1 Ã@E|E~1•K@E}E1´E~170E€1•K€E}E�1´E€165E‚1´E|170Eƒ3@D§5”¨¹E„1 pE…1Á€E„E†1ŸÁ@E…E‡1ŸÁ@E†Eˆ1ŸÁ@E‡E‰1fEˆ168EŠ1•K‚EˆE‹1¤EŠ170EŒ1ŸÁ€E‡E�1@EŒEŽ1€EŒE�1 ÀE†E�1dE�UE‘1Ê€E�E’1•K@E‘E“1¤E’NE”1΀E‘E•1dE”tanhE–1Žp€E”E—1û€E–E˜1tE—1E™1¤E—tEš1ŸÁ€E…E›1ŸÁ@EšEœ1ŸÁ@E›E�1tEœ170Ež1•K€EœEŸ1´Ež170E 1ŸÁ€E›E¡1@E E¢1€E E£1 ÀEšE¤1Î@E£E¥1dE¤TE¦1Žp€E¤E§1 ÀE¦E¨1dE§UE©1¤E§NEª1¤E£tE«9 LL     E¬@D©7渿0'='=@æThe 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. (ä+æE­-+E®)E¬~ 0,0,128E¯-E°)E¬~ 0,0,128E­E±-E²)E¬~ 0,0,128E¯E³-E´)E¬~ 0,0,128E±Eµ-E¶)E¬~ Symbol0,0,128E³E·-E¸)E¬~ 0,0,128EµE¹-Eº)E¬~ 0,0,128E·E»-E¼)E¬~ 0,0,128E¹E½-E¾)E¬~ 0,0,128E»E¿-EÀ)E¬~ 0,0,128E½EÁ-EÂ)E¬~ 0,0,128E¿EÃ-EÄ)E¬~ Symbol0,0,128EÁEÅ-EÆ)E¬~ 0,0,128EÃEÇ-EÈ)E¬~ 0,0,128EÅEÉ-EÊ)E¬^ 0,0,128EÇEË-EÌ)E¬~ 0,0,128EÉEÍ-EÎ)E¬~ 0,0,128EËEÏ-EÐ)E¬~ 0,0,128EÍEÑ-EÒ)E¬~ 0,0,128EÏEÓ-EÔ)E¬~ 0,0,128EÑEÕ-EÖ)E¬~ 0,0,128EÓE×-EØ)E¬~ 0,0,128EÕEÙ-EÚ)E¬~ 0,0,128E×EÛ-EÜ)E¬~ 0,0,128EÙEÝ-EÞ)E¬~ 0,0,128EÛEß-Eà)E¬~ 0,0,128EÝEá-Eâ)E¬~ 0,0,128EßEã-Eä)E¬~ 0,0,128EáEå-Eæ)E¬~ 0,0,128EãEç-Eè)E¬~ 0,0,128EåEé-Eê)E¬~ 0,0,128EçEë-Eì)E¬~ 0,0,128EéEí-Eî)E¬~ 0,0,128EëEï-Eð)E¬~ 0,0,128EíEñ-Eò)E¬~ 0,0,128EïEó-Eô)E¬~ 0,0,128EñEõ-Eö)E¬~ 0,0,128EóE÷-Eø)E¬~ 0,0,128EõEù-Eú)E¬n Arial0,0,128E÷Eû-Eü)E¬@ Symbol0,0,128EùEý-Eþ)E¬@ Arial0,0,128EûEÿ-F)E¬@ Arial0,0,128EýF-F)E¬@ Symbol0,0,128EÿF-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¬@ 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