Sta$s$cal physics (microcanonical ensemble) Introduc$on. Introduc$on 18/09/2014. W.J. Briels. How do we tell one molecule from another?

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1 8/9/4 a$s$cal hysics icocaoical eseble W.J. iels Iouc$o How o we ell oe olecule fo aohe? Iouc$o How o we ell oe olecule fo aohe? hey have iffee eegy seca

2 8/9/4 I. oy How o we ell oe acoscoic syse fo aohe? I. oy How o we ell oe acoscoic syse fo aohe? hey have iffee eegy seca I. oy How o we ell oe acoscoic syse fo aohe? hey have iffee eegy seca

3 8/9/4 I. oy How o we chaaceize seca? Oe saes accoig o iceasig eegies a give V I. oy How o we chaaceize seca? oa$o fo acoscoic syses: Φ Φ V gai Φ Θ I. oy ales isei cysal Ieal gas e Φ 6π / e V Φ e π 6π

4 8/9/4 I. oy Defie eoy l Φ ales isei cysal Ieal gas e l l 6π / e V l e l π 6π I. oy Defie eoy l Φ ales isei cysal Ieal gas e l l 6π / e V l e l π 6π II. heoyaics ee you have eve hea abou heoyaics 4

5 8/9/4 II. heoyaics ee you have eve hea abou heoyaics How oes eegy chage whe I chage eoy volue o ube of a$cles? V Wha oes i ea? V V V II. heoyaics egy chages a cosa ube of a$cles close syse II. heoyaics I classical echaics aiaba$c eas chage aaee Coclusio: aiaba$c eas a cosa eoy 5

6 8/9/4 6 II. heoyaics egy chage a cosa ube of a$cles V V V e egy chage a cosa ube of a$cles a cosa eoy e II. heoyaics Mahea$cs V V V V V II. heoyaics Ieal gas e V e / l π V v V V V V Choose v V V

7 8/9/4 II. heoyaics eeaue: ig syse i heal coac wih sall boy of ieal gas a easue eeaue of ieal gas Lae: o wo syses a i heal coac V V Coclusio: V II. heoyaics Defie: µ ial esul V µ µ V II. heoyaics oy is eesive V V eeaue essue a cheical oe$al ae iesive V V µ V 7

8 8/9/4 esiviy II. heoyaics V V V V µ V V µ II. heoyaics Gibbs- Duhe V µ V V µ µ V µ o hee iesive vaiables ae ieee Ieezzo III. Ievesible ocesses a equilibiu Φ Θ Φ 8

9 8/9/4 Ieezzo III. Ievesible ocesses a equilibiu Φ Θ Φ I he logaih oly he las e coibues l Φ l Ieezzo III. Ievesible ocesses a equilibiu Φ Θ Φ I he logaih oly he las e coibues l Φ l ale haoic cysal M M M M Φ M III. Ievesible ocesses a equilibiu oaeous ocesses ae iuce by eovig cosais Cosaie cosaie heally isula$g heally couc$g 9

10 8/9/4 III. Ievesible ocesses a equilibiu oaeous ocesses ae iuce by eovig cosais Cosaie heally isula$g cosaie heally couc$g c u u c u c III. Ievesible ocesses a equilibiu oaeous ocesses ae iuce by eovig cosais Cosaie heally isula$g cosaie heally couc$g c u u c u c oy iceases wih soaeous ocesses Δ u c III. Ievesible ocesses a equilibiu alysis a oea$oal cieio c u u ove ossible echage eegies

11 8/9/4 III. Ievesible ocesses a equilibiu alysis a oea$oal cieio u III. Ievesible ocesses a equilibiu alysis a oea$oal cieio u a a a e u σ III. Ievesible ocesses a equilibiu alysis a oea$oal cieio u a a a e σ a a a e u σ L ρ πσ

12 8/9/4 III. Ievesible ocesses a equilibiu alysis a oea$oal cieio u a a a e σ a a a e u σ ρ L πσ L u a a l l l ρ πσ III. Ievesible ocesses a equilibiu alysis a oea$oal cieio a a u c chage eegy u$ll eoy is aial III. Ievesible ocesses a equilibiu quilibiu cieio: ai$oal eisibu$o of eegy volue a a$cles wo chage eoy V µ µ V V µ µ Isolae syse: V V

13 8/9/4 III. Ievesible ocesses a equilibiu quilibiu cieio: ai$oal eisibu$o of eegy volue a a$cles wo chage eoy µ µ V µ µ III. Ievesible ocesses a equilibiu lea$ve cieio fo cosa eeaue: u syse lus heosa i isola$o Δ Δ Δ h h h Δ / Δ h Δ Δ Δ Δ IV. obabili$es eie: we s$ll iscuss isolae syses ll of he saes ay be obseve wih equal obabiliy

14 8/9/4 IV. obabili$es eie: we s$ll iscuss isolae syses ll of he saes ay be obseve wih equal obabiliy o obsevable efie a # saes a a a a e a / eesively use by isei IV. obabili$es eie: we s$ll iscuss isolae syses egoic hyohesis: easuee of obsevable a a a yiels ha you iels 4

15 8/9/4 a$s$cal hysics caoical eseble W.J. iels I. Caoical eseble. obabili$es u syse i huge heosa heosa yse I. Caoical eseble. obabili$es u syse i huge heosa heosa yse ea he oal i.e. syse lus heosa as beig isolae

16 8/9/4 I. Caoical eseble. obabili$es u syse i huge heosa heosa yse obabiliy o fi he syse i sae h o o h ea he oal i.e. syse lus heosa as beig isolae o I. Caoical eseble. obabili$es obabiliy o fi he syse i sae e h h o h h Δ h Δ / I. Caoical eseble. obabili$es obabiliy o fi he syse i sae h o h h Δ h h Δ / e h h h h h Δ Δ... h h h h h Δ Δ...

17 8/9/4 I. Caoical eseble. obabili$es obabiliy o fi he syse i sae h o h h Δ h h Δ / e h h h h h Δ Δ... h h h h h Δ Δ... e / I. Caoical eseble. heoyaics o o e h h Δ e Δ h h h h / / / / e e Q I. Caoical eseble. heoyaics o e o o h h Δ e Δ h h h h / / / / e h h l Q e Q

18 8/9/4 I. Caoical eseble. heoyaics o o e h h Δ e Δ h h h h / / / / e o h h h h l Q lq e Q I. Caoical eseble. heoyaics o o e h h Δ e Δ h h h h / / / / e o h h h h l Q lq e Q lq I. Caoical eseble. heoyaics alea$ve veage eegy β e Q egy vaia$os l l Q β β Defie eoy l β l 4

19 8/9/4 C. lucua$os I. Caoical eseble obabiliy o obseve : β e Q shaly iceases wih e β shaly eceases C. lucua$os I. Caoical eseble obabiliy o obseve : β e Q shaly iceases wih e β shaly eceases he ouc has a aiu he wih of he isibu$o aou he aiu is σ I. Caoical eseble C. lucua$os Calcula$o of wih of eegy isibu$o σ σ l Q β 5

20 8/9/4 6 I. Caoical eseble Calcula$o of wih of eegy isibu$o s$ae σ l β σ Q Q l β β β σ Wih α α σ C. lucua$os I. Caoical eseble L e e e Q ρ πσ σ / / / e L Q ρ πσ l l l l l as befoe D. ac o ico caoical eseble II. Ieac$g syses ei classical a$$o fuc$o Φ h Q z... e Iegae ove oea Z Q Λ h π Λ Φ Z e heal e oglie legh Cofigua$o iegal

21 8/9/4 7 II. Ieac$g syses Φ Φ e e veages of fuc$os which oly ee o cofigua$os: Φ Z e his hols fo Caesia cooiaes ohewise you us iouce Jacobias II. Ieac$g syses Z V V V Λ l l l µ oiae Z Z Z Z Z l l l l Λ Z Z V V l l µ. Cheical oe$al II. Ieac$g syses oe$al eegy u Φ Φ u u e e e e V Z Z V β β β β Φ Φ veage of ove all cofigua$os a all osi$os u e β. Cheical oe$al

22 8/9/4 µ l ρλ II. Ieac$g syses. Cheical oe$al u / l e Wo efoe o big a a$cle fo ifiiy io he syse II. Ieac$g syses. Va e Waals Ieac$o oe$al Diaee σ σ ϕ 4ε 6 σ. Va e Waals II. Ieac$g syses Divie sace io lile cubes of size Δ σ Z cof e βφ cof Δ oiae Lea- Joes by squae well wih iaee σ a ieac$o age less ha σ Oly cofigua$os wih zeo o oe a$cle e cube Oly eaes eighbou ieac$os $ll oo ifficul 8

23 8/9/4. Va e Waals Φ z ε M Z e βφ M M II. Ieac$g syses Mea fiel aoia$o: vey allowe cofigua$o has he sae aveage eegy Cofigua$o iegal Δ ube of allowe saes M V / Δ Dis$guishable a$cles II. Ieac$g syses. Va e Waals M lλ l z ε lδ M M Δ l zεδ Δ V V l / II. Ieac$g syses. Va e Waals M lλ l z ε lδ M M Δ l zεδ Δ V V l / a V b V a zεδ / b Δ / 9

24 8/9/4 III. Coase gaiig eieally: liie esoluio iulaioally: liie couig owe Coceioally: elage uesaig III. Coase gaiig eiology eaie cooiaes aicles liiae cooiaes ah aicles {... } q { q... M} ah Cofigua$o iegal Z Z M III. Coase gaiig Φ e M qj q M Φ q e M M ae cae of Jacobia J q M

25 8/9/4 III. Coase gaiig Cofigua$o iegal Φ Z M M e Φ q q qj Z M M M e M q J ae cae of Jacobia o ow o I will o elicily e$o he Jacobia assuig ha i is icooae io he oe$al accoig o M q Φ l M M q J q Φ III. Coase gaiig Ieezzo H M h Q M M M M e he eally owlegibles aog you will say ha his is chilish; he a$$o fuc$o is ivaia ue caoical asfoa$os. III. Coase gaiig Ieezzo H M h Q M M M M e he eally owlegibles aog you will say ha his is chilish; he a$$o fuc$o is ivaia ue caoical asfoa$os. We less owlegibles ow ha axe iega$g ove oea we ge he evious esul bac

26 8/9/4 Defie III. Coase gaiig M Φ q l q e he Z e M is he fee eegy of he eliiae egees of feeo i he fiel of he oe$al ue o he eaie egees of feeo. III. Coase gaiig o$ce: ssuig we ae able o calculae a eese eacly he we s$ll ge he eac fee eegy fo ou coase gaie syse; III. Coase gaiig o$ce: ssuig we ae able o calculae a eese eacly he we s$ll ge he eac fee eegy fo ou coase gaie syse; oeove aveages of fuc$os of he coase eaie egees of feeo calculae wih he coase oe$al ae eac.

27 8/9/4 ha you iels

28 8/9/4 Dyaical ocesses W.J. iels I. oe acoscoic heology. hea elaa9o oulus H e y v γh ess G γ y I. oe acoscoic heology. hea elaa9o oulus ecial cases e sai γ γ δ γ γ Θ

29 8/9/4 I. oe acoscoic heology. hea elaa9o oulus ecial cases e sai γ γ δ γ γ Θ y γ G δ γ G I. oe acoscoic heology. hea elaa9o oulus ecial cases e sai γ γ δ γ γ Θ y y H γ H γ G δ γ G e sai ae γ γθ y γ G Θ γ G I. oe acoscoic heology. hea elaa9o oulus ecial cases e sai γ γ δ γ γ Θ y y H γ H γ G δ γ G e sai ae γ γθ y γ G Θ γ G η li y γ G

30 8/9/4 I. oe acoscoic heology. hea elaa9o oulus Oscillaoy shea γ γ cos γ γ si I. oe acoscoic heology. hea elaa9o oulus Oscillaoy shea γ γ cos γ γ si y [ G si G cos ] γ G G si G G cos I. oe acoscoic heology. Dissia9o Wo efoe e seco by eeal foce e s yhγ veage wo efoe e cycle π / V γ γ V G π / y

31 8/9/4 4 I. oe acoscoic heology C. Kaes- Koig Causaliy: he ue iega9o bouay i G γ he lowe iega9o bouay i cos / si G π G G I. oe acoscoic heology C. Kaes- Koig Mahea9cs u i covegece facos whe eee si G G π cos G G π cos si G G π cos si G π π G II. Liea esose. o- equilibiu f s ;.. Geealize obabiliy isibu9os i cofigua9o sace o becoe 9e eee ie eeece esuls fo ie eee oe9es ;

32 8/9/4 5 II. Liea esose. o- equilibiu f s Z e eq eq / ; Φ equilibiu ; eq eq G ; ; ; G oa9o II. Liea esose. elaa9o eae syse ; Θ Φ Φ Z e ; β β Φ e Z β β Φ is oe i euba9o sile ah [ ] Z Z β [ ] [ ] ; e Z β β β Φ [ ] eq ; β II. Liea esose. elaa9o veage value of qua9y [ ] G eq ; β [ ] β valuae ; eq G ie coella9o fuc9o

33 8/9/4. elaa9o esose fuc9o II. Liea esose Φ Θ o Φ [ β ] Φ β [ ] [ ] Φ β III. hea elaa9o e y s γ H Wo efoe by eeal foce uig se sai e w s y γ H w Φ γ y eˆ... γ y eˆ Φ Φ γ y γ y y V y III. hea elaa9o obabiliy isibu9o ave se sai γ y eˆ... γ y eˆ eq e Z βφ βγ e βγ V y eq eq βγ Vy hea elaa9o oulus y G y V G γ y y 6

34 8/9/4 IV. Coase gaiig eieally: liie esoluio iulaioally: liie couig owe Coceioally: elage uesaig IV. Coase gaiig Cosie a aicle wih give iiial osiio a iiial velociy. lo is ah fo iffee iiial cofiguaios a velociies of he bah ie you will ge a isibuio of osiios a velociies eiology IV. Coase gaiig eaie cooiaes aicles liiae cooiaes ah aicles ah {...} q { q... M} {...} {... M} Goal Deive equaio of oio fo aicles givig coec isibuios wihou elicily ivolvig he bah 7

35 8/9/4 8 IV. Coase gaiig il q H q H H H q q ie evoluio Hailoia echaics IV. Coase gaiig Hailoia q q q H Φ ah Hailoia ; q q q H Φ il q H q H H H q q ie evoluio Hailoia echaics IV. Coase gaiig e il il oal soluio

36 8/9/4 eaae foces e il e e IV. Coase gaiig il il il il Φ e il il il aveage foce flucuaig foce veage foce Φ il e IV. Coase gaiig { βh q ; } l q e veage foce a 9e lucua9g foce e il IV. Coase gaiig 9

37 8/9/4 lucua9g foce e il oa9o IV. Coase gaiig Mae use of il e e e ile il il il lucua9g foce il e e IV. Coase gaiig il il il e ile il e il il e lucua9g foce il e e IV. Coase gaiig il il il e ile il e il il e is o he 9e evolve ii9al ao foce bu

38 8/9/4 esul so fa IV. Coase gaiig Φ il e il esul so fa IV. Coase gaiig Φ il e il valuae Liouville oeao a efo soe ahea9cal aiula9os Φ β M Meoy lucua9o- issia9o ha you iels

39 8/9/4 ochasc yaics W.J. iels I. Lagevi ewo Φ Φ Φ Φ I. Lagevi Lagevi a

40 8/9/4 I. Lagevi Lagevi Φ a a a δ I. Lagevi o eeal foce a / / a e e / / a e e / a a e ao foce: a Lae es: e / e / a a I. Lagevi a a Maov Cδ C C lae es C e / e / / δ a a δ lucuao- issiao

41 8/9/4 I. Lagevi a Φ θ Φ θ θ I. Lagevi lae es / e a / e a / e a isei / / e e a a I. Lagevi M β Geealize Lagevi M ς ς

42 8/9/4 Geealize Lagevi I. Lagevi ς M Maov ς δ M δ Lagevi II. ow ow LeH ha sie veage ove ieval Δ Φ a Φ a Δ Δ Δ Δ II. ow Φ ow LeH ha sie a Δ Δ Δ igh ha sie Δ Φ a Φ ΔX Δ X Δ Δ Θ 4

43 8/9/4 5 II. ow Θ Θ Θ Δ Δ Φ Δ X X II. ow efiee [ ] Δ Δ Δ Δ II. ow efiee [ ] Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ

44 8/9/4 II. ow efiee Φ ΔX Δ Δ Θ X Δ Δ Δ Φ ΔX X Δ Δ Δ Θ Φ Δ ΔX Δ Δ Θ X Φ ΔX Δ X X Δ Δ Θ lu II. ow heoiffusio o eeal foce ow Δ ΔX Δ Θ X J c c X X eco e J J X J X X D X c X D X X c X X lu II. ow heoiffusio o eeal foce ow Δ ΔX Δ Θ X J c c X X J c X c X c Dc J D oe X X 6

45 8/9/4 Geealise Lagevi olve ewo s equaios of oio a icely wie he esul β M Meoy aes a olyes wih eoy liiae all egees of feeo ece hose of he ceal ooe of each olye. he cause of eoy ie Iouce vaiables o ee ac of he sae of he bah 7

46 8/9/4 he cause of eoy sa olyes oally I he sall scale siulaio he efoaio of he eliiae egees of feeo is coelae wih he ece islacees of he aicles he cause of eoy sa olyes oally I he sall scale siulaio he efoaio of he eliiae egees of feeo is coelae wih he ece islacees of he aicles icoially aicles ove wih hei ose i he wi a hei hai i he bac o because of ieia bu because of eaglees wih ohe sas he cause of eoy ie Iouce vaiables o ee ac of he sae of he as ij ij ew ew ij ij ij ij 8

47 8/9/4 aid Kee ac of he egee of a iig ij ; ais bewee each ai: { } equilibiu: { ; ais} ee eegy: ij i j s e Φ α ij ij [ ] ij aid qs. of oo i i [ i i ] ai [ ij ij ] aij ij ow Osage aid qs. of oo i i α i [ i i ] ai [ ij ij ] aij ij j [ ] ij ij i ij ow Osage asie foce 9