From the seventeenth th century onward it was realized that t material systems could often be described by a small

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Egbert Wade
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1 Itroducto to statstcal mechacs. The macroscoc ad the mcroscoc states. Equlbrum ad observato tme. Equlbrum ad molecular moto. Relaxato tme. Local equlbrum. Phase sace of a classcal system. Statstcal esemble. Louvlle s theorem. Desty matrx statstcal mechacs roertes. Louvlle s-ema ll equato. ad ts 1

2 Itroducto to statstcal mechacs. From the seveteeth th cetury oward t was realzed that t materal systems could ofte be descrbed by a small umber of descrtve arameters that were related to oe aother smle lawlke ways. These arameters referred to geometrc, dyamcal ad thermal roertes of matter. Tycal of the laws was the deal gas law that t related roduct of ressure ad volume of a gas to the temerature of the gas.

3 Beroull (1738 Joule (1851 Clausus (1857 Krög (1856 C. Maxwell (1860 J. Loschmdt (1876 L. Boltzma (1871 H. Pocaré (1890 J. Gbbs (190 Plack (1900 Lagev (1908 Comto (193 Debye (191 Ferm (196 TEhrefest T. Este (1905 Smoluchowsk (1906 Paul (195 Bose (194 Thomas (197 Drac (197 Ladau (197 3

4 Eergy States Ustable: fallg or rollg Stable Metastable: t low-eergy erch Fgure 5-1. Stablty states. Wter (001 A Itroducto to Igeous ad Metamorhc Petrolog Pretce Hall.

5 We work wth systems whch are equlbrum How do we defe equlbrum? The system ca be Mechacal, Chemcal ad Thermal equlbrum We call all these three together as Thermodyamc Equlbrum. Whch meas : all the eergy states are equally accessble for all the artcles How do we dstgush betwee Classcal l ad Quatum systems? 5

6 Mcroscoc ad macroscoc states The ma am of ths course s the vestgato of geeral roertes of the macroscoc systems wth a large umber of degrees of dyamcally freedom (wth ~ 10 0 artcles for examle. From the mechacal ot of vew, such systems are very comlcated. But the usual case oly a few hyscal arameters, say temerature, the ressure ad the desty, are measured, by measof whch h the state t of the system s secfed. A state defed ths cruder maer s called a macroscoc state or thermodyamc state. O the other had, from a dyamcal ot of vew, each state of a system ca be defed, at least rcle, as recsely as ossble by secfyg all of the dyamcal varables of the system. Such a state s called a mcroscoc state. 6

7 Proertes of dvdual molecules Posto Molecular geometry Itermolecular forces Proertes of bulk flud (macroscoc roertes Pressure Iteral Eergy Heat Caacty Etroy Vscosty

8 What we kow Soluto to Schrodger equato (Ege-value roblem h Wave fucto Allowed eergy levels : E 8 U E m Usg the molecular artto fucto, we ca calculate average values of roerty at gve QUATUM STATE. Quatum states are chagg so radly that the Quatum states are chagg so radly that the observed dyamc roertes are actually tme average over quatum states.

9 Defto ad Features the Thermodyamc Method Thermodyamcs s a macroscoc, heomeologcal theory of heat. Basc features of the thermodyamc method: Mult-artcle hyscal systems s descrbed by meas of a small umber of macroscocally measurable arameters, the thermodyamc arameters: V, P, T, S (volume, ressure, temerature, etroy, ad others. ote: macroscoc objects cota ~ atoms (Avogadro s umber ~ 6x10 3 mol 1. The coectos betwee thermodyamc arameters are foud from the geeral laws of thermodyamcs. The laws of thermodyamcs are regarded as exermetal facts. Therefore, thermodyamcs s a heomeologcal theory. Thermodyamcs s fact a theory of equlbrum states,.e. the states wth tmedeedet (relaxed V, P, T ad S. Term dyamcs s uderstood oly the sese how oe thermodyamc arameters vares wth a chage of aother arameter two successve equlbrum states of the system.

10 Classfcato of Thermodyamc Parameters Iteral ad exteral arameters: Exteral arameters ca be rescrbed by meas of exteral flueces o the system by secfyg exteral boudares ad felds. Iteral arameters are determed by the state of the system tself for gve values of the exteral arameters. ote: the same arameter may aear as exteral oe system, ad as teral aother system. Itesve ad extesve arameters: Itesve arameters are deedet of the umber of artcles the system, ad they serve as geeral characterstcs of the thermal atomc moto (temerature, chemcal otetal. Extesve arameters are roortoal to the total mass or the umber of artcles the system (teral eergy, etroy. ote: ths classfcato s varat wth resect to the choce of a system.

11 Iteral ad Exteral Parameters: Examles A same arameter may aear both as exteral ad teral varous systems: System A System B T Cost M P P = Cost V = Cost V Exteral arameter: V Exteral arameter: P, P = Mg/A Iteral arameter: P Iteral arameter: V, V = Ah

State Vector ad State Equato Alcato of the thermodyamc method mles that the system f foud the state of thermodyamc equlbrum, deoted X, whch s defed by tme-varat state arameters, such as volume,

12 State Vector ad State Equato Alcato of the thermodyamc method mles that the system f foud the state of thermodyamc equlbrum, deoted X, whch s defed by tme-varat state arameters, such as volume, temerature ad ressure: ( V, T, P The arameters (V,T,P T are macroscocally measurable. Oe or two of them may be relaced by o-measurable arameters, such teral eergy or etroy. ote that oly the mea quatty of a state arameter A s tme-varat, see the lot. X A mathematcal relatosh that volve a comlete set of measurable arameters (V,T,P T Ps called the thermodyamc state equato f ( VTP,,, 0 Here, ξ s the vector of system arameters

13 Averagg The hyscal quattes observed the macroscoc state are the result of these varables averagg the warratable mcroscoc states. The statstcal hyothess about the mcroscoc state dstrbuto s requred for the correct averagg. To fd the rght method of averagg s the fudametal rcle of the statstcal method for vestgato of macroscoc systems. The dervato of geeral hyscal lows from the exermetal results wthout cosderato of the atomc-molecular structure s the ma rcle of thermodyamc aroach. 13

14 Averagg Method Probablty of observg artcular quatum state P Esemble average of a dyamc roerty E E P Tme average ad esemble average U lm E t lm E P

15 Thermodyamcs ad Statstcal Mechacs Probabltes 15

16 Par of Dce For oe de, the robablty of ay face comg u s the same, 1/6. Therefore, t s equally robable that ay umber from oe to sx wll come u. For two dce, what s the robablty that the total wll come u, 3, 4, etc u to 1? 16

17 Probablty To calculate the robablty of a artcular outcome, cout the umber of all ossble results. The cout the umber that gve the desred outcome. The robablty of the desred outcome s equal to the umber that gves the desred outcome dvded by the total umber of outcomes. Hece, 1/6 for oe de. 17

18 Par of Dce Lst all ossble outcomes (36 for a ar of dce. Total Combatos How May , , 3+1, , 4+1, +3, , 5+1, +4, 4+,

19 Par of Dce Total Combatos How May 7 1+6, 6+1, +5, 5+, 3+4, , 6+, 3+5, 5+3, , 6+3, 4+5, , 6+4, , Sum = 36 19

20 Probabltes for Two Dce Total % Prob % 0

21 Probabltes for Two Dce Dce Probab blty umber 1

22 Mcrostates ad Macrostates Each ossble outcome s called a mcrostate. The combato of all mcrostates that gve the same umber of sots s called a macrostate. The macrostate that cotas the most mcrostates s the most robable to occur.

23 Combg Probabltes If a gve outcome ca be reached two (or more mutually exclusve ways whose robabltes blt are A ad B, the the robablty blt of that outcome s: A + B. Ths s the robablty of havg ether A or B. 3

24 Combg Probabltes If a gve outcome reresets the combato of two deedet evets, whose dvdual robabltes are A ad B, the the robablty of that outcome s: A B. Ths s the robablty of havg both A ad B. 4

25 Examle Pat two faces of a de red. Whe the de s throw, what s the robablty blt of a red face comg u?

26 Aother Examle Throw two ormal dce. What s the robablty blt of two sxes comg u? 1 1 ( (

27 Comlcatos s the robablty of success. (1/6 for oe de q s the robablty of falure. (5/6 for oe de + q = 1, or q = 1 Whe two dce are throw, what s the robablty of gettg oly oe sx? 7

28 Comlcatos Probablty of the sx o the frst de ad ot the secod s: q Probablty of the sx o the secod de ad ot the frst s the same, so: 10 ( ( 1 q

29 Smlfcato Probablty of o sxes comg u s: ( 0 qq The sum of all three robabltes s: ( + (1 + (0 = 1 9

30 Smlfcato ( + (1 + (0 = 1 ² + q + q² =1 ( + q² =1 The exoet s the umber of dce (or tres. Is ths geeral? 30

31 Three Dce ( + q³ = 1 ³ + 3²q + 3q² + q³ = 1 (3 + ( + (1 + (0 = 1 It works! It must be geeral! ( +q =1 31

32 Bomal Dstrbuto Probablty of successes attemts ( + q = 1! P ( q!(! where, q = 1. q 3

33 Thermodyamc Probablty The term wth all the factorals the revous equato s the umber of mcrostates t that t wll lead to the artcular macrostate. It s called the thermodyamc robablty, w. w!!(! 33

34 Mcrostates The total umber of mcrostates s: w w True robablty P ( For a very large umber of artcles w max 34

35 Mea of Bomal Dstrbuto P ( P where ( q P! ( where q P!!( ( P P ( ( : otce 35

36 Mea of Bomal Dstrbuto P P ( ( P P ( ( q P ( ( q 1 1 (1 ( 36

37 Stadard Devato ( ( P 37

38 Stadard Devato P P ( ( q q ( ( 1 q q 1( ( ( 1 q q q 1 ( ( ( 38

39 Stadard Devato q q q ( ( ( q q q ( ( 39

40 For a Bomal Dstrbuto q q 40

41 Cos Toss 6 cos. Probablty of heads:! 6! 1 1 P( q!(!!(6! 6! 1 P (!(6!

42 For Sx Cos Bomal Dstrbuto Probab blty Successes 4

43 For 100 Cos Bomal Dstrbuto Proba ablty Successes 43

44 For 1000 Cos Bomal Dstrbuto Probablty Successes 44

45 Multle Outcomes w!!!! 1 3!! 45

46 Strlg s Aroxmato For large : l! l l w! l! l! l! l! ( l l w l l w l ( l l! 46

47 umber Exected Toss 6 cos tmes. Probablty blt of heads: 6! 6! 1 1 P( q!(!!(6! 6 6! 1 P(!(6! umber of tmes heads s exected s: = P( 47

48 Zero Low of Thermodyamcs Oe of the ma sgfcat ots thermodyamcs (some tmes they call t the zero low of thermodyamcs s the cocluso that every eclosure (solated from others system tme come to the equlbrum state where all the hyscal arameters characterzg the system are ot chagg tme. The rocess of equlbrum settg s called the relaxato rocess of the system ad the tme of ths rocess s the relaxato tme. Equlbrum meas that the searate arts of the system (subsystems are also the state of teral equlbrum (f oe wll solate them othg wll hae wth them. The are also equlbrum wth each other- o exchageby eergy ad artcles betwee them. 48

49 Local equlbrum Local equlbrum meas that the system s cosst from the subsystems, that by themselves are the state of teral equlbrum but there s o ay equlbrum betwee the subsystems. The umber of macroscoc arameters s creasg wth dgresso of the system from the total equlbrum 49

50 Classcal hase system Let (q 1, q... q s be the geeralzed coordates of a system wth degrees of freedom ad ( 1... s ther cojugate momet. A mcroscoc state tt of the system s df defedby secfyg the values of (q 1, q... q s, 1... s. The s-dmesoal sace costructed from these s varables as the coordates the hase sace of the system. Each ot the hase sace (hase ot corresods to a mcroscoc state. Therefore the mcroscoc states classcal statstcal mechacs make a cotuous set of ots hase sace. 50

51 Phase Sace t t1 1 Phase sace,..., 1,, 3,...,, r1, r1, r3 r r

52 Phase Orbt If the Hamltoa of the system s deoted by H(q,, the moto of hase ot ca be alog the hase orbt ad s determed by the caocal equato of moto H q q H 1 (=1,...s (1.1 P Phase Orbt Costat eergy surface H(q,=E (1. H ( q, E Therefore the hase orbt must le o a surface of costat eergy (ergodc surface. 5

53 - sace ad -sace Let us defe - sace as hase sace of oe artcle (atom or molecule. The macrosystem hase sace (-sace s equal to the sum of - saces. The set of ossble mcrostates ca be reseted by cotues set of hase ots. Every ot ca move by tself alog t s ow hase orbt. The overall cture of ths movemet ossesses certa terestg features, whch are best arecated terms of what we call a desty fucto (q,;t. Ths fucto s defed such a way that at ay tme t,, the umber of reresetatve ots the volume elemet (d 3 q d 3 aroud the ot (q, of the hase sace s gve by the roduct (q,;t d 3 q d 3. Clearly, the desty fucto (q,;t symbolzes the maer whch the members of the esemble are dstrbuted over varous ossble mcrostates at varous stats of tme. 53

54 Fucto of Statstcal Dstrbuto Let us suose that the robablty of system detecto the volume dddqd 1... d s dq 1... dq s ear ot (,q equal dw (,q= (q,d. The fucto of statstcal dstrbuto (desty fucto of the system over mcrostates the case of oequlbrum systems s also deeds o tme. The statstcal average of a gve dyamcal hyscal quatty f(,q s equal f f (, q ( q, ( q, ; t d ; t d 3 3 qd qd 3 3 (1.3 The rght hase ortrat of the system ca be descrbed by the set of ots dstrbuted hase sace wth the desty. Ths umber ca be cosdered as the descrto of great (umber of ots umber of systems each of whch has the same structure as the system uder observato coes of such system at artcular tme, whch are by themselves exstg admssble mcrostates 54

55 Statstcal Esemble The umber of macroscocally detcal systems dtb dstrbuted td alog admssble mcrostates wth desty defed as statstcal esemble. A statstcal esembles are defed ad amed by the dstrbuto fucto whch characterzes t. The statstcal average value have the same meag as the esemble average value. A esemble s sad to be statoary f does ot deedd exlctly l o tme,.e. at all tmes t 0 (1.4 Clearly, for such a esemble the average value <f> of ay hyscal quatty f(,q wll be deedet of tme. aturally, the, a statoary esemble qualfes to rereset a system equlbrum. To determe the crcumstaces uder whch Eq. (1.4 ca hold, wehave to make arather study of the movemet of the reresetatve ots the hase sace. 55

56 Lovlle s theorem ad ts cosequeces Cosder a arbtrary "volume" " the relevat rego of the hase sace ad let the "surface eclosg ths volume creases wth tme s gve by t d (1.5 where d d(d 3 q d 3. O the other had, the et rate at whch h the reresetatve ots flow outofthevolume (across the boudg surface sgveby σ ρ( ν dσ (1.6 here v s the vector of the reresetatve ots the rego of the surface elemet d, whle ˆ s the (outward ut vector ormal to ths elemet. By the dvergece theorem, (1.6 ca be wrtte as 56

57 or dv ( v d (1.7 where the oerato of dvergece meas the followg: 3 dv ( v 1 (1.8 ( q ( q I vew of the fact that there are o "sources" or "sks" the hase sace ad hece the total umber of reresetatve ots must be coserved, we have, by (1.5 ad (1.7 dv( v d d t d (1.9 t t dv( v d 0 t (

58 The ecessary ad suffcet codto that the volume tegral (1.10 vash for arbtrary volumes s that the tegrated must vash everywhere the relevat rego of the hase sace. Thus, we must have t dv( v 0 (1.11 whch s the equato of cotuty for the swarm of the reresetatve ots. Ths equato meas that esemble of the hase ots movg wth tme as a flow of lqud wthout sources or sks. Combg (1.8 ad (1.11, we obta 3 dv ( v ( q ( 1 q 58

59 0 3 3 q ( q q q q t (1.1 The last grou of terms vashes detcally because the equato of moto, we have for all, q q q q q q H H (, (, (1.13 q q q From (1.1, takg to accout (1.13 we ca easly get the Louvlle equato equato 3 ρ ρ ρ ρ 0 1 H q ρ, t ρ ρ q ρ t ρ ( where {,H} the Posso bracket.

Further, sce (q, ;t t, the remag terms (1.1 may be combed to gve the «total» tme dervatve of. Thus we fally have d,h0 (1.15 dt t Equato (1.15 embodes the so-called Louvlle s theorem.

60 Further, sce (q, ;t t, the remag terms (1.1 may be combed to gve the «total» tme dervatve of. Thus we fally have d,h0 (1.15 dt t Equato (1.15 embodes the so-called Louvlle s theorem. Accordg to ths theorem (q 0, 0 ;t 0 =(q,;t or for the equlbrum system (q 0, 0= (q,, that meas the dstrbuto fucto s the tegral of moto. Oe ca formulate the Louvlle s theorem as a rcle of hase volume mateace. t t= 0 0 q 60

61 Desty matrx statstcal mechacs The mcrostates quatum theory wll be characterzed by a (commo Hamltoa, whch may be deoted by the oerator. At tme t the hyscal state of the varous systems wll be characterzed by the corresodet wave fuctos (r,t, where the r, deote the osto coordates relevat to the system uder study. H Let k (r,t, deote the (ormalzed wave fucto characterzg the hyscal state whch the k-th system of the esemble haes to be at tme t ; aturally, k=1,... The tme varato of the fucto k (t wll be determed by the Schredger equato 61

62 H k k ( t ( t (1.16 Itroducg a comlete set of orthoormal fuctos, the wave fuctos k (t may be wrtte as k k k ( t a ( t ( k (1.18 a ( t ( t d here, * deotes the comlex cojugate of whle d deotes the volume elemet of the coordate sace of the gve system. Obvously eough, the hyscal state of the k-th system ca be descrbed equally well terms of the coeffcets. The tme varato of these coeffcets wll be gve by 6

63 k * k * a ( t ( t d H ( t d k * k = H am( t m d m = Hma k m( t m (1.19 where H m H m d * (1.0 The hyscal sgfcace of the coeffcets a k ( t s evdet from eq. (1.17. They are the robablty amltudes for the k-th system of the esemble to be the resectve states ; to be ractcal the umber a k ( t reresets the robablty that a measuremet at tme t fds the k-th system of the esemble to be artcular state. Clearly, we must have 63

64 a k ( t 1 (for all k (1.1 ( t We ow troduce the desty oerator as defed by the matrx elemets (desty matrx m ( t 1 k k* a ( t a ( t m k1 (1. clearly, the matrx elemet m (t s the esemble average of the quatty a (ta * m (t whch, as a rule, vares from member to member the esemble. I artcular, the dagoal elemet m (t s the esemble average of the robablty, a k ( t the latter tself beg a (quatum- mechacal average. 64

65 Equato of Moto for the Desty Matrx m (t Thus, we are cocered here wth a double averagg rocess - oce due to the robablstc asect of the wave fuctos ad aga due to the statstcal asect of the esemble!! The quatty m (t ow reresets the robablty that a system, chose at radom from the esemble, at tme t,, s foud to be the state. I vew of (1.1 ad (1. we have k a ( t m ( t k k* am ( t a ( t k 1 (1.3 Let us determe the equato of moto for the desty matrx m (t. 65

66 1 t k k * k k * m ( a m ( t a ( t a m ( t a ( t 1 = = k 1 l k k * k * k * H mla l ( t a ( t a m ( t H la l ( t l l 1 H ( t ( t H k ml l ml l = (H H m (1.4 Here, use has bee made of the fact that, vew of the Hermta character of the oerator, Eq.(1.4 may be wrtte as Ĥ H * l =H l l. Usg the commutator otato, t, H 0 (1.5 66

67 Ths equato Louvlle-ema s the quatum-mechacal aalogue of the classcal equato Louvlle. Some roertes of desty matrx: Desty oerator s Hermta, + = - The desty oerator s ormalzed Dagoal elemets of desty matrx are o egatve 0 Rereset the robablty of hyscal values 1 67

2. Independence and Bernoulli Trials

2. Independence and Bernoulli Trials . Ideedece ad Beroull Trals Ideedece: Evets ad B are deedet f B B. - It s easy to show that, B deedet mles, B;, B are all deedet ars. For examle, ad so that B or B B B B B φ,.e., ad B are deedet evets.,