Phase Space p t 2 2 t1 1 Phase space p,..., 1, p2, p3,..., p, r1, r1, r3 r r
Phase Orbt If thehamltonan of the system s denoted by H(q,p), the moton of phase pont can be along the phase orbt and s determned by the canoncal equaton of moton p H q q H p (=1 1,2...s) (1.1) 1) P Phase Orbt Constant energy surface H(q,p)=E (1.2) H ( q, p) E Therefore the phase orbt must le on a surface of constant energy (ergodc surface). 2
- space and -space Let us defne - space as phase space of one partcle (atom or molecule). The macrosystem phase space (-space) s equal to the sum of - spaces. The set of possble mcrostates can be presented by contnues set of phase ponts. Every pont can move by tself along t s own phase orbt. The overall pcture of ths movement possesses certan nterestng features, whch are best apprecated n terms of what we call a densty functon (q,p;t). Ths functon s defned n such a way that at any tme t, the number of representatve ponts n the volume element (d 3 q d 3 p) around the pont (q,p) of the phase space s gven by the product (q,p;t) d 3 q d 3 p. Clearly, the densty functon (q,p;t) symbolzes the manner n whch the members of the ensemble are dstrbuted over varous possble mcrostates at varous nstants of tme. 3
Functon of Statstcal Dstrbuton Let us suppose that the probablty of system detecton n the volume ddpdqdp 1... dp s dq 1... dq s near pont (p,q) equal dw (p,q)= (q,p)d. The functon of statstcal dstrbuton (densty functon) of the system over mcrostates n the case of nonequlbrum systems s also depends on tme. The statstcal average of a gven dynamcal physcal quantty f(p,q) s equal f f ( p, q) ( q, ( q, p ; t ) d p; t) d 3 3 qd qd 3 p 3 p (1.3) The rght phase portrat of the system can be descrbed by the set of ponts dstrbuted n phase space wth the densty. Ths number can be consdered as the descrpton of great (number of ponts) number of systems each of whch has the same structure as the system under observaton copes of such system at partcular tme, whch are by themselves exstng n admssble mcrostates 4
Statstcal Ensemble The number of macroscopcally dentcal systems dtbtd dstrbuted along admssble mcrostates wth densty defned as statstcal ensemble. A statstcal ensembles are defned and named by the dstrbuton functon whch characterzes t. The statstcal average value have the same meanng as the ensemble average value. An ensemble s sad to be statonary f does not dependd explctly l on tme,.e. at all tmes (1.4) t 0 Clearly, for such an ensemble the average value <f> of any physcal quantty f(p,q) wll be ndependent of tme. aturally, then, a statonary ensemble qualfes to represent a system n equlbrum. To determne the crcumstances under whch h Eq. (1.4) can hld hold, wehave to make arather study of the movement of the representatve ponts n the phase space. 5
Lovlle s theorem and ts consequences Consder an arbtrary "volume" " n the relevant regon of the phase space and let the "surface enclosng ths volume ncreases wth tme s gven by t d (1.5) where d d(d 3 q d 3 p). On the other hand, the net rate at whch h the representatve ponts flow out of the volume (across the boundng surface ) sgvenby σ ρ( ν n )dσ (1.6) here v s the vector of the representatve ponts n the regon of the surface element d, whle s the (outward) nˆ unt vector normal to ths element. By the dvergence theorem, (1.6) can bewrtten as 6
Statstcs of Multpartcle Systems n Thermodynamc Equlbrum The macroscopc thermodynamc parameters, X = (V,P,T, ), are macroscopcally observable quanttes that are, n prncple, functons of the canoncal varables,.e. f f( p1, p2,..., ps, q1, q2,..., qs), 1,2,..., n, n s ( f, f,..., f ) ( V, P, T,...) X 1 2 n X X ( p, p,..., p, q, q,..., q ) 1 2 s 1 2 s However, the specfcaton of all the macroparameters X does not determne a unque mcrostate, p p ( X), q q ( X) Consequently, on the bass of macroscopc measurements, one can make only statstcal statements about the values of the mcroscopc varables.
Statstcal Descrpton of Mechancal Systems Statstcal descrpton of mechancal systems s utlzed for mult-partcle problems, where ndvdual solutons for all the consttutve atoms are not affordable, or necessary. Statstcal descrpton can be used to reproduce averaged macroscopc parameters and propertes of the system. Comparson of objectves es of the determnstc and statstcal approaches: Determnstc partcle dynamcs Statstcal mechancs Provdes the phase vector, as a functon of tme Q(t), based on the vector of ntal condtons Q(0) Provdes the tme-dependent probablty densty to observe the phase vector Q, w(q,t), based on the ntal value w(q,0)
Statstcal Descrpton of Mechancal Systems From the contemporary pont of vew, statstcal mechancs can be regarded as a herarchcal multscale method, whch elmnates the atomstc degrees of freedom, whle establshng a determnstc mappng from the atomc to macroscale varables, and a probablstc mappng from the macroscale to the atomc varables: Mcrostates Macrostates (p,q) X k k determnstc conformty probablstc conformty
Dstrbuton Functon Though the specfcaton of a macrostate X cannot determne the mcrostate (p,q) = (p 1,p 2,,p s ; q 1,q 2,,q s ), a probablty densty w of all the mcrostates can be found, w p, p,..., p ; q, q,..., q ; t 1 2 s 1 2 s or abbrevated: wpqt (,, ) The probablty of fndng the system n a gven phase volume G: The normalzaton condton: WGt (, ) wpqtdpdq (,, ) ( pq, ) G w( p, q, t) dpdq 1
Statstcal Ensemble Wthn the statstcal descrpton, the moton of one sngle system wth gven ntal condtons s not consdered; thus, p(t), q(t) are not sought. Instead, the moton of a whole set of phase ponts, representng the collecton of possble states of the gven system. Such a set of phase ponts s called a phase space ensemble. If each pont n the phase space s consdered as a random quantty wth a partcular probablty ascrbed to every possble state (.e. a probablty densty w(p,q,t) s ntroduced n the phase space), the relevant phase space ensemble s called a statstcal ensemble. p t = t 2 : G 2 G volume n the phase space, occuped by the statstcal ensemble. t = t 1 : G 1 q
Statstcal Averagng Statstcal average (expectaton) of an arbtrary physcal quantty F(p,q), s gven most generally by the ensemble average, F() t F( p, q) w( p, q,) t dpdq ( pq, ) The root-mean-square fluctuaton (standard devaton): ( F) F F 2 The curve representng the real moton (the expermental curve) wll mostly proceed wthn the band of wdth 2Δ(F) F True value F F ( F ) For some standard equlbrum systems, thermodynamc parameters can be obtaned, usng a sngle phase space ntegral. Ths approach s dscussed below. t
Ergodc Hypothess and the Tme Average Evaluaton of the ensemble average (prevous slde) requres the knowledge of the dstrbuton functon w for a system of nterest. Alternatvely, the statstcal average can be obtaned by utlzng the ergodc hypothess n the form, Here, the rght-hand hand sde s the tme average (n practce, tme t s chosen fnte, though as large as possble) F 1 t F F p (), q () d, t t 0 Ths approach requres F as a functon of the generalzed coordnates. Some examples Internal energy: U H( p, q) 2Ek Temperature: T k U du Change of entropy: S U T vl Gas dffuson constant: t D 3 B T 2 2 T 1 1 F C T V dt Here, E k mean knetc energy per degree of freedom v mean velocty of molecules l mean free path of gas partcles
Law of Moton of a Statstcal Ensemble A statstcal ensemble s descrbed by the probablty densty n phase space, w(p,q,t). It s mportant to know how to fnd w(p,q,t) at an arbtrary tme t, when the ntal functon w(p,q,0) q0) at the tme t = 0sgven gven. In other words, the equaton of moton satsfed by the functon w(p,q,t) s needed. p w(p,q,t 1 ) w(p,q,t qt 2 ) w(p,q,0) Γ 2 Γ 1 Γ 0 q The moton of of an ensemble n phase space may be consdered as the moton of a phase space flud n analogy to the moton of an ordnary flud n a 3D space. Louvlle s theorem clams that 0 1 2... Due to Louvlle s theorem, the followng equaton of moton holds 2s w H w H w [ H, w], [ H, w] (Posson bracket) t 1 q p p q
Equlbrum Statstcal Ensemble: Ergodc Hypothess For a system n a state of thermodynamc equlbrum the probablty densty n phase space must not depend explctly on tme, w t Thus, the equaton of moton for an equlbrum statstcal ensemble reads 0 [ H, w] 0 A drect soluton of ths equaton s not tractable. Therefore, the ergodc hypothess (n a more general form) s utlzed: the probablty densty n phase space at equlbrum depends only on the total energy: w ( p, q ) H ( p, q, a ) otes: the Hamltonan gves the total energy requred; the Hamltonan may depend on the values of external parameters a = (a 1, a 2, ), besdes the phase vector X. Ths dstrbuton functon satsfes the equlbrum equaton of moton, because [ H, ( H )] 0 Exercse: Check the above equalty.
Ensemble Method Ensemble? : Infnte number of mental replca of the system of nterest Large Reservor (const.t) All the ensemble members have the Same,V.T Energes can be exchanged but molecules cannot. Current = 20 but nfnty
Two postulates Long tme average = Ensemble average at nfnty t tme E1 E2 E3 E4 E5 In an ensemble, the systems of enembles are dstrbuted unformly (equal probablty or frequency) Ergodc Hypothess Prncple of equal a pror probablty
Averagng Method Probablty of observng partcular quantum state P n Ensemble average of a dynamc property n E E P Tme average and ensemble average U lm E t lm n E P
Calculaton of Probablty n Ensemble Several methods are avalable Method of Undetermned multpler : :
Maxmzaton of Weght Most probable dstrbuton Weght W! n! n! n!... 1 2 3! n!
Ensembles Mcro canoncal ensemble: E,V, Canoncal ensemble: T,V, Constant pressure ensemble: T,P, Grand canoncal ensemble: T,V,μ 21
Canoncal Ensemble: The canoncal ensemble occurs when a system wth fxed V and and the system s at constant temperature (connected to an nfnte heat bath). In the canoncal ensemble, the probablty of each mcrostate s proportonal to exp (- βem).
E 1 E EE 2 1 Systems 1 and 2 are weakly coupled such that they can exchange energy. What wll be E 1? E, EE E EE 1 1 1 1 2 1 BA: each confguraton s equally probable; but the number of states that gve an energy E 1 s not know. 23
E, E E E E E 1 1 1 1 2 1 E E E E E E ln, ln ln 1 1 1 1 2 1 ln E 1, E E1 Energy s conserved! 0 de E 1 =-de 2 1, V 1 1 ln E E 2 1 ln 1 E 1 0 E 1 1 E 1,V 1 2,V 2 Ths can be seen as ln 1 E 1 ln 2 EE 1 an equlbrum E1 E2 condton, V, V ln E E 1 2 1 1 2 2 V, 24
Entropy and number of confguratons Conjecture: S ln 25
S k ln E B Wth k B = 1.380662 10-23 J/K In thermodynamcs, the absolute (Kelvn) temperature scale was defned such that S E V, 1 T But we defne: n 1 de TdS-pdV d ln EE E V, 26
And ths gves the statstcal defnton of temperature: k B E 1 ln E T In short: V, Entropy and temperature are both related to the fact that we can COUT states. Basc assumpton: 1. leads to an equlbrum condton: equal temperatures 2. leads to a maxmum of entropy 3. leads to the thrd law of thermodynamcs 27
umber of confguratons How large s? For macroscopc systems, super-astronomcally large. For nstance, for a glass of water at room temperature: 10 210 25 Macroscopc devatons from the second law of thermodynamcs are not forbdden, but they are extremely unlkely. 28
Canoncal ensemble 1/k B T Consder a small system that can exchange heat wth a bg rese ln E E E E E E E E ln ln ln Hence, the probablty to fnd E : P E E E E E k T E E exp E k T B EE exp E k T j expe k T P E B j j B j B Boltzmann dstrbuton 29
Averagng Method Probablty of observng partcular quantum state P n Ensemble average of a dynamc property n E E P Tme average and ensemble average U lm E t lm n E P
Thermodynamcs What s the average age energy e of the system? E Compare: EP E FT 11 T E E exp E exp E j j ln e p E ln Q ln exp VT,, Hence: F ln Q VT,, kt B 31
Canoncal Partton Functon Q e j E j
The Boltzmann Dstrbuton Task : Fnd the domnatng confguraton for gven and total energy E. Fnd Max. W whch satsfes ; n dn 0 E t En E 0 dn
Method of Undetermned Maxmum weght, W Multplers Recall the method to fnd mn, max of a functon d ln W 0 lnw dn 0 Method of undetermned multpler : Constrants should be multpled by a constant and added to the man varaton equaton.
Method of undetermned multplers Method of undetermned multplers ln ln E dn dn dn dn W W d 0 ln dn E dn W 0 ln E W 0 E dn
n n W ln ln ln j j j n n n n n W ) ln ( ln ln 1 ln 1 ln 1 ln ln n n n j j j j j j n n n n n n n 1 ln 1 ln ) ln ( j j j j j n n n n n lnw n n n W ln 1) (ln 1) (ln ln
ln 0 E n E e n E j j e e n E j j j e 1 j e E j Boltzmann Dstrbuton E E j e e n P (Probablty functon for energy dstrbuton) j
Canoncal Partton Functon Boltzmann Dstrbuton P n e e E E j e Q Canoncal Partton Functon j E Q e j E j
Canoncal Dstrbuton: Prelmnary Issues One mportant prelmnary ssue related to the use of Gbbs canoncal dstrbuton s the addtvty of the Hamltonan of a mechancal system. Structure of the Hamltonan of an atomc system: kn H E 1 2 U 1 2 ( p, p,..., p ) ( r, r,..., r ) p ( ) (, ) (,, )... 2 W1 r W2 r rj W3 r rj rk 2m, j, j, k j Here, knetc energy and the one-body bd potental tlare addtve,.e. they can be expanded nto the components, each correspondng to one partcle n the system: kn kn kn E ( p 1, p 2,..., p ) E ( p 1) E ( p 2)... W1( r ) W1( r1) W1( r2)... Two-body and hgher order potentals are non-addtve (functon Q 2 does not exst), W ( r ) W r r W ( r, r )... Q ( r)... Q ( r )..., 2 j j 2 j 2 2 j, j
Canoncal Dstrbuton: Prelmnary Issues Thus, f the nter-partcle nteracton s neglgble, W W... E W kn 2 3 1 the system s descrbed by an addtve Hamltonan, p H h, h W ( ) 2 1 r 2m Here, H s the total Hamltonan, and h s the one-partcle Hamltonan. For the statstcal descrpton, t s suffcent that ths requrement holds for the averaged quanttes only. The mult-body components, W >1, cannot be completely excluded from the physcal consderaton, as they are responsble for heat transfer and establshng the thermodynamc equlbrum between consttutve parts of the total system. A mcromodel wth small averaged contrbutons to the total energy due to partcle-partcle nteractons s called the deal gas. Example: partcles n a crcular cavty. Statstcally averaged value W 2 s small: 3 partcles: 5 partcles: tot tot W 0 049 E 2 0.049 W2 0.026026 E
Canoncal Dstrbuton Suppose that system under nvestgaton Σ 1 s n thermal contact and thermal equlbrum wth a much larger system Σ 2 that serve as the thermostat,or heat or heat bath at the temperature T. From the mcroscopc pont of vew, both Σ 1 and Σ 2 are mechancal systems whose states are descrbed by the phase vectors (sets of canoncal varables X 1 and X 2 ). The entre system Σ 1 +Σ 2 s adabatcally solated, and therefore the mcrocanoncal dstrbuton s applcable to Σ 1 +Σ 2, 1 w( p1, q1; p2, q2) E H( p1, q1; p2, q2) ( E) Assume 1 and 2 are number of partcles n Σ 1 and Σ 2 respectvely. Provded that 1 << 2, the Gbbs canoncal dstrbuton apples to Σ 1 : Thermostat T Σ 1 2 Σ 1 Σ 2 1 2 H( p, q) 1 kt wpq (, ) e ( pq, ) ( p1, q1) Z
CanoncalDstrbuton: Partton Functon The normalzaton factor Z for the canoncal dstrbuton called the ntegral over states or partton functon s computed as H( pqa,, ) 1 kt Z e dpdq 3 (2 )! ( pq, ) H ( pr, ) 1 kt e d 3 1... d d 1... d (2 )! p p r r ( pr, ) Σ 2 Thermostat T 1 Σ 1 2 Before the normalzaton, ths ntegral represents the statstcally t t averaged phase volume occuped by the canoncal ensemble. The total energy for the canoncal ensemble s not fxed and n prncple t may The total energy for the canoncal ensemble s not fxed, and, n prncple, t may occur arbtrary n the range from to (for the nfntely large thermostat, 2 ).
Partton Functon and Thermodynamc Propertes The partton functon Z s the major computatonal characterstc of the canoncal ensemble. The knowledge of Z allows computng thermodynamc parameters of the closed sothermal system (a V, external parameter): Free energy: (relates to mechancal work) Entropy (varety of mcrostates) Pressure Internal energy FTV (, ) ktlnz F S kln ZT ln Z T T V F P kt ln Z VT V 2 U FTSkT ln Z T These are the major results n terms of practcal calculatons over canoncal ensembles. Class exercse: check the last three above formulas wth the the method of thermodynamc potentals, usng the frst formula for the free energy.
Thermodynamc Propertes and Canoncal Ensemble Internal Energy U Q U E, V ( qs) E P, V E e E 1 Q ( qs) E 1 Q ln Q Q e E, V
Thermodynamc Propertes and Canoncal Ensemble Pressure at state Small Adabatc expanson of system dx ( w ) ( de P ) PdV F dx E V F dx PdV w dv de F E V
Thermo recall (2) Frst law of thermodynamcs de TdS pdv Helmholtz Free energy: F ETS df SdT pdv FT 1 F F F F T 1T T 1T T F TS E 46
We have assume quantum mechancs (dscrete states) but we are nterested n the classcal lmt exp 2 1 p d d exp 3 h! p 2 m 1 3 h Volume of phase space (partcle n a box) E r U r 1 Partcles are ndstngushable! Integraton over the momenta can be carred out for most system 3 3 2 2 2 2 p p m dp exp dp exp 2 m 2 m 47
Defne de Brogle wave length: 1 2 2 h 2 m Partton functon: 1 Q, V, T d exp 3! r U r 48
Example: deal gas 1 Q, V, T d expu r 3! r 1 V d 1 3 3! r! Free energy: Pressur e: V F ln 3! 3 ln ln ln 3 ln V F P V T V Energy: E F 3 3 2 k T B 49
Ideal gas (2) Chemcal potental: = F T,V, j F ln 3 ln V ln 3 ln 1 IG 0 ln 50
Ensembles Mcro canoncal ensemble: E,V, Canoncal ensemble: T,V, Constant pressure ensemble: T,P, Grand canoncal ensemble: T,V,μ 51
Summary: Canoncal ensemble (,V,T) Partton functon: 1 Q, V, T d exp 3! r U r Probablty to fnd a partcular confguraton P Free energy exp U F ln Q VT,, 52
Thermodynamc Propertes and Canoncal Ensemble Pressure at quantum state P P Pressure PP E 1 E P Pe Q Q V ln Q E e V, Q V 1 ln Q P lnv Probablty 1 E E e Equaton of State n Statstcal Mechancs
Thermodynamc Propertes and Canoncal Ensemble w q du Entropy rev rev PdE E dp P E d du w q du ) ( rev w PdV dv E P de P rev dp Q PdP E dp w dv dv V d ) ln ln ( 1 PdP Q ln 1 ) (
Thermodynamc Propertes and Canoncal Ensemble Entropy q q PdE rev rev 1 d ( ln PdP q P ln P ) TdS d( U ln Q) TdS rev S S S k ln Q U / T S0 k ln Q U / T k P ln P k ln P The only functon that lnks heat (path ntegral) and state property s TEMPERATURE. 1/ kt
Summary of Thermodynamc Propertes n Canoncal Ensemble U S ln Q kt ( ) lnt ln Q kln Q ( ) lnt V, V, ln Q ln Q H kt( ) V, ( ) T, All thermodynamc propertes lnt lnv Can be obtaned from A kt ln Q PARTITIO FUCTIO G ln Q ) lnv ktln Q ( T, ln Q kt T, V, j
Ensembles Mcro canoncal ensemble: E,V, Canoncal ensemble: T,V, Constant pressure ensemble: T,P, Grand canoncal ensemble: T,V,μ 57
Constant pressure smulatons: V, E, E E V V 1/k B T PT,P,T ensemble p/k B T Consder a small system that can exchange volume and energy wth a bg reservor ln ln ln V V, E E ln V, E E V E V V E EE, V V E pv ln EV kt kt, B B Hence, the probablty to fnd E,V : P E, V EE, exp E V V pv jk, jk, E E V V j, k exp Ej pv k exp E pv 58
Thermo recall (4) Frst law of thermodynamcs Hence and d E T d S p d V + d 1 S = EV, T S V T, p T 59
,P,T,, ensemble (2) In the classcal lmt, the partton functon becomes 1 3,, exp d exp Q P T dv PV U r! r The probablty to fnd a partcular confguraton: r, V r, exp P V PV U r 60
Grand canoncal smulatons:, E, E E 1/k B T μ,v,t VT ensemble -μ/k B T Consder a small system that can exchange partcles and energy wth a bg reservor ln ln ln, E E ln, E E E E EE, E ln E kt kt, B B Hence, the probablty to fnd E, : P E, EE, exp E jk, jk, E E j, k exp Ej k k exp E 61
Thermo recall (5) Frst law of thermodynamcs Hence and d E T d S p d V + d 1 S = EV, T S TV, T 62
μ,v,t, ensemble (2) In the classcal lmt, the partton functon becomes exp Q, V, T d expur 3! r 1 The probablty to fnd a partcular confguraton:, r exp, r P U r 63
Classcal Statstcal Mechancs It s not easy to derve all the partton functons usng quantum mechancs Classcal mechancs can be used wth neglgble error when energy dfference between energy levels (E) aresmaller thank kt. However, vbraton and electronc states cannot be treated wth classcal mechancs. (The energyspacngs areorder of kt)
Phase Space Recall Hamltonan ofewtonanmechancs H ( r H, p ) KE(knetc energy) PE(potental energy) p ( r, p ) U ( r1, r2,..., r 2m H p r H r p Instead of takng replca of systems (ensemble members), use abstract phase p space compose of momentum space and poston space (6) Average of nfnte phase space )
Ensemble Average U 1 lm E( ) d lm 0 np ( ) E( ) d P ( d ) Fracton of Ensemble members n ths range (d) d) Usng smlar technque used for Boltzmann dstrbuton b P ( ) d exp( H / kt ) d... exp( H / kt ) d
Canoncal Partton Functon Phase Integral T... exp( H / kt ) d Canoncal Partton Functon Q... c exp( H / kt ) d Match between Quantum and Classcal Mechancs c lm... exp( E / kt )... exp( H / kt) d!h T ) For rgorous dervaton see Hll, Chap.6 c 1 F
Canoncal Partton Functon n Classcal Mechancs Q 1... exp( H / kt ) d! h F
Example : Translatonal Partton Functon for an Ideal Gas H ( r H ( r, p, p 3 2 p H 2m ) KE(knetc energy) PE(potental energy) p ) U ( r, r2,..., r 2m 1 ) o potental energy, 3 dmensonal space. Q 1... exp( 3! h 1! h 3 exp( 1 2 mkt 2! h 3 / 2 p 2m V 2 p 2m ) dp 3 ) dp... dp V 0 1 dr dr 1 2 dr dr... dr 3 1
Sem Classcal Partton Functon The energy of a molecule s dstrbuted n dfferent modes Vbraton, Rotaton (Internal : depends only on T) Translaton (External : depends on T and V) Assumpton 1 Hamltonan operator can be separated nto two parts (nternal + center of mass moton) H op H CM op H nt op Q exp( E CM E kt nt ) exp( E kt CM ) exp( E kt nt ) Q Q CM (, V, T ) Qnt (, T )
Sem Classcal Partton Functon Internal parts are densty ndependent and most of the components have the same value wth deal gases. Q nt (,, T ) Qnt (,0, T ) For solds and polymerc molecules, ths assumpton s not vald any more.
Sem Classcal Partton Functon Assumpton 2 For T > 50K, classcal approxmaton can be used for translatonal part.
H CM Q! 3 p 2 x p 2 y 2m p 2 z U ( r, r,..., r p 3 2 2 2 1 x y z 3... exp( ) dp... 3! h 2mkT Z 1 p 2 p ) ( U / kt) dr 3 1/ 2 2 h 2 mkt Z... ( U / kt) dr1 dr2... dr 3 Confguraton Integral Q 1 3 nt Q! Z For non central forces (orentaton effect) 1 Z d... ( U / kt) dr dr... dr 1 2 3 d... d 1
Canoncal Ensembles After adopton of the ergodc hypothess, t then remans to determne the actual form of the functon φ(h). Ths functon depends on the type of the thermodynamc system under consderaton,.e. on the character of the nteracton between the system and the external bodes. We wll consder canoncal ensembles of two types of systems: 1) Adabatcally solated systems that have no contact wth the surroundngs and have a specfed energy E. The correspondng statstcal ensemble s referred to as the mcrocanoncal ensemble, and the dstrbuton functon mcrocanoncal dstrbuton. 2) Closed sothermal systems that are n contact and thermal equlbrum wth an external thermostat of a gven temperature T. The correspondng statstcal ensemble s referred to as the canoncal ensemble,, and the dstrbuton functon Gbbs canoncal dstrbuton. Both systems do not exchange partcles wth the envronment.
Mcrocanoncal Dstrbuton For an adabatcally solated system wth constant t external parameters, a, the total t energy cannot vary. Therefore, only such mcrostates X can occur, for whch H ( p, q, a) E constant Ths mples (δ Drac s delta functon) and fnally: wpq (, ) E H( pqa,, ) E, a 1 wpq (, ) E H( pqa,, ) ( Ea, ) where Ω s the normalzaton factor, pq ( Ea, ) EH ( pqa,, ) dpdq pq ( pq, ) (P,T,V, ) Wthn the mcrocanoncal ensemble, all the energetcally allowed mcrostates have an equal probablty to occur.
Mcrocanoncal Dstrbuton: Integral Over States The normalzaton o factor Ω s gven by ( Ea, ) ( Ea, ) E a Phase volume where Γ s the ntegral over states, or phase ntegral: ( E, a ) 1 f (2 )! 1 (2 ) f! H ( p, q, a) E dpdq d (, ) 1 1 E H( pq, ) d p... d p d q... d q ( pq, ) 0, x 0 1, x 0 ( x), - Planck's constant, j - number of DOF per partcle Γ(E,a) represents the normalzed phase volume, enclosed wthn the hypersurface of gven energy determned by the equaton H(X,a) = E. Phase ntegral Γ s a dmensonless quantty. Thus the normalzaton factor Ω shows the rate at whch the phase volume vares due to a change of total energy at fxed external parameters.
Mcrocanoncal Dstrbuton: Integral Over States The ntegral over states s a major calculaton characterstc of the mcrocanoncal ensemble. The knowledge of Γ allows computng thermodynamc parameters of the closed adabatc system: 1 S 1 S k ln, T, P E ( E, V) V E (These are the major results n terms of practcal calculatons over mcrocanoncal ensembles.)
Summary: mcro canoncal ensemble (,V,E) Partton functon: 1 3,, d d, Q V E H E h! p r p r Probablty to fnd a partcular confguraton Free energy P 1 S ln Q V,, E 78
Mcrocanoncal Ensemble: Illustratve Examples We wll consder one-dmensonal llustratve examples of computng the phase ntegral, entropy and temperature for mcrocanoncal ensembles: Sprng-mass harmonc oscllator Pendulum (non-harmonc oscllator) We wll use the Hamltonan equatons of moton to get the phase space trajectory, and dthen evaluate the phase ntegral.
Harmonc Oscllator: Hamltonan Hamltonan: general form H T( p) U( x) Knetc energy T 2 p 2m k m Potental energy U kx 2 2 x Potental energy s a quadratc functon of the coordnate (dsplacement form the equlbrum poston) The total Hamltonan H ( p, x) p kx 2m 2 2 2 Parameters: 21 21 m 10 kg, k 25 10 /m
Harmonc Oscllator: Equatons of Moton and Soluton Hamltonan and equatons moton: Parameters: 21 21 m10 kg, k 2510 /m 2 2 p kx H H p H ( x, p) x, p x, p kx 2m 2 p x m Intal condtons (m, m/s): x(0) 0.2, x (0) 0 x (0) 0.4, x (0) 0 x(0) 0.6, x (0) 0
Harmonc Oscllator: Total Energy Total energy: E H( x, p) Const (at any x( t), p( t)) E 1 x(0) 0.2 x (0) 0 E 2 E 3 x(0) 0.4 x (0) 0 x(0) 0.6 x (0) 0 E 0.510 J, E 2.010 J, E 4.510 J 21 21 21 1 2 3
Phase ntegral: Harmonc Oscllator: Phase Integral 1 ( E) A( E), A( E) E H( x, p) dxdp E H(, j ) 2 2 ( x, p) x p x p 0 j0 2 x / step for x, 2 p / step for p, number of ntegraton steps x max p max A 1 A 2 A 3 3 1 0.9510 2 3.8010 3 8.54 10 For the harmonc oscllator, phase volume grows lnearly wth the ncrease of total energy. 12 12 12
Harmonc Oscllator: Entropy and Temperature 23 Entropy: S kln, k 1.3810 J/K 22 S 1 3.81 10 J/K 22 S2 22 S3 4.1110 J/K 4.00 10 J/K 1 S Temperature: T E We perturb the ntal condtons (on 0.1% or less) and compute new values The temperature s computed then, as 1 1 S S 2 kn T ( benchmark: T E ) E E k T T T 1 2 3 36.3 K 145.1 K 326.5 K E and S.
Pendulum: Total Energy Total energy: 2 p E H(, p) mglcos Const (at any ( t), p( t)) 2 2ml E 1 (0) 0.3 (0) 0 E 2 (0) 1.8 (0) 0 E 3 (0) 3.12 (0) 0 p l - angular momentum E 1.8710 J, E 0.4510 J, E 1.9610 J 21 21 21 1 2 3
Phase ntegral: Pendulum: Phase Integral 1 ( E) A( E), A( E) E H(, p) ddp E H(, j ) 2 2 (, p ) p p 0 j0 2 x / step for, 2 p / step for p, number of steps max p max A 1 A 2 A 3 1 0.410 2 11.210 3 21.4 10 For the pendulum, phase volume grows O-lnearly wth the ncrease of total energy at large ampltudes. 12 12 12
Pendulum: Entropy and Temperature Entropy: S kln, 23 k 1.3810 J/K 22 S 1 3.6810 J/K 22 S2 22 S3 4.15 10 J/K 4.24 10 J/K 1 S Temperature: T E We perturb the ntal condtons (on 0.1% or less) and compute new values The temperature s computed then, as 1 S S 2 kn T ( benchmark: T E ) E E k T 1 T T 2 3 63K 6.3K 153.8 K 96.0 K E and S.
Summary of the Statstcal Method: Mcrocanoncal Dstrbuton 1. Analyze the physcal model; justfy applcablty of the mcrocanoncal dstrbuton. 2. Model lndvdual d partcles and dboundares. 3. Model nteracton between partcles and between partcles and boundares. 4. Set up ntal condtons and solve for the determnstc trajectores (MD). 5. Compute two values of the total energy and the phase ntegral for the orgnal and perturbed ntal t condtons. 6. Usng the method of thermodynamc parameters, compute entropy, temperature and other thermodynamc parameters. If possble compare the obtaned value of temperature wth benchmark values. 1 S 1 S kln, T, P E ( E, V ) V E 7. If requred, accomplsh an extended analyss of macroscopc propertes (e.g. functons T(E), S(E), S(T), etc.) by repeatng the steps 4-7.
Free Energy and Isothermal Processes Free energy, also Hl Helmholtz hlt potental tls of mportance for the descrpton of sothermal processes. It s defned as the dfference between nternal energy and the product of temperature and entropy. F U TS Snce free energy s a thermodynamc potental, the functon F(T,V,, ) ) guarantees the full knowledge of all thermodynamc quanttes. Physcal content of free energy: the change of the free energy df of a system at constant temperature, represents the work accomplshed by, or over, the system. Indeed, df du TdS SdT du TdS W df SdT W W Isothermal processes tend to a mnmum of free energy,.e. due to the defnton, smultaneously to a mnmum of nternal energy and maxmum of entropy.