STATISTICAL MECHANICAL ENSEMBLES 1 MICROSCOPIC AND MACROSCOPIC VARIABLES PHASE SPACE ENSEMBLES. CHE 524 A. Panagiotopoulos 1

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1 CHE 54 A. Panagotopoulos STATSTCAL MECHACAL ESEMBLES MCROSCOPC AD MACROSCOPC ARABLES The central queston n Statstcal Mechancs can be phrased as follows: f partcles (atoms, molecules, electrons, nucle, or even lvng cells obey certan mcroscopc laws wth specfed nterpartcle nteractons, what are the observable propertes of a macroscopc system contanng a large number of such partcles? Examples of mcroscopc and macroscopc varables are gven below for /6th mole of an n- component monoatomc gas (approx. 0 3 molecules obeyng classcal mechancs. Mcroscopc varables 3x0 3 postons (x,y,z 3x0 3 veloctes (x,y,z Macroscopc varables n+ ndependent thermodynamc varables - e.g. for n,,t PHASE SPACE The multdmensonal space defned by the mcroscopc varables of a system. n the example above, t would be a 6x0 3 -dmensonal space, wth ndependent varables (r,p (r,r,r 3,...,r ;p,p,p 3,...,p where bold symbols ndcate vectors, r s the poston and p the momentum (p mu. Momenta are used rather than veloctes, because the classcal and, especally, the quantum equatons of change are more elegantly wrtten n terms of momenta. The evoluton of such a system n tme s descrbed by ewton's laws: dr du u ; ( r, r, r3..., r m dt dt r where U s the total potental energy of the system, the gradent of whch s mnus the force. For a much smpler system, a one-dmensonal harmonc oscllator, phase space s two dmensonal, wth coordnates the poston and the momentum varables. ESEMBLES An ensemble s a collecton of all mcrostates of a system, consstent wth the constrants wth whch we characterze a system macroscopcally. For example, a collecton of all possble states of Some materal n ths secton s derved from Chap. 3, D. Chandler, "ntroducton to modern statstcal mechancs", Oxford Unversty Press, 987.

2 CHE 54 Ensembles the 0 3 molecules of gas n the contaner of volume wth a gven total energy U s a statstcal mechancal ensemble. For the one-dmensonal harmonc oscllator wth a gven energy, the phase space s a crcular trajectory n poston and momentum space. ERGODC HYPOTHESS Expermental measurements on any macroscopc system are performed by observng a system for a fnte perod of tme, durng whch the system samples a very large number of possble mcrostates. n order to connect the measured propertes wth the propertes calculated from statstcal mechancs, we have to assume that: For suffcently long tmes, a macroscopc system wll evolve through (or wll come arbtrarly close to all mcroscopc states consstent wth the macroscopc constrants we mpose n order to control the system. n other words, expermental measurements (performed by tme averages and ensemble averages are equvalent. The "ergodc hypothess" s more than just a hypothess. t s a general property of almost all real systems composed of a large number of partcles. The ergodc hypothess s equvalent to the postulate of classcal thermodynamcs that there exst stable equlbrum states fully characterzed by n+ ndependent macroscopc varables. ote that the ergodc hypothess does not state anythng about the relatve probabltes of observng gven states - t just states that all states wll eventually be observed. A general property F wll thus obey the followng rule: Fobserved Σ P v F v <F> probablty of value of property ensemble average fndng the system F n mcrostate v n mcrostate v MCROCAOCAL ESEMBLE: COSTAT U,, Basc Postulate of Statstcal Mechancs: For an solated system at constant U, and all mcroscopc states of a system are equally lkely at thermodynamc equlbrum. f Ω(,U s number of mcrostates wth energy U, then the probablty of mcrostate v, accordng to the postulate above, s P v /Ω (,U The concept of densty of states s really of quantum mechancal orgn. However, t can be ntroduced wthout resort to quantum mechancs by assumng that a gven contaner s parttoned

3 CHE 54 Ensembles 3 nto many small compartments, and the veloctes of molecules are dscretzed n terms of the possble drectons and magntudes. Example: Balls n a box Let us consder a very smple example, namely a box wth 0 slots, each contanng a ball that can be at the bottom of the slot (wth energy 0, or at one of two hgher levels, wth energes + and +, respectvely. The number of mcrostates for ths system depends on the overall energy of the box. There are: 0 state wth energy 0; 0 states wth energy +; states wth energy + and so on. As you can see, the number of mcrostates for even a smple system ncreases rapdly wth the energy of the system. Defnton of Entropy Let us defne a quantty S, such that: S k B lnω(,u where k B s a constant (to be later dentfed as Boltzmann's constant, k B R/ A.380x0-3 J/K, where A s Avogadro's umber, A 6.03x0 3 mol -. S has the followng propertes:. S s extensve: f we have two ndependent subsystems, A and B, then SA+B k B ln(ωa+b k B ln(ω A Ω B k B lnω A + k B lnω B S A +S B The reason for ths s that, for ndependent subsystems, each mcrostate of system A can be combned wth a mcrostate of system B to gve a mcrostate of the combned system. For mxng two fluds, we only get the above expresson f we assume that the partcles n the systems are ndstngushable. f partcles were dstngushable addtonal "states" would be avalable to the combned system resultng from the possblty of exchangng the postons and momenta of partcles n dfferent regons. Although the ndstngushablty of partcles s really of quantum mechancal orgn, t was ntroduced ad hoc by Gbbs before the development of quantum mechancs, precsely to make statstcal-mechancal entropy an extensve property.. S s maxmzed at equlbrum: For a system wth nternal constrants (e.g. nternal rgd walls or barrers to energy transfer, the number of possble mcrostates s always less than the number of mcrostates after the constrants are removed. S (,U > S (,U; nternal constrants

4 CHE 54 Ensembles 4 To see ths second property, consder the box wth partcles of the example above, and thnk of any constrant to the system at a gven energy (say U+. An example of a "constrant" would be to have that the frst fve slots have exactly unt of energy. The number of mcrostates n ths case s (5x55, less than the 55 states avalable to the unconstraned system. n concluson, S has the same propertes as the entropy. Statement ( above s the mcroscopc statement of the Second Law of thermodynamcs. From the Fundamental Equaton of thermodynamcs, du Td S Pd + μ d d S du + T P T d μ T d [] The second form of the fundamental equaton (the "Entropy Representaton" s the most useful for Statstcal Mechancs. Snce (from equaton S T ln Ω k T B β [] The symbol β s commonly used n statstcal mechancs to denote the nverse temperature. Example (cont.: Balls n a box n the example above, we can now defne the temperature of the box from equaton. ntally, as the energy ncreases, the temperature ncreases, as expected. However, somethng strange happens to the system at energes greater than +0 - can you guess what that s? What s the sgn of the temperature for energes > +0? s ths physcally reasonable? CAOCAL ESEMBLE: COSTAT,T Let us frst obtan the equlbrum condton for two systems, and, that are placed n thermal contact, so that they can exchange energy. The number of mcrostates avalable to the combned system must be a maxmum at equlbrum, snce the combned system s under condtons of constant and U dscussed prevously. Mathematcally, the condton for equlbrum s that

5 CHE 54 Ensembles 5 S total s maxmum δs total 0 δs + δs S S 0 δu + δu 0 Snce the total system s solated, δu + δu 0. Combnng wth the above condton, we obtan S S T T n other words, systems that can exchange energy much have the same temperature at equlbrum. ow, let us assume that one of the two systems s much larger than the other, so that t effectvely acts as a "constant-temperature bath." The total system (small system + bath s agan consdered under U condtons. ow consder the small system at a gven mcrostate ν wth energy U ν. The energy of the bath s U B U - U ν. The bath can be n any of Ω(U B Ω(U - U ν mcrostates. Snce the probablty of the total system beng n any partcular combnaton of mcrostates wth a gven total energy s the same, the probablty of fndng the small system n state ν s P ν Ω (U - U ν exp (ln(ω (U - U ν We can expand lnω around Ω(U gven that U ν s much smaller than U: ln Ω ln ν ν + ( Ω( U U ln( Ω( U U hgher ordr terms and substtutng back n the expresson for P ν usng lnω/ β /k B T, P ν exp ( - βu ν [3] Ths s a very mportant result. n words, we are fndng that the probablty of each mcrostate n the canoncal ensemble (constant, T s proportonal to the exponental of the energy dvded by the temperature. n order to fnd the absolute probablty of each mcrostate, we need to make sure that the sum of all the probabltes s one. The normalzaton constant for ths s called the "canoncal partton functon," Q. Q(, T exp( β [4] all mcrostates U The summaton over mcrostates s performed over all energes and partcle postons. For the smple example of the box wth partcles we have been followng, the partton functon at a gven nverse temperature β would be: Q(β + 0exp(β + 55exp(β +

6 CHE 54 Ensembles 6 Where the frst term n the summaton comes from the sngle energy state wth U 0, the second from the 0 states wth U + and so on. For ths smple system, both volume and number of partcles are fxed, so they do not appear n the summaton - however, n the general case the partton functon would be a functon of both and. Once the partton functon s defned, the probablty of each mcrostate can now be wrtten explctly: P ν exp( βu ν Q [5] Therefore, n the canoncal ensemble, a general property F s gven by all mcrostates F exp( βu < F > [6] exp( βu all mcrostates t s possble to relate dervatves of lnq to thermodynamc propertes. For example, ln Q ( Q / β β Q U exp( βu ν ν, exp( βu ν One can also calculate the averaged squared fluctuaton of energy n the canoncal ensemble: Pν U ν ( Pν ν ( δu ( U U ( Q / β Q ( Q / β Q ln Q β β usng the defnton of the heat capacty, C , we get T, ( δ U k T [7] B C Ths s a remarkable result! We have obtaned a relatonshp between the thermodynamc parameters of a system and the sze of the spontaneous fluctuatons. t s nterestng to note that the heat capacty of the system, C, s an extensve varable (grows lnearly wth the sze of the system. Ths mples that the relatve magntude of the spontaneous fluctuatons grows as:

7 CHE 54 Ensembles 7 / ( δu / B ( k T C / [8] For a macroscopc system, O(0 3, ths s a very small number: the energy of an deal gas system at equlbrum wth a room-temperature bath s constant to roughly part n 0. However, for small systems typcally used n smulatons, 00-,000, so that typcal fluctuatons are 0%-3%. Q can be dentfed wth a famlar thermodynamc functon. To do ths, let us wrte k B ln Q k B ln β exp( U ν [9] ν As just shown, the relatve fluctuatons n energy for a macroscopc system are very small. We can approxmate the sum n equaton 9 by summng just the domnant terms. All of these wll have U ν . There are Ω( such terms (snce ths s the number of mcrostates at that U, and thus k B lnq k B ln(ω ( exp(-β k B lnω ( - / T S - U / T - A / T [0] Ths should have been expected. n the mcrocanoncal (const. U ensemble the mportant functon Ω s such that k B lnω S. Recall that S s the functon maxmzed at const. U. n the canoncal (T ensemble, the thermodynamc functon beng mnmzed s A, or equvalently -A / T s maxmzed. We see a smlar relatonshp of -A / T wth k B lnq. The approxmaton s exact at the "thermodynamc lmt" (for an nfnte system. The table below summarzes the connectons between "classcal" and "statstcal" thermodynamc propertes dscussed thus far. Const. and U Const., and T Classcal Statstcal Classcal Statstcal Entropy, S s maxmzed at equl. k B lnω S Helmholtz Energy, A s mnmzed at equl. k B lnq -A / T Example (Chandler 3.4 Consder a system of dstngushable ndependent partcles, each of whch can exst n one of two states separated by energy ε. We can specfy the state of the system, ν, by lstng ν ( j n, n,..., n j,..., n, n 0 or The total energy of the system for a gven state s

8 CHE 54 Ensembles 8 Frst, we can obtan thermodynamc propertes of ths model from the mcrocanoncal ensemble. The number of mcrostates wth energy mε s: So that (usng Strlng s approxmaton, : β lnω E lnω ε m ( ( ln( m++ lnm ε ( mln( m+ mlnm ε m ε ln m m ε ln m βε ln m Equvalently,, whch gves E 0 at T0 (only ground state s populated and Eε/ as T (all states are equally lkely. The same result could have been obtaned n the canoncal ensemble: The exponentals factor nto an uncoupled product: The nternal energy E s ensemble., exactly the same as the result n the mcrocanoncal

Lecture 4. Macrostates and Microstates (Ch. 2 )

Lecture 4. Macrostates and Microstates (Ch. 2 ) Lecture 4. Macrostates and Mcrostates (Ch. ) The past three lectures: we have learned about thermal energy, how t s stored at the mcroscopc level, and how t can be transferred from one system to another.