Physics 212: Statistical mechanics II Lecture I

Transcription

1 Physcs 212: Statstcal mechancs II Lecture I A theory s the more mpressve the greater the smplcty of ts premses, the more dfferent knds of thngs t relates, and the more extended ts area of applcablty. Therefore the deep mpresson that classcal thermodynamcs made upon me. It s the only physcal theory of unversal content whch I am convnced wll never be overthrown, wthn the framework of applcablty of ts basc concepts. A. Ensten Let us start wth an overvew of the materal to be covered ths semester. You, the student, should by now have had a sold undergraduate semester of basc statstcal mechancs, ncludng such concepts as entropy and the laws of thermodynamcs together wth applcatons to deal classcal and quantum gases. There s a great deal more to the story, ncludng both new fundamental concepts and a wealth of applcatons. Many of the terms and deas n ths overvew may be qute unfamlar, but should become much clearer durng the course of the semester. A note on textbooks: I chose the newsh book by Kardar as requred because t covers the most ntellectually challengng topcs of the course and has a good selecton of problems. For a modern perspectve on basc materal, I suggest the book by Sethna n Oxford s Master s Degree seres. Part I (nonequlbrum statstcal mechancs) and Part II (ntroducton to RG and crtcal phenomena) are relatvely standard materal. Good specalzed books for part 2 are by Cardy and by Goldenfeld. Part III (selected applcatons of parts I and II) wll consst largely of recent examples for whch there s not a good textbook. Some general texts for Part I whch contan knetc theory are K. Huang, Statstcal mechancs. R. Balescu, Equlbrum and nonequlbrum statstcal mechancs L. D. Landau and E. M. Lfshtz, Statstcal mechancs and Physcal knetcs A useful reference on the fluctuaton-dsspaton theorem s R. Kubo, M. Toda, N. Hashtsume, Statstcal physcs II (Nonequlbrum) A nce small book for Part II (crtcal phenomena and RG) s J. Cardy, Scalng and renormalzaton n statstcal physcs Others are S.-K. Ma, Statstcal mechancs N. Goldenfeld, Lectures on phase transtons and the renormalzaton group, whch has some useful materal on dynamc phenomena. The frst few lectures wll study how entropy, temperature, and other thermodynamc equlbrum quanttes emerge from mcroscopc equatons of moton (the problem of generalzed knetc 1

2 theory or hydrodynamcs). Understandng the approach to thermodynamc equlbrum at a hgh level of rgor s a dffcult mathematcal challenge, but here we wll be content wth the standard physcs arguments datng back to Boltzmann, Gbbs, and others. Understandng knetc theory wll let us understand a bt better where the followng quanttes come from (here we take the entropy to be dmensonless): S = log Ω = p log p kt = ( ) S 1 (1) E Let s start wth the entropy S. In the above formulas for S, Ω s the number of equally lkely states n the frst expresson, and p are the event probabltes of a general probablty dstrbuton n the second expresson. For mcroscopc equatons of moton whch are tme-reversal nvarant, t seems perplexng that there can be emergence of rreversblty. For example, consder partcles trapped n the left half of a box at t = 0 (see fgures at end). From the behavor of real systems, one knows that when the partcles are released, ther densty tends to spread over tme untl a unform equlbrum dstrbuton of partcles s reached. How s ths consstent wth the Poncare recurrence theorem, whch predcts that eventually the partcles wll all return at the same tme to the left sde? The frst part of Part I wll work on the mportant classcal example of the classcal gas wth bnary collsons, and gve a mcroscopc dervaton of the Naver-Stokes equaton. In the next lecture or two we wll see how entropy ncreases n ths smple problem (Boltzmann s H theorem ). One example of the subtletes of hydrodynamcs, usng ths dlute flud as an example, s that there s not just one equlbraton tme : nstead there s a rapd tme on whch local densty, mean velocty, and temperature become well defned (relaxaton to a local thermal dstrbuton), and then a much longer tme to reach global thermal equlbrum. A clue for why these three quanttes take a whle to reach overall equlbrum s that they correspond to three quanttes (partcle number, momentum, and energy) that have conservaton laws. Hydrodynamcs s an example of the mportance of feld theores (.e., descrptons n terms of contnuous functons of space-tme) and coarse-granng, whch can be defned as averagng over unmportant degrees of freedom to focus on the essentals. We wll study hydrodynamcs as a detaled example of ths; Kardar uses phonons n solds as an example where calculatons are easer because the theory (neglectng anharmoncty) s lnear. If t s assumed that a system remans close to thermal equlbrum, a much more general pcture of weak nonequlbrum physcs can be based upon the concept of lnear response. Consder thermodynamc equlbrum of a suspenson of heavy partcles n a flud, wth a constant gravtatonal feld pullng the heavy partcles downward. The heavy Brownan partcles undergo collsons wth the molecules of the flud, and mcroscopcally undergo a random walk as a result (gnorng for the moment the gravtatonal feld). Ths sort of Brownan moton wll be the subject of at least one lecture toward the end of Part I. We expect the equlbrum densty n the gravtatonal feld to go as ρ(h) exp( βmgh). (2) At equlbrum, there are two fluxes of heavy partcles whch should exactly balance each other: gravty pulls partcles downward, whle the partcle gradent leads to a dffusve flux upward. 2

3 Let s calculate the gravtatonal effect frst. Frst consder one heavy partcle at zero temperature, fallng under gravty. Now we need to nvoke a result from hydrodynamcs. The free fall downward of the heavy partcle n a vscous flud has a termnal velocty, whch for a sphercal Brownan partcle of radus η s v term = F 6πrη = mg 6πrη (I beleve that ths s called the Stokes law.) Here η s the vscosty of the flud. In a moment we ll say somethng about why the vscosty enters here. Dffuson predcts a current drected upward: j D = D ρ = Dρβmg (4) Note that the strength of gravty drops out. Equate these two, to get D = 1 6πrβη Now the deeper sgnfcance of ths relaton s that the rght-hand sde can be related to the velocty correlaton functon of the Brownan partcle n equlbrum: as a result, D = (3) (5) dt v(0) v(t) (6) Ths s a fluctuaton-dsspaton relatonshp n that the left sde descrbes a dsspatve lnear response (dffuson) whle the rght sde descrbes dynamcal fluctuatons n thermal equlbrum. Many mportant transport quanttes, such as most conductvtes n solds, are dsspatve lnear responses and can be calculated The so-called Ensten relaton (5) for Brownan moton dates from around We wll cover the Kubo formula and possbly other results from the 50s and 60s whch nvole quantum-mechancal versons of the same concept: weak nonequlbrum propertes lke the dffuson constant can be expressed n terms of correlatons of related quanttes n the equlbrum system. The Kubo formula relates the electrcal conductvty of a quantum system to ts current-current correlaton functon, and s essental for many current problems n physcs and electrcal engneerng. Part II of the course wll start wth the modern theory of contnuous phase transtons. You should by now be aware of the lqud-gas transton (bolng) of water. There s a lne of frstorder transtons n the phase dagram (a transton s frst-order f there s a dscontnuous frst dervatve of free energy such as pressure). Ths lne termnates at a crtcal pont C (see fgure), where the transton s second-order and only the second dervatves of the free energy are dscontnuous. Along the frst-order lne near C, we can wrte an equaton for the densty dfference between the lqud and gas phases, whch goes to zero at C: ( ) T β TC ρ L ρ G (7) T The crtcal exponent β s found expermentally to be qute smlar for many dfferent lqud-gas transtons, even though the transton temperature T C vares from substance to substance. Its numercal value from experment and theory s about β (8) 3

4 Now consder a smple model of a ferromagnet. As was dscovered a long tme ago (I beleve n Chna n the 4th century BC), a magnetzed pece of ron loses ts ferromagnetsm above some crtcal temperature T C. For smplcty, let s consder an Isng or unaxal magnet, where the local magnetc moment ponts along ether ẑ or ẑ. Now below T C the magnet develops a spontaneous magnetc moment, ether up or down. The phase dagram n the (H, T ) plane of such a magnet, where H s an appled magnetc feld, also has a lne of frst-order transtons, termnatng n the pont (0, T C ) where the transton s second-order. Along the frst-order lne, there s a magnetzaton dfference M M, and near the crtcal pont ths scales as a power-law: ( ) T β TC M M. (9) Once agan, the value of β seems to be constant for dfferent magnetc substances, as long as they are unaxal (.e., have only dscrete up-down symmetry, rather than the contnuous symmetry of rotatons n the plane or n 3-D). The crtcal temperature T C vares wdely, however. Its numercal value from experment and theory s about T C β (10) The equalty of crtcal exponents lke β across wdely dfferent problems n physcs s qute surprsng, and was only understood n the 1970s through the renormalzaton group (RG) theory of phase transtons. Ths theory ncludes the concept of unversalty classes determned by symmetry and dmensonalty. The lqud-gas transton n 3D and the unaxal magnet n 3D share, near the crtcal pont C, the same symmetry and dmensonalty, and hence have the same crtcal propertes. The renormalzaton group s a way to handle problems wth physcs on a contnuous range of length scales (no separaton of scales ). The theory of second-order phase transtons may sound very specalzed. There are many other examples of RG beyond phase transtons, ncludng dynamcal problems lke nterfacal growth/evoluton (balance of dffuson and stochastcty), percolaton/polymers/other soft condensed matter problems, many examples n hgh-energy physcs, and possbly turbulence (normal and MHD). There are even RG applcatons to problems n ecology and economcs. We wll try to cover a few of these applcatons toward the end of the course. Now the overvew ends and the course materal begns. As a prelmnary to our study of knetc theory, let s recall the followng smple dervaton of the Boltzmann factor exp( βh), whch may be famlar from your undergraduate course n statstcal mechancs. (Ths β = 1/kT has nothng to do wth the crtcal exponent β defned above!) Consder a subsystem weakly coupled to a large number of other dentcal systems, n such a way that energy s exchanged but the energy levels are unmodfed. Then the number of confguratons wth ν 1 systems n energy E 1, ν 2 n energy E 2, and so on, s P = N! ν 1!ν 2!ν 3!... (11) Now we can ask: whch dstrbuton of energy s the most probable? Strlng s asymptotc approxmaton s log(n!) n(log n 1). (12) 4

5 Now P and therefore log P wants to be maxmzed subject to the constrant of fxed total energy and fxed N. ν = N, e ν = E (13) Then maxmzng (see note on method of Lagrange multplers below) log P α ν β e ν (14) becomes for one partcular and then takng the dervatve wth respect to ν gves ν log ν + ν αν βe ν (15) log ν + α + βe = 0 ν = e α βe. (16) It remans to determne α and β from the constrant equatons. We know that the total number of systems should be N, whch lets us wrte e α e βe = N. (17) So the percentage of systems n state ν s ν N = e βe e βe = e βe Z. (18) Here Z exp( βe ) s the partton functon of a sngle subsystem. Now β s stll undetermned. The total energy of the system s constraned to be It s left as an exercse to show that ν e = NZ 1 e e βe = E total. (19) d log P de total β = β, (20) whch confrms that β = (kt ) 1, snce the left sde of the above s equal to (kt ) 1 by the defnton of fundamental temperature. A nce descrpton of the sharpness of the probablty dstrbuton of system energes s n Chapter 2 of the undergraduate textbook Thermal Physcs, by Kttel and Kroemer. We can ask about how good ths model of subsystems really s, and what the above assumptons have to do wth entropy. For the classcal gas wth bnary collsons, to be studed n the next lecture, we ll fnd that wthout bnary collsons, entropy s constant; wth collsons, entropy ncreases. Hence nformaton s lost somewhere n the model of two-body collsons. Complete nformaton about a classcal system of N partcles s contaned n the N-body dstrbuton functon f(t, x 1, p 1, x 2, p 2, x 3, p 3,..., x N, p N ). In realty one has to make do wth much less nformaton about a macroscopc system. The fundamental object for much of Part I s the one-body dstrbuton functon f(t, x, p), whch we defne so that ts ntegral over a volume of phase 5

6 space gves the expected number of partcles n that volume of phase space. Ths functon already contans the local partcle densty n(t, x) = f(t, x, p) dp (21) as well as the local mean energy and other unversal quanttes. We wll see examples later of how two-partcle correlatons not contaned n f can stll affect the evoluton of f and hence of the local densty and other physcal quanttes. Mathematcal nterlude: Let me revew a quck example of the method of Lagrange multplers used above, to make clear ts geometrc content. We wll be usng ths method a few more tmes durng the course. suppose we want to maxmze a functon f(x, y) of two varables, subject to a constrant g(x, y) = C. If the constrant were not nvolved, then local mnma or maxma or saddle ponts ( statonary ponts ) of f would satsfy f = ( x f, y f) = 0. (22) Now let us consder the effect of the constrant. One can search for a statonary pont of f subject to the constrant by rememberng that f ponts n the drecton of most rapd change of f. From now on I ll say mnmum n place of statonary pont, although n practce one may want to verfy that statonary ponts that are found usng ths technque are actually mnma. We want to fnd a pont where movng n the drecton f, n order to change f, would force a change n the value of g, therefore volatng the constrant. In other words, we want f g, or f = α g (23) for some number α. In components the above equaton becomes that x f = α x g y f = α y g. (24) Now note that these are exactly the same equatons we would have obtaned from searchng from an unconstraned mnmum of the functon f αg. At ths pont, because of ntroducng α, we have two equatons for three unknowns (α and the two coordnates (x 0, y 0 ) of the mnmum), so the problem s underdetermned. But recall that we stll have the constrant equaton g(x 0, y 0 ) = C, so the problem has been reduced to solvng three equatons for three unknowns. The multple-constrant, multple-varable generalzaton of the above s a lttle bt more dffcult to pcture geometrcally: now the gradent of the functon must le n the subspace spanned by the gradents of the constrants. Wth a functon of N varables and M constrants, one ntroduces M free multplers lke α above, and wnds up wth N + M equatons for the unknowns. 6