6-1 CHAPTER 6 BOLTZMANN'S H-THEOREM In the atter part o the nneteenth century, Ludwg Botzmann amost snge-handedy estabshed the ed now nown as statstca mechancs. One o hs maor contrbutons, and undoubtedy the most controversa, was hs H-theorem. Ths wor was pubshed n 1872 wth the ntent o showng that the second aw o thermodynamcs derves rom the aws o mechancs. Whe t s now generay agreed that the attempted proo was unsuccessu, the controversy attendng ths wor was utmatey beneca to the burgeonng ed o statstca mechancs because t orced the worers to thn through and rene the statstca and probabstc concepts that were ntroduced by Botzmann. 6.1 THE H-THEOREM H = n δx δy δz δp δp δp (6-1) Botzmann began by denng the uncton H or a dute gas comprsed o spherca partces 1 where s a dstrbuton uncton whch determnes the number o partces n ocated n the spata regon δx δy δz and havng momentum n the range δp x δp y δp z through the reaton n = (x, y, z, x y z p, p, p,t) δ x δ y δ z δ p δ p δ p (6-2) x y z The term δx δy δz δp x δp y δp z s denoted δv µ and s reerred to as the "voume" o a ce n 6-dmensona µ-space. The ces occupy equa "voumes" o µ-space. Each partce has sx degrees o reedom and coud be competey speced by a pont n µ-space. Thus, a quantty o gas contanng N partces can be represented by a swarm o N ponts n µ-space and the dstrbuton uncton tes us how these N ponts are parttoned among the ces o µ-space. The summaton n Eq. (6-1) s taen over a o the ces n µ-space. As ndcated n x y z 1 Ths dervaton cosey oows R. C. Toman, The Prncpes o Statstca Mechancs, Oxord Unversty Press, London, 1938, pp. 134-140.
Eq. (6-2), the dstrbuton uncton coud depend upon poston, momentum, and tme. The uncton H can be restated as 2 n n H = N n + constant (6-3) N N I n /N coud be taen as the probabty o a partce beng ound n the th ce o µ-space, we coud wrte Eq. (6-3) as H = N P n P constant (6-4) + The rst rght-hand term o Eq. (6-4) woud appear to be reated to the statstca mechanca entropy 3, but t must be remembered that the atter quantty reers to an equbrum state and thereore the P 's shoud be the ce occupaton probabtes when the equbrum dstrbuton prevas. Our substtuton o n /N or P mpes that the oowng reatonshp between H and S s vad ony as equbrum s approached 6-2 or S H = - + constant (6-5) dh ds - = (6-6) The tme dervatve o S can thereore be obtaned rom the tme dervatve o H whch n turn depends on the change n wth tme. Partce cosons provde the mechansm or changes n and when moecuar chaos s assumed, t can be shown that 4 d H d t 0 (6-7) 2 3 4 See Appendx 6A or detas. Compare wth Eq. (2-10). For detas see Appendx 6B.
6-3 As per Eq. (6-6) ths resuts n d S 0 (6-8) d t Thus, H can never ncrease and Eq. (6-5) s vad, S can never decrease. These dervatves become zero at equbrum where together orward and reverse cosons zero out and the Maxwe-Botzmann dstrbuton prevas. 5 6.2 THE PARADOXES The scentc communty was ntay septca o Botzmann's resut and a engthy controversy ensued. 6 For the most part, the attacs on the H-theorem centered on two eatures whch are now nown as paradoxes assocated wth the names o Zermeo and Loschmdt. Zermeo's paradox, aso nown as the recurrence paradox, cas Botzmann's resut nto queston because, accordng to a theorem proved by Poncaré: any mechanca system o xed energy and voume obeyng the aws o cassca mechancs (the type o system consdered by Botzmann) must eventuay return arbtrary cose to ts nta state. Thus, Botzmann's H shoud eventuay ncrease as the system returns to ts nta state and thereore H can not be sad to never ncrease. It turns out that the ey word here s eventuay; Poncaré's theorem speces a nte recurrence tme, but estmates show ths tme to be astronomcay arge. Thus, whe acceptng the vadty o Zermeo's cam, Botzmann was abe to argue that because o very ong recurrence tmes, no one woud ever observe a system wth ncreasng H and thereore hs H-theorem woud not conct wth experence. Botzmann aso used the argument that the Second Law s statstca n nature and stated that whe systems wth ncreasng H were possbe, ther occurrence coud be assgned an extremey ow probabty. There does not seem to be a snge precse statement o the paradox attrbuted to Loschmdt. In hs ony pubshed statement on the subect, 5 Dened by Eq. (2-4). 6 A detaed account s provded by S.G. Brush, The Knd o Moton We Ca Heat, North-Hoand Pubshng Co., Amsterdam, 1976, pp. 598-612.
Loschmdt wrote 7 "...the entre course o events w be retraced at some nstant the veoctes o a ts parts are reversed." Botzmann n hs response assocated the obecton wth Loschmdt and because the two were coeagues, there seems to be tte doubt that they had dscussed the matter. The gst o Loschmdt's obecton was that Botzmann had used Louve's equaton, whch s based on the cassca equatons o moton, to descrbe the behavor o hs system. Because these cassca equatons are symmetrc n tme and because the use o statstca methods ntroduces no asymmetry, t s not reasonabe to expect the asymmetrc behavor o Botzmann's H. Thus, Loschmdt's paradox mght be smped to: How s t possbe to obtan tme-asymmetrc behavor or a system that obeys tme-symmetrc equatons? Ths queston was not setted decsvey unt van Kampen 8 n 1962 stated that the moecuar chaos assumpton used by Botzmann n determnng coson requences provdes the source o the tme-asymmetrc behavor. Thus, moecuar cosons are represented by a Marovan process where the hstory o the system s orgotten or gnored and does not determne uture behavor. It s easy to demonstrate that a smpe Marovan process shows the same type o tme-asymmetrc behavor as Botzmann's H. We w use the amous dog-and-eas probem. There are two dogs A and B that share a tota o N eas seray numbered rom 1 to N. Aso, there s an urn contanng bas numbered rom 1 to N rom whch bas are drawn at random. When a ba s drawn, the number s read and the correspondng ea umps rom the dog t s on to the other dog. The ba s then returned to the urn, mxed wth the other bas, and the drawng contnues. The stuaton n whch a eas are ntay on dog A has been computed or the case o N=500. The progress o the game s shown on Fg. 6-1 where the number o eas on each dog s potted versus the number o events (draws). A random number generator smuated the drawng o bas. Aso potted on Fg. 6-1 s a quantty abeed "Entroea" dened as 6-4 "Entroea" = -( A n A + B n B ) 7 bd. 8 N.G. van Kampen n Fundamenta Probems n Statstca Mechancs, E.G.D. Cohen ed., North-Hoand Pubshng Co., Amsterdam, 1962, pp. 173-182.
6-5 Fgure 6-1. Dogs and eas a Marovan process. where A and B are the ractons o the tota number o eas on dogs A and B respectvey. As suggested by Eq. (6-3), "Entroea" s anaogous to -H. Note that despte mnor uctuatons, "Entroea" ncreases to an asymptotc vaue n much the same way as the H-theorem seems to predct or the entropy. Ths s characterstc o a Marovan process where each event s determned soey n terms o probabty, but there s no way "Entroea" can be reated to the thermodynamc entropy whch s dened n terms o a reversbe heat eect. Thus, rreversbty s not expaned by the H-theorem, but sneas nto the dervaton through the necessary but deceptvey nnocent assumpton o moecuar chaos.
6-6 6.3 COMMENTARY When consderng the mport o Botzmann's H-theorem, there are two maor areas o dsappontment: the nabty to drecty reate H to the entropy except very near the na or equbrum state, and the aure to denty the source o rreversbty. It has aready been noted that Eq. (6-5) s vad ony at equbrum, ater H has ceased to change. We can thereore expect that the resut ds/dt 0 appes ony near equbrum and may wsh to nqure as to the type o process or whch H (or rather dh/dt) has been evauated. In obtanng the resut stated by Eq. (6-7), the dstrbuton uncton was assumed ndependent o spata poston and thereore we must recognze that we have been consderng a macroscopcay homogeneous gas. We then as whether our cacuated change n H corresponds to an observabe change n state or whch we expect the entropy to change. We concude that a change o state, occurrng n an soated system n whch the gas remans macroscopcay homogeneous, s dcut to magne. In ths regard, ter Haar 9 n hs treatment o the H-theorem states that " a dstrbuton ders apprecaby rom the equbrum dstrbuton, the return to equbrum s qute rapd." Hs estmate o the reaxaton tme or ths process n a gas at 300 K and atmospherc pressure s 10-9 seconds whch ceary ndcates that the changes we have consdered or H do not correspond to observabe changes n entropy. Thus, the H-theorem has tte to do wth entropy and many demonstrates the stabty o the equbrum dstrbuton. An approach empoyng dstrbuton unctons and the moecuar chaos assumpton produces good resuts when apped to the cacuaton o transport propertes and t seems reasonabe that Botzmann's H, measurabe, woud ndeed be ound to never ncrease. Aso, t s we nown that the Sacur-Tetrode equaton, a resut o equbrum statstca mechancs, correcty predcts entropy changes between we-dened changes o state o an dea gas. Yet, n spte o these successes n the equbrum and non-equbrum reams, the orgn o or the contrbutve mechansm or rreversbty has not been ound. Some have proposed a cosmc rather than a oca orgn. 10 9 D. ter Haar, Eements o Thermostatstcs, Hot, Rnehart, and Wnston, New Yor, 1966, 2nd ed., pp. 20-22. 10 See, or exampe, B. Ga-Or, n A Crtca Revew o Thermodynamcs, E.B. Stuart, A.J. Branard, and B. Ga-Or eds., Mono Boo Corp., Batmore, 1970, pp. 445-472.
6-7 The aure o moecuar theory to provde a proo or expanaton o the second aw may be due to ncompatbe descrptons o reaty. 11 In the moecuar vew, deaty s ound n the moton o partces whch, as we have seen, s reversbe and operates n a tmeess ashon ndependent o human presence. On the other hand, deaty n thermodynamcs s represented by the reversbe process whch never occurs naturay and can be approached ony through the nterventon o a human agent n reducng rcton and potenta gradents. Thus, Botzmann's H-theorem may have aed because congruence s not possbe n the mappng o one vew o reaty onto the other. 11 See Sec. 9.3
6-8 APPENDIX 6A Smpcaton o H The uncton H can be shown to be reated to the statstca mechanca entropy. We begn by rewrtng Eq. (6-2) = n / δ v µ and substtutng ths resut nto Eq. (6-1) to obtan H n = δv µ n n δv µ δv µ whch reduces to or ( n n n n nδv ) H = µ ( n n n N nδv ) H = µ Because the ce voume δv µ s constant, the second rght-hand term n ths equaton s constant. A bt o agebrac manpuaton yeds = N n n n + [ N n N N nδv ] H µ N N where the rght-hand braceted term s a constant.
6-9 APPENDIX 6B The eect o bnary cosons upon H Beore determnng dh/dt, t s convenent to regard as a contnuous uncton n µ-space and restate Eq. (6-1) H =... n dvµ Because ntegraton s over a sx coordnates o µ-space, the ntegra s a uncton ony o tme. Thus, we may wrte dh d d =... + dvµ dt n (6-9) Rememberng that dv µ s the number o partces n a ce o derenta sze and notng that the ntegra o over a o µ-space smpy resuts n the tota number o partces, we nd the second term o the ntegra to be zero and wrte Eq. (6-9) as dh d =... n dvµ (6-10) Further, we assume to be ndependent o poston, we can ntegrate over the spata coordnates x, y, and z to obtan dh d =V n dω (6-11) where dω s the range o momenta, dp x dp y dp z. In order to determne the sgn o dh/dt, we must attempt a descrpton o the coson process and begn by rewrtng Eq. (6-2). n = δ vδ ω (6-12) We have assumed to be ndependent o poston and may sum over a voume eements, δv, to obtan the number o partces, N, n the contaner that have momentum n the range δω. N =V δ ω (6-13) We now consder a coson between two partces wth nta momenta n the ranges δω and δω to produce partces wth momenta ranges δω and δω 1.
6-10 Ths type o coson, whe conservng momentum, w decrease N and N and ncrease N and N and w occur wth a requency proportona to the product N N. Thus, rom Eq. (6-13) we wrte the requency, Z, as Z = C (6-14) where C s the coson constant. From Z we can determne the rate o change o the number o partces n these momentum ranges dn dn = dn = - = dn C = - (6-15) Usng Eq. (6-13) to evauate these dervatves we obtan dn d - = -V δ ω = C dn d - = -V δ ω = C dn d = V δ ω = C dn d = V δ ω = C We now wrte Eq. (6-11) as a summaton (6-16) dh dt = d n V δω (6-17) dt Usng Eq. (6-16) to repace the term n parenthess, we nd the contrbuton o cosons to the summaton to be or - C n - C n +C n +C n C n Next, we consder the reverse coson where partces wth nta momenta n the ranges δω and δω code to produce partces wth momenta n the ranges δω and δω. The requency o these cosons, Z, s Z = C
6-11 where the coson constant C s the same as or the orward coson as can be shown by the appcaton o Louve's equaton. By the same reasonng we can wrte or the contrbuton o reverse cosons to Eq. (6-17) C n For ths par o orward and reverse cosons, the contrbuton to the summaton s C ( - ) n Ths expresson has the orm, C(x-y)n y/x, whch can be shown to aways be negatve or zero or postve vaues o x and y. Because the 's are aways postve, the contrbuton to the summaton due to a par o orward and reverse cosons w be zero or negatve. Because t can be shown or spherca partces that a possbe bnary cosons occur n orward and reverse pars, the summaton o Eq. (6-17) w be comprsed o a arge number o terms that are ether negatve or zero. We can then state that dh 0 dt