Thermodynamic constraints on fluctuation phenomena

Transcription

1 hermodynamc constrants on luctuaton phenomena O J E Maroney he Centre or me and he School o Physcs Unversty o Sydney NSW 2006 Australa and Permeter Insttute or heoretcal Physcs 3 Carolne St N, Waterloo, ON, N2L 2Y5, Canada (Dated: December, 2009 he relatonshps between reversble Carnot cycles, the absence o perpetual moton machnes and the exstence o a non-decreasng, globally unque entropy uncton orms the startng pont o many textbook presentatons o the oundatons o thermodynamcs. However, the thermal luctuaton phenomena assocated wth statstcal mechancs has been argued to restrct the doman o valdty o ths bass o the second law o thermodynamcs. Here we demonstrate that luctuaton phenomena can be ncorporated nto the tradtonal presentaton, extendng, rather than restrctng, the doman o valdty o the phenomenologcally motvated second law. Consstency condtons lead to constrants upon the possble spectrum o thermal luctuatons. In a specal case ths unquely selects the Gbbs canoncal dstrbuton and more generally ncorporates the salls dstrbutons. No partcular model o mcroscopc dynamcs need be assumed. PACS numbers: a, a I. INRODUCION he exstence o a globally unque entropy as a uncton o thermodynamc state, whch s non-decreasng n tme, s one o the central tenets o classcal phenomenologcal thermodynamcs[, 2]. By contrast, the meanng o entropy wthn the context o statstcal mechancs seems to dey consensus(see [3, 4] or examples. Snce the start o statstcal mechancs there has been concern that the exstence o luctuaton phenomena leads to volatons o the second law o thermodynamcs. hs may lead to decreases n entropy, the exstence o perpetual moton machnes or maybe even the nablty to dene an entropy at all. Maxwell s demon represents a persstent strand o thought experments dedcated to explorng these possbltes[5 7]. Most attempts to construct a second law o thermodynamcs or statstcal mechancs nvolve one o two strateges: restrct the doman o valdty o the classcal statement (usually to relable, contnuous processes so as to exclude luctuaton phenomena; or to attempt to derve a new second law wthn the doman o statstcal mechancs. Here we nvestgate the possblty o a thrd approach: to extend the doman o the phenomenologcal second law to nclude, constran, and predct the extent o the luctuaton phenomena, whch reduces to the more amlar verson luctuaton phenomena are absent. We nd that such an extenson seems, n prncple, possble, and that wth addtonal work t s possble to dene an entropy uncton consstent wth ths. Some possble relatonshps o ths luctuaton second law to conventonal statstcal mechancs can be nerred. he approach o the paper s as ollows. Secton 2 brely revews the equvalence o the Kelvn, Clausus Electronc address: o.maroney@usyd.edu.au and Carnot versons o the second law o thermodynamcs. Secton 3 then proposes an extenson o the Kelvn verson, to ncorporate luctuaton phenomena. Logcally equvalent generalsatons o the Clausus and Carnot versons are deduced, and some constrants are deduced about the orm o the extended second law. Secton 4 revews the dervaton o an entropy uncton and shows when the exstence o a luctuaton entropy uncton can be deduced. Fnally Secton 5 consders some relatonshps to statstcal mechancal entropes, ncludng the Gbbs and salls[8] entropes. II. PHENOMENOLOGICAL SECOND LAW extbook versons o the Second Law o hermodynamcs (see, or example, [9, 0], when expressed n terms o heat lows and heat baths, take orms such as: Kelvn: No process s possble whose sole result s the extracton o heat rom a heat bath and ts converson to work. Clausus: No process s possble whose sole result s the transer o heat rom one heat bath to another heat bath at a hgher temperature. Carnot Heat Engne: No heat engne operatng between heat baths at temperatures < 2 can operate at an ecency n E exceedng the ecency o a reversble heat engne: n E n CE = 2 Carnot Heat Pump: No heat pump operatng between heat baths at temperatures < 2 can operate at an ecency n P exceedng the ecency o a reversble heat pump: n P = 2 2 Demonstraton o the logcal equvalence o each par o these statements can easly be ound n a textbook such as [0]. he equvalence s typcally proven by the

2 the second law. he problem arses that luctuaton phenomena, such 2 as Brownan moton, do, n prncple, volate all these statements o the second law, when probablstc processes are allowed. Attempts to dene a moded second law wll typcally restrct the doman o valdty. It may be suggested that the second law only apples to the thermodynamc lmt o an nnte number o atoms where luctuatons become neglgble, or t may be suggested that the second law only apples to contnuous or relable processes: (an P = Q p n W p CP (bn E = W e n Q e CE FIG. : Relable Heat Pumps and Engnes means o dagrams such as n Fgure 2. hs dagram shows the combnaton o heat engne and heat pumps beng used to attempt volatons o the Kelvn and Clausus statements. Fgure 2(a shows that a heat pump can operate wth ecency n p = Q c W p > = Q c W c, then n combnaton wth a reversble heat engne operatng at n CE = W c Q c there s a net converson o W c W p > 0 heat rom the lower temperature heat bath nto work, volatng the Kelvn statement. Smlarly Fgure 2(b shows a heat engne operatng wth ecency n e = W c Q e > n CE = W c Q c can be combned wth a reversble heat pump operatng at = Q c W c could transer heatq c Q e > 0 rom a colder to hotter heat bath wthout requrng work, thus volatng the Clausus statement. It should be noted that ths demonstraton (a (b FIG. 2: Equvalences o Volatons o Second Laws requres a number o usually unstated assumptons, such as the absence o negatve temperatures. In partcular, the equvalence requres t to be physcally possble to construct a reversble heat engne or pump. For example, t were not physcally possble to buld a heat engne whose ecency could reach that o a theoretcal reversble heat engne, then t would not necessarly ollow that a real heat pump exceedng the Carnot ecency could volate the Kelvn or Clausus versons o No relable process s possble whose sole result s the extracton o heat rom a heat bath and ts converson to work. No process s possble wth probablty one, whose sole result s the extracton o heat rom a heat bath and ts converson to work. No contnuously operatng process s possble whose sole result s the extracton o heat rom a heat bath and ts converson to work. No process s possble whose sole result s, on average, the extracton o heat rom a heat bath and ts converson to work. Restrctng the doman o valdty n ths way, however, proves unable to provde answers to many nterestng questons about the thermodynamc consequences o luctuaton phenomena. Can systems wth a nte number o atoms be used to contnuously, relably convert heat to work? I a process can succeed wth probablty less than one, how much work can be extracted? I a process only operates or a nte amount o tme how much work can be extracted? Can t be arbtrarly large? Can a process exst whch can extract an arbtrarly large quantty o work wth probablty arbtrarly close to one whle stll alng on average due to catastrophc alure when t does al? hs can be llustrated by consderng a hypothetca amly o processes, parametersed by N >. Process N wll, wth probablty N, generates N unts o work rom heat, but wth probablty N t requres N 2 unts o work to be dsspated. he mean work produced s regardless o the value o N, but as N arbtrarly large amounts o work are produced wth probablty arbtrarly close to one. Even more extreme examples can easly be constructed. Such a amly o processes satses several o the restrcted laws above, but does not accord wth our experence o luctuaton phenomena. Suppose or all real heat engnes n e n max < n CE. All that could be mpled would be that the ecency o real heat pumps were bounded by n p but > n n max n max CP. Note that such a heat pump, wth n p >, would not be possble to operate reversbly as a heat engne.

3 III. FLUCUAIONS AND HE SECOND LAW In ths Secton the man argument o the paper wll be explored. Rather than ollow the path o the modcatons n Secton II, restrctng the doman o valdty o the second law so as to exclude luctuaton phenomena, t wll nstead be expanded to nclude luctuaton phenomena. Fluctuatons wll be treated as beng probablstc processes, occurrng wth probablty less than one. he moded law should set a constrant upon the sze o luctuatons that can occur, and should reduce to the luctuaton-ree second law when only determnstc processes occur. he proposed modcaton to the phenomenologcal second law s based upon nothng more than the observaton that the greater the sze o the luctuaton, the less probable ts occurrence. From ths t s proposed that, or a gven sze o luctuaton, there s a maxmum possble lkelhood o t occurrng: here s no cyclc process 2, whose sole result s the extracton o a quantty o heat, Q, rom a heat bath at temperature, and ts converson to work, whch can occur wth probablty p, unless: 3 One trval soluton would be: (Q, = 0, Q > 0. hs would correspond to all luctuatons beng orbdden. At the other extreme, (Q, =, Q would mply one could get arbtrarly close to any sze o luctuaton, at any probablty. hs s a more restrctve condton than the mean converson o heat to work over cycle beng negatve, although t does mply t. he proo o ths s straghtorward. I there exsts a process whch can produce a postve expectaton value or producton o work over a sngle cycle, then repeatng that cycle a large number o tmes produces an expectaton value as large as one lkes, wth a gaussan spread around that mean. he probablty that any gven quantty o work can be exceeded becomes close to one. Hence any process whch can produce a postve expectaton value or work wll, on repeated applcaton, exceed any uncton <. hs knd o luctuaton - extractng work rom a sngle heat bath - wll be called a Kelvn luctuaton, and be represented as n Fgure 3(a, showng W work beng extracted rom a heat bath at temperature. p (Q, ( where s a uncton whose propertes wll be deduced rom nternal consstency. he denton s such that t s assumed or any gven Q and there exsts an actual physcal processes whch can get arbtrarly close to occurrng wth probablty (Q,. I not, then there must exst a lower value o that should have been used nstead. It s possble to mmedately note some propertes o : as the uncton bounds a probablty, t cannot become negatve; t s always possble to dsspate work as heat; there s a process that extracts Q > Q wth probablty p, then by also dsspatng work W = Q Q, there s a process that extracts Q wth probablty p. hese mmedately constran the uncton: (Q, 0 (2 (Q, = Q 0 (3 (Q, (Q, Q > Q (4 he last condton mples that s also a derentable uncton o Q, then 0 Q (5 Q 2 When dscussng probablstc cycles, a cyclc process wll mean a process whch returns to ts orgnal state wth probablty p, but wth probablty p may end up n a derent state to ts startng pont. (a (b FIG. 3: Kelvn and Clausus Fluctuatons he equvalence o Kelvn luctuatons to other knds o luctuatons wll now be demonstrated. A. Kelvn and Clausus Fluctuatons A Clausus luctuaton, as n Fgure 3(b, wll denote the spontaneous transer o Q work rom a heat bath at to a heat bath at 2 > occurrng wth a maxmum probablty C (Q,, 2. One way to acheve a Clausus luctuaton s gven n Fgure 4(a, combnng a Kelvn luctuaton wth a relable Carnot pump operatng at ecency = Q W = 2 2. hs can occur wth probablty (W,, so C (Q,, 2 cannot be less than ths: C (Q,, 2 (W, = ( Q,. A Kelvn luctuaton can smlarly (Fgure 4(b be created rom a Clausus luctuaton, by allowng the heat Q rom the Clausus luctuaton to drve a relable Carnot engne at ecency n CE = W Q = 2. hs mples C (Q,, 2 (W, = (Qn CE, and n CE =

4 4 (a (b FIG. 4: Convertng Kelvn and Clausus Fluctuatons (a FIG. 6: Kelvn Fluctuatons and Fluctuaton Heat Pump (b establshes ( Q C (Q,, 2 =, = (Qn CE, ( ( = Q, 2 (6 Carnot heat engne gves (Q, P (W, n P,, 2 Substtutng Q = W (n P gves P (W, n P,, 2 = ( ( np W, Fgure 7(a augments the Carnot heat pump wth ( (an P = Q W > (bn E = W Q > n CE (a (b FIG. 5: Fluctuaton Heat Pumps and Engnes B. Kelvn, Clausus and Heat Pump Fluctuatons A luctuaton heat pump (Fgure 5(a s a heat pump that s able to operate wth a hgher ecency than a reversble Carnot heat pump, but only wth a probablty less than one o success. he maxmum probablty o success, P (W, n P,, 2 o achevng ecency n P = Q W > can be deduced ether rom the Kelvn luctuaton law (Fgure 6 or the Clausus luctuaton law (Fgure 7. In the Fgure 6(a, creatng a luctuaton pump wth ecency n P = Q W >, by augmentng the behavour o a regular Carnot pump wth a Kelvn luctuaton shows P (W, n P,, 2 (Q,. In Fgure 6(b, creatng a Kelvn luctuaton o sze Q, by extractng the heat pumped by luctuaton heat pump at ecency n P = Q W >, and usng t to drve a FIG. 7: Clausus Fluctuatons and Fluctuaton Heat Pum Clausus luctuaton o sze Q to create a luctuat pump o ecency n P = Q W >. Now usng the w extracted rom a Carnot engne to drve a luctuat heat pump, gves a Clausus luctuaton n Fgure 7 Combned P (W, n P,, 2 = C (Q,, 2 wth Q W (n P, so P (W, n P,, 2 = C (W (n P,, 2 It can be easly conrmed that ths s consstent wth relatonshp C (Q,, 2 = ( Q,. C. Kelvn, Clausus and Heat Engne Fluctuat Smlarly, a luctuaton heat engne (Fgure 5(b a heat engne that can operate wth a hgher ece than a reversble Carnot heat engne, but only wt probablty less than one o success.

5 D. Heat Pumps and Engnes Augmentng a Carnot heat engne wth a Kelvn luctuaton o sze Q, Fgure 8(a, creates a luctuaton heat engne, whle usng the heat pumped by a regular Carnot pump to drve a luctuaton heat engne, Fgure 8(b, creates an equvalent Kelvn luctuaton. Gvng the maxmum probablty achevable or a luc- It s now possble to compare the expressons 5 or E (Q, n E,, 2 and P (W, n P,, 2 drectly hs ( gves E (Q, n E,, 2 = P (W, n P,, 2 n W P = Q (n E n CE. o conrm consstency ths can also be derved rom the dagrams n Fgure 0. In Fgure 0(a, a luctuaton heat engne, oper- (a FIG. 8: Kelvn Fluctuatons and Fluctuaton Heat Engnes tuatng heat engne to extract heat Q rom a heat bath at temperature 2, wth ecency n E = W Q > n CE, depostng the remander n a heat bath at temperature < 2 as E (Q, n E,, 2, the dagrams quckly yeld Q = Q(n E n CE and the relatonshp (b E (Q, n E,, 2 = (Q(n E n CE, (9 Fgure 9 provdes the equvalent analyss or Clausus luctuatons, now creatng a Clausus luctuaton by drvng a regular Carnot pump wth the work extracted by a luctuaton heat engne. As Q n CE = Q(n E n CE (a (b FIG. 0: Fluctuaton Heat Pumps and Engnes atng at n E = W e Q e mproves the ecency o a Carn heat pump, by usng some o the pumped work to r turn a hgher proporton o the heat nto work, to cr ate a luctuaton heat pump, wth ecency n P = Q W In Fgure 0(b, a luctuaton heat pump, wth e cency n P = Q p W p mproves the ecency o a Carn heat engne to create a luctuaton heat engne wth e cency n E = W e Q e. It can readly be conrmed th ( W np = Q (n E n CE. E. Heat and emperature (a FIG. 9: Clausus Fluctuatons and Fluctuaton Heat Engnes (b ( ne E (Q, n E,, 2 = C (Q,, 2 n CE (0 Agan, ths s consstent wth the relatonshp between C and. here remans sx dagrams or luctuatons nvolvn two heat baths. hese dagrams determne the relato shp between Kelvn luctuatons at derent temper tures. Fgure shows how a Kelvn luctuaton ca be converted to an equvalent Kelvn luctuaton at hgher or lower temperature, by usng a Carnot pump engne. hs supples heat rom a second bath to repla the heat obtaned rom the luctuaton. he overall pr cess s then a Kelvn luctuaton rom the second he bath. From Fgure (a, t can be seen that the probabl o obtanng a Kelvn luctuaton o sze Q 2 at temper ture 2 cannot be less that the probablty o obtann a Kelvn luctuaton o sze Q at temperature, pr vded Q = Q 2 2. (Q 2, 2 (Q, ( Fgure (b shows the reverse process, or wh (Q 2, 2 (Q,, so (Q, = (Q 2, 2 wh

6 heat engne, n 6 CE, wth probablty p, unless: ( Q p (n E n CE (a FIG. : Kelvn Fluctuatons at Derent emperatures Q = Q 2 2. Wrtng = 2 ths leads to (Q, = (Q,. As ths must hold or all and 2, and so or all ( Q (Q, = (2 he remanng our dagrams are essentally the same as the dagrams n Fgures 4, 8(b and 6(b, except they nvolve a Kelvn luctuaton rom the hgher temperature heat bath. Comparson o these processes agan leads to Equaton 2. (b F. Fluctuaton Frendly Second Law Combnng the result rom Secton III E, wth those rom Sectons III A to III D, t s now possble to state the luctuaton compatble generalzatons o the ormulatons o the Second Law o hermodynamcs gven n Secton II Kelvn: here s no process, whose sole result s the extracton o a quantty o heat, Q, rom a heat bath at temperature, and ts converson to work, whch can occur wth probablty p, unless: ( Q p Clausus: here s no process, whose sole result s the extracton o a quantty o heat, Q, rom a heat bath at temperature, and ts transer to a heat bath at temperature 2 >, whch can occur wth probablty p, unless: ( ( p Q 2 Heat Engne: here s no cyclc process, operatng solely as a heat engne between heat baths at temperatures 2 >, whch can extract a quantty o heat, Q, rom the hotter heat bath, wth ecency n E exceedng that o a relable, reversble Heat Pump: here s no cyclc process, operatng solely as a heat pump between heat baths at temperatures 2 >, whch can use a quantty o work, W, wth ecency n P exceedng that o a relable, reversble heat engne,, wth probablty p, unless: ( ( W np p hese our ormulatons are logcally equvalent, n the same manner that the our ormulatons o the luctuaton-ree second law gven n Secton II are logcally equvalent. G. Kelvn-Clausus nequalty. hese our ormulatons can be expressed n the same way. Combnng a sngle luctuaton wth Carnot pumps and engnes connectng heat baths at multple temperatures reveals that there s a more general ormulaton o the luctuaton laws. Just as all our o the normal phenomenologcal laws may be seen as specal cases o the law: here s no process, whose sole result s the extracton o quanttes o heat, Q, rom heat baths at temperatures, convertng the net heat extracted nto work, unless: Q 0 (3 so all o the luctuaton laws are specal cases o: here s no process, whose sole result s the extracton o quanttes o heat, Q, rom heat baths at temperatures, convertng the net heat extracted nto work, whch can occur wth probablty p, unless: p Q (4 he general ormulaton should make clear the role that Carnot cycles plays wthn the dervaton o the specc luctuaton laws. Carnot pumps and engnes connectng a number o derent heat baths are able to reversbly move heat between them n any combnaton provded the net eect s Q = 0. Any gven luctuaton can thereore be converted nto another luctuaton, nvolvng derent heat baths, but whch has the same value o Q.

7 H. Combnng luctuatons he next stage s to consder combnng luctuatons, by dagrams nvolvng more than one luctuaton. As t turns out, only two dagrams, Fgure 2 are requred to deduce the general relatonshp. (a (b FIG. 2: Combnng Kelvn Fluctuatons In Fgure 2(a there s a sngle Kelvn luctuaton resultng n Q + Q 2 heat extracted rom a heat bath at temperature. One possble way o ths happenng s two ndependent processes occur, each rom heat baths at temperature, resultng n two separate Kelvn luctuatons, extracted Q and Q 2 heat, respectvely. Fgure 2(b gves a process by whch Q +Q 2 can be extracted, so the mnmal probablty o a Kelvn luctuaton o that sze cannot be less that the probablty o the two ndependent luctuatons both occurrng: ( Q + Q 2 ( Q ( Q2 (5 As ths must happen or all Q, Q 2, the luctuaton law must satsy the general unctonal nequalty 3 (x + y (x(y (6 hs leads drectly to the general equaton Q ( Q (7 that would also be deduced rom consderng dagrams wth multple luctuatons and wth Carnot pumps and engnes operatng between multple heat baths. hs property n tsel can be used to demonstrate that, there exsts some x = x 0 > 0 such that (x 0 = 0 then t must be the case that x > 0, (x = 0,.e. luctuatons must be possble at all scales, they 7 are possble on any scale. Intutvely ths should be obvous: provded a small luctuaton can occur wth a nonzero probablty, p, then accumulatng n such luctuatons nto a luctuaton n tmes large s always possble wth probablty p n. Any sze o luctuaton may occur wth small, but non-zero probablty, provded n s large enough. I t were the case that accumulatng small luctuatons was the optmum process or obtanng a large luctuatons, then: Q = ( Q (8 hs requres (x + y = (x(y. Provded s a contnuous uncton, ths has a unque soluton: ( Q = e λ Q (9 where λ s a unversal constant whose value would need determnng expermentally to be the recprocal o Boltzmann s constant: λ = k. It s, perhaps, surprsng that such a amlar uncton wthn statstcal mechancs mght be obtaned rom the purely phenomenologcal arguments ollowed here! Unortunately, there seems no strong reason to demand that a large luctuaton cannot, n prncple, be more probable than gettng an equvalent szed luctuaton through the accumulaton o a large number o small luctuatons. It may, on the arguments consdered so ar, smply be the case that large luctuatons can spontaneously occur, wth a hgher probablty. Equaton 9 s not the only possblty. he restrctons on the orm o (x are (x 0 (20 (x = x 0 (2 x 0 x > 0 (22 (x + y (x(y x, y > 0 (23 Other unctons whch could satsy all these requrements nclude:. (x = + n a nx n (24 wll satsy all the condtons speced whenever n!a n m!l!a m a l or all n = m + l. Specc cases nclude: (a n!a n = m!l!a m a l. hs leads to a n = (a n e (x = e a x n! (25 3 hs may be converted nto a more amlar orm usng F (x = ln[(x] to get F (x + F (y F (x + y. In passng, t may also be noted that (x s derentable, then t can be shown rom Equaton 6 that (x (x (0 and (0 (0 2. (b For all n >, let a n = 0 (x = + a x (26

8 (c I some (x that satses the condtons, then g(x = n (x, wth n > wll satsy the condtons, so q (x = ( + a x /a wth 0 a. 2. An even slower allng uncton such as l (x = + ln( + ax can also satsy the requrements. IV. FLUCUAIONS AND ENROPY (27 (28 In Secton III t was shown that the Kelvn-Clausus- Carnot versons o the second law, ormulated n terms o cyclc processes and heat baths, can be generalsed n a consstent way to nclude luctuaton phenomena. However, phenomenologcal thermodynamcs does not become genunely powerul untl Equaton 3 s used to dene a non-decreasng, global uncton o state called entropy. Wth luctuatons possble, t s clear that any such globally dened uncton o state can decrease wth some probablty. In ths Secton t s shown that t s stll possble to dene a meanngul entropy uncton, wth a relatonshp to the luctuaton law n Equaton 4. A. Phenomenologcal Entropy he Kelvn-Clausus nequalty: here s no process, whose sole result s the extracton o quanttes o heat, Q, rom heat baths at temperatures, convertng the net heat extracted nto work, unless: Q 0 mmedately mples that, there exsts a process, whose sole result s to transorm state A, nto state B, whle extractng quanttes o heat,, rom heat baths at temperatures, then there s no process whose sole result can be to transorm state B nto state A, whle extractng quanttes o heat,, rom heat baths at temperatures, unless: + j 0 (29 It s a straghtorward mathematcal constructon (see Appendx to show ths mples the exstence o a nonempty convex 4 set o unctons o state, {S θ (X}, whch 4 For any two S, S {S θ (X} then or any 0 p, t s the case that ps + ( ps {S θ (X}. each satsy the ollowng condton: I there exsts a process, whose sole result s to transorm state A nto state B, whle extractng quanttes o heat,, rom heat baths at temperatures, then S θ (A S θ (B 8 (30 he unctons S θ (X wll be reerred to as thermodynamc entropes. he expresson o the phenomenologcal second law, n terms o these thermodynamc entropes, s here exst unctons o the thermodynamc state {S θ (X}, such that or any two thermodynamc states A and B, there s no process, whose sole result s to transorm state A nto state B, whle extractng quanttes o heat, rom heat baths at temperatures, unless or all S θ (X. S θ (A S θ (B (3 In an adabatc process, no heat s extracted or generated n any heat bath, so ths requres S θ (A S θ (B. As ths result must also hold or processes whch transorm B nto A, then S θ (B S θ (A (32 hs must hold or all processes, so the set {S θ (X} s bounded by the processes whch maxmse the quanttes and. I the two states A and B can be connected by a reversble cycle, then the maxmum reached s + = 0 (33 n whch case the entropy derence between the two states s xed to be the same value or all unctons n {S θ (X}: S θ (B S θ (A = = (34 I all states can be connected by reversble cycles, then there s a sngle uncton, unque up to an addtve constant. It s mportant to note that reversblty s requred or the unqueness o the entropy uncton, but s not necessary to prove the exstence o a non-decreasng set o entropy unctons.

9 B. Fluctuaton Entropy law he exstence o the luctuaton law does not prevent the dervaton o the exstence o the thermodynamc entropy unctons {S θ (X}. her sgncance s restrcted to relable (.e. probablty one processes. Unortunately t does not mmedately ollow that a luctuaton law can be deduced constranng the probablty o a reducton n thermodynamc entropy. An essental stage n the deducton o a law relatng entropy to luctuatons, s the dentcaton o an approprate nequalty or closed cycles ncorporatng any two states, such as Equaton 29, but or cycles nvolvng luctuatons. Such an nequalty cannot be drectly obtaned rom the luctuaton law. he luctuaton law, Equaton (4, mples that, there exsts a process, whose sole result s to transorm state A, nto state B, whle extractng quanttes o heat,, rom heat baths at temperatures, and whch can occur wth probablty p AB, then there s no process whose sole result can be to transorm state B nto state A, whle extractng quanttes o heat,, rom heat baths at temperature, whch can occur wth probablty p BA, unless: p AB p BA Invertng the uncton gves: (p AB p BA + + (35 (36 However, the relatonshp (x + y (x(y, when nverted, yelds (pq (p + (q (37 and ths does not allow the deducton o a sutable nequalty.. Relable Paths o proceed urther, t s necessary to consder relable paths between A and B. Let q (AB be the heat gener- 9 ated n heat baths at temperatures, or a process that can occur wth probablty one, and whose sole eect, apart rom extractng heat rom heat baths and convertng them to work, s to transorm state A nto state B. It ollows that there s no process, whose sole result s to transorm state B nto state A, whle extractng quanttes o heat rom heat baths at temperatures, whch can occur wth probablty p BA, unless (p BA q (AB + (38 Smlarly, q (BA s the heat generated n heat baths at temperatures, or a process that can occur wth probablty one, whose sole eect, apart rom extractng heat rom heat baths and convertng them to work, s to transorm state B nto state A, then there s no process, whose sole result s to transorm state A nto state B, whle extractng quanttes o heat rom heat baths at temperatures, whch can occur wth probablty p AB, unless (p AB + It s mmedately possble to deduce both that q (AB + q (BA (39 q (BA 0 (40 (by usng a process or whch ether p AB = or p BA = and that + q (BA + q (AB + (p AB + (p BA (4 Equaton 40 mples the exstence o the thermodynamc entropes {S θ (X}, as beore. Equaton 4 mples the exstence o a convex set o unctons o state {S φ (X}, whch wll be called the luctuaton entropes, and whch all satsy + q (AB (p AB S φ (B S φ (A (p BA + q (BA (42 q (AB q (BA In order to narrow down the range o permssble entropes, the terms and should each be as large as possble, subject to the constrant o Equaton 40. hs produces the ollowng entropy luctuaton law:

10 Let be the heats extracted rom heat bath at temperatures, by a process, whch occurs wth probablty one, whose sole other result s to transorm state A nto state B, and whch maxmses the value o over all such processes. here exsts sngle valued unctons o state {S φ (X}, such that, there exsts a process occurrng wth probablty p, whose sole result s to transorm state A nto state B, whle extractng quanttes o heat,, rom heat baths at temperatures, then S φ (A S φ (B+ (p (43 o restrct these to a unque uncton S φ (X requres that there exst cycles 5 or whch (p AB + (p BA = 2. Reversble Paths + + (44 I t s the case that the equalty n Equaton 40 s met, then Equaton 4 takes the orm (p AB + (p BA and + (45 (p AB S φ (B S φ (A (p BA (46 hs s not sucent to ensure S φ (B S φ (A s unque. However, n ths case, t s possble to deduce the exstence o the globally unque thermodynamc entropy rom the relable paths and S θ (X {S φ (X}. 0 I both S θ (X and S φ (X are unquely dened, then (p AB + (p BA = 0, n whch case S θ (X = S φ (X. However, n general, the thermodynamc entropes {S θ (X} are not restrcted to a sngle globally unque uncton, then there may exst S θ (X / {S φ (X}. It s worth notng that Equaton 48 mples (p AB + (p BA 0 (49 I, on the other hand, Equaton 44 holds or luctuaton cycle, or whch also requres q (AB + q (BA < 0, then ths (p AB + (p BA > 0 (50 + In other words, there exst any luctuatons rom state A to state B, and vce versa, that can dene a unque luctuaton entropy derence S φ (B S φ (A when combned wth a relable but rreversble cyclc path between A and B, then t must be the case that there are no relable, reversble cyclc paths between states A and B. he exstence o a globally unque S φ (X that s not smultaneously a globally unque S θ (X would mply relable, reversble processes cannot exst. Relable, reversble cycles mply an entropy luctuaton law: here exsts sngle valued unctons o state {S φ (X}, such that, there exsts a cyclc process, occurrng wth probablty one, operatng between states A and states B, wth a zero net extracton o heat over the cycle, then or any other process, occurrng wth probablty p, whose sole result s to transorm state A nto state B, whle extractng quanttes o heat,, rom heat baths at temperatures, then S φ (A S φ (B + (p (5 S θ (B S θ (A = = j (47 and there s a globally unque thermodynamc entropy S θ (X {S φ (X} or whch Equatons 38 and 39 gve (p AB S θ (B S θ (A (p BA (48 5 When dealng wth luctuatons, a cycle s a process or whch the system starts n state A wth certanty, reaches the state B wth probablty p AB, and then the condtonal probablty or returnng to state A, gven that t reached state B, s P BA. 3. Exponental Statstcs Fnally, note that the luctuaton law takes the exponental orm dscussed n Secton III H, then (pq = (p + (q (52 so Equaton 36 leads mmedately to (p AB + (p BA + j j j (53

11 hs gves Equaton 48 wthout needng the exstence o relable paths. hs mples there exsts a convex set o luctuaton entropes {S η (X} {S φ (X} satsyng (p AB S η (B S η (A (p BA j (54 Unquely denng an S η (X entropy would requre j j (p AB + j (p BA = 0, but ths does not necessarly unquely dene ether S φ (X or S θ (X. In ths case, however, a unque S φ (X does exst then t s necessarly equal to a unque S θ (X, and vce versa. V. FROM FLUCUAIONS O SAISICAL MECHANICS he possble relatonshp o the luctuaton spectrum to statstcal mechancs wll now be brely explored. It wll be assumed throughout ths Secton that a globally unque entropy S(X = S θ (X = S φ (X can be determned, and only a sngle heat bath at temperature wll be used. he entropy luctuaton law now takes the orm: here exsts a sngle valued uncton o state S(X, such that or any process, occurrng wth probablty p, whose sole result s to transorm state A nto state B, whle extractng quanttes o heat,, rom heat baths at temperatures, then S(A S(B + (p Q(AB (55 Suppose the system s n an ntal state, wth entropy S, nternal energy E, and s subject to a process durng whch t luctuates to state wth probablty p. Durng the course o the process, heats Q are generated n heat baths at temperatures and requres work W to be perormed. By conservaton o energy, the nternal energy o state s E = E + W Q (56 By the entropy luctuaton law, the entropy o state must obey S S + (p Q (57 hs equaton must hold or each possble luctuaton away rom the ntal state, so that S ( p S + (p Q (58 necessarly holds. he orm o ths constrant s very suggestve o entropy unctons that occur n statstcal mechancs. j j A. Maxmal Fluctuatons he denton o the uncton s such that there must exst some process or whch the equalty n Equaton 57 s met: S = S + (p Q (( = S E ( S E W p (59 (60 However, there s no guarantee that a sngle process can exst whch acheves the maxmum luctuaton or every possble outcome. I such a process dd exst, then S = ( p S + (p Q (6 would hold. hs smlarty to statstcal mechancs s brought even closer under two condtons:. I a set o maxmal luctuatons occur whch do not generate heat, on average, then p Q = 0. he entropy ormula then becomes: S = p ( S + (p (62 2. I a set o maxmal luctuatons can take place, wthout requrng external work to be perormed (W = 0 then: (( p = S E ( S E (63 or p = ( F F, where F = S E (64 F = S E (65 B. Example luctuaton laws Let us consder the unctons rom Secton III H. (x = + n a nx n (a e (x = e ax. hs generates the amlar Gbbs canoncal statstcs. e (p = a ln p (66 S = p = Z e e a F / p S a p ln p (67 wth Z e = e a F/ = e a F / (68

12 (b (x = ( + a x (p = a ( p (69 S = p S a (N (70 p = Z ( + a βf (7 wth N the number o dstnct states n the summaton, Z = (+a F and β = /( Z. (c q (x = (+a x /a. hs generates statstcs smlar to the salls non-extensve entropes. q (p = ( p a (72 a S = p S a p a (73 p = Z q ( + a βf /a (74 wth Z q = ( + a F/ /a and β = /( Z a q. 2. he slowly allng uncton l (x = ( + ln( + ax yelds l (p = (e (p (75 a S = p S + p e (p (76 a a p = Z l ( + ln( + a βf /Z l (77 wth Z l = + ln( + a F/ and β = e Z l / VI. CONCLUSION Startng rom the physcal ntuton that larger thermal luctuatons must be less probable than smaller luctuatons, we have suggested a luctuaton law that states that or any gven sze o luctuaton, there s a nontrval maxmum probablty o t occurrng. hs smple suggeston proves surprsngly rutul. he equvalence o the Kelvn, Clausus and Carnot ormulatons o the phenomenologcal second law o thermodynamcs s shown to naturally generalse to the luctuaton law, and urther constran t to be o the orm: here s no process, whose sole result s the extracton o quanttes o heat, Q, rom heat baths at temperatures, convertng the net heat extracted nto work, whch can occur wth probablty p, unless: p Q (78 2 wth the uncton urther constraned by the requrement Q ( Q (79 I the underlyng dynamcs s ound to be such that larger luctuatons can only occur through the accumulaton o smaller luctuatons, then ths requres the uncton to have the exponental orm: Q = e λ ( Q (80 It s nterestng to note that the phenomenologcally motvated approaches o Szlard and o sza and Quay[, 2] to statstcal mechancs derve the canoncal dstrbuton by makng a smlar assumpton (see also [3]. We have urther shown that the deducton o the exstence o a non-decreasng thermodynamc entropy uncton o state may stll be ollowed, to derve a luctuaton entropy uncton o state. Under a smlar knd o crcumstance or whch the thermodynamc entropy can be deduced to be globally unque, then the luctuaton entropy can be deduced to be globally unque. Furthermore, the thermodynamc and luctuaton entropes are both globally unque, then they are necessarly dentcal (up to an addtve constant. hs holds out hope that more rgorously axomatc developments o the thermodynamc entropy, such as that o Leb and Yngvason[], may be generalzed n a smlar manner to ncorporate luctuaton phenomena. Some possble orms o the entropy luctuaton law have been nvestgated. he exponental orm naturally produces the Gbbs canoncal dstrbuton or thermal luctuatons. Non-extensve entropes, such as the salls entropy, can also be seen to arse naturally n ths approach. Further nvestgaton s needed to explore the consstency o derent unctons. In partcular, the requrement that the mean heat extracted over a cycle s Q non-postve, 0, may be expected to urther constran whch unctons are admssble. APPENDIX: ENROPY FUNCIONS FOR IRREVERSIBLE CYCLES Suppose there exsts a path dependant quantty, Ω λ AB (a property o a partcular path λ, n a state space, rom state A to state B well dened or all paths λ, states A and states B, or whch: λ, λ Ω λ AB + Ω λ BA 0 (A. and that there exsts at least one path rom each A to each B or whch the correspondng value o Ω s nte, so that n λ [ Ω λ AB ] <. hen there exsts a non-empty convex set o unctons o state {S(X}, such that or all paths λ and states A and B: S(A S(B + Ω λ AB (A.2

13 Proo: Dene Ω AB = n λ [ Ω λ AB ]. So Ω λ AB Ω AB. As Ω BA < and Ω AB Ω BA, then Ω AB >. By denton, the mnmum value o Ω gong rom A to C cannot be more than the value gong rom A to C va a path ncludng B: and S +A (X S +A (Y = Ω AX Ω AY S A (X S A (Y = Ω XA + Ω Y A 3 (A.2 (A.3 Ω AC Ω AB + Ω BC (A.3 It ollows that or any A, so Ω AC Ω AB Ω BC (A.4 Ω AB Ω AC Ω BC (A.5 Ω AC Ω BC Ω AB (A.6 Ω BC Ω AC Ω AB (A.7 Dene the set o unctons o state {S Y (X} by S +A (X = Ω XA S A (X = Ω AX (A.8 (A.9 S +A (X S +A (Y Ω XY Ω λ XY (A.4 Ω Y X Ω λ Y X (A.5 S A (X S A (Y Ω XY Ω λ XY (A.6 Ω Y X Ω λ Y X (A.7 and t s then easly demonstrated that or any dstrbuton Y w(y =, w(y 0, that the weghted uncton o state S(X = w(y S Y (X Y (A.8 hese are clearly well dened, nte unctons o state, and they exst, so the set {S Y (X} s not empty. Note that as Ω XX = 0: Ω XY = S +Y (X S +Y (Y (A.0 = S X (X S X (Y (A. satses as S(A S(B Ω λ AB (A.9 Ω λ Y X Ω Y X = S +X (X S +X (Y S A (X S A (Y S +Y (X S +Y (Y = Ω XY Ω λ XY (A.20 Note, that the set { Y w(y S Y (X} does not necessarly nclude all the unctons whch satsy the nequalty o Eq. (A.2. It only demonstrates the exstence o a non-empty set o such unctons. It s now a trval matter to show rom Equaton A.20 that, whenever the equalty n Equaton A. can be reached, that all unctons n the set { Y w(y S Y (X} (ndeed, all unctons satsyng Equaton A.2 wll gve the same entropy derence between states A and B. By extenson, the equalty n Equaton A. can be reached or all pars o states, then there s a sngle uncton, S(X, unque up to an addtve constant. ACKNOWLEDGMENS I would lke to thank Harvey Brown, John Norton, ony Short, Jos Unk and Steve Wensten or dscussons and suggestons that have nluenced the development o ths paper, and an anonymous reeree or helpul comments. Research at the Permeter Insttute or heoretcal Physcs s supported n part by the Government o Canada through NSERC and by the Provnce o Ontaro through MRI. [] E. H. Leb and J. Yngvason, Physcs Reports 30, (999. [2] J. Unk, Studes n Hstory and Phlosophy o Modern Physcs 32, 305 (200. [3] D. P. Sheehan, ed., Frst Internatonal Conerence on Quantum Lmts to the Second Law (Amercan Insttute o Physcs, [4] G. P. Beretta, A. Ghonem, and G. Hatsopoulos, eds., Meetng the Entropy Challenge (Amercan Insttute o Physcs, [5] J. Earman and J. D. Norton, Studes n the Hstory and Phlosophy o Modern Physcs 29, 435 (998. [6] J. Earman and J. D. Norton, Studes n the Hstory and Phlosophy o Modern Physcs 30, (999. [7] H. S. Le and A. F. Rex, eds., Maxwell s Demon 2: Entropy, classcal and quantum normaton, computng (IoP, 2003, SBN [8] C. salls, Journal o Statstcal Physcs 52, 479 (988.

14 [9] E. Ferm, hermodynamcs (Dover, 937, (Dover publcaton 956. [0] C. J. Adkns, Equlbrum hermodynamcs (Cambrdge Unversty Press, 968, 3rd ed. [] L. Szlard, Zetschrt ur Physk 32, 753 (925, reprnted n [4] pg wth Englsh translaton pg [2] L. sza and P. M. Quay, Annals o Physcs 25, 48 4 (963, reprnted n [5]. [3] O. J. E. Maroney, ArXv e-prnt servce (2007, arxv.org://quant-ph/ [4] B.. Feld and G. Wess-Szlard, eds., he Collected Works o Leo Szlard. Scentc Papers, vol. (MI Press, 972, SBN [5] L. sza, Generalzed hermodynamcs (MI, 977.