Josiah Willard Gibbs and his Ensembles

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1 Josiah Willard Gibbs and his Ensembles K P N Murthy P r o lo g u e J o sia h W illa rd G ib b s w a s a m o n g st th e la st o f th e in - te llec tu a l g ia n ts o f th e cla ssic a l era o f p h y sic a l sc ie n c es. H e w a s b o rn in N e w H a v en, C o n n e cticu t, U S A, w h ere h is fa th e r w a s a p ro fe sso r a t Y a le U n iv e rsity 's D iv in ity sch o o l. K P N Murthy is currently a professor at the School of Physics, University of Hyderabad. His main research interests are in statistical physics, Monte Carlo methods, molecular dynamics, radiation transport, random walks, regular and anomalous diffusion and non-linear dynamics and chaos. Keywords Gibbs, thermodynamics, statistical mechanics, Gibbs ensembles, vector algebra. G ib b s stu d ied a t Y a le a n d o b ta in e d h is d o c to ra l d eg ree in th e y e a r H is th e sis w a s O n th e F o rm o f th e T eeth o f W h eels in S p u r G ea rin g. It is in d e ed re m a rk - a b le th a t th o u g h G ib b s k ick e d o h is c a re er w ith a p ra c - tic a l p ro b lem in e n g in e e rin g, in la te r y ea rs h e p u rsu e d a n d c o n trib u ted to so m e o f th e m o st a b stra c t id e a s in th e h isto ry o f sc ien c e. H is w a s th e rst to re c eiv e a d o c - to ra te in en g in e e rin g a n d th e se c o n d in sc ie n c e, a w a rd e d in th e U n ited S ta tes o f A m e ric a. H e th e n jo in e d Y a le C o lleg e a n d ta u g h t L a tin fo r tw o y e a rs a n d n a tu ra l p h i- lo so p h y fo r o n e y ea r. In th e y e a r , G ib b s w e n t to E u ro p e fo r stu d ies w h ere h e ca m e u n d er th e in u e n ce o f G u sto v R o b e rt K irch h o a n d H erm a n n L u d w ig F e r- d in a n d v o n H e lm h o ltz. T h e th re e y e a rs h e sp e n t in E u - ro p e w ere a lm o st th e o n ly tim e h e ev e r sp e n t a w a y fro m h is h o m eto w n. O n re tu rn h e w a s a p p o in te d a p ro fe sso r o f m a th e m a tic a l p h y sic s; th is p o st w a s w ith o u t sa la ry p a rtly b e c a u se G ib b s h a d n ev e r p u b lish e d! G ib b s re m a in ed in Y a le u n til h is d ea th in th e y ea r H e led a n u n e v e n tfu l life. H e n e v e r m a rrie d a n d liv e d w ith h is siste r a n d b ro th er-in -la w ; h is b ro th er-in -la w w a s a lib ra ria n in Y a le. H e n e v e r tried to so c ia liz e. H e p u b - lish e d sp a rsely. T h is w a s ch ie y b e ca u se G ib b s w a s a p e rfe ctio n ist a n d stro v e h a rd fo r a v e ry h ig h d e g re e o f rig o u r, ec o n o m y o f w o rd s a n d c la rity in p resen ta tio n. 12 RESONANCE July 2007

2 A lso h e to o k g re a t ca re to tie u p, a s m u ch a s p o ssib le, a ll th e lo o se e n d s b e fo re e v e n th in k in g o f w ritin g u p h is w o rk fo r p u b lic a tio n. E a r ly W o r k o n T h e r m o d y n a m ic s T h e e a rly w o rk o f G ib b s w a s o n th e rm o d y n a m ic s. A s th e n a m e su g g e sts, th e rm o d y n a m ic s d e a ls w ith h e a t a n d w o rk. T h is su b je ct w a s ch ie y m o tiv a te d b y m a n 's d e - sire to ex tra c t w o rk fro m h e a t. M o re g e n era lly th e rm o - d y n a m ic s e x p la in s th e b eh a v io u r o f m a c ro sc o p ic sy ste m s o n th e b a sis o f e m p irica l la w s. T h e rm o d y n a m ic la w s a re d irec tly d e d u c ed fro m e x p e rim en ts. A sim p le ex a m p le is th e id e a l g a s la w, a lso k n o w n a s C h a rle s' la w o r B o y le's la w. A cc o rd in g to it, th e p ro d u ct o f p re ssu re a n d v o l- u m e o f a g iv en q u a n tity o f a n id e a l g a s (a n y d ilu te g a s, sa y a ir, ca n b e c o n sid e red a s a n id e a l g a s) is a co n sta n t a t c o n sta n t te m p era tu re. (S ee a rtic le b y V K u m a ra n in th is issu e.) G ib b s' e a rly w o rk [1 ] w a s o n g eo m e trica l rep re se n ta tio n o f th e rm o d y n a m ic q u a n titie s su ch a s e n e rg y, v o lu m e a n d e n tro p y. T h e ch a n g e o f th e rm o d y n a m ic sta te o f th e sy s- te m is g iv en b y a su rfa ce in th e se th re e d im e n sio n s. T h e ta n g en t a t a n y p o in t o n th e su rfa c e w ill in d ic a te th e te m - p e ra tu re a n d p re ssu re o f th e m a cro sc o p ic sy ste m w h e n it is in th e p a rtic u la r th e rm o d y n a m ic sta te re p re se n te d b y th e p o in t. F ro m th e sh a p e o f th e su rfa ce G ib b s d ed u ce d g e o m e tric a lly th e c o n d itio n s fo r eq u ilib riu m a n d c rite ria fo r sta b ility. J a m e s C lerk M a x w e ll w a s so fa sc in a te d b y th e sim p lic ity a n d eleg a n c e o f th e g e o m e tric a l m e th o d th a t h e co n stru c ted a m o d e l fo r th e th e rm o d y n a m ic su r- fa c e fo r w a ter, a p la ste r ca st o f w h ich h e p resen te d to G ib b s. T h is p la ste r ca st is o n p e rm a n e n t e x h ib itio n a t th e S lo a n e P h y sics L a b o ra to ry in N e w H a v e n. T h e p a p e rs th a t G ib b s p u b lish ed d u rin g { , o n equ ilibriu m o f h ete rog en eo u s su b sta n ces [2 ] la id th e fo u n - d a tio n fo r ch e m ica l th e rm o d y n a m ic s. T h is w o rk is o n e RESONANCE July

3 Physicists were familiar with the role of temperature in establishing thermal equilibrium and of pressure in establishing mechanical equilibrium. Gibbs showed that equality of chemical potential establishes diffusional equilibrium. o f th e g re a te st a ch ie v e m en ts in th e p h y sica l sc ien c e s o f th e n in e tee n th c en tu ry. In th e se p a p ers G ib b s a n - n o u n ce d h is in v e n tio n o f th e c o n c e p t o f ch em ica l p o - te n tia l. P h y sic ists w e re th en fa m ilia r w ith th e ro le o f te m p era tu re in esta b lish in g th erm a l e q u ilib riu m a n d o f p re ssu re in e sta b lish in g m e ch a n ic a l e q u ilib riu m. G ib b s sh o w e d th a t eq u a lity o f ch e m ica l p o te n tia l e sta b lish e s d i u sio n a l e q u ilib riu m. H e a lso w ro te o f w h a t w e n o w c a ll `G ib b s' p h a se ru le ' fo r d escrib in g eq u ilib ria a m o n g st d i e ren t p h a se s o f m a tter. V e c t o r A lg e b r a, O p tic s, a n d F o u r ie r S e r ie s In th e y ea rs { G ib b s w o rk e d o n v e cto r a n a ly - sis. T h e d o t a n d cro ss p ro d u cts th a t w e a re fa m ilia r w ith w e re in v e n ted b y G ib b s. H e co m b in e d th e id ea s o f W illia m R o w a n H a m ilto n o n q u a te rn io n s a n d th o se o f H e rm a n n G Äu n th e r G ra ssm a n o n th e th e o ry o f e x te n sio n, to p ro d u c e th e m a th e m a tic a l e ld o f v e c to r a n a ly sis. H e rig h tly h e ld th e m o d ern v ie w th a t a v ec to r is a n en tity b y itse lf a n d sh o u ld n e v e r b e c o n fu se d fo r its c o m p o - n e n ts. G ib b s' E lem en ts o f vecto r a n a lysis w a s p rin te d p riv a te ly in N ew H a v en in a n d E a rly in h is ca ree r G ib b s re co g n ize d a n d a sse rte d th a t p h a se tra n sitio n o c cu rs tru ly in in n ite sy stem s. A n - o th e r in tere stin g p ie c e o f m a th e m a tica l w o rk th a t G ib b s d id re la te th e c o n v e rg e n c e o f a F o u rie r se rie s fo r a p iec e - w ise c o n tin u o u s a n d d i e ren tia b le fu n c tio n. T h e F o u rie r se rie s a p p ro x im a tio n d isp la y s a n o v e rsh o o t in th e le ftsid ed in te rv a l a n d a sy m m e tric u n d e rsh o o t in th e rig h t- sid ed in terv a l. T h e h e ig h t o f th e o v e rsh o o t o r th e d e p th o f th e u n d e rsh o o t d o e s n o t d ec re a se w ith in c re a se o f th e n u m b e r o f te rm s in th e su m m a tio n, a co u n ter-in tu itiv e re su lt. T h is b e h a v io u r, n o w k n o w n a s G ib b s p h e n o m - e n o n, is d e scrib e d in th e tw o b rie f le tte rs th a t G ib b s su b m itte d to th e e d ito r o f N a tu re in D e c em b e r a n d A p ril B e tw e e n a n d , G ib b s p u b - lish e d v e p a p e rs [3 ] co m p a rin g e le ctro m a g n e tic th e o ry 14 RESONANCE July 2007

4 w ith e la stic th eo rie s a n d sh o w ed u n a m b ig u o u sly th a t th e e m p iric a l p h e n o m e n a in o p tic s c a n b e e x p la in e d o n th e b a sis o f e lec tro m a g n e tic th eo ry o f M a x w e ll a n d n o t b y th e e la stic th e o rie s. A s p o in te d o u t ea rlier G ib b s w a s ex tre m e ly c a re fu l w h e n it c a m e to p u b lic a tio n. H e in sisted o n rig o u r a n d c o m - p letio n so m u ch so th a t h is m o n u m e n ta l w o rk o n E lem en ta ry p rin cip les in sta tistica l m ech an ics [4 ] w a s p u b - lish e d ju st o n e y ea r b e fo re h is d e a th. It c o n ta in s th e re su lts o f h is re se a rch ca rrie d o v e r a p e rio d o f n ea rly th irty y e a rs. G ib b s { T h e F o u n d e r o f S t a t is t ic a l M e c h a n ic s W e c a n sa y th a t G ib b s a lo n g w ith B o ltz m a n n, M a x w e ll a n d E in ste in fo u n d e d th e su b je ct o f S ta tistica l M ech a n - ics. S ta te d in sim p le te rm s, S ta tistic a l M e ch a n ic s h elp s u s c a lcu la te th e m a c ro sc o p ic p ro p e rtie s o f a n o b jec t fro m th o se o f its m icro sco p ic c o n stitu e n ts { a to m s a n d m o le - c u le s { a n d th e ir in tera c tio n s. T o th e stu d e n ts o f sta tistic a l m ech a n ics G ib bs is a h o u se -h o ld n a m e. T h e y co m e to k n o w o f h im th ro u g h G ib b s' en sem b le s u p o n w h ich is b a se d th e e n tire e d i ce o f sta tistic a l m ech a n ics. T o a p p re c ia te G ib b s' c o n trib u tio n to w a rd fo u n d in g o f sta - tistic a l m e ch a n ic s w e h a v e to sta rt w ith m a n k in d 's ea rly e o rts to u n d ersta n d th e n a tu re o f m a tte r a n d o f h ea t. A t o m is m o f E a r ly T im e s A n cien t m a n, irresp e ctiv e o f w h ich c iv iliz a tio n h e b e - lo n g e d to { th e In d ia n o r th e G re e k, th e M o h ist o r th e M a y a n { m u st h a v e d e n ite ly sp ec u la te d o n th e p o ssib ility o f tin y in v isib le a n d in d iv isib le p a rticles a sse m b lin g in v e ry la rg e n u m b ers in to a v isib le c o n tin u u m o f so lid s a n d liq u id s a n d a n in v isib le co n tin u u m o f a ir th a t su r- ro u n d u s. T h e In d ia n s h a d a n a m e fo r it: A n u 1. T h e G re ek s c a lled it a to m { th a t w h ich c a n n o t b e cu t. T itu s 1 The Indian school of atomism, dating back to 600 BC, talks of attribute-less particles combining in pairs to form dyads. A dyad is also imperceptible, though it has acquired an attribute of two-ness. It requires three dyads to combine and form a triad. The triad is perceptible; it has attributes that can be observed and measured. RESONANCE July

5 L u c re tiu s C a ru s (9 4 { 5 5 B C ) m u se d o n th e n a tu re o f th in g s a n d w ro te a six -b o o k lo n g p o e m c a lle d D e R e- ru n m N a tu ra. In it, h e w rite s o f m a tte r, m a d e o f a to m s th a t o n e c a n n o t se e. A c c o rd in g to h is v erse s a ll th e n a t- u ra l p h en o m en a w e se e a ro u n d a re c a u sed b y in v isib le a to m s m o v in g a ro u n d ra n d o m ly h ith e r a n d th ith e r, try - in g o u t a ll p o ssib le fo rm s a n d m o v e m en t in th e c o u rse o f in n ite tim e a n d e v e n tu a lly se ttlin g d o w n in to a d isp o - sitio n th a t w e se e n o w. T h ere w a s n o ro le fo r G o d in th e sch em e o f th in g s. A to m ism o f th e v e ry e a rly tim e s w a s in h ere n tly a n d e rc ely a th e istic. P erh a p s th is ex p la in s w h y it lo st fa v o u r a n d la n g u ish e d in to o b liv io n fo r o v e r tw o th o u sa n d y e a rs. R e v iv a l o f K in e t ic T h e o r y Bernoulli rightly concluded that pressure is the force exerted (per unit area) by a very large number of randomly moving molecules bouncing off the wall. T h e rev iv a l c a m e p erh a p s in th e se v e n te e n th c en tu ry w ith th e w o rk o f G a lile o, T o rric elli, P a sc a l, B o y le, D a n ie l B e rn o u lli, C h a rle s, G a y -L u ssa c, J o se p h B la ck, J a m e s W a tt, a n d J o h n D a lto n a n d sev era l o th ers. R o b e rt B o y le m o d e led a ir a s a co llec tio n o f sp rin g s in x ed p o sitio n s; th e sp rin g s re sist c o m p ressio n (w h ich e x p la in s a ir p ressu re ) a n d e x p a n d in to a v a ila b le sp a ce. D a n ie l B e rn o u lli w en t a step fu rth e r a n d p ro p o se d a b illia rd b a ll a to m m o d e l. B e rn o u lli's b illia rd b a ll a to m m o v e s fre ely in sp a ce a n d w h e n it b o u n ce s o th e w a ll o f th e co n ta in e r it e x erts a tin y fo rc e. B ern o u lli rig h tly c o n c lu d e d th a t p re ssu re is th e fo rc e ex e rted (p e r u n it a rea ) b y a v ery la rg e n u m b e r o f ra n d o m ly m o v in g m o lec u les b o u n c in g o th e w a ll. It is n o t d i± c u lt to u n d ersta n d su ch a k i- n e tic th eo ry in th e c o n te x t o f a to m ic m o tio n s g iv in g rise to p re ssu re. B u t p h y sicists h a d d i± c u lty in c o m p re h e n d - in g k in etic th eo ry o f h e a t: a to m ic m o tio n s g iv in g rise to h e a t { b e it u n d u la tin g m o tio n a b o u t x e d p o sitio n s lik e B o y le im a g in e d o r free m o tio n in th e a v a ila b le sp a c e o f th e co n ta in e r lik e B ern o u lli m o d e le d. T h is d i± cu lty is p e rfe ctly u n d e rsta n d a b le sin c e it w a s k n o w n th a t h e a t c o u ld b e tra n sm itte d th ro u g h v a c u u m { lik e th e h e a t fro m th e su n. H e n c e h ea t c a n n o t b e a p ro p e rty o f a 16 RESONANCE July 2007

6 su b sta n c e; it h a s to b e a su b sta n c e b y itse lf. A n to in e - L a u ren t d e L a v o isie r c a lled th e su b sta n ce `C a lo ric'. It w a s fo u n d th a t C a lo ric u id a lw a y s o w e d fro m h ig h e r to lo w er tem p e ra tu re s. H ea t e n g in es th a t p ro d u c e d lo - c o m o tio n fro m b u rn in g o f co a l sta rte d d o ttin g th e E u - ro p ea n c o u n try sid e in th e la te e ig h tee n th c en tu ry. C a r n o t a n d H is H e a t E n g in e S a d i C a rn o t w a s in trig u e d b y th e v ery id e a o f a h e a t e n g in e th a t m a n a g e d to d o so m e th in g th a t e v e n th e a lm ig h ty n a tu re c o u ld n o t d o. A h e a t e n g in e c o n v e rts h e a t in to m o v e m en t. In n a tu re w e n d th a t it is th e m o v e m e n t w h ich d u e to fric tio n g en e ra te s h ea t a n d n o t th e o th e r w a y. T h e re is n o p h en o m en o n lik e u n -frictio n o r a n ti-fric tio n w h ich w o u ld sp o n ta n eo u sly re -a ssem b le h e a t b a ck in to a m o v em e n t. C a rn o t c a m e to th e b rillia n t c o n c lu sio n th a t m e re p ro d u ctio n o f h ea t is n o t su ± cien t to give birth to th e im pellin g po w er; it is n ecessa ry th a t th ere sh o u ld be co ld ; w ith o u t it, h ea t is u seless. C a rn o t a rg u ed th a t if a c erta in q u a n tity q o f c a lo ric su b sta n ce fa lls fro m a b so lu te te m p e ra tu re T 1 to z e ro th e n th e w o rk p ro d u c ed w o u ld b e W = q. S in c e th e ca lo ric u id fa lls o n ly to a n ite te m p era tu re T 2 (0 < T 2 < T 1 ), o n ly th e p ro p o rtio n a l fra c tio n, (T 1 T 2 )= (T 1 0 ), o f q sh o u ld e q u a l th e w o rk p ro d u c e d. H e n c e th e e ± cien cy o f a h e a t e n g in e c a n n o t e x ce e d 1 (T 2 = T 1 ), w h ere T 1 is th e te m - p e ra tu re o f th e h e a t so u rce (th e b o ile r) a n d T 2 th a t o f th e h e a t sin k (th e ra d ia to r). T h u s e v en a n id ea l h e a t e n g in e h a s a n e ± cie n cy le ss th a n u n ity. T h e K in d o f M o tio n W e C a ll H e a t V e ry so o n it w a s rea lize d th ro u g h th e m e ticu lo u s e x p e r- im e n ts o f R u m fo rd, M a y e r a n d J o u le th a t h e a t is n o t a su b sta n c e. H e a t is a c tu a lly en erg y o r m o re p re cise ly e n erg y in tra n sit. It is lik e w o rk w h ich is a lso e n e rg y in tra n sit. O n c e w e id en tify h e a t w ith e n erg y, C a rn o t's n d in g b ec o m e s in trig u in g. It a m o u n ts to sa y in g th a t Heat is actually energy or more precisely energy in transit. RESONANCE July

7 h e a t c a n n o t b e c o m p le te ly c o n v erte d in to w o rk w h e rea s w o rk c a n b e c o n v e rted co m p le tely in to h e a t { a n im - b a la n c e in n a tu re's sch e m e. T h is th e rm o d y n a m ic irre - v e rsib ility { th e o n e -w a y n a tu re o f en e rg y co n v e rsio n { is w h a t w e n o w c a ll th e S ec o n d L a w o f T h e rm o d y n a m ic s. 2 Extremely slow; the process looks almost static. 3 A reversible process is one in which infinitesimal change in the external condition will cause a reversal of the process. R u d o lf C la u siu s c a m e to k n o w o f C a rn o t's w o rk a d e c a d e la ter. H e fe lt th a t C a rn o t's b a sic c o n c lu sio n a b o u t e x - tra c tin g w o rk fro m h e a t w a s c o rre ct a n d c o n sid e re d it a s o f g re a t fu n d a m e n ta l im p o rta n ce. H e c a lle d it C a rn o t's p rin cip le o r th e S e co n d la w o f th erm o d y n a m ic s. B u t th e n C la u siu s re je cte d C a rn o t's `c a lo ric ' re a so n in g. B y th e n h e k n ew th a t h e a t w a s a k in d o f m o tio n { ce a sele ss a n d ra n d o m { o f th e in v isib le a to m s a n d m o le cu le s. T o e x p la in C a rn o t's p rin cip le in th e c o n te x t o f th e em e rg in g p ictu re o f h e a t a s e n e rg y in tra n sit, C la u siu s in v e n te d a n e w th erm o d y n a m ic v a ria b le ca lled e n tro p y, d e n o te d b y th e sy m b o l S. H e rep h ra se d C a rn o t's p rin c ip le a s d S 0 in a n y th erm o d y n a m ic p ro c e ss. In th e a b o v e re - la tio n, e q u a lity o b ta in s w h e n th e p ro c ess is q u a si-sta tic 2 a n d a lso re v e rsib le 3. T h u s th e S e c o n d la w o f th e rm o d y - n a m ics ca p tu re s a n e sse n tia l tru th a b o u t m a c ro sco p ic b e h a v io u r { n a m e ly it is n o t tim e re v ersa l in v a ria n t. T h e re is a d e n ite d irec tio n o f tim e { th e d ire ctio n o f in cre a sin g en tro p y. P h y sic ists a re p u z z le d b y th e S e c o n d la w. H o w d o es it a rise? A n a to m, th e c o n stitu e n t o f a m a c ro sc o p ic o b - jec t, o b e y s N ew to n 's la w s. N ew to n ia n d y n a m ics is tim e re v e rsa l in v a ria n t: Y o u c a n n o t tell th e p a st fro m th e fu tu re ; th e re is th e d e te rm in ism { th e p re se n t h o ld in g b o th th e e n tire p a st a n d th e en tire fu tu re. T h e a to m s, in d iv id u a lly, o b ey th e tim e rev ersa l in v a ria n t N ew to n ia n d y n a m ic s; h o w e v er, th e ir co llec tiv e b eh a v io u r se em s to b re a k th e tim e sy m m e try. T h u s th e tw o p illa rs o f th eo retic a l p h y sic s { N ew to n ia n m ech a n ics a n d th e rm o d y n a m ic s se e m to sta n d in co n - tra d ic tio n. T h e fo rm er is tim e re v e rsa l in v a ria n t w h ile 18 RESONANCE July 2007

8 th e la tter is n o t. H o w d o w e co m p reh e n d th is m icro - m a c ro d ich o to m y? In th e sy n th e sis o f a m a c ro sco p ic o b je ct fro m its m ic ro sco p ic c o n stitu e n ts w h e n a n d w h y d o es th e tim e -re v e rsa l in v a ria n c e b re a k d o w n? T h is is a q u e stio n th a t h a u n ted th e sc ien tists th e n, h a u n ts u s n o w a n d m o st a ssu re d ly sh a ll h a u n t u s in th e fu tu re, n e a r a n d fa r. B o lt z m a n n a n d H is T r a n s p o r t E q u a t io n B o ltz m a n n h a d a n e x tra o rd in a ry c o u ra g e to su g g e st th a t th e tim e a sy m m e tric m a c ro sco p ic p h en o m e n a c a n b e e x - a c tly d e riv e d fro m th e tim e -sy m m etric m icro sc o p ic la w s. In d ee d h e d eriv e d a n o n lin e a r tra n sp o rt e q u a tio n sta rtin g fro m N e w to n 's e q u a tio n s o f m o tio n. T h e so lu tio n o f th e B o ltzm a n n tra n sp o rt eq u a tio n { th e so c a lle d H -fu n ctio n { is tim e a sy m m e tric. T h e n eg a tiv e o f H c a n b e id e n ti e d w ith en tro p y. H a lw a y s d e c rea se s w ith tim e u n til it re a ch e s a m in im u m w h en th e sy ste m eq u i- lib ra te s. T h e sta te m en t th a t th e d e riv a tiv e o f H w ith re sp e c t to tim e is a lw a y s le ss th a n o r e q u a l to ze ro is c a lle d `B o ltzm a n n H -th e o re m '. B u t v e ry so o n B o ltz - m a n n re a liz e d th a t th e sto ssza h la n sa tz { c o llisio n n u m - b e r a ssu m p tio n { o f M a x w e ll, w h ich h e em p lo y e d, sto le in a n elem e n t o f sto ch a stic ity in to h is o th erw ise p u re ly d y n a m ic a l d e riv a tio n o f th e tra n sp o rt e q u a tio n. B u t th e n h e co n te n d e d co rrec tly th a t h is H -th eo re m is v io - la ted o n ly w h e n th e m a cro sc o p ic sy ste m sta rts o fro m so m e sp e cia l in itia l co n d itio n w h ich a re e x trem e ly sm a ll in n u m b er. F o r a n o v erw h elm in g ly la rg e n u m b e r o f in i- tia l c o n d itio n s th e d y n a m ic a l e v o lu tio n d o e s o b e y th e H -th e o rem. In o th e r w o rd s th e ty p ica l b eh a v io u r o f a m a c ro sc o p ic sy stem is in v a ria b ly c o n siste n t w ith th e H th e o re m. S u b se q u e n tly, B o ltz m a n n fo rm u la ted th e en tire p ro b - le m in sta tistica l te rm s a n d g a v e a d e n itio n o f en tro p y a s p ro p o rtio n a l to th e lo g a rith m o f th e n u m b e r o f m i- c ro sta te s o f a m a c ro sc o p ic sy ste m. L e t u s d ig ress a little Boltzmann defined entropy as proportional to the logarithm of the number of microstates of a macroscopic system. RESONANCE July

9 GENERAL ARTICLE b it an d try to u n d erstan d w h at on e m ean s b y an en sem - b le, a m icrostate an d a m acrostate. G ibbs Ensem bles C onsider a sim ple experim ent of tossing of a coin. Let th e p rob ab ility of H eads be p and oftailsbe q = 1 p. Thesamplespace S forthisexperim entconsistsoftwo outcom es fh ;Tg. L et u s toss a coin M tim es an d collect the results in a set denoted by. T hus M is th e num ber of elem ents of. Let m H denote the num ber of tim es H eads appears in. W e say th at th e set is a n en sem b le if p = m H =M. In oth er w ord s an ou tcom e of the sam ple space appears in the ensem ble as often as to re ect correctly its p rob ab ility. T h e size M of the en sem b le is taken to b e su ± cien tly large so th at each outcom e of the experim ent appears in the ensem ble a certain n u m b er of tim es p rop ortion al to its p rob ab ility. If p = 0:75 then an exam ple of an ensem ble of size four is = fh ;T;H;Hg, w h ere th e ou tcom e Headsoccurs th ree tim es an d th e ou tcom e Tailsoccurs once consistent w ith th eir p rob ab ilities. = fh ;T;H;H;T;H;H;Hg is also an en sem b le b u t of size eigh t. W e n d th at th reefourth of is H eads and one-fourth is tails. Noticewe do not really need to m ake any assum ption about how to con stru ct an en sem b le. W e can con stru ct it b y a d eterm in istic algorith m, a sto ch astic algorith m or b y actu ally carry in g ou t th e ex p erim en t a very large nu m b er tim es. L et u s n ow con sid er an ex p erim en t of tossin g N identical coin s. A n ou tcom e of th e ex p erim en t con sists of a strin g ofh eads and Tails. T h e len gth of th e strin g is N.Letus calleach outcom e a m icrostate and denote it by the sym - bol!.letn(!)bethenumberofheads in m icrostate!. T he value ofn d i ers in gen eral from on e m icrostate to an oth er. In th e ex am p le con sid ered hn i = N p. If w e consider the case of p = 1=2, then all the 2 N microstates are equally probable and hn i = N=2. T hus in th e m ach in ery of statistical m ech an ics d evelop ed by 20 RESONANCE July 2007

10 GENERAL ARTICLE G ibbs,corresponding to each therm odynam ic variable, w e id en tify a statisticalm ech an icalran d om variab le. W e th en con stru ct a G ib b s en sem b le of realization s of th e random variable. T he average of the random variable calculated over the ensem ble gives the value of the corresponding therm odynam ic variable. T h e ab ove h as to b e con trasted w ith B oltzm an n 's form ulation. T o each value of the m acroscopic variable w e attach an en trop y d e n ed as p rop ortion al to th e logarith m of th e n u m b er of m icrostates associated w ith th at v a lu e. F o r ex a m p le, in th e co in to ssin g p rob lem if th e m acroscopic variable denoting the num b er of H eads in a toss of N coins takes a value n th en th e nu m b er of m icrostates associated w ith it is given b y The average of the random variable calculated over the ensemble gives the value of the corresponding thermodynamic variable. (n ;N )= N! n!(n n )! ; (1) wherewehavetakenp = 1=2. In B oltzm ann's form u- lation th e sy stem takes th at valu e of th e m acroscop ic variable n for w h ich en trop y is m axim u m. (n ;N )is maximum forn = N=2. H ence in B oltzm ann's form u- lation also,n=2 is the value ofthe corresponding therm odynam ic variable. W e can de ne B oltzm ann entropy as N! S B = k B lo g (2) n!(n n )! w h ich is d i eren t fro m G ib b s en trop y S G = N lo g (2). A quick calculation w illshow that the probability ofn di ering from N=2 b y m ore th an an arb itrarily sm all fraction of N=2 goes to zero w hen N is la rg e. F o r ex - am ple, the probability for n to lie ou tsid e th e interval N=2 ²;N =2+ ² is0.002 w hen ² is o n e p ercen t o f N=2for N = In the study ofm acroscopic system s w e shall be dealing w ith N of th e ord er ThusforlargeN, B oltzm an n en trop y an d G ib b s en trop y h ave for p ractical purposes the sam e value. RESONANCE July

11 Gibbs approach to statistical mechanics was to generalize Newtonian mechanics to arbitrary, though strictly finite, number of degrees of freedom. The energy of the system is a fluctuating quantity which when averaged over a Gibbs ensemble of microstates yields the corresponding thermodynamic energy. G ib b s' a p p ro a ch to sta tistic a l m e ch a n ics w a s to g e n e r- a liz e N ew to n ia n m e ch a n ic s to a rb itra ry, th o u g h stric tly n ite, n u m b e r o f d eg re es o f fre e d o m. T o th is e n d h e k n it p o sitio n a n d m o m en tu m in to a sin g le fa b ric ca lled `p h a se sp a ce '. L et m e e x p la in th is b y co n sid erin g a n iso la te d sy ste m o f N m o le cu le s c o n n ed to a v o lu m e V w ith a to ta l en e rg y E. E a ch m o le c u le is sp e c i e d b y th re e c o o r- d in a te s o f p o sitio n a n d th re e co o rd in a te s o f m o m e n tu m. T h e sy ste m is th u s sp e c i e d b y a p o in t in a 6 N d im e n - sio n a l p h a se sp a c e. S in ce th e m o le cu le s in th e sy ste m a re c o n sta n tly in m o tio n c o llid in g w ith e a ch o th e r a n d w ith th e w a lls o f th e c o n ta in er, th e p h a se sp a ce p o in t re p re - se n tin g th e sy stem is in ce ssa n tly in m o tio n. C o n sid e r a c o lle c tio n o f a la rg e n u m b er o f m e n ta l c o p ie s o f th e sy ste m, a ll id e n tic a l m a cro sc o p ica lly { h a v in g th e sa m e v a lu e o f E, V a n d N { b u t m a y d i er in th e ir m ic ro sco p ic d e ta ils. T h is c o lle ctio n o f sy ste m s is c a lled a G ib b s e n - se m b le. T h e m e m b e rs o f th is e n se m b le a re re p re se n te d b y a d istrib u tio n o f p o in ts in th e p h a se sp a c e. E a ch p h a se sp a c e p o in t e v o lv es a s p er N ew to n ia n d y n a m ics. T h e n a sim p le c a lc u la tio n o f th e n u m b er sy ste m s ly in g w ith in in n ite sim a l lim its o f p h a se sp a c e y ie ld s th e b a sic e q u a tio n { th e L io u v ille's e q u a tio n { o f c la ssic a l sta tistic a l m e ch a n ic s. D e p e n d in g o n th e n a tu re o f th e co n stra in ts, w e g e t d iffere n t G ib b s e n se m b les. W e sa w a b o v e a n ex a m p le o f a G ib b s en sem b le d e sc rib in g a n iso la te d sy stem. T h is is c a lle d m ic ro ca n o n ic a l en sem b le, d e te rm in e d b y E, V a n d N. O n th e o th er h a n d if w e x th e te m p era tu re, v o lu m e a n d th e n u m b er o f p a rticle s w e g e t a ca n o n i- c a l e n se m b le, w h ich d e scrib e s a c lo se d sy ste m. It is in th e rm a l co n ta c t w ith th e su rro u n d in g s { c a lle d th e h e a t b a th. It g iv e s en e rg y to th e h ea t b a th o r ta k es e n e rg y fro m it. T h u s th e e n e rg y o f th e sy ste m is a u c tu a tin g q u a n tity w h ich w h e n a v e ra g ed o v er a G ib b s en sem b le o f m icro sta te s y ield s th e c o rre sp o n d in g th erm o d y n a m ic e n erg y. T h e fra c tio n a l n u m b e r o f e n trie s in th e G ib b s' 22 RESONANCE July 2007

12 c a n o n ic a l en sem b le th a t a re in a m ic ro sta te! is p ro - p o rtio n a l to e x p [ E (! )= k B T ] w h e re E (! ) is th e e n e rg y o f!, T is th e te m p era tu re a n d k B is a c o n sta n t n a m e d a fte r B o ltzm a n n. T h is is c a lle d `c a n o n ic a l d istrib u tio n '. W e c a n c a lc u la te th e a v e ra g e o f e n erg y d e n o te d b y he i o v e r a G ib b s en sem b le. T h e n he i o f sta tistic a l m e ch a n - ic s g iv e s th e th e rm o d y n a m ic e n erg y, o fte n d en o te d b y U, i.e. U he i. O th e r th e rm o d y n a m ic p ro p erties c a n a lso b e c a lcu la te d b y a sim ila r a v e ra g in g o f th e ir sta - tistic a l m e ch a n ic a l c o u n te rp a rts o v e r a G ib b s' c a n o n ic a l e n se m b le. If w e a llo w b o th en e rg y a n d n u m b e r o f m o le cu le s o f th e sy ste m to u c tu a te (th e su rro u n d in g s co n stitu te a h e a t b a th a s w e ll a s p a rtic le b a th ) w e g e t a g ra n d c a n o n ic a l e n se m b le fo r w h ich th e v o lu m e, tem p era tu re a n d ch e m - ic a l p o te n tia l a re th e relev a n t v a ria b les. S t a t is t ic a l M e c h a n ic s o f M a x w e ll, B o lt z m a n n, G ib b s a n d E in s t e in A n a tu ra l q u e stio n th a t a rise s re la te s to th e o rig in o f p ro b a b ility in G ib b s sta tistic a l m ech a n ic s. It is p re - c ise ly in th is c o n te x t th a t G ib b s fo rm u la tio n d i ers fro m th a t o f M a x w ell, B o ltz m a n n a n d E in ste in. M a x w e ll w a s p erh a p s th e rst to re c o g n iz e th e n e e d fo r a sta - tistic a l a p p ro a ch to k in etic th eo ry. H is d e riv a tio n o f th e p ro b a b ility d istrib u tio n o f th e sp ee d o f m o n o a to m ic n o n -in te ra c tin g g a s m o lec u les is a n e x erc ise in in g e n u - ity a n d eleg a n c e. T h e p ro b a b ility is p ro p o rtio n a l to e x p o n e n tia l o f k in e tic e n e rg y. T h is w a s la ter g e n e ra l- iz e d b y B o ltz m a n n w h o in c lu d ed p o te n tia l e n e rg y a risin g d u e to e x te rn a l fo rc es a n d d u e to in te rn a l in te ra c tio n s a m o n g st th e m o le c u le s a n d sh o w e d th e p ro b a b ility o f a m ic ro sta te is p ro p o rtio n a l to e x p [ E = k B T ], w h ere E is th e to ta l en erg y o f th e m ic ro sta te, T is th e te m p era - tu re a n d k B is th e B o ltz m a n n c o n sta n t. T h e p ro b a b ility d istrib u tio n is n o w k n o w n a s M a x w e ll{ B o ltzm a n n d istrib u tio n. M a x w e ll c a te g o ric a lly sta ted th a t th e S e co n d Gibbs formulation differs from that of Maxwell, Boltzmann, and Einstein in the context of probability in statistical mechanics. RESONANCE July

13 4 For an interesting account of Maxwell s demon and other demons see [6]. la w o f th e rm o d y n a m ic s is sta tistic a l in n a tu re a n d h e n ce th e re is a n o n -z ero p ro b a b ility o f it b ein g co n tra v e n ed. In fa c t h e co n stru c ted a d e m o n { n o w ca lle d M a x w e ll's d e m o n { th a t v io la tes th e S e co n d la w 4. T h u s fo r M a x w ell, p ro b a b ility a rises a s a n ex tra a ssu m p tio n in sta tistic a l m ech a n ics, fo r d e sc rib in g m a cro sco p ic b e h a v io u r. F o r B o ltz m a n n th e p ro b a b ility is o f d y n a m ic a l o rig in. A sin g le d y n a m ic a l tra jec to ry o f a n e q u ilib riu m iso la te d sy ste m v isits a ll th e p h a se sp a c e p o in ts ly in g o n a co n - sta n t en e rg y su rfa c e. T h e p ro b a b ility o f n d in g th e sy s- te m in a re g io n o f its p h a se sp a c e is th e fra ctio n o f o b - se rv a tio n tim e th e d y n a m ica l tra je c to ry sp e n d s in th a t re g io n. E in ste in 's fo rm u la tio n fo llo w s th e m e th o d s o f k in e tic th e o ry o f g a se s. In E in ste in 's sta tistic a l m e ch a n ic s, th e sy ste m m u st n e c essa rily b e fo rm e d b y a v ery la rg e n u m - b e r o f m o lec u les. T h e re su lt o f a m e a su rem e n t m u st b e id en ti ed w ith tim e a v e ra g e. T h ere fo re h e c o n stru cts a tim e -e n se m b le. T h e a v e ra g e o v e r th e tim e e n se m b le is e q u a ted to p h a se a v era g e. 5 Boltzmann was perhaps the lone champion of molecular kinetic theory of heat. He had to contend with the criticism and ridicule from the most influential and vociferous of the German speaking community the so called energeticists led by Ernst Mach and Wilhelm Ostwald. The energeticists did not approve of molecular composition of matter and the kinetic theory. For them energy was the only fundamental entity. They dismissed with contempt any attempts to describe energy or energy transformation in more fundamental atomistic terms or kinetic picture. Only in the year 1905, the reality of atoms and molecules was established unambiguously by Einstein in his work on Brownian motion. In G ib b s fo rm u la tio n, p ro b a b ility en te rs n o t a s a n a d - d itio n a l h y p o th e sis n o r a s a c o n seq u e n c e o f th e fo rm u - la tio n b u t a s a p a rt o f th e d a ta o n in itia l c o n d itio n s o f a n e n se m b le o f m e ch a n ic a l sy stem s. W h e n a p p lie d to a sy ste m o f a la rg e n u m b e r o f m o le c u le s, G ib b s fo rm u la - tio n re d u c es to th o se b a se d o n k in etic th e o ry. S tric tly h is fo rm u la tio n d o es n o t req u ire a n y a ssu m p tio n o n th e m o le c u la r co m p o sitio n o f m a tte r. W e sh o u ld re a liz e th a t in th e tim e s o f G ib b s, k in e tic th e o ry d id n o t re ce iv e g e n - e ra l a c c ep ta n c e 5. G ib b s fo rm u la tio n h in g e s o n co m b in - in g th e c o n g u ra tio n a n d m o m e n tu m in to a sin g le p h a se a n d in te g ra tin g o v e r a p h a se -sp a c e d e n sity d escrib in g e q u ilib riu m G ib b s e n se m b le. T h e p h a se sp a c e d e n sity is u n ifo rm o n a c o n sta n t e n e rg y su rfa c e fo r a m icro c a n o n - ic a l e n se m b le a n d is a n e x p o n e n tia l fu n ctio n o f e n e rg y fo r a c a n o n ic a l e n se m b le, a s w e h a v e see n e a rlie r. 24 RESONANCE July 2007

14 E p ilo g u e T h e rig o u r, ec o n o m y, g e n e ra lity a n d e le g a n c e o f G ib b s fo rm u la tio n o f sta tistic a l m ech a n ics is b o rn e o u t b y th e fa c t th a t th e en tro p y u c tu a tio n th e o re m 6 d e riv e d b y G a lla v o tti a n d C o h e n [7 ] h a s b e e n sh o w n a s a G ib b s p ro p erty. F o r G ib b s, sc ien ti c re c o g n itio n w a s slo w in co m in g. T h is w a s m a in ly b ec a u se G ib b s w a s n o t a p ro p a g a n d ist fo r h is o w n w o rk ; n o r, fo r th a t m a tte r, w a s h e o n e, fo r sc ie n c e. B u t e v e n tu a lly h is w o rk sp o k e fo r h im a n d h is fa m e g re w slo w ly a n d ste a d ily. H e w a s ele cte d a m e m b e r o f se v era l im p o rta n t sc ien ti c a c a d e m ie s o f th e w o rld lik e th e A m e rica n P h ilo so p h ic a l S o c iety, th e D u tch S o cie ty o f S cien ce s in H a a rle m, th e R o y a l S o c ie ty o f S c ien c e s in G Äo ttin g en, th e R o y a l In stitu tio n o f G re a t B rita in, th e C a m b rid g e P h ilo so p h ic a l S o c ie ty a n d th e R o y a l S o cie ty o f L o n d o n, to n a m e o n ly a fe w. T h e A m erica n A ca d - e m y o f B o sto n a w a rd e d h im th e R u m fo rd m e d a l in a n d th e R o y a l S o cie ty g a v e h im th e C o p ley m e d a l in , `th e h ig h est h o n o u r E n g lish S c ie n c e h a s to b e - sto w '. W h en E in stein w a s a sk e d w h o m h e c o n sid e red to b e th e m o st p o w e rfu l th in k ers h e h a d k n o w n, h e rep lied, \...L o ren z...bu t I n ever m et W illa rd G ibbs; perh a p s, h a d I d o n e so, I m igh t h a ve p la ced h im besid e L o ren z". L e t m e e n d b y q u o tin g M a x P la n ck : \... G ibbs w ill ever be recko n ed a m o n g th e m o st ren o w n ed th eo retica l p h y sicists of all tim es". 6 The fluctuation theorem of Cohen and Gallavotti arises in the context of nonequilibrium steady systems. The entropy refers to phase space contraction rate. Suggested Reading [1] Gibbs first two papers were entitled Graphical methods in thermodynamics of fluids and A method of Geometrical Representation of the Thermodynamic Properties of Substances by means of Surfaces, in The Scientific papers of J. Willard Gibbs, (2 vols); edited by H A Bumstead and R Gibbs Van Name, Longmans, Green and Co., New York, 1906; Dover Reprint, New York, RESONANCE July

15 Address for Correspondence K P N Murthy* School of Physics University of Hyderabad Hyderabad Andhra Pradesh, India. kpnmsp@uohyd.ernet.in * On deputation from Materials Science Division Indira Gandhi Centre for Atomic Research Kalpakkam Tamilnadu, India [2] J W Gibbs, Trans. Conn. Acad., Vol.III, p.108, Oct May 1876; p.343, May 1877 July [3] J W Gibbs, Note on the Electromagnetic Theory of Light, Am. J. Sci., Vol.23, p.262, p.460, 1882; Vol.25, p.107, 1883; Vol.35, p.467, 1888; Vol.37, p.129, [4] J W Gibbs, Elementary principles in statistical mechanics, Yale University Press, [5] M Guillen, An unprofitable experience: Rudolf Clausius and the Second law of thermodynamics in Five Equations that changed the world: the power and poetry of Mathematics, Hyperion, New York, p.165, [6] H S Leff and A F Rex (Eds), Maxwell s Demon: Entropy, Information and Computing, Adam Higler, Bristol, H S Leff and A F Rex (Eds.), Maxwell s Demon, Princeton Univ. Press, Princeton, H S Leff and A F Rex (Eds.), Maxwell s Demon 2: Entropy, Classical and Quantum Information, Computing, Institute of Physics, J D Collier, Two faces of Maxwell s demon reveal the nature of irreversibility, Studies in History and Philosophy of Science, p.257, June D L Hogenboon, Maxwell s Demon: Entropy, Information, Computing, American Journal of Physics, p.282, March H S Leff, Maxwell s Demon, Power and Time, American Journal of Physics, pp , Feb [7] G Gallavotti and E G D Cohen, Dynamical ensembles in non-equilibrium statistical mechanics, Phys. Rev. Lett., Vol.74, p.2694, G Gallavotti and E G D Cohen, Dynamical ensembles in stationary states, J. Stat. Phys., Vol.80, p.931, D Ruelle, Smooth dynamics and new theoretical ideas in non-equilibrium statistical mechanics, J. Stat. Phys., Vol.95, p.393, [8] [9] H A Bumstead, Josiah Willard Gibbs, American Journal of Science, XVI(4), [10] R C Cantelo, J Willard Gibbs, a brief biography and a summary of his contributions to chemistry, Can. Chem. Metallurgy, Vol.8, p.215, [11] L P Wheeler, Josiah Willard Gibbs, The History of a Great Mind, Yale University Press, [12] R J Seeger, J Willard Gibbs, American mathematical physicist par excellence, Pergamon Press, [13] Jagdish Mehra, Josiah Willard Gibbs and the Foundations of Statistical Mechanics, Foundations of Physics, Vol.28, p.1785, RESONANCE July 2007