Josiah Willard Gibbs and his Ensembles - Indian Academy of Sciences

GENERAL ARTICLE
Josiah Willard Gibbs and his Ensembles
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
Keywords
dynamics, radiation
transport, random walks,
regular and anomalous
diffusion and non-linear
dynamics and chaos.
Gibbs, thermodynamics, statis-
tical mechanics, Gibbs en-
sembles, vector algebra.
P r o lo g u e
K P N Murthy
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.
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 1 8 6 3 . 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 ilo
so p h y fo r o n e y ea r. In th e y e a r 1 8 6 6 , 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 rd
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 1 9 0 3 .
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 .
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GENERAL ARTICLE
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 lu
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 ste
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 rfa
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 1 8 7 6 { 1 8 7 8 , 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
13

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.
GENERAL ARTICLE
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 t ic s , a n d F o u r ie r S e r ie s
In th e y ea rs 1 8 8 0 { 1 8 8 4 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 1 8 8 1 a n d 1 8 8 4 .
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 tsid
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 1 8 9 8
a n d A p ril 1 8 9 9 . B e tw e e n 1 8 8 2 a n d 1 8 8 9 , 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

RESONANCE July 2007
GENERAL ARTICLE
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 a n 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 ibbs 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 rro
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 combin-
ing in pairs to form dyads. A
dyad is also imperceptible,
though it has acquired an at-
tribute of two-ness. It requires
three dyads to combine and
form a triad. The triad is percep-
tible; it has attributes that can
be observed and measured.
15

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.
GENERAL ARTICLE
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 eru
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 tu
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
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 in
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

RESONANCE July 2007
GENERAL ARTICLE
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 giv e 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 t io 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 rim
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.
17

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.
GENERAL ARTICLE
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.
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

RESONANCE July 2007
GENERAL ARTICLE
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 ilib
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 itia
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 ic
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.
19

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 reefou
rth 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 (! )bethenumberofH eads 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

RESONANCE ⎜ July 2007
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
− (n ;N )=
N !
n !(N ¡ n )!
; (1)
wherewehavetakenp = 1=2. In B oltzm ann's form ulation
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
maximumforn = N=2. H ence in B oltzmann's formu lation also,N=2is the value ofthe corresponding therm
odynam ic variable. W e can de¯ne B oltzmann entropy
as
S B = k B lo g
·
N !
n !(N ¡ n )!
¸
(2)
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 = 10 5 . In the study ofm acroscopic system s w e shall
be dealing w ith N of th e ord er 10 25 .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.
The average of the
random variable
calculated over the
ensemble gives
the value of the
corresponding
thermodynamic
variable.
21

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.
GENERAL ARTICLE
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 ra
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 rd
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 ic
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

RESONANCE July 2007
GENERAL ARTICLE
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 liz
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.
23

4 For an interesting account of
Maxwell’s demon and other de-
mons see [6].
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.
GENERAL ARTICLE
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 ste
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 .
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

E p ilo g u e
Suggested Reading
RESONANCE July 2007
GENERAL ARTICLE
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 1 9 0 1
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
1 9 0 1 , `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 ibb s; 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 ibb s 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
o f a ll tim es" .
[1] Gibbs first two papers were entitled ‘Graphical methods in thermody-
namics 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, 1961.
6 The fluctuation theorem of
Cohen and Gallavotti arises in
the context of nonequilibrium
steady systems. The entropy
refers to phase space contrac-
tion rate.
25

Address for Correspondence
K P N Murthy*
School of Physics
University of Hyderabad
Hyderabad 500 046
Andhra Pradesh, India.
Email:
kpnmsp@uohyd.ernet.in
* On deputation from
Materials Science Division
Indira Gandhi Centre for
Atomic Research
Kalpakkam 603 102
Tamilnadu, India
GENERAL ARTICLE
[2] J W Gibbs, Trans. Conn. Acad., Vol.III, p.108, Oct. 1875–May 1876;
p.343, May 1877–July 1878.
[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, 1889.
[4] J W Gibbs, Elementary principles in statistical mechanics, Yale Uni-
versity Press, 1902.
[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, 1995.
[6] H S Leff and A F Rex (Eds), Maxwell’s Demon: Entropy, Information
and Computing, Adam Higler, Bristol, 1990.
H S Leff and A F Rex (Eds.), Maxwell’s Demon, Princeton Univ. Press,
Princeton, 1990.
H S Leff and A F Rex (Eds.), Maxwell’s Demon 2: Entropy, Classical
and Quantum Information, Computing, Institute of Physics, 2003.
J D Collier, Two faces of Maxwell’s demon reveal the nature of
irreversibility, Studies in History and Philosophy of Science, p.257,
June 1990.
D L Hogenboon, Maxwell’s Demon: Entropy, Information, Comput-
ing, American Journal of Physics, p.282, March 1992.
H S Leff, Maxwell’s Demon, Power and Time, American Journal of
Physics, pp.135–142, Feb. 1990.
[7] G Gallavotti and E G D Cohen, Dynamical ensembles in non-equilib-
rium statistical mechanics, Phys. Rev. Lett., Vol.74, p.2694, 1995.
G Gallavotti and E G D Cohen, Dynamical ensembles in stationary
states, J. Stat. Phys., Vol.80, p.931, 1995.
D Ruelle, Smooth dynamics and new theoretical ideas in non-equilib-
rium statistical mechanics, J. Stat. Phys., Vol.95, p.393, 1999.
[8] http://www.mlahanas.de/Physics/Bios/WillardJGibbs.html
[9] H A Bumstead, ‘Josiah Willard Gibbs’, American Journal of Science,
XVI(4), 1903.
[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,
1924.
[11] L P Wheeler, Josiah Willard Gibbs, The History of a Great Mind, Yale
University Press, 1952.
[12] R J Seeger, J Willard Gibbs, American mathematical physicist par
excellence, Pergamon Press, 1974.
[13] Jagdish Mehra, Josiah Willard Gibbs and the Foundations of Statis-
tical Mechanics, Foundations of Physics, Vol.28, p.1785, 1998.
26 RESONANCE July 2007