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On the passage to equilibrium
Volum(~ 33A. n u m b e r ;{
PHYSICS
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2
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The H-dependence of ,:r at 4 . 2 K i s a l s o sh(}\~,t] in fig. 1. T h e s a t u r a t i o n lllOlllent of crs : 138 e m u g r a m . o b t a i n e d f r o m m a g n e t i z a t i o n d a t a at 1.5 K. i s m u c h h i g h e r t h a n t h a t r e p o r t e d by B u s c h et a l . [6]. T h e g e n e r a l s h a p e s o f our magnetization and differential susceptibility curves are, however, s i m i l a r to earlier w{}rk on powdered sl)ecimens
PARAMAGNE TIC
K68 ~66
19 O c t o l } e r
l a t t e r ( p r e s u m a b l y l e s s p u r e ) s a m p l e s h o w e d n{} ~ - a n o m a l y It t, a l t h o u g h dBl dH d e c r e a s e d mark e d l y at t h e t r a n s i t i o n .
~tl3
H II [IOO]
°7o
LFTTEIIS
8
T ~'2 ( K x/~ }
Fi~. -!. \ ' a r i a t i o u ~)1 H t (inccrn:ll licidl \\ith 7<~,'2-'. [?C!{,FCH(,~ ',~ y i e l d s /It(0) 72 ± l k O e ( i n t e r n a l f i e l d ) , w h e r e t h e q u o t e d e r r o r i s l a r g e l y d u e to u n c e r t a i n t i e s in t h e e s t i m a t e of t h e d e m a g n e t i z i n g f i e l d . T h e d a t a f o r H t a g r e e c l o s e l y w i t h t h e r e s u l t s of ultrasonic measurements on the same sanq}le. Earlier ultrasonic work[51 on another sample which had a room temperature r e s i s t i v i t y of -- 10 - 2 9, - e r a g a v e s i m i l a r r e s u l t s , although d H t dT:~/'2 i w a s s o m e w h a t h i g h e r . M e a s u r e m e n t s of tile d i f f e r e n t i a l s u s c e p t i b i l i t y in t h e
ON
THE
Ii] ,J.W. Battle~ :m(i G, I.:.l<',urctt P h \ s . lle\'. !;I (1!)70) 3021. 121 II. Falk. P h s s . I{c\. i:g5 tl !idl! .\135,2. [;',] F. I3. A n d e r s ( m and H.l~,c!:~lldm. Ph\,s. llc~. I:;i; {l!l{;t) A1 0(b,, l~j J . F e d u r :met 1:. P.vttc. Ph\,-,, / l c \ . l{;s (1.91P,) d i l l {:;I Y. Shal)ir:~ and T. B. th,{,d t)h\>. [,{,tWrs 31A iI}!% 3',1. !i;i G. B u s c h . l ' . J u m ) d . P. Sct~\~ 4~ ( ) . \ ' o g t and F. tiuliigcP. P h y s . I;CELL'P,'-; i) (l'.)(i]) 7. [~[ I, S.,Jac()}}s :rod S. [}, Sil\'('rstcitl P h y s . I{c\. [A~ttt~p.1;] (19(;t} 272.
PASSAGE
TO
EQUILIBRIUM
S. SIMONS
(pHC,'H B l a r v ('r~f!, :,', . M i L ' Eu(/ H(m(i. l ~md,m. ! A
lleceived
i 5 S{?l)tcmbeP l!)7~1
It is l)ointcd out that for v a r i o u s s y s t e m s , qu~mtitics equal o r c l o s c I 5' celatc,i ~,. ~ucce, s s i x e t i m e , i c r i v a t i v e s of the entrOl)y altern',~te in s i g d u r i n g the p a s s a g e to e q u i l i b r i u m .
It i s k n o w n t h a t t h e t r a n s i t i o n to e q u i l i b r i u m for an isolated system is characterized by the l i m e v a r i a t i o n of tile e n t r o p y S b e i n g s u c h t h a t dSd/
0.
(1)
It i s t h e p u r p o s e of tile p r e s e n t c o m n m n i c a t i ( m to point out that for various systems which can be described by linear equations, the above relation 154
(1) m a y b e e x t e n d e d to a s c r i b e a s p e c i f i c s i g n t,, S01), w h e r e ill c e r t a i n c a s e s S0z) d e n o t e s d J l S " d P z. w h i l e i n o t h e r s it d e n o t e s a q u a n t i t y c l o s e l y r e l a t e d to t h i s d e r i v a t i v e , ; in f a c t (-P~S( n ) f o r ,z i. Consider
0
(2~
f i r s t a h o m o ~ ~ , ~ m o ~ . ~ s y s t e n ~ of p : ~ P
Volume 33A, number 3
PHYSICS LETTERS
t i c l e s - gas m o l e c u l e s , e l e c t r o n s or phonons whose d i s t r i b u t i o n function f ( k , t ) is sufficiently c l o s e to an e q u i l i b r i u m d i s t r i b u t i o n F ( k , T) f o r the Boltzmann equation to take i ts l i n e a r f o r m ~/~ t = M~
(3)
w h e r e f = F - (OF/OE) ~ (k,t) and M is the c o r r e sponding i n t e g r a l c o l l i s i o n o p e r a t o r . Then using the s t a n d a r d definitions of entropy [e.g. 1], it is found that if only leading t e r m s in ~ a r e r e t a i n e d
dS/dt = F ( ~
~ dp/~t)
(4)
w h e r e F is a p o s i t i v e constant and (0,~) = f (OF/~E) O(k)~V ( k ) d k . With this definition of s c a l a r product it is known that M is both H e r m i t i a n and p o s i t i v e s e m i - d e f i n i t e ; s e e Chapman and Cowling [2] and Ziman [3]. By r e peated d i f f e r e n t i a t i o n with r e s p e c t to t i m e of eq. (4), t o g e t h e r with the u s e of eq. (3), r e s u l t (2) is obtained w h e r e S (n) = dnS/dtn. F o r an i s o l a t e d inhomogeneous s y s t e m of p a r t i c l e s the above p r o o f b r e a k s down as the B o l t z mann equation (3) is then s u p p l e m e n t e d by a t e r m v" g r a d $ and the o p e r a t o r v" g r a d is not H e r m i t i a n . C o n s i d e r , h o w e v e r , such a s y s t e m of p a r t i c l e s which is such that the t e m p e r a t u r e d e p a r t s in s o m e g e n e r a l fashion by an amount O(r,t) f r o m the m e a n t e m p e r a t u r e To. Then if 10! << To, it is shown by Landau and Lifshitz [4] that
OS_ 1 ~t
f K m n ~0 ~0 d~To 2 ~x m ~x n
w h e r e Krn n is the t h e r m a l conductivity t e n s o r , while 0 s a t i s f i e s the conduction equation
(5)
19 October 1970
~0
020
C ~t- = Krnn Oxm Oxn
(6)
w h e r e C is the s p e c i f i c heat. It may be shown f r o m eqs. (5) and (6) that dnS/dt n can be e x p r e s s e d in the f o r m
dnS/df n = s(n) + R(n) w h e r e S(n) is a volume i n t e g r a l throughout the s y s t e m involving sp at i al d e r i v a t i v e s of 0, and R (n) i s a surface i n t e g r a l o v e r the boundary of the s y s t e m . On making u s e of the s y m m e t r y and p o s i t i v e definite n a t u r e of Kmn, the r e s u l t (2) may be shown to be t r u e . A proof s i m i l a r to the above may a l s o be give~ f o r the e a s e of p a r t i c l e diffusion within an i s o lated s y s t e m . H e r e dS/dt = / 3 f I g r a d c ]2 dT w h e r e fi is a p o s i t i v e constant and the p a r t i c l e c o n c e n t r a t i o n c ( r , t ) s a t i s f i e s the equation
a c/O t = DV2c with D the diffusion coefficient. Full d e t a i l s of the work outlined h e r e will be published e l s e w h e r e .
References [1] L. D. Landau and E. M. Lifshitz, Statistical physics (Pergamon 1958). [2] S. Chapman and T. G. Cowling, The mathematical theory of non-uniform gases (Cambridge 1952). [3] J. M. Ziman. Electrons and phonons (Oxford 1960). [4] L. D. Landau and E. M. Lifshitz, Theory of elasticity (Pergamon 1959).
155
PHYSICS
EuTe
72 : % ~,
w
i
txl F-
%,
CANTED :
z_
ii
!{~. 71.
64
W e a r e i n d e b t e d to T. IL R e e d | o r t h e EuT{: s i n g l e c r y s t a l a n d t , E..1. M c N i f t ,Jr. |{}1" v a l u a b l e assistance.
62 O
2
4
6
l{tTl~
The H-dependence of ,:r at 4 . 2 K i s a l s o sh(}\~,t] in fig. 1. T h e s a t u r a t i o n lllOlllent of crs : 138 e m u g r a m . o b t a i n e d f r o m m a g n e t i z a t i o n d a t a at 1.5 K. i s m u c h h i g h e r t h a n t h a t r e p o r t e d by B u s c h et a l . [6]. T h e g e n e r a l s h a p e s o f our magnetization and differential susceptibility curves are, however, s i m i l a r to earlier w{}rk on powdered sl)ecimens
PARAMAGNE TIC
K68 ~66
19 O c t o l } e r
l a t t e r ( p r e s u m a b l y l e s s p u r e ) s a m p l e s h o w e d n{} ~ - a n o m a l y It t, a l t h o u g h dBl dH d e c r e a s e d mark e d l y at t h e t r a n s i t i o n .
~tl3
H II [IOO]
°7o
LFTTEIIS
8
T ~'2 ( K x/~ }
Fi~. -!. \ ' a r i a t i o u ~)1 H t (inccrn:ll licidl \\ith 7<~,'2-'. [?C!{,FCH(,~ ',~ y i e l d s /It(0) 72 ± l k O e ( i n t e r n a l f i e l d ) , w h e r e t h e q u o t e d e r r o r i s l a r g e l y d u e to u n c e r t a i n t i e s in t h e e s t i m a t e of t h e d e m a g n e t i z i n g f i e l d . T h e d a t a f o r H t a g r e e c l o s e l y w i t h t h e r e s u l t s of ultrasonic measurements on the same sanq}le. Earlier ultrasonic work[51 on another sample which had a room temperature r e s i s t i v i t y of -- 10 - 2 9, - e r a g a v e s i m i l a r r e s u l t s , although d H t dT:~/'2 i w a s s o m e w h a t h i g h e r . M e a s u r e m e n t s of tile d i f f e r e n t i a l s u s c e p t i b i l i t y in t h e
ON
THE
Ii] ,J.W. Battle~ :m(i G, I.:.l<',urctt P h \ s . lle\'. !;I (1!)70) 3021. 121 II. Falk. P h s s . I{c\. i:g5 tl !idl! .\135,2. [;',] F. I3. A n d e r s ( m and H.l~,c!:~lldm. Ph\,s. llc~. I:;i; {l!l{;t) A1 0(b,, l~j J . F e d u r :met 1:. P.vttc. Ph\,-,, / l c \ . l{;s (1.91P,) d i l l {:;I Y. Shal)ir:~ and T. B. th,{,d t)h\>. [,{,tWrs 31A iI}!% 3',1. !i;i G. B u s c h . l ' . J u m ) d . P. Sct~\~ 4~ ( ) . \ ' o g t and F. tiuliigcP. P h y s . I;CELL'P,'-; i) (l'.)(i]) 7. [~[ I, S.,Jac()}}s :rod S. [}, Sil\'('rstcitl P h y s . I{c\. [A~ttt~p.1;] (19(;t} 272.
PASSAGE
TO
EQUILIBRIUM
S. SIMONS
(pHC,'H B l a r v ('r~f!, :,', . M i L ' Eu(/ H(m(i. l ~md,m. ! A
lleceived
i 5 S{?l)tcmbeP l!)7~1
It is l)ointcd out that for v a r i o u s s y s t e m s , qu~mtitics equal o r c l o s c I 5' celatc,i ~,. ~ucce, s s i x e t i m e , i c r i v a t i v e s of the entrOl)y altern',~te in s i g d u r i n g the p a s s a g e to e q u i l i b r i u m .
It i s k n o w n t h a t t h e t r a n s i t i o n to e q u i l i b r i u m for an isolated system is characterized by the l i m e v a r i a t i o n of tile e n t r o p y S b e i n g s u c h t h a t dSd/
0.
(1)
It i s t h e p u r p o s e of tile p r e s e n t c o m n m n i c a t i ( m to point out that for various systems which can be described by linear equations, the above relation 154
(1) m a y b e e x t e n d e d to a s c r i b e a s p e c i f i c s i g n t,, S01), w h e r e ill c e r t a i n c a s e s S0z) d e n o t e s d J l S " d P z. w h i l e i n o t h e r s it d e n o t e s a q u a n t i t y c l o s e l y r e l a t e d to t h i s d e r i v a t i v e , ; in f a c t (-P~S( n ) f o r ,z i. Consider
0
(2~
f i r s t a h o m o ~ ~ , ~ m o ~ . ~ s y s t e n ~ of p : ~ P
Volume 33A, number 3
PHYSICS LETTERS
t i c l e s - gas m o l e c u l e s , e l e c t r o n s or phonons whose d i s t r i b u t i o n function f ( k , t ) is sufficiently c l o s e to an e q u i l i b r i u m d i s t r i b u t i o n F ( k , T) f o r the Boltzmann equation to take i ts l i n e a r f o r m ~/~ t = M~
(3)
w h e r e f = F - (OF/OE) ~ (k,t) and M is the c o r r e sponding i n t e g r a l c o l l i s i o n o p e r a t o r . Then using the s t a n d a r d definitions of entropy [e.g. 1], it is found that if only leading t e r m s in ~ a r e r e t a i n e d
dS/dt = F ( ~
~ dp/~t)
(4)
w h e r e F is a p o s i t i v e constant and (0,~) = f (OF/~E) O(k)~V ( k ) d k . With this definition of s c a l a r product it is known that M is both H e r m i t i a n and p o s i t i v e s e m i - d e f i n i t e ; s e e Chapman and Cowling [2] and Ziman [3]. By r e peated d i f f e r e n t i a t i o n with r e s p e c t to t i m e of eq. (4), t o g e t h e r with the u s e of eq. (3), r e s u l t (2) is obtained w h e r e S (n) = dnS/dtn. F o r an i s o l a t e d inhomogeneous s y s t e m of p a r t i c l e s the above p r o o f b r e a k s down as the B o l t z mann equation (3) is then s u p p l e m e n t e d by a t e r m v" g r a d $ and the o p e r a t o r v" g r a d is not H e r m i t i a n . C o n s i d e r , h o w e v e r , such a s y s t e m of p a r t i c l e s which is such that the t e m p e r a t u r e d e p a r t s in s o m e g e n e r a l fashion by an amount O(r,t) f r o m the m e a n t e m p e r a t u r e To. Then if 10! << To, it is shown by Landau and Lifshitz [4] that
OS_ 1 ~t
f K m n ~0 ~0 d~To 2 ~x m ~x n
w h e r e Krn n is the t h e r m a l conductivity t e n s o r , while 0 s a t i s f i e s the conduction equation
(5)
19 October 1970
~0
020
C ~t- = Krnn Oxm Oxn
(6)
w h e r e C is the s p e c i f i c heat. It may be shown f r o m eqs. (5) and (6) that dnS/dt n can be e x p r e s s e d in the f o r m
dnS/df n = s(n) + R(n) w h e r e S(n) is a volume i n t e g r a l throughout the s y s t e m involving sp at i al d e r i v a t i v e s of 0, and R (n) i s a surface i n t e g r a l o v e r the boundary of the s y s t e m . On making u s e of the s y m m e t r y and p o s i t i v e definite n a t u r e of Kmn, the r e s u l t (2) may be shown to be t r u e . A proof s i m i l a r to the above may a l s o be give~ f o r the e a s e of p a r t i c l e diffusion within an i s o lated s y s t e m . H e r e dS/dt = / 3 f I g r a d c ]2 dT w h e r e fi is a p o s i t i v e constant and the p a r t i c l e c o n c e n t r a t i o n c ( r , t ) s a t i s f i e s the equation
a c/O t = DV2c with D the diffusion coefficient. Full d e t a i l s of the work outlined h e r e will be published e l s e w h e r e .
References [1] L. D. Landau and E. M. Lifshitz, Statistical physics (Pergamon 1958). [2] S. Chapman and T. G. Cowling, The mathematical theory of non-uniform gases (Cambridge 1952). [3] J. M. Ziman. Electrons and phonons (Oxford 1960). [4] L. D. Landau and E. M. Lifshitz, Theory of elasticity (Pergamon 1959).
155