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Statistical theory on the thermodynamic evolution-criterion of Glansdorff and Prigogine
Volume 24A, number 7
PHYSICS LETTERS
the g e n e r a l i z e d t r a n s i t i o n m a t r i c e s off the e n e r g y s h e l l (tXv(~): k ¢ #, v ~ ~) as well as those on the energy shell (txu(#): e i t h e r of (X, u) = #). In a si m i l a r fashion (fz D) (C) can be d e s c r i b e d in t e r m s of the g e n e r a l i z e d t r a n s i t i o n m a t r i c e s . The p r e s e n t f o r m u l a t i o n is valid whether or not the s y s t e m is s u b j e c t e d to a constant magnetic field. If the m o m e n t u m r e p r e s e n t a t i o n is used for the case of B = 0 eqs. (1), (2), (4) and (5) can be g r e a t l y simplified.
STATISTICAL
27 March 1967
Details will be published e l s e w h e r e .
References 1. c . c . Chen and S. Fujita, J. Phys. Chem. Solids 28 (1967). 2. R. Kubo, J. Phys. Soc. Japan 12 (1957) 570. 3. E. Merzbacher, Quantum mechanics (John Wiley and Sons, 1961) p.492.
T H E O R Y ON T H E T H E R M O D Y N A M I C EVOLUTION-CRITERION OF GLANSDORFF AND PRIGOGINE F. S C H L ( ) G L
Technische Hochschule Aachen Received 17 February 1967
The thermodynamic evolution-criterion of Glansdorff and Prigogine is derived from a statistical theory. The starting-point is the fact that during the evolution a quantity cannot increase which gives a measure for the knowledge of the observer.
Let Pi be the p r o b a b i l i t y - d i s t r i b u t i o n over the m i c r o s t a t e s i of a p h y s i c a l s y s t e m which may be in i n t e r a c t i o n and in m a t e r i a l exchange with its e n v i r o n m e n t . In p r e v i o u s p a p e r s [1] it was shown that K(p,P°) =~. Pi (lnPi - l n p °) $
(1)
is the entropy produced in the i n t e r i o r of the s y s tem by the change of the state Pi to the e q u i l i b r i u m - s t a t e pO. This quantity is positive and v a n i s h e s only if all Pi a r e equal to pO. The quantity K m e a s u r e s the excess-knowledg'e contained in the d i s t r i b u t i o n Pi if po c o r r e s p o n d s to the a p r i o r i expectations based on o b s e r v a t i o n s of the e n v i r o n m e n t only. It should be d i s t i n g u i s h e d from the difference of entropy in e i t h e r state and was i n troduced by R4nyi [2] as " I n f o r m a t i o n s g e w i n n " (gain of information). The knowledge of the expectation values M v of a set of o b s e r v a b l e s defines the d i s t r i b u t i o n Pi = exp (~ - ~ M i ~
(2)
by the r e q u i r e m e n t of m a x i m a l e n t r o p y [3]. (Greek i n d i c e s o c c u r i n g twice a r e to be s u m m e d . ) M v a r e t h e r m a l e x t e n s i v e p a r a m e t e r s or f l u c t u a tion q u a n t i t i e s as well. Between both t h e r e is no
p r i n c i p a l difference. If both d i s t r i b u t i o n s in (1) a r e of the type (2) the e n t r o p y - p r o d u c t i o n p e r unit time b e c o m e s
P = - R ( p , p o) = / v x ~
(3)
with "fluxes" and "forces"
iv =~v
x v = ~ v - x° •
(4)
Now let Pi be a s t a t i o n a r y state of the s y s t e m with c e r t a i n t i m e - i n d e p e n d e n t boundary conditions. Let f u r t h e r P i + 6 P i b e a neighbouring state which d u r i n g the t e m p o r a l evolution u n d e r fixed boundary conditions goes o v e r into Pi. During this p r o c e s s the knowledge of the o b s e r v e r over the e n v i r o n ment r e m a i n s unchanged. The e x c e s s - k n o w l e d g e over the s y s t e m itself which is m e a s u r e d by the gain of i n f o r m a t i o n cannot i n c r e a s e . T h e r e f o r e during the p r o c e s s one gets
R(p + 8p, p ) --< 0 .
(5)
F o r states of the type (2) this gives the evolutionc r i t e r i o n of Glansdorff and P r i g o g i n e [4]:
lU6Xu = 5xP > / 0 .
(6)
In the s p e c i a l case where the s y s t e m a p p r o x i m a t e l y allows a d e s c r i p t i o n by localized t h e r m a l 393
Volume 24A, number 7
PHYSICS LETTERS
q u a n t i t i e s e n t r o p y - p r o d u c t i o n can be w r i t t e n in the f o r m (7)
P = f d r Iv(r) Xv(r) .
Local v e c t o r i a l f l o w - d e n s i t i e s ] v a r e connected with the r a t e f l u x e s I v by
entropy of a m i x t u r e of i s o t o p e s depends on wethe r the o b s e r v e r can distinguish the components or not. Yet in the v a r i a t i o n a l c r i t e r i o n [6] the s u r f a c e i n t e g r a l in (9) in f a c t has no influence b e c a u s e it is fixed by the boundary conditions and (6) g i v es
5xP = f
(8)
I v = 1VIv = co v - v j v
27 March 1967
d r ( ~ V S X v + j V v s x v) >1 0 .
(I0)
and one can w r i t e P = - f drv(jVXv)
+ fdr(wVXu+]VVXv)
.
(9)
In this f o r m the f i r s t i n t e g r a l is a p u r e s u r f a c e contribution of e n t r o p y - f l o w . It may look confusing that in the d e s c r i p t i o n by ] u i n s t e a d by I v the quantity P which was i n t r o d u c e d as i n t e r n a l e n t r o p y - p r o d u c t i o n contains now e n t r o p y - f l o w through the s u r f a c e . Yet this fact shows that in a c o n s i s t e n t s t a t i s t i c a l t h e o r y e n t r o p y - p r o d u c tion as well as entropy depend on the knowledge of the o b s e r v e r . If we go o v e r f r o m the d e s c r i p tion w h e r e only the I v a r e used to a d e s c r i p t i o n by j v we have to i n t r o d u c e local c o r r e l a t i o n . That m e a n s a change of d i s t r i b u t i o n s and knowledge. The situation is analogous to Gibbs' paradox. The
ON
THE
POSSIBILITY OF IN A T O M I C
M. Ya. AMUSIA, V. V. A F R O S I M O V ,
The author w i s h e s to thank Dr. A. Stahl and Dr. R. Bausch f o r d i s c u s s i o n s .
References
1. F. Schlbgl, Z. Physik 191 (1966) 81; Z. Physik 198 (1967) 559. 2. A.R~nyi, Warscheinlichkeitsrechnung, Berlin (VEB Deutscher Ve;lag der Wissenschaften,1966). 3. E. T. Jaynes, Phys. Rev. 106 (1957) 620. 4. P. Glansdorff and I. Prigogine, Physica 30 (1964) 351; I. Prigogine, Non-equilibrium thermodynamics, variational techniques and stability (The University of Chicago Press, 1966).
OBSERVING ELECTRON
Yu. S. G O R D E E V ,
MANY-BODY SHELLS N.A. C H E R E P K O V
EFFECTS
and S. I.S H E F T E L
Ioffe P h y s i c o - T e c h n i c a l Institute, USSR A c a d e m y o f Sciences, L e n i n g r a d
Received 23 February 1967
We propose the analysis of inelastic scattering of keV-electrons by atoms as a function of the energy transferred in collision to reveal the role of many-body effects in an atom.
R e c e n t p r o g r e s s in m a n y - b o d y t h e o r y and in e x p e r i m e n t s on a t o m i c c o l l i s i o n s r a i s e the q u e s tion of the c o n t r i b u t i o n f r o m m a n y - b o d y e f f e c t s in a t m i c e l e c t r o n s h e l l s . A t h e o r e t i c a l study m e e t s with c o n s i d e r a b l e c a l c u l a t i o n a l d i f f i c u l t i e s , while e x p e r i m e n t a l s tu d i e s m e e t the difficult r e q u i r e m e n t of an unambiguous i n t e r p r e t a t i o n of the r e s u l t s . R e c e n t l y [1] m o d e l s have b e e n p r o p o s e d , involving both s i n g l e - and m a n y - e l e c t r o n e x c i t a tion, to explain the r e c e n t l y o b s e r v e d d i s c r e t e e n e r g y l o s s e s in s i n g l e c o l l i s i o n s of two a t o m s . H o w e v e r , the e x p e r i m e n t a l data a r e not sufficient to choose between t h e s e e x c i t a t i o n s and an a n a l y 394
p~
Fig. 1. s i s of o t h e r e x p e r i m e n t s w h e r e m an y - b o d y ef f e c t s might show up [e. g. 2] does not allow us to d e c i d e e i t h e r . We need thus e x p e r i m e n t s which would allow us to i n t e r p r e t the data unambiguous. ly. Such an e x p e r i m e n t might be a study of the
PHYSICS LETTERS
the g e n e r a l i z e d t r a n s i t i o n m a t r i c e s off the e n e r g y s h e l l (tXv(~): k ¢ #, v ~ ~) as well as those on the energy shell (txu(#): e i t h e r of (X, u) = #). In a si m i l a r fashion (fz D) (C) can be d e s c r i b e d in t e r m s of the g e n e r a l i z e d t r a n s i t i o n m a t r i c e s . The p r e s e n t f o r m u l a t i o n is valid whether or not the s y s t e m is s u b j e c t e d to a constant magnetic field. If the m o m e n t u m r e p r e s e n t a t i o n is used for the case of B = 0 eqs. (1), (2), (4) and (5) can be g r e a t l y simplified.
STATISTICAL
27 March 1967
Details will be published e l s e w h e r e .
References 1. c . c . Chen and S. Fujita, J. Phys. Chem. Solids 28 (1967). 2. R. Kubo, J. Phys. Soc. Japan 12 (1957) 570. 3. E. Merzbacher, Quantum mechanics (John Wiley and Sons, 1961) p.492.
T H E O R Y ON T H E T H E R M O D Y N A M I C EVOLUTION-CRITERION OF GLANSDORFF AND PRIGOGINE F. S C H L ( ) G L
Technische Hochschule Aachen Received 17 February 1967
The thermodynamic evolution-criterion of Glansdorff and Prigogine is derived from a statistical theory. The starting-point is the fact that during the evolution a quantity cannot increase which gives a measure for the knowledge of the observer.
Let Pi be the p r o b a b i l i t y - d i s t r i b u t i o n over the m i c r o s t a t e s i of a p h y s i c a l s y s t e m which may be in i n t e r a c t i o n and in m a t e r i a l exchange with its e n v i r o n m e n t . In p r e v i o u s p a p e r s [1] it was shown that K(p,P°) =~. Pi (lnPi - l n p °) $
(1)
is the entropy produced in the i n t e r i o r of the s y s tem by the change of the state Pi to the e q u i l i b r i u m - s t a t e pO. This quantity is positive and v a n i s h e s only if all Pi a r e equal to pO. The quantity K m e a s u r e s the excess-knowledg'e contained in the d i s t r i b u t i o n Pi if po c o r r e s p o n d s to the a p r i o r i expectations based on o b s e r v a t i o n s of the e n v i r o n m e n t only. It should be d i s t i n g u i s h e d from the difference of entropy in e i t h e r state and was i n troduced by R4nyi [2] as " I n f o r m a t i o n s g e w i n n " (gain of information). The knowledge of the expectation values M v of a set of o b s e r v a b l e s defines the d i s t r i b u t i o n Pi = exp (~ - ~ M i ~
(2)
by the r e q u i r e m e n t of m a x i m a l e n t r o p y [3]. (Greek i n d i c e s o c c u r i n g twice a r e to be s u m m e d . ) M v a r e t h e r m a l e x t e n s i v e p a r a m e t e r s or f l u c t u a tion q u a n t i t i e s as well. Between both t h e r e is no
p r i n c i p a l difference. If both d i s t r i b u t i o n s in (1) a r e of the type (2) the e n t r o p y - p r o d u c t i o n p e r unit time b e c o m e s
P = - R ( p , p o) = / v x ~
(3)
with "fluxes" and "forces"
iv =~v
x v = ~ v - x° •
(4)
Now let Pi be a s t a t i o n a r y state of the s y s t e m with c e r t a i n t i m e - i n d e p e n d e n t boundary conditions. Let f u r t h e r P i + 6 P i b e a neighbouring state which d u r i n g the t e m p o r a l evolution u n d e r fixed boundary conditions goes o v e r into Pi. During this p r o c e s s the knowledge of the o b s e r v e r over the e n v i r o n ment r e m a i n s unchanged. The e x c e s s - k n o w l e d g e over the s y s t e m itself which is m e a s u r e d by the gain of i n f o r m a t i o n cannot i n c r e a s e . T h e r e f o r e during the p r o c e s s one gets
R(p + 8p, p ) --< 0 .
(5)
F o r states of the type (2) this gives the evolutionc r i t e r i o n of Glansdorff and P r i g o g i n e [4]:
lU6Xu = 5xP > / 0 .
(6)
In the s p e c i a l case where the s y s t e m a p p r o x i m a t e l y allows a d e s c r i p t i o n by localized t h e r m a l 393
Volume 24A, number 7
PHYSICS LETTERS
q u a n t i t i e s e n t r o p y - p r o d u c t i o n can be w r i t t e n in the f o r m (7)
P = f d r Iv(r) Xv(r) .
Local v e c t o r i a l f l o w - d e n s i t i e s ] v a r e connected with the r a t e f l u x e s I v by
entropy of a m i x t u r e of i s o t o p e s depends on wethe r the o b s e r v e r can distinguish the components or not. Yet in the v a r i a t i o n a l c r i t e r i o n [6] the s u r f a c e i n t e g r a l in (9) in f a c t has no influence b e c a u s e it is fixed by the boundary conditions and (6) g i v es
5xP = f
(8)
I v = 1VIv = co v - v j v
27 March 1967
d r ( ~ V S X v + j V v s x v) >1 0 .
(I0)
and one can w r i t e P = - f drv(jVXv)
+ fdr(wVXu+]VVXv)
.
(9)
In this f o r m the f i r s t i n t e g r a l is a p u r e s u r f a c e contribution of e n t r o p y - f l o w . It may look confusing that in the d e s c r i p t i o n by ] u i n s t e a d by I v the quantity P which was i n t r o d u c e d as i n t e r n a l e n t r o p y - p r o d u c t i o n contains now e n t r o p y - f l o w through the s u r f a c e . Yet this fact shows that in a c o n s i s t e n t s t a t i s t i c a l t h e o r y e n t r o p y - p r o d u c tion as well as entropy depend on the knowledge of the o b s e r v e r . If we go o v e r f r o m the d e s c r i p tion w h e r e only the I v a r e used to a d e s c r i p t i o n by j v we have to i n t r o d u c e local c o r r e l a t i o n . That m e a n s a change of d i s t r i b u t i o n s and knowledge. The situation is analogous to Gibbs' paradox. The
ON
THE
POSSIBILITY OF IN A T O M I C
M. Ya. AMUSIA, V. V. A F R O S I M O V ,
The author w i s h e s to thank Dr. A. Stahl and Dr. R. Bausch f o r d i s c u s s i o n s .
References
1. F. Schlbgl, Z. Physik 191 (1966) 81; Z. Physik 198 (1967) 559. 2. A.R~nyi, Warscheinlichkeitsrechnung, Berlin (VEB Deutscher Ve;lag der Wissenschaften,1966). 3. E. T. Jaynes, Phys. Rev. 106 (1957) 620. 4. P. Glansdorff and I. Prigogine, Physica 30 (1964) 351; I. Prigogine, Non-equilibrium thermodynamics, variational techniques and stability (The University of Chicago Press, 1966).
OBSERVING ELECTRON
Yu. S. G O R D E E V ,
MANY-BODY SHELLS N.A. C H E R E P K O V
EFFECTS
and S. I.S H E F T E L
Ioffe P h y s i c o - T e c h n i c a l Institute, USSR A c a d e m y o f Sciences, L e n i n g r a d
Received 23 February 1967
We propose the analysis of inelastic scattering of keV-electrons by atoms as a function of the energy transferred in collision to reveal the role of many-body effects in an atom.
R e c e n t p r o g r e s s in m a n y - b o d y t h e o r y and in e x p e r i m e n t s on a t o m i c c o l l i s i o n s r a i s e the q u e s tion of the c o n t r i b u t i o n f r o m m a n y - b o d y e f f e c t s in a t m i c e l e c t r o n s h e l l s . A t h e o r e t i c a l study m e e t s with c o n s i d e r a b l e c a l c u l a t i o n a l d i f f i c u l t i e s , while e x p e r i m e n t a l s tu d i e s m e e t the difficult r e q u i r e m e n t of an unambiguous i n t e r p r e t a t i o n of the r e s u l t s . R e c e n t l y [1] m o d e l s have b e e n p r o p o s e d , involving both s i n g l e - and m a n y - e l e c t r o n e x c i t a tion, to explain the r e c e n t l y o b s e r v e d d i s c r e t e e n e r g y l o s s e s in s i n g l e c o l l i s i o n s of two a t o m s . H o w e v e r , the e x p e r i m e n t a l data a r e not sufficient to choose between t h e s e e x c i t a t i o n s and an a n a l y 394
p~
Fig. 1. s i s of o t h e r e x p e r i m e n t s w h e r e m an y - b o d y ef f e c t s might show up [e. g. 2] does not allow us to d e c i d e e i t h e r . We need thus e x p e r i m e n t s which would allow us to i n t e r p r e t the data unambiguous. ly. Such an e x p e r i m e n t might be a study of the