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The metric structure of irreversible thermodynamics
Z theor. BioL (1988) 132, 125-126
L E T T E R TO T H E E D I T O R
The Metric Structure of Irreversible Thermodynamics A p a p e r by Peusner (1986) came to our attention because of the notable similarity between his claims (of a Euclidean geometry of fluctuation-dissipation space) and what we established with mathematical rigour in our sequence of papers on the phenomenological calculus (see References), most of which were published in the
Journal of Theoretical Biology. Table 1 shows some of the direct correspondences between the formulation of our 1982 p a p e r (Richardson et al., 1982) and Peusner's 1986 paper. Note also that in Peusner there are inconsistencies and substantial confusion in the contra- and co-variant indices. TABLE
1
Richardson & Louie
Peusner
a; Fi R=aiFi L° = ai'aJ
x/g.i; ai 6 = aix/g,
ii
(sum over i)
go X i = g~ie~ (sum over j)
J' = L;JFi = lal 2 = J ' . F ,
2~s
= I¢12 = 2×.~,
Note: in the R i c h a r d s o n - L o u i e formulation, the Einstein s u m m a tion convention of s u m m i n g over repeated upper and lower indices is used. In Peusner's formulation, the variances are confused.
We would like to present a representative sample of the m a n y mathematical errors in Peusner's p a p e r - - s o m e minor and others quite substantial. (1) In his equation (13), X~ = - a A S / a a ; , but in equation (14), the minus sign conveniently disappears. This confusion in sign has serious consequences. Since entropy is maximal at equilibrium, AS is negative definite, not "larger than zero" as stated by Peusner at the top of p. 135 and in equation (23). (2) In order that (go) be a metric tensor consistent with Peusner's formulation, the basis would have to be
{~/g--~~i,, ~/~i2} and not {i~,i2}
as ( m i s l e a d i n g l y )
implied,
n o r {1~,1~2} as c l a i m e d
(after
equations (17) and (18)). (3) At the bottom of p. 133, Peusner should have become alerted to his inconsistent usage of variances: note a~ in his earlier equations and off in AS = X • at. (4) In Section 4, Peusner confuses c o m p o n e n t s with basis vectors. He appears to be performing a G r a m m - S c h m i d t process on his c o m p o n e n t s ~;. The 125 0022-5193/88/090125+02 $03.00/0
© 1988 Academic Press Limited
126
A. H. LOUIE AND 1. W. R I C H A R D S O N
confusion p r o b a b l y arose because H o f f m a n & K u n z e (1961) (Peusner's reference) used ai to d e n o t e vectors. It is not our p u r p o s e here to analyse Peusner's p a p e r in detail. We w o u l d simply like to point out that "the Euclidean g e o m e t r y o f fluctuation-dissipation s p a c e " was well established by us several years earlier. Since then, the theory o f the p h e n o m e n o l o g i c a l calculus o f description space (as we call the theory) has been extended both mathematically (to R i e m a n n i a n manifold, Hilbert tensor spaces . . . . ) and in terms o f physical and biological applications ( q u a n t u m mechanics, aging, relativity . . . . ). In particular, in a second 1982 p a p e r (Louie et al., 1982) we constructed a g e o m e t r y consistent with irreversible t h e r m o d y n a m i c s , where the constitutive parameters {a i} lead to the metric tensor {L~i}, from which the O n s a g e r s y m m e t r y relations are immediately established. The metric form that c o r r e s p o n d s to the Second Law is a direct c o n s e q u e n c e o f the construction o f a positive definite norm. As a contrast, Peusner assumes his (go) is a metric by "physical r e q u i r e m e n t " (top o f p. 135). We urge the readers to consult our sequence o f papers (loc. cit. and Louie, 1983; Louie & Richardson, 1983; R i c h a r d s o n & Louie, 1983, 1986; Louie & Richardson, 1986) for a solid f o u n d a t i o n o f the metric g e o m e t r y o f irreversible t h e r m o d y n a m i c s . In particular, it should b e c o m e clear that the p r o p e r (and unique) metric associated with Peusner's fluctuation " - A S " is given in our f o r m u l a t i o n by the response vector fl = a~J i with the scalar "fluxes" J i = ai/x/2 and g~ = a;.aj. It s h o u l d be noticed that the characterization o f a dissipative system by a response tensor, FI, rather than by a vector, R, allows the consideration o f vector (not merely scalar) forces and fluxes. Finally, it must be pointed out that equation (32) o f R i c h a r d s o n (1980) should read FI:FI = (a ~. ai)(F~.F~); the subsequent identification with the dissipation function is then trivial. 86 Dagmar Avenue,
A. H. LOUIE
Vanier, Ontario, K 1 L 5 T4, Canada
908 Middle Avenue, Apt. O, Menlo Park, C A 94025, U.S.A.
I. W. RICHARDSON
(Received 22 September 1987, and in revised form 25 January 1988)
REFERENCES HOFFMAN, K. & KUNZE, R. (1961). Linear Algebra. Englewood Cliffs, NJ: Prentice-Hall. LoutE, A. H. (1983). Bull. Math. Biol. 45, 1029. LOUIE, A. H. & RICHARDSON, 1. W. (1983). Math. Modelling 4, 555. I-ouxE, A. H. & RICHARDSON, 1. W. (1986). Math. Modelling 7, 227. LouIE, A. H., RICHARDSON, 1. W. & SWAMINATHAN,S. (1982). J. theor. Biol. 94, 77. PEUSNER, L. (1986). J. theor. Biol. 122, 125. RICHARDSON, I. W. (1980). J. theor. Biol. 85, 745. RrCHARDSON, I. W. & LOUIE, A. H. (1983)../. theor. Biol. 102, 199. R~C'HARDSON, I. W. & LOUIE, A. H. (1986). Math. Modelling 7, 211. RICHARDSON, |. W., LOUIE, A. H., & SWAMINATHAN,S. (1982). J. theor. Biol. 94, 61.
L E T T E R TO T H E E D I T O R
The Metric Structure of Irreversible Thermodynamics A p a p e r by Peusner (1986) came to our attention because of the notable similarity between his claims (of a Euclidean geometry of fluctuation-dissipation space) and what we established with mathematical rigour in our sequence of papers on the phenomenological calculus (see References), most of which were published in the
Journal of Theoretical Biology. Table 1 shows some of the direct correspondences between the formulation of our 1982 p a p e r (Richardson et al., 1982) and Peusner's 1986 paper. Note also that in Peusner there are inconsistencies and substantial confusion in the contra- and co-variant indices. TABLE
1
Richardson & Louie
Peusner
a; Fi R=aiFi L° = ai'aJ
x/g.i; ai 6 = aix/g,
ii
(sum over i)
go X i = g~ie~ (sum over j)
J' = L;JFi = lal 2 = J ' . F ,
2~s
= I¢12 = 2×.~,
Note: in the R i c h a r d s o n - L o u i e formulation, the Einstein s u m m a tion convention of s u m m i n g over repeated upper and lower indices is used. In Peusner's formulation, the variances are confused.
We would like to present a representative sample of the m a n y mathematical errors in Peusner's p a p e r - - s o m e minor and others quite substantial. (1) In his equation (13), X~ = - a A S / a a ; , but in equation (14), the minus sign conveniently disappears. This confusion in sign has serious consequences. Since entropy is maximal at equilibrium, AS is negative definite, not "larger than zero" as stated by Peusner at the top of p. 135 and in equation (23). (2) In order that (go) be a metric tensor consistent with Peusner's formulation, the basis would have to be
{~/g--~~i,, ~/~i2} and not {i~,i2}
as ( m i s l e a d i n g l y )
implied,
n o r {1~,1~2} as c l a i m e d
(after
equations (17) and (18)). (3) At the bottom of p. 133, Peusner should have become alerted to his inconsistent usage of variances: note a~ in his earlier equations and off in AS = X • at. (4) In Section 4, Peusner confuses c o m p o n e n t s with basis vectors. He appears to be performing a G r a m m - S c h m i d t process on his c o m p o n e n t s ~;. The 125 0022-5193/88/090125+02 $03.00/0
© 1988 Academic Press Limited
126
A. H. LOUIE AND 1. W. R I C H A R D S O N
confusion p r o b a b l y arose because H o f f m a n & K u n z e (1961) (Peusner's reference) used ai to d e n o t e vectors. It is not our p u r p o s e here to analyse Peusner's p a p e r in detail. We w o u l d simply like to point out that "the Euclidean g e o m e t r y o f fluctuation-dissipation s p a c e " was well established by us several years earlier. Since then, the theory o f the p h e n o m e n o l o g i c a l calculus o f description space (as we call the theory) has been extended both mathematically (to R i e m a n n i a n manifold, Hilbert tensor spaces . . . . ) and in terms o f physical and biological applications ( q u a n t u m mechanics, aging, relativity . . . . ). In particular, in a second 1982 p a p e r (Louie et al., 1982) we constructed a g e o m e t r y consistent with irreversible t h e r m o d y n a m i c s , where the constitutive parameters {a i} lead to the metric tensor {L~i}, from which the O n s a g e r s y m m e t r y relations are immediately established. The metric form that c o r r e s p o n d s to the Second Law is a direct c o n s e q u e n c e o f the construction o f a positive definite norm. As a contrast, Peusner assumes his (go) is a metric by "physical r e q u i r e m e n t " (top o f p. 135). We urge the readers to consult our sequence o f papers (loc. cit. and Louie, 1983; Louie & Richardson, 1983; R i c h a r d s o n & Louie, 1983, 1986; Louie & Richardson, 1986) for a solid f o u n d a t i o n o f the metric g e o m e t r y o f irreversible t h e r m o d y n a m i c s . In particular, it should b e c o m e clear that the p r o p e r (and unique) metric associated with Peusner's fluctuation " - A S " is given in our f o r m u l a t i o n by the response vector fl = a~J i with the scalar "fluxes" J i = ai/x/2 and g~ = a;.aj. It s h o u l d be noticed that the characterization o f a dissipative system by a response tensor, FI, rather than by a vector, R, allows the consideration o f vector (not merely scalar) forces and fluxes. Finally, it must be pointed out that equation (32) o f R i c h a r d s o n (1980) should read FI:FI = (a ~. ai)(F~.F~); the subsequent identification with the dissipation function is then trivial. 86 Dagmar Avenue,
A. H. LOUIE
Vanier, Ontario, K 1 L 5 T4, Canada
908 Middle Avenue, Apt. O, Menlo Park, C A 94025, U.S.A.
I. W. RICHARDSON
(Received 22 September 1987, and in revised form 25 January 1988)
REFERENCES HOFFMAN, K. & KUNZE, R. (1961). Linear Algebra. Englewood Cliffs, NJ: Prentice-Hall. LoutE, A. H. (1983). Bull. Math. Biol. 45, 1029. LOUIE, A. H. & RICHARDSON, 1. W. (1983). Math. Modelling 4, 555. I-ouxE, A. H. & RICHARDSON, 1. W. (1986). Math. Modelling 7, 227. LouIE, A. H., RICHARDSON, 1. W. & SWAMINATHAN,S. (1982). J. theor. Biol. 94, 77. PEUSNER, L. (1986). J. theor. Biol. 122, 125. RICHARDSON, I. W. (1980). J. theor. Biol. 85, 745. RrCHARDSON, I. W. & LOUIE, A. H. (1983)../. theor. Biol. 102, 199. R~C'HARDSON, I. W. & LOUIE, A. H. (1986). Math. Modelling 7, 211. RICHARDSON, |. W., LOUIE, A. H., & SWAMINATHAN,S. (1982). J. theor. Biol. 94, 61.