ON ONSAGER'S PRINCIPLE OF MICROSCOPIC REVERSIBILITY *) · 2018-10-22 · b) Microscopie rèversibility. Suppose that the ai are even functions ofthe velocities.of the' individual

by H. B. 'G. CASIMIR 530.162

R 14.,

ON ONSAGER'S PRINCIPLE OF MICROSCOPICREVERSIBILITY *)

'-.Summary " "After a short synopsis of Onsagcr' s theory of reciprqcal relauons in

. irreversible processes, the theory is applied to a....number of simpleexamples, In the first place we consider the thermomolecular pressuredifference; this example will also offer an opportunity of discussingthe "quasi-thermostatic" methods. In the secondplace the conductionof heatis studied and it is shown that Onsagcr's' relation leads to~ Oi L[i1;] = 0 rather thanto L[ih]' = 0: Finally we discuss the' con- .duetion of electricity by first deriving a relation of symmetry for anarbitrary four-pole from whicha symmetry relation for thc conduc-tivity tensor is then easily deduced. .J

1. 'Lntrod uctioti. There exist a numbe~ ofrelations pertaining to irreversible processes---:-thesymmetry of the tensor-of heat conduction andrKe Iv in+s relations,

- between thermoelectric quantities are among the best known examples -, that appear to be universally valid although they cannot be proved bythermodynamics or by considerations on macroscopie symmetry. Yet when ,,''a theory of the 'irreversible process based on aparficular model is workedout, these relations are always confirmed. N. B 0h r 1)'was the first to demon-strate this clearly in the case of K el vin' s relations and he found that theirvalidity was ultimately owing to the circumstance that the .fundamentalequations governing the. motion of individual particles 'àre symmetric withrespect to past and future, or, mathematically speaking, that they areinvariant under a transformation t~(-t). More recently On sag er ê] has triedto show that a general class of reciprocal relations can be ,derived from thisprinciple of microscopie reversihility without having recourse to any partic-ular modeland he has' applied his theory to numerous examples, including ,;- those mentioned above, thus adding a new and very fundamental principleto macroscopie thermodynamics, But although we do' not doubt. the essen-tial truth of Onsager's ideas, his application of these ideas to particularcases is, in our opinion, not always entirely satisfactory, In this paper we shallendeavour to treat a few simple cases somewhat more explicitly, but forcompleteness sake we shall first give a short summary, of the generaltheory. In this way we shall also have an opportunity of discussing theessential assumptions involved' in Onsager's analysis **). .

# i • I '

'2:. Generol Theory" ", , __a) Theory of flltctu,atidns. 'L~t us consider an adiabatically insulated system~ I

*jThis paper was also published in R~v. Mod. Physics 17, 34,3, 194,5. ,**)The whole subject was extensively discussed in a colloquium organized during the

spring of 1944 by the "Nederlandsche Natuurkundige Verecniging"; the author, would like' to express his thanks to many of those present, especially to Prof.H. A. Kramers and Mr. B. D. H. Tellegen for v~luable di~cussions.

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186 H: B. C. CASIllIIR ,

and a number of variables Xi '(i = 1, ... , n) which in the equilibrium stateassume values xoi• We put "

Xi = Xoi• + ai. -

The ~ntropy in a state diffe~ing from the equilibriu~ state will be S '80'+- L1S, where L1S is of the for~.

L1S = - 1/2 ~ SiI, ai a",i,k '

(1) _

. where Sik is a positive definite form. The probability distribution for the,ÇJ.i is given by -,

ekdS d 1 d';W ( 1 ') d 1 d . a ." a

a , ... ,aB a ... aB = f f kAS d' 1 d B... ea ... a(2)

where 1Gis Boltzmánn's constant. Define

yi = :4 Sik a" ,k

(3)

then it is easily shown that-- - ' Iyi al = k iji.

(ij~- 0 whenever ï =t=- I; ij~= 1)According to the fundamental notions of statistical mei:hanic~ this averagemay be interpreted either as an average over a microcanonical ensembleof systems, or as a time average for one single system.

By. solving eq. (3) for ii: ' "

(4)

ai = :4Si! YI,I

'where SUis the reciprocal matrix to Sij, we find

ai ai = k Sij.

(5)

(6)

The' variables yl defined by'. (3) will in the following be designed asconjugate, variables. .'

, -b) Microscopie rèversibility. Suppose that the ai are even functions of thevelocities .of the' individual particles, so that they are invariant when t is"replaced by =t, The fact that the future behaviour of a systém havingspecified values of ai at a time t, on the average, is identical with its pastbehaviour, can be expressed by the equation

(7)

the suffi~es denoting that the val~es remain fixed,' so that th~' average istaken over -a section of the microcanonical ensemble corresponding to .these values. Multiplyiûg by' al (t) and taking :the average over all possiblevalues of al (t), .", a" (t) we find '

(8)

where the average .is now again over the total microcanonical ensemble andmayalso be interpreted as an ayerage over t., ' .: .'

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ON .ONSAGER'S PRINCIPLE OF IIUCROSCOP~C REVERSIBILITY· 187

. , 'Similarly; if a number of variables {JI< are odd functions of the, velocities,

we have . .(9) .{Jl (t + 'i) fI' (I) ..... fi" (;) = - {Jl (t.o...!... 'ij -fl' (I) •• ;'•• -fl:' (I)

and again we fin?---,---,--,-----,---,--{Jl (t) {Jlt (t + 7:) = {J}' (t) {Jlt (t - -c). . (~O)

Let us now combine a number of even quantities ai (i = 1, ... , 11) anda number of odd quantities {J}' (It = 1, ... , 11). Then

, al' (t + -C)a'·(I) •...• a" (I);, fI' (I). ''', r" (I) = ak. (t - -c)"' {Il a" (1);-,8' (I) -fiV (I) (11).

{J.u (t+ 'i)", (I) ..... a" (I); fI' (I). : ... fI" (s) = -{J,u (t-.)", (I) un (I);-fl' (I) -{>" (I)

from which it follows..(12)s . al (t) {J" (t + r) = _al (t) {J" (t--c).

As an illustration we may quote the brownian motion of a galvanometer:if we call the deflection a and the angular velocity {J these quantities willsatisfy eq. (12). . .If a magnetic field is present the equations (8), (10) and (12) are no

longer valid, but must be replaced by .

(8')al (t) ak (t + .,H) = al (t) uk (t -'., -H),and so on ..~) Regression of fluctuations. Let us transform the macroscopie equations,which are of the form . .

. (13)

into -the formái = .2pik YI'

k '.'"

We assume that the same equations also describe the average behaviour ,of fluctuations in the following sense: there exists a time interval+rj suchthat for • > -Cl' but 'i ~ T (where T is the time in which, according to ourequations, a.disturbance. of equilibrium is appreciably reduced) " . ,

I (14) .

.' . ai (t + -c),,1 (I)..... all (I) _ai (t) = + • .2pik yk (t).'. k

(15)

This assumption calls for some comment. In the first place it is not, Iegiti-. mate to write ái instead of ~ai (t + .)- ai (t) ~/. for it is a consequence ofmicroscopie reversibility that the mean value of this derivative is zero. Thetime interval 'il is required to' establish a state of steady flow and the con-dition that .1be small as compared with the time in which the deviation·from equilibrium is appreciably reduced imposes evidently some sort of limi-tation on the mechanics of the system. It should be borne in mind, however,,that many applications of the macroscopie .equations are based on essen-.·tially the same condition. Further it is by no means evident that a set ofequations, applying originally to deviations large compared with. fluctua- " ..tions, can also be applied to the average behaviour of these fluctuations~themselves. Of course, the fact, that the macroscopie equations are linearpartly justifies an extrapolation to very small deviations, but in principle .one may imagine a pseudo-linearity holding only at reasonably large ampli- .

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188 H. n. G. CASIMIR •

tudes. The acceptance of eq. (15) is really a new hypothesis and althoughthe same hypothesis is made in the theory of brownian motion, we do notthink .:that it ,can rigorously be proved without referring in some way oranother to kinetic theory. The' author feels satisfied, however, that it willhord in all cases that. can be treated by means of the usual method basedon the equation of Maxwell and Boltzmann. I

. Once eq. (15) is accepted, the analysis is very simple. Multiplying bya' (t) and taking the average we find

al (t) }ai (t + .)- ai (t)~ = •pil. kIn the same way

'ai (t) ~a! (t + .)- al (t)~ _: ~ pli. Tc

But according to eq. (8) the ~eft-hand sides of these equations are equal;·hence

pli = pil. . (16)

This is Onsager's fundamental relation. The same relation holds if themacroscopie equations are expressed in terms of variables of the" {J-type":

. pl.1I = p!t!.. '. (17)If both even and od'd"q~antities enter into the equations the situation isslightly more complicated. In this case the entropy, being an even function,is a sum of two terms .

'Ll S . -_1/2 ~ :E s; ai·ak + :E SI.,. (JI- {J" {

and also the conjugate quantities fall into two groups, that are.even andodd in time: . '.'

Y· - 'Ç' S'k a" y - ",,'S {Ju'" - ~ .,. J. - "': J.,lt '.

The macroscopie equations ar~

ái. :Epi'" y". + :E pi}, YI.m J.

fj;. = :E pl.m ym + :E pl -,11y.1Im /'

and .a similar procedure as before leads to:pil =v".pi.,. = pilI.,-r" = pl.i.

Ifa magne~ic field is present, these equations have to be replaced bypil(H) , p~i(~H) (16')

. and so on.

3. Thermomolecular pressure difference; relation. to pseudo-thermosuuic, methods

". As a first example, we shall discuss a case wh~re the, application of thegeneral. theory is perfectly simple and straightforward. We consider avessel containing an ideal gas and divided into two equal compartments,~y a wall with a small hole. Let n1 and n2 be the number of grammolecules

.: III the two compartments and '1'1: ,T2 the. re.spective temperaturès.

AS = _.Cv + R a 2 _ 1 2' 2Cv 'LJ . Tt 1 TtT2. Cva2 + nT ala2•

The conjugate quantItIes are'

. . CL'+ R Cv . (R . Cv )Yl = 2 al _ 2 -T a2 = 2, - on2 _ T oT2 ,• Tt Tt . Tt2 Cv 2 'oT9

Y2=._ 7,T al +-- a2 = 2 T2-'" nT2 CvLet us now write down the equations for the transport through the hole inthe wall:' . - .

s : "A _ç: -Ó. B oT2un2 = un2+ , T2'

. '. oT20P2 = Q OTt2+ W T2 '

, I

ON ONSAGER'~ PRINCIPLE OF MICROSCOPIC REVERSIBILITY 189

In equilibriumnl __:__n2= n; Tl = T2 ' T,

The fluctuations of'nu~bers and teniperatures are subject to the conditions

, anI + on2 = 0; 0 (ni Tl + Tt2 T2) == . .0.. .

The second condition, which, exprcsses the fact that the total energy is. constant, leads to

n (oTI + oT2) + on2 (oT2 _ oTt) = o.. . , .

Starting from the well-kn?wn expression

S = n Cv log T +.R log V _ n R log n

a simple calculation yields

~v (oT2P .: !!: (on2)2,'. n

We shall now introduce a new set of variables: -. I

LIS -.

al = on2 '\'

a2 = o'U2 = (on2' T+ oT'2,'n) Cv.Then

The first equation expresses that a flow of gas may be caused both by a 'difference of concentration and by a difference of temperature (since weassume that a linear approximation is valid, we have èJTI= -èJT2 and \T2 -"Tl = 2èJT2); the second equation states that energy is transported bythe moving molecules, but also by conduction of heat, ' 'Transforming to conjugate variables: _. ~

/'

. _!An ,,+' ! (AnTC. + B),al _ 2 R YI ' , 2, R, , Y2 ,

" -=-! AnQ + ! (ATtTCvQ + BQ + ')a2 - 2 R YI 2 ,R , W Y2' ", "

_.'

190 . • ou," B. G. CASBlIR

We can now.apply Onsager's general equation:

: AnQ = AnTCv+ BR R '

or. B Q'::__TCvA = R n.

Suppose that a difference of temperature is artificially maintained, butthat we wait until ,Jü = O. Then we have

" . B ~T~n=---, A T2

Or

nQ/T- Cv. ~T.

R T~n

Ons age r' s theory gives a generalrelation between the difference of coneen-tration caused by a difference of temperature and the energy carried bymoving molecules. Of course the value of Q can only he derived from a. kinetic theory: If the diameter of, the hole is small compared with thefree path, a ·simple calculation gives:

1Q = .(Cv+ "2 R) T,

leading to./

or.P = Const. -v T ,

. whichisKnudsen'swell-knownformula.For·alarge orifice on the other'hand .

Q- (Cv+R) Tand it follows·

PI = P2''. Our' example affords also an interesting illustration of the quasi-thermo-\ static procedure used by several authors. Let us. write our equations inthe form . . '.

äl = all Yl + a12 Y2 ,. ä2= a21Yl + a22 Y2 ..

'Consider firs~ a stationary state with ä1= 0, then·ai2 .

Yl = - ~Y2'. all

. Nextwe carry out' a virtual displacement ~al-(inour case: we transport a.~ number of molecules) and we assume that the quantity ~a2; which is dis-

-placed along with ~al'~sthe same as if Y2 were .zero, so that -,

a21 ~~a2 = - uai,all. •

which means here that ~U -.Q ~n. It; is now assumed that the change of. entropy ~ue to the virtual displacement vanishes:

/ ,

The addition of an antisymmetrie tensor to Lik has no physical consequencesas long as eq. (18) is fulfilled, and it is therefore not to be expected ,thatOnsager's theory correctly applied willlead to the result L(il'l =t(Lik-Lki)=0. To apply Onsager's theory we must begin with a discussion oftemperatUre fluctuations. Let us consider a solid of total volume V anddivide it into a' number of cells Vs. Be T, the temperature in each respective cell, To being the equilibrium temper~ture and C the specific heatper unit volume, then the change of entropy willbe given by,

Th Th ~, JC" 1 f ,1 f·LlS=~Vs TdT=~V'To Cd'{'-To2..EVs (T-To)CdT+ ... =

I·

..;ON ONSAGER'S PRINCIPLE OF lIUCROSCOPTC REVJ!:RSIBILITY 191 :

al2 = a21•

Evidently this way of arriving at the relation ~f symmetry cannot be justified- at all by thermodynamic theory. Since, however, all applications of quasi'

thermostatic reasoning are based on essentially the same equations, it cannow be unde~stood why this procedure has usually Ied to correct results.

4. Conduction ·of heat in crystalsThe' equations for the conduction of heat in crystals can be written in '

the form Wi = ...E Lik Ok T;I<

here <h, denotes the derivative with respect to XI< and ~e write xl' X2, Xafor the cartesian coordinates x, y, z. NO"W this system is certainly not of theform (14), for in a three-dimensional case w, is not thè time derivativeof a thermodynamic variable. As a matter of fact w, is not even uniquelydefined, for only its divergence has a direct macroscopical physical "meaning. This means that we may add to Lil, a quantity Pik, as long as

..E ai (pil, Ok T) = ~,~ai (pik) a" T +PiT, èkOkT~ = 0;il" .I ik ".

This equation is satisfied for an arbitrary distribution of temperature, if

andpik __:_-:- phi...EOipik = O.i

..(18)

To To To

where we have put T.- To = Lt T'; , The variable 'conjugate to zlT, isthus found to be (CfTo2) Vs,Lt T. and .eq. (4) gives '

,. , - To2

Vs Lt t: Lt r.= k C a.,Let us sum th~s equation over a number of cells ~i, then

...E VSi zl TSi LI'f'r = 0, when Si"* T for all SiSi " . .-

, ...E VSi LtTSi LIT, = kT.,2/C, when Si . r for one Si.Si

.. ' )

\192 . H. B. G. CASlmR

Passing to the limit ofvery small cells .

. fjfLlT (x, y, z) LlT (x', y', z') dx dy dz ~ ~~~o2j~'

the upper or lower result holding according as x', y', z' is outside or insidethe range of integration. This can also be written in the symbolical form

:LlT (x; y, z) zl T (x', y', z') , (TJTo2jC) 6 (x--x') ó (y-y') (3 (z-z').'Ve introduce the Fourier components of LlT:

(1' 3/ rr' .LlT (x) = 2n)' 2JJj ei (kxl a (k) dk] dk2 dk3

with

a (k) = (~)3;2 rjT e-i(kXl LlT (x) dx dy.dz,2n JJ ~ ,

and, LI T being real .a (k) = a (--;-k)* .

. It is easily seen that a (k) and á (k)* CjTo2 are conjugate variables satis-fying the equation

a (k) a (k')* = (kTo2jC) r5\(kl-kl') (3 (l'2-k2') 15(k3 -7'a').~ext we determine the rate ~f change of a (k):

(1' 3/ fI" .Ct (k) . _).'2 I e-i(k x] zl T (x) dx dy dz =2n u u .

':' = (2~F2jJJe-i(kxl. ~ !.':. àn (Lllm àn; Ll T) cl;\:dy dz.,

. / . ·We assume that Lilf is a-function of x, y, z which vanishes at infinity. Bypartial integration we. find: '. \' . .

à .(k) = (2~rj2~jfJ [~ :: ~kll'kmLllm + ikn(àrnLmll) qe-i(kXj 'LlTclxclydz

Introducing the Fourier integral for LlT

\' à' (kJ = (2~r!·2~JjJ'fJJ s (-ku k",Lnm-ikn (à",LIl;n») ~i(!k:-kj.X)dx cly dz.

, (2~r/2a (k') clkl' dk2i dka'-

= (2~r/2~fff !m~-k,.kmLnm(k-k')-(km'-k:").k,,Lnm(k-k:) {a(k') clkI'dk2' clk3',.~ . . .

where Lnm (k) is the Fourier component of Lnm, or, finally

... á. (k) = (2~r/2~JfJ-- !min ~m'u: (k-k') a (k') clki d~2dka,' ..We can now use On~ager's relation of symmetry bearinginmind that theconjugate variable to Q, (k) is proportional to a (k)* or a.(-k): .

~ kn' i; Ln": (k+ k') =~km' kn u; '(k+ k'),nm nm

or~ km kn' Llnm] (k + k') = O. 'nll'

This mu~t hold for arbitrary km, whence .'~ K" L[nm] (K) = 0 ~"Now .we have

L[nm] (K)·=. (2~r/2JJJ e-i ,~~X) ~[nm] (x) dx dy dz'.,-

ON ONSAGER'S PRINCIPLE OF MICROSCOPIC REVERSIBILITY 193,

Putting k + k" K so that k", K _,.k, we find: ,". , 'I km (Kn __: lin) L[nm) (K) , ~ km K; Linm] (K) = 0 .

mn' mn'

and

Therefore we find ~timately .2: Oin L[nm) = 0 .m'

I '.

This equation can be obtained 'somewhat more directly though less rigor-ously by using the LlTinstead of their Fourier components and.making aliberal use of £5-functions. We have: -

" ','. . .} ,(eLl T = ~ 0" I u; OmZIT \'= .nm - ' - )

.fff'- ~ 0,,' ~~/(X-X') £5(y-y') ~(z-z')~ u; (x') Om'LI T '(x') dx' dy' &'=nm '. . ..

= !IJ 2, Om' ~inm(x') 0,,' [£5(x-x'). ~ (y-y') £5(;-z')]~ LI T (x') dx' dy' dz',nm

where'om' denotes differentiation with respect to Xm'., This is of the form

LIT (x) = JJJ K(x, x') LIT (x') dx' dy' dz', , '

and Onsager's principle gives:

~ Om' [Ln;n (x') On' ~'£5(xl~Xl') £5(X2 - x2') £5(x3 - x3') ~] = ',. nm _ '0 . ". '.

= ~ e, [Ln';' (x) 0,,' ~0 (x1-:-,X1') ~ (X2 - x2') ~ (xa - x3') ~]: .nm

In order to draw à conclusion from this equation we multiply by an arbi-trary function f (x') and integrate. We obtain: '~ ,

X 0" (Ln:'.Omf) = 2: Om (L"m On!)nm rim

ornm nm

and, f being arbitrary,~ 0" L)nm] = O."

Our result is, that an antisymmetrie component of La, has no observablephysical consequences whatever and it is permissible to püt .it equal tozero. Itmust be zero if.we ~gree that Lik is zero in vacuum and that LUl fora given .suhstance does not 'depend on the shape of the sample. Though itwould of course be very foolish not to accept this convention it is interestingto note that Ûnsager's relation by itself does not compel us to do so.. '.

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194 II. B. G. CASIlIUR

An analysis shnilar to that .given above applies to all cases where' a cur-rent, the divergence of which has' a physical meaning, is given in terms of

. the gradient of à thermodynamic variable. ,',

5.' Conduction of electricity, "

In agreement ~th the remárk at the end of the preceding section, theconduction of electricity in solids, might he- treated on similar lines as the

, , conduction ofheat. There is, however, an additional complication inasmuchas the divergence of the current gives the rate of change of the chargedensity, whereas the current is détermined by the gradient of the electricpotential which is in general a «:.omplicated function of the charge. There-fore we prefer to use a different method. We consider a sample of a con-ducting solid with four leads attached at the points A; B, C, D. If we res-trict ourselves to situations where the current 11entering at:A. is equalto the current going out at B, and the current 12, entering at C is equal tothe current going, out at D, and if V1 and V2 be the poten'tial differences ...between A and B and between C and D., then there "ill hold a set ofequations:

VI = 411 11+ A12 12,

V2 = A 21 11+ A22 12,

or11= all VI + a12 V2,

12= a2~ VI + a22 V2•

I , Let us ,now suppose that a large capacitor Cl is .connected to A and Band similarly a capacitor C2 to C and I? Then 11 and 12 are the deri-vatives -of the charges' on these capacitors and moreover the potentialsare; apart from a 'constant factor, the conjugate variables to these charges.Therefore On~sage1:"'s'relatio,n gives: -' ','

a12. = a21 or A12 . .A21•'I'his equation of symmetry will hold for an arbitrary fourpole as long as nomagnetic field is p~esent .. In that case:

a12 (H) = a21 (-H).. I

It is interesting to note that our 'equations may also be written as

11-;-.hll VI + b12 J2,V2 == b2I VI ~~ b22 12,

N0'Y if C and D are not connected to a capacitor but to a self-inductor,the magnetic flux in this inductor, which is, proportional to 12, may beregarded as a dynamical variable and V2 is its derivative with respect totime. So we can again write down an equation of symmetry. But now 12is an' odd quantity', VI an even quantity, hence ' .

b12= -b21•

in accordance with what is found by direct" transformation. Thè author isindebted to Mr.T'e.Il eg en for drawing his attention tothis case.which madehim aware of the neoessity-of making a distinction between even and odd, quantities in the formulation of the general theory. ' ., We sháll now investigate the symmetry relations for the tensor of electric

'resistance: 'Ei = ~ Rik ik',

•• .l>

r

ON ONSAGER'S PRINCIPLE OF MICROSCOPIC REVERSIBILITY 195

where Ei are the components of the 'electric field strength and iT, the com-ponents of the current density. We shall arrive at the following theorem:if for a certain substance a r.elation aik = aki holds for all possible fourpoles, .then we have also Ril;=Ru, To establish this theorem we consider a thinrectangular plate in the xy-plane, with i~s corners at (0, 0), (0, 1), (b, 0),(b, 1) and with leads C and D attached to the sides parallel to the z-axisat points with arbitrary but equal x-coordinate xc, the leads A .and Bbeing similarly attached to the sides parallel to the y-axis at points withy-coordinat~ YA' We pave': -- '.' -

Vi = An 11.+ A12 12

V2 = A2l 11 + A22 12- /

. with

but. on the other handb •

VI = J (Rn i1+ R12 i2) Y=YA dx.o

Now

and hence: . b

VI = R12 12 + Rn Ui1 dx ~Y-:'YA •o

Specialising to' the case 11= 0' ". . . b

A12 = R12 + (Rl~iI2) ) J i1dx k=o.. 0 Y=YA

In the same way w~ can calculate V2 and our symmetry relation becom~sb .' I'

~Ji1dxh.=o ~f i2 dyh.=oRI2 + Rn Y=YA = R

21+~R22 o. ' '"t=xc.

12 11

This equation holds for arbitrary values. of Xc and yA' Let us then' supposethat we write it down for n.m. a:rangements correspondin_g to ' ,

Xc = 0,bln, 2bln, .:.. (n-l) bInyA = 0, 11m, 211m, .... (m:_l) 1lm

and let us sum the result:I • _ b

R12 +Rn 4~J i~dx II, =0. ,/ m,n 0 Y=.rA

• 'I

112mn = R21 +R22 2,' ~ J t« clXII.=om,n 0 X=XC

On the left-hand side we shall first carry out the summation over m whichleaves the distribution of current unchanged, since this depends only on theposition of the points C and D through which 12 enters and goes out.' Butin the Iimirof very, high m, the expression (l/m) 4(m) i1 is proportional. to J i1dy and therefore it is zero when 11= 0. In the same way we find that,the sum over n at the right-hand side vanishes, which establishes the desired, .' . Iresult. . '" " '.It is of some intérest to note that the symmetry of one specialfourpole

is not sufficient for deriving the symmetry of the conductivity tensor: it is'only the symmetry of all fourpoles which enables us to obtain this result:

196 H. n. G. CASIltIIR

Further one might be .astonished that we arrive here'at once at the sym-metry relation and not at the relation

f~?i R[ik] = O. . \The reason is that our analysis tacitly assumes that R;k is identical for the . ~. whole series of m.n fourpoles and that,Rik = 0 in vacuum. If we had chosento treat also the conduction of heat by using "thermal fourpoles" we shouldalso at once have found L[ik] = O. '. .If a magnetic field is present we have.

Ril, (H) = Rh. (-H) , '- -a relation which was first given by Meixner 3).This result mayalso be expressed by the statement that the symmetric

tensor R(ik) is an even function of H, the antisymmetrie tensor. R[ik] anodd function of H. Introducing the axial vector R (where RI = R[23] andso on) we can write .

, Ei = ~R(ik)'ik + [I X R],k J

The total electsic field strength is given by a symmetric resistance tensorwhich is an even function of H and a Hall vector which is an odd functionof H,The situationwas discussed by Gel~ritsen and the author 4) in con-nection with the conductivity of bismuth in a magnetic field 'and we referto their note for further details. .

Eindhoven, February 1945

REFERENCES

1) N. Bo·hr. Studier over Metall~rnes Elektronteori, Copenhagen 1913.2) L. Onsager. Phys. Rev. 37, 405, 1931. 38, 2265, 1931.3) J. Meixne;. Ann. d. Phys. 40, 165, 1941.4) H. B. G. Cnaim ir; arid A. N. Gerritsen, Physica 8, U07, 1941.

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