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Macroscopic and microscopic reversibility
?+fAcR0sc0Pxc
AND *3acRo.sccPIc
Rzlvmsam
n?e%uaPatiotl~patioIl thcom is one of the fundamrntai LhcCWcmsin lirsar thamodynvnics of proccsra. A derivation oa the basis of sta&ical m&a&s has baa given by C&n and Welton [I]; it was ntbscqucntly simpli%cd by Kubo [IO& A thamodynamic derivation of this theorem has been presented by Callen and Grcme (m. [SD. This theorem rdates the comlation funaions of the %uctuations of cxrensiw varia& to the admittance matrix which uxmats the intensive vaziabIcs with their tha-modynamical& coojuga!e extcssi~e variabk. If, in addition, microscopic revcrsib12ity is invoked, then the comhtioa matrix has certain symmetfy pro@cs which, through the %uctuafioadissipation theorrm, arc rc%cctcd in corxsponding symmetry propcrtia of the admittaae lIm&k
properdcs of the adrniaanoc matrix are also obtained ift description variabies is used, if the OnsagerZasimir resipmcd rdatioas (PJ, (tl3. [3D are applied and if, &tally, the internal variabics are eliminated (11). The importance of the CaUen-Granc theorem, supplemented by microscopic rcvtibilty, lits in the faa that it can be fonnrJa& witboot considering cxplidtly internal variables as parametar of state As an example we mention the admittan= matrix of an daxr4 RCf-nerwork which is symmetric incspcc&ivc of the detaiIs of the structure of the nerworlr and of tbt charges and the currents through the various elements of the nczwo& Thuz tins theintcxsting question whethcrthesymme~properiia oftheadmittancz matrix occurring in the relation between extensive and intensive time varying variabkr oLa be derived from a principle which is puniy macroscopic and in which - hopcfuIly some kind of invariance a* time reversal, which should be of a macroscopic charact~, Nap a pan. llL first aswer to this problem has been given by Day (41 and by Gunin 163. who r+ These
symmetry
with intcmd
38
1. Mm
thmrskts to the constitutive equationsof linear viscoei&c materials wioh therma1 aspects being included by Gutin. We s&ail give in the scqncl a gencrakatkm of their msuits to other contitutive equations of the afmefkt type and we hail also make full ust of the powehli passivity property of suchcojstitutive equationsIn partiallar ‘h following points wiil be emphashd and treated: I. Appticatioas are possible amoag o&en in Linear elastic, tknnal ektricJ magnetic rehtioa with various moss eEciS Z The occurring constitutive equations are linear passive transformations; therefore the rich mathematical theory of linear passive transfbrmations (or Syncms) is applicable. 3. In addition to the time reversed procemes introduced by Day we shaJ3 also consider cyck proctsxt and the time reversed cyclic procmses. rnbothcasesthistimerevcSalis of a macroscopic nature and wiU therd~re be termed macrowpic time rcvusaL s&ml
4. Tjrme reversal win be properly defined for both types of procases and the oo~urzc~tcc of even as well as of odd parameters of state and of a st-pcrposcd constant megnetic 8dd wiil k taken into muot in *&isd&&ion. The distinction betuxen even and odd paxameters of state is impaative if one’wishcs to introduce the concept of macroscopic revem= ibilityatall. 5. Al~ough physically of minor imporran~ complex iinut pzJrive transformarioas will be inciuded. 7&y zqairc an intercsthg rukbition of time wusal. 6. It win k shoan that the postufate that a certain functional is unchanged under properly d&nod time reversal leads to the same symmetry prop&es of the admiw matrix of the amstitutive quations as the &xnutiow5issipation thccmm plus microscopic time reversal revcrsibiiity. Thus the conncctioa between microscopic and -pit mauoscopic reversibility are obtained from microwiU be cstablirhui and two kinds of scupic revcrsiity.
7. The results obtained for Iinear constitutive quations can easiiy be generaikd to Iiear structures built up from materials with linear ecnstitutive equations; thus such strucnms aiso exhibit macroscopic revcrsiiility as do *&e constitutive quatiom, 8. A dose relatioa between macroscopic revexsiiiiity and a certain raiprocity tbctnun canberzstaMished. Whrsk rmacmcopic revmibl2ity an be designated as an actual priaciplc remah stiU an open question. AU the results obtained in this paper refer to the l&r domah For noa-bar constitutivc equations we have so far no check by a simpie examptc, much less a microscopic theory, which would prove macroscopic revSWiry. lkrefon scopic revemibihty should be regarded within the mentioned Iimitations, but not yet as a geneal prkipik
A rather general set of Sinear constitutivc equations will be given which describes various kinds of rhxation phenomena: mechanical rclaxazioa, temperature daxatiou,
MAcItosa3Pfc
and the mpame
tedor
AND MICROSCOPlC RmERslBILITY
or me peeror
having sixtan compoaam ea& The fundamental inqualiiy of tbcnnod~mics (p3b [Ia of dectro-magnaic fields 1151.has now the simpk form both vedors
gcncntized to tbc prcsma
where the dot between f atrd g indicates the scalar product. The Iincar coostitutive equations express the fate vector g(r) as a fuactionaJ of &e
J. MEIfCNEiZ
40
islinear, imuiant~against any timeuandation and which sati+ the pppirirywnditiun (23) for all f(r) from a rusaabiy chomz class of fuoeriom; sncb axastitlnive CquaLiolu f(f) - g(t) arc called IineLv ~assin! ?n&wTar~~ or ~~;rorsirrjpsrrmx stimulus vector f(r) which
fia
sectionwe shall givethe d&tion v== (I% [91, VD and we shall
and some important prom of linear also introduce tile two w-Tunction?t we parsive adopt a more general point of view by admitting complex-vaiued stimu5 f(z) md xcIa this
spa== ,rfrr DBIXIIOX I. roc&sC,
A vector valued function f(t) with n componcats /I(t) is mid to bJors (m=O.1,2..,) if its compoaass
l.ared&eloa--cOst
.
A functional transformation f(z) - g(r) is said to em
D IInrrp
passire~anX 1, it assigns to cvuy f(f) e C, a vcaor-valued function g(r) E C, of the w with I! ampoaalts k(t); 2 the sapsporition principle holds; 3. the u2nsformatioa is k&ant under aXl t&c shifts; 4. for ail stimuli in C, ;b: passivity propeq Re _i, ,gI ~O)s&?
3 0
golds. The star * denotes tb conjuga? compla TEEOSW 1.
Emy
g(r) - Af(r)+@lp)+B
linear passk
MCXI
WI rtal3 value
qszem f(z) -+ g(r) has a repmwuatim
j f(r)&+ -. +
i P(t-r)f(s)&+ -9
where A, B are non-nega!ibc d&tie
Remitian
P(t) -
j [P(o)-P(t-s)]~(s)f&, -.
m!ricrs,
7 Pm d@(m) -0
u ir a HcrmiiliM mafAx, md
(331
with a mrri.r e(u). For epery 9ecror x, the /imction ji’P(r)x is a positive dq%&ejanction wifh mm raiuerao and the fktaion f g4(~)s is a real, boundc& non-drcreusing fun&w in - m < w e co which is conrinuol~ta: w = 0 and has fke the value zero. l 2% matrix @(a)
MACROSCOPXC AND MICXZOSCOPXC REVERSIBhzry
41
1.
2 3.
A rtid~ f(tj in ct r
I’(p) -pA+ht+~B+R(p), W9
(33) (3.10)
- M(O) + (17hARl -
d ~~~trix R(p) will be called the cbe propcrtks R(1) - R(r)*,
residual part
limpR(p)
(3.1 q
Rep > 0.
_i,~d@(t&
* 0,
of the odmit:ace
Iim~J?cp)
msttix Y(p); it hy
- 0.
(3.12)
1. MElxNER
42 The
ammu
matrixY(p) isa positive maeix vector I holomorphic
the individual tanu
in (34,
which acans that the function ji* Y(p)x is for tvciy iu Rep > 0 aud has that a non-negative real pan Also
PA, io, $ B, R(p) arc positive maties.
The kfr manher in the inequaIity (3.1) wii be dcsiguatcd as the W-/Mczio~Z because in t.hc PM sumemin reMon it it, apart from a factor I/P, the work trausf’ by the surf&z sznzzm to unit volume of the rzfcrcna state during the intcrml -QP c I ( 7. Ftoa TImrun I the foPowing result m be acrivad
We ag’x~e that f(-r) is also in &; then in (4.1) and (42) we can choose 7 - CDand obtain, now omitting the parameter r, the FV,-famajoa?i W, (f(t)) :-
Re j f(r)*. g(z)& -Q
j&VBF(O?+f j (1 +mz)$(m)’ &P(m)F(o),
(43
r-‘-f(r)&.
(4.4)
-0
WkC
F(m)
-
j
-a
No mtricdcms on the linear passive systems must he made, in order tit the W,-fune tional k d&nai For cxampic, it i! not ncceary that for the admitted s&nuG f(r) the response g(r) approaches a hitc ihit as I 4 a~, It is dikat with the IV,-functioml we are now going to define and which pert&s to qdk pA cyclicprocas is d&cd & such a way that the szimulus and the respcmse are periodic. Butas~utw~alwayrtostan.atsometimc~tmayalsostartatr---aPbutiD~~ a manner &at (3.19 holds); it can be periodic at hcst after some time r,, which we may take to - 0. We the&on choose a stimulus in c. which mm not be spaSed < 0 but which, in 0 c I < CI),is f(r)= C(qP, k
m+o.
' '. .
in - co e I (4.5)
MAcRoscoP
c
AND
MIcRoscoPxc
REvERsIB1m
The set of numbers k may also be inbite; but then appropriate conditions on the as& totic behavior of the ol as k - f ca must te imposed in order to acre approptiatt CCQ. vcrgence properties. But it s&m to consider the tite cast With such a cboiee of the stimulus the response b in gcncral not p&o& for 2 )r 0. But for some subclass of the IineZr passive SyStemSit IIWy become aqmptotic3Ily pdodc sf-+ 0o.Wbca we speak ofaqck proce wc always mean the asymptotic pehdic process fkom t to t+22+ with f -, a, ic after t5t transients have died 0uL Sufadent conditions for the cxktena: of a qdic prw with a szhuSu~ of the propary (45) &we call them ColtditionSs-arc ([la: 1. #(a) is absohatcly corJtinuous; 2 Q(m) exists waywhere, is continuog and in every 5nite interval of W W&tion; is. as a function of e, for all real 0 absolutdy iotcgabft 3. [@‘(a+&-ol(o-e)],” ia-c(eCL, when r>Q Under thex conditions the tit Y(iw) exists for all rcaJ w f 0 ;md the rnponse to the stimolIxs (49 is g(r) - B[ j f(S)&- z (Lkw)“&j + & Y(hJ)~c-+o(r), -m where 0(l) 4 0 as t + Q). This is a simple gawrakation of Thxem 52 in (la Atcnnwithk-Ointhespmin(4~hadtokomined,bec;lure~g(r)mi~ contain a tam which is aqmptotically proportionaI to t uniat B I 0.
(4.6)
We intmdwc now the Fyrfonctionalby #+%I-
W, (f(r)} : = Iim Rc W
I W’ - P(s) &. (4.7) I. In the special case of the purr strewstrain relation W, {f(r)) is apart from a factor l/T+ the work mnsf&cd by the surfaoe stresses to unit VOIUWof the refcmxce statr during oue puiod 2+0 afla the trans&ts have died ou& With f(t) from (4.5) and g( 2) from (4.6) the W&xnai~d assumesthevalue (4.g) QearIy, the vdua of f(t) in -co
(43)
The l&t result can be derived from IlSj, Theorem 43.
1. MpxNEIl
44
Atfintwcmnnaplainwhataruo~dbyatimercverscdnimnlurLcetfCr) be a stimulus in cs 50 tbzt f( - t) is also in E2. For tkc time revc& stimafur we cmnot jaas take the proczs f( -r). We snn~ also bear in mind that in physical systems there are vahbics and panmetus which arc odd under time remsaL & a rcprcscatative for odd ;Llld this is the most important casc-+a const2nt salpcrpo#l pamnetm we cIl_ maptic inductioa The proper d:finitioo of zicmcopic rmmi-ty is as foJ.Ious: D-ON. The the termed sttnuhcr to a pmccss f(t) taking place in the Iinar pusive _qstan with constant sqeqmed ~~~gncticinduction B+ is the stinmiau d(-t)’ taking pb in the iiucar passive a%lzna with the opposite value -B+ of the mag%tic inductioa The matrix z is a diagosai mwix with diagoaal dcmcW,s + 1 or - 1 asordhg as the rcspstive c.cmpooctn off is CISZor odd with rcspcct to time revrml; f(t) aad f(- f) aro both to be &osco in cz. Tllc transition to the uxljngatc ampIcx value in the time rot t& format f&o0 that o&y such simple rsUIt#saE rcvcn&stimuiusisrrca=ssary obtained (3 is we9 known that th time reversal opcr;itor in wave zrechks also changes
the wave function into its coajugatc compkx vztru). It is important to note that the mpow td the stimubxs f(t) is not simply rcfattd to the response to the time revencd srimotus Ir(-t)-. From (43) vx obraia W, (f(t), B-jW, (d(-t)‘, -B’) = ;~(O)‘[&B+)- rB(-B+)c]F(O)+ + $ _i, (1 +G)f(w)’ We 2sc ilow in a pcsiZio0 to cnunti
~Q(uJ, B+)- t&m,
tbc WphanwL
-B+)r]F(o).
(5.X)
a
Wl-T~~ Lrt borS f(t) and f(-t) be in 2;. 37xen the WI-$mctiona& for the process f(t) and its time rercxsedproccs “i( - t)* me &h&f nicK two funuioMt arc eq4 fir LG pomirtld f(t) if md o@ Q B(B+) -
&-B+)L,
#d)(m,B+) = r&w,
-B+)i
(-m
<.o < ca).
(52)
Proo/: The *-~-assertion foIIons hmdiatiy from (5.1). For the proof of the *oniy iT ass&on we chmsc a szimdus f(r) so that-51~ (4.4)-
MACROSCOPlC AND M.KROSCOP!C REVERSW
with
2tt
aad f(4)
2r’bi~
axmalltvectbt&Tba
arc dcariyin
ea.we d&e
t%(a) =
now the fuction x(a) by
(1+w~* u~@(w,B+)- r&u,
-B+)e]a
(SJ)
withthcnormati2atonX(O)-Oandwith x(o) - f[X(w+O)+X(ueO)),
(S-6)
Siacc @(a~, fB+) is continuous at pt = 0.90 is X(W). The egunliry of the W&Duiol& in the left lzlcmber of (5.1) yiads with (43)
ur X(0+) - X(0) - 0 ._ j &cr, 0 -0 0 for~points~,toCcontinuity?nd~~~~for~cr~inO((p,~aJ~virayoC (S&L We d&te now a new ftanczioo F(W) as the F(-co) in (53). and obtain in the same - Oia-as
We restrict BOWour attention to linear passive systems whicharcsubjat to the con&as d&n& in scctioo 3. tions S, both for B+ and -B+ and turn to the cycfic prThe first problem is now how to define the mverscd c/de. Given 2 szimulus f(r) in cx which in 0 g I e ca has the special form (43 with some w f 0, we introduce the function f1(r) - f(2x/o-
f) 0 7
.4@-
(Odtsi2z/cm).
(6.1)
Hcretoforr: we have coasiderrd q&al definitions of time reven& szimuli which 14 to the W, and IV,-thcormrr BCCAUXthe passivity property (3.1) of Iinur passive systcxm
MACROSCOPIC
AID
MXCXOSCOPXCREVERSIBILITY
47
symm&ic in the stimulus and reqow components, it is ‘also psiiie to consi& g(r) as the stimdus and f(r) as the respond. If f(t) is time reversed according to one of the @en two dchitions, g(z) is in gcnemI not time reversed. Also, if g(r) is takai as the stimu!us and is time rcvcmcd according to one of the given two dhitions, then the rcspoav f(r) is in gamzI not time reversed. That is to say, if we postulate equality of the respahe HMmctioaaIs for this stimulus g(r) and its time reversed stimdy we may obtain additioaai properties of the admittance matrix Y@, B+). In both casts we obtain fomdy the same result (4.5). but in the second GUC with @(w. B+) being the spscval matxix of . the impdance ma&ix Y(o,B+)-‘. we can proceed CVUImore gendy by clloosillg some compotits of tbc vcdot f(r) aad the remaining componcnt~ of the vector g(r) as the stimuhas, and the other v ua~~ of the vccton f(r) aad g(r) as the raponsc More precisely, we introda diagonal projatioa matrix 1 with A2 = dandd&casstimuhsandrcqoase
is
f(r; 1) - JJO)+ Cl- L)g(r),
(7.1)
LO; It) - (1 -MO+
c12)
telr).
lk &armation f( t; 1) - *( 2; A) exists for aU f(r; I) in T2 if (7.X), (72) and (33) can be solved for g(r, A) as a linear functionaI of f( r, A). The neczssaq and safE&nt condition for that is easily obtained if a eouatiom are written in Laph * dct[l+(l-%)qplrpll + 0. is Iincas and tims imaria! and suisda
This zmls5ocmation
Re j fir, l)‘.g(t, -*
A)& = Re 5 fv)*gtOdr --
transforms and WI
the passivity comiitioo 3r 0
(an rral XI.
(7.9
For the proof (7-I) and (72) are introduced and use is made of P - A; tbe rest foIlowr from (3.1). For stimuii f(r, 1) in c2 which vanish ia z < 0 oat obtains, in analogy to (3-Q, in terms of Lapfase aa&orms ie; wte& the inrrnirtmce
nutt*
2) - 0;
&,
1). (
04
I(p; A) is gjve~ by
I@; A) -
[X- f+ mp)J[l+(l~
i)Y(p)j-’
[A+ Y(p)(f-
1+ Y(jY)i).
I)]-‘[r-
(7.8
For A- 1 it rcdum to the admittana mat& for f = 0 to the ixtpcdana matrix. In all diagonal projectors f it is a positive mafrk The reprcscntatioa (3.9) to (3.12) of a positive ma&x also applies to the immittaaa m;luix I&; A, B+). We ootc the relations .
I@; 1. B’) - pA(1, B+)+iu(rt, B*)+ +(A.
B+)+.R(p: A, B’),
(7.7)
1. MPXNER
48
R(1; 1, P)
-
J@;A, B+)*;
SipR(y; ta
E&R@:1, B+)
1, B+) : 0,
I
0.
cI%;
FromtbcrrsuloiaScaionsSanddwc:obtainimmediatcty~foUowingi 1. If rhr W,-#tmctioMkfor allpwmssef f(z, ~)andttipropalytLntrcnrsrdpwacncs are qui-it is un&mmd that f( ft. 1) E c2 and that L is aqy diagonal pwjeuor+an (7.3),, (7-U) Md (7.13) hold n&h rhese PIajcaorr 2 r the W&m&mats fat all &ic processes f(r, A) and zh& pwHy rime rzIcIsd cycfic pwcesses me eqd-it is mdcrttood thar f(t, L) E c* ad rhur 1 is any &izgwd prajccroF-then (7.12) und (7.13) hp’d m% r&e ptojccrort. The problem is LIOWto invcstigata to what extcot (7.10) can be proved if V.12) ad (7.13) are assumed to hold for all diagonal pmjazton i. No use will be made of (52). It will aIs suf6ct to exploit tba consequences of ~.u) for the special projaton & - da- d-d,
(no summation
oyct a)
cf.14
which have only one zero on the diagonal WC assume that the diagonal efemenk I?of Y are invertible, irz other words that none of r&n is identically zero. If Y, %w zro in Rep > 0, then the Pti and the Y, (i I 1,2, . . . . n) would be constank satis@ing
?dKlCOSCOPlc
AND bfICJWScOPfC
?Y+Pm = 0. Wecho&nowinparh!ara
rqmseatations
rt’rl@; B+)-‘,
4&;
4 S+) -
I,&;
1, B+) - -Y&;
L(P;
4 B+) -
$15)
B+) ~n(p; B+)“,
m6)
K&P; B+)Y,,@;B+)-‘, -
-%a
(7.11)
h*
03)
(3‘9) and (7.7) will be writtea RP; B*) - $x@;
0;
49
- 1 and obtain from (7.6) G~!I k = 2,3; ..‘, m
d9d4 The
REVERSIBEJlY
1, B+) - $k(p;
.
B+) + R(p; B+),
cts)
1, B+)+R(p;
clro)
1, B+)
~I(P;l,B+)~Y(P;B+~etcTbeX@;%,B+)arepoiynomiasinpofdcgra~Z We take now the qwion (7.!s) in tile form fX&; 1
A B+)+%,(p;
and sttbtracz f’tu it the equation for B+. Denoting
1, B’)
II+X1,@; 1 t7w B+)+&(P;
B+)
- I
wIti& ii obtained from (721) w&n -B+ is s&&&d
XO@P;1, B+)- LY(&,(~X&
f, -B+)
- JXo@; 5 B’),
c122)
we obtain by virtue of (7.X2) and afta eliminating the R(p; 1, ET) by Cr.!91 asd using mr).
+
$dX,,(p;d, B+)dX,,@;
B+)Y,,(p;
B*)+c9X*,@; B-) - 0.
033)
?fdx,,@, B+) I 0, then Y,,(p;B+)ismninB*. Letmoow assomt that dX,, (P, B+) et 0; thea 6X”,(p, A, P) f 0 and Yxr@, B*) is a solution of (723) and therefoe has avayspahlform. Without proof WT note: Unakr fhc aaZitiona1 assumption tkt rk A(1, B+), a(l, B+). B(t, B+) are ho~omu@k in B+ tn a connected domein which with B+ ah c0ntah.s -B+, r/u fMaion Y,, @p;B+) ir glen in B+ with tk only exception that Yll(p, B+) * AlIp, or tit1 tw B,Jp wz’tk -41 3 4 4,
3 4 al1 =md.
Now we turn to the equations (7.16) and (7:17’) and assume hat Yll(p, B+) is &a in B+. We write (7.17) witb -BC in piace of B’, multiply this equation by rll(L) t&A) and subtract it from (7.16). Due to (720), (7.12) and (7.18) WCobtain with k - 2.3, ..l, I
dX,,(p; I, B+)Y,,(p; B+) - 4X,&;
B+).
(72s)
1. MEtXNE%
50
So we have either dX,,@; i, B+) = 0
and SX,,@; 8‘) - 0 or Y&p; B*) is tk ratio of two poiynomials in p of degree < 2 Only the SccoDdcase needs furtim mnsidaation. Tk equatioll LobuinedinaJimilarwayas~24).It~~tosctthatthezcrosandpobofY~’,,(p;B+) anocclxonlyon*iInaginarypaxk
Either aU d-ccxfBckws vatsisttor P%%(p,B+) takes auc of the fdowing
TEXBEM4.
Ginn an admittance mutrix Y(p; B+)
whirh is mui for
forrrrr
psi&e
p. hdb-
B’ ad -B’, anti has IW&god &nzaxr of tk forms (727). Zf the imnrittmrcl matrices I(p; I, B+) d&red /mm sub an a&nir~ctnce mopjlc by (7.6) exist /or al.t projeczors (7.14) and their residwl parts R(p; 1, B+) Ame the prom (‘?.U), then (7.10) and (7.~1) hofc. r Y(p; B+) is scalar, hoZomorphic in u domain W&&J cunnects B* wtd -B+, md has a residuaIpart which b an even fmction of B+. tkn I’@; B+) is M evenfmction of B+ U&S Y(p, B’) is prvpoMor& to p or lo pda or is an w morphic ia a &main
OoIutQnt,
which cmneets
..
It is assumed in tbc analysis of (7Z) that B+ 8 0, or thaw Ysr(pr B+) is not iadqa& at of B+. Otbmvisc (723) is triviqlly satisfied without any fi@cr consequence; kcsw of ax,,(p) = 0 and dX,,@, 1) = 0. The analysis of the dX,, and of the dXkI, rrrnainr valid and the first part of Theorem 4 remains tnte while the secondpast beama an aapty
statemen% The excepted admittance u3untcr-exampIcs. Che h
matrices nay aIso have the pmpcrq
(7.10). but thee arc
MACR~PIC
n
AND MIC%O5CDPlC REVEkStBILIW
reversibility when appkl to the stimuli f(i, A) with any 1 Bothtypcsof epic fnm (7.I4)--~ then speak of full woscopic revusibiiity-therefore lead to t& symmetry pro(7.10) except for a Smau &class of admktance ma* &Redprodtyaod
me
-* while w far m which kad from an quiliirium state to au quiliiriuat aatc and eycticphave been considered for ma& elements only, we now s&n wntemplate the colTMpooding situation in elastic uzatakl bodia and by dctitioa, unstmkd bomoWecbooxasarekuastatetheunnrrsud, gmeomstat+attnnperatracrofamatcrialbodyandpermitonly~d~ from this state which may, however, vaty with the position vector s and with time 1. Ektromagnetic fields are exdudcd Thar only the 5~3 ten compoaen6 in (23) aud (24)~~icvantWeintcgiatc(ZS)ovathc~~~u~ofthebodyaMiobtaio
Rx, 1) -
(t(x, 0. Q@. 01,
‘3~. r) - (b,
0. &,
0).
(82)
W&E x deaotcs a surfacz point of the body, T rT)/r-l,thef,&ilc~alx~ tioas and r,& = a,&, with t.bc arta demcot do; the do, are the compoamts of the ivc~ element vector, aad us(x, I) is the surfice stress tensor. The q[s. t) denotes the velocity Sdd of the surfpoints and 4,(x, t) is the ou~ard normal component of the beat ff~c vustor through the surface la the derivation of (8.1) from (23) use is made of the e Inwa
The summation amventioa is adopted. We expu that the koowlaige of the Sdd F(x. I), with the specifkation that the proa starts at r = -a~ from the qtibrium state 7 = 0, tl = 0, determines tbc motion and the heat flux in the bdy and on its surface In paxxicuiar, we see &a& due to (8.1). the relation F(x, t) -, G(x. r) coostituta a passive system with the surfaoc integral in (8.1) defining the scalar product of the fields F(x. t) and G(x, t). It is also linear if we rcstrkt ourscivcs to small departures from the original state, and time iovarian= is evident. if there is no other action oa the body. 73ercfore there exists a representation (Hackenbroth [7D which, when expressed in Laplace transforms (3.1). reads 6 (I, p) - j Y(x, x’; p) t(x’. p) do’.
(8.4)
s -
J. MUXNER
on x and i.
me admittane operator Y(x, it p) is a 4 x 4 matrix which also dcpds Itisa’positivegcntofw&ichmunsthp
Rcfdufdo’ &x)*Y(x:,
Y;p)&i> > 0
WI
for all vector fields 4%)wtich are absolutely squareintegrable over cbe body.Thisadmittanceo~torgcncraIirestheco~ofapositive~ From the theory of
is Id for positive p.
linear pvsivcsystems one dduao,
qtp)
=pA+$B+a+Rt,p).
condduing
sprf?a of
t,b
that Y(x,d;p)
(8-W
In~cumpi!:ancompoaentsofFandofOarrev~withscspeabtime~~ Mamoswpic memimty yidcb mx, i;p) Cmsidc+g
(84,
- R(i.
x:&S
R(p) * k(p).
ml)
(8.7), (S-II), (S-12), (8.13) and (8.14), UT obtain m,~;p)-d~,i)
- W*%3)+tiv% m-a
-
(8.18)
xP)+a.
Frh macroscopic revm%Zty yidds, hoi+zver, a(x, i) - 3 and a - 0 apar! from admittance amrices q(p) of one of the forms (727) and apart from admittmoz operatom macroscopic revcrsiiiby indodcs alI vandonnod stimuii p(l;i;p)ofsimiJarfonrnFulI =drapo= 3(x, r; L(x)) = I(x)F(x, r)+ [Z- Z(x:)lG(x. 2). (g-19) Gfx. r, 3x))
= II-Vx)W(x,
0+W)G(x,&
(839
with r(x) being a diagonal matrb who55 diagoaai ekn4mts A&x) m the valac 1 en thtpartDIofthebody~~aodnoootbtwmplemeotaypartDPofitfhc~ fonrredstimrrli~amixedboundayvalue~vtblcminthescnscthatachcompo~ i of F(x, r) is a givea fanaion of position and time on the mrfact domsinc L$ and tbc mqceive ampmcm of G(x, I) are given furaions of Htioa and time OLIthe wmp&mcntary pm @, wbiic the compooam of F(x, t) on fl and tbc wmpo~eoo of G(x, t) 00 LJ, (r’- 1,2,-) waszimtc t!KrcqwmsC lbe statement a(x, i) = 0 as a cmsqTJa~ pie lVvm%~ is made of full mby analogy to ot = 0, bat is. lacking a direct proof, given as a wajeurc Hmrva, a(x, 9) -Ocankdcrivedfromq(p)~ ij@)asenrraciataIia~~!. Weamsidcr aow two wmpkx stimuli P”(x, I) and F*‘(x, r), They prodncc response Gc*)(x, z) aud Gt*‘(x, r), aad a ~WCCSSinside the body aitb &r, I), #@(r. f), o(I)@. 2). $“(r, f), +(r. t) (r’ - 1.2 and r is tbe position v-or to points inside th body% By USCof the bafance oquatioag the apmsion L - f [W(x,
t)Fr”(X, t)-&“(X,
r)Tc*‘(x, z)] ch
(83)
caObCtaDdonwdinto
With the assumption of mail dcpanuxes from the original state I& hst inttgal caa be nglcctal and ,(” and @*’ cm be replaced by their values e* in the ori@iaji StatC, thus aegkring terms which are small of third order. We asstme now thaz Fc*‘(x, I) ad P*‘(x, t) WC two wmpkx stimuli 4th the same time depcndeoce @‘, Rep > 0. Due to the Iinevity of all LIB%under the precnt asrumptioa,
s
1. hEDU4ER
G(‘Vx. 1). p(‘)(r, tX r<“(r, t), r”“(r, ?), @(r, I) will have the same time dependen= bcCawcor+“-J+,(822)~IxoSto L =
$[y’)(r, f)cp(“(r, t)-
Bazausc of the assumed tbc ad (833) t = $d~$drr+(~‘,
y”(r,
f)cp<“(r, r)jg+ by.
and,
(823)
dcpendcncc we can use (8.4) and (8.5) to obtain from (82ij
I)[~(x,
x’;P)-
Y(x’, x;p~jF~(x,
2)
= S~p”‘(r,f)t~(P)-9~)1~(~)(~II,t)e+
N.
(824)
n¶is~&siIxun~tbc THEOREM5. I/ rk adnittw ma&ix of rk constitutive eqzwtia~ (8.10) it not of ON of zhr zprciol fomzs in (727). thenfun mucroscopicreversibiiily qpiieti to tkse aw sztii~eeqwtion.s of u materid elmem enwaim q(p) = i(p) and L = 0. Beaua tk nfmdi P*(x,z) (i - 1,Z) me arbitrary in ikir &p&cl on 5 it fwtkr /O&WS ihm fix,i;p)
-
Y(r’,%P)-
@u)
Gwuerseiy, if L - 0 for df SZirndi F(X.
f, then t) (i - 1,2) with time &pt&nce (825) iroldr and i/L - 0 for ail &bodies, howmr, smdl, thea 9(p) = $@) and thr cons25 Wi9e eqmZims of a materiaf element suotwy fUn ~WOSWpiC revs. The apation (821) with L 7 0 is cakd the reciproeirr ihoran. It takes thir mmc from the following property: Let a(x - x co? be a disttiiutioa over the surface of thebody
with d(x-r’0)
= 0 if xf P(s,
x@ and jb(x-x@) I) - d(x-x(~TP
do = I. Assume the’ spcdrl (i A 1.2;
stimuli
Rep > 0).
(SW
fbea &*‘(I(“, Rccipruity ekUrodynami~
r)(Q - &‘(ti”,
r)nt.
theorems of a related type have been givg previousiy ‘m ekstidry for materials with dispersion and losses (11% f1IQ.
9. The f%lnatioadbipation
theorem and miauscopic
(827) and in
rcvcrribill~
We now tntn back to the general stimulus vector f(r) in (2.3) and its response vator g(l) in (24). We multiply g(r) by eV*whcrr Y is M arbitrary volume and intrgratc from -Q up to time r. For small dcparturcs from the rcfcrcnce state we cm replace e(r) by e+ and obtain
If the stimulus f(t) vanishes at all tires, then in macroscopic terms uz have e(r) - 0. In microscopic terms we have, however, statistical fluctuations of the components of &(I) about their values in the reference state. According to the fluctuation-dissipation
MAcRoscoPIc
&f&-yare lekL&
tk#vwt
f(z)
4
g(r&
We d&c
AND
M1cR0scoP1C REvERsmnrrY
to the admittana the wrrU.ioa
mafrixY(p) of thc’p&Yc
.
55
linear tral&omIation
matrix
(93 whercg,,Q) is now the time depcndcncc of the statistkil fktuatioo of the variable g,, about the rtfdna state and ( > denotes au ensemble avuagr Thedactuation dissipation S(r)
-
c4rYl(OBYIO+m
-=-(12oD (93) zssxiatcdwit.bthelioearpoJsive Jystclnf(r)~~z)ret (3.9) aad (3.11). In the quantum mechanical formulatioa of the tluctuation tbeorcm tkrcis adigbxm0di6cati0a of (93),tbe oaumna of a scaiar factor (see [lb [IQ which b of no iaffrmc on our ES&S on the symmetry propdia of the residual part of tbc . odrmttaac= ma&i% It should also be noted that the ductuatio~ of the time iate@ of the heat 4oa q(s) may not be bounded, or that the intcgzd in (93) may be divcrgcnt. This izsmdmm compficatiozs which, however, m be 1~110ycd ‘I5 &x&sarion dissipation theorem as fonnuiatai in (93) has nothing to do with Ininoscopic m, 5&y. Rather, this is a separate and indepmdeat statanmt TO fmmrjarc microscopic revasibility we consider our material in a magnetic 5dd with the magartic induction 13. of -B+,thatbtoscry,~considaa~car~e~ withtbepnrnmctcrBwhicf,~~~thcvaluaB+or-B+w~rrfcftotwodiITarnt stats. ?beo rP quantities and functions that ctie linear payi= ryffcrn =Y dqald Gn w. Mcmswpk mwrsibiIiy is aprcsscci by Z!ICpzopaty
~whcre4i(~)isthespccBalnwix
AF(t, B+) - tR(r, of t&e am&rion
mat&
-B*)t
j-co
< t < co)
(PI, [211, @2D. By (93) this properry is refk%i
br(w, B’) = cb(w , -B+)r and from tbc representation
(-co
in the rr;iation
<(D 4 cc)
(3.11) of the residual adnakancz
Iz(p, B’) - rR(p, -B+)r
(9.4)
(9.3
matrix WCarrive at
(RIP ) a)-
(9.6)
lbpr the Pucruation-dissipation theorem comb&d with mi~pic m-9’ does not yield any syrnn?etry properties for A, u, B in (3.9) beyond those &a kloh which were that A, 8, B axz Hermitian mauias with A, B bein& morrova, non-I3QN.h deiinite and P hich hold for all linear passive systems, but which arc in genera! not of a type similar to (9.6). Anyhow, the result (9.6) is quivaknt to the consequence of ma=Pic reverw’bility (SZ?)~or (65) according to what type of macroscopic revcrsiiility is considered The Buctuatioodissipation theorem combined with macroscopic revenibility is e*haustcd with the result (K& or (6.4, but does not lead to full macroscopic mciu’biiity. Some further results QO be deduced for special cw uith additional argurncnt~.
JUEIXNER . .
56
Coos&r the special linear passive system f(1) - g(z) d&cd by physical point of view the system is said to have the prom ifforallf(~)e~~whichzrrcanstaatint,O,wilhVf .
by (2.3) and @A). Guided of approach to ctquihbriurn
- 0 ia I 3 0, the rcspciosc g(r)
approaches tcfo as 1 - 00 and if, moreover, the time integral of g(t) approaches a constant value as f 4 03 (but possibly except for the time integrat of the heat flow). Then one Qw prove that B - 0 which yie!ds (52), as a trivial consqucne Furthermore, one has P(t) - 0 as z -+ CDand that the principal minor of the matrix 4; which does not n&ate a ampormt of q or a component of Vf propaty Ia
WI2
quiliirium
or
to any of the components of f(2) or g(r), rcs~cclively, has the
.
(W.
the components of 5 g(r)&, with the cxcrptioo of the heat flow illtegraf -0
arr linear functions of the comp&ats
of f(O), Icaving out the component
of V+.
K one
rquim that these linear laws are in~t Under the mcr5& the0 the property (54a of (6.3) ~a k verified for the matrix components of A u%ch occur the&n. The full symmetry (7.10) of the admittauce matrix can k derived ftom the Oasag~Casimir reciprocal rciatiolls. They arc applicable if the state of the material ekmcnt at time I is spccikd ‘by the vector f(r) in (3) and by a complete set of even and odd internal variables whose vaks refer to the same time r. The quations of tkmn~odynvnio of imdblc proocsscJ can. for small departures from an qui&ium rcfautcr: Sate, be linearized and &mstitu:e a Joallcd linear thermodynamic formal&n. An cxampk is given in [zo]. The a~pticjtion of the Onsagtr-Casimir reciprocal rciations is straightforward. The eiirnination of the internal vtidzks then leads to a functional rdatioa f(r) - g(r) (see (2.2) and (2.4)) with an adrnitcncc matrix of the property (7.10). (For such an e!iminatioa see p-al. p. 484 and [19b This demonstrates that the Onsager-Cruimir reciprocal mkuions yield more iaforxnation than the fluctuation-dissipjtion t&mm combined aith microscopic rcvcnibiSity. That the Iattcr yields restricted results is due to the fact that only the fluctuatioos of cxtcb sivt quantities arc really mcaniagful. This prohibits the cxchztge of variables which has been discussed at the beginning of Section 7. The principk of full macroscopic reversibility is therefore a consquence of the Ousagcr-Casimir reciprocal rciati01~ REFERENCB (11Gkn. H. B.. and T. A. Wefton. Phyr RC-C. 83 (19s1). 34. [2) Cdkn. H. B., and R. F. Greene, ihid. 86 (1952). 702 131 Casimir.H. 8. G.. Rev. Mod. Php. 17 (1945). 343. (4 DRY. W. A.. Ad. RahuJ hi&. A&. JO (1971). (3 Gm, R. F.. and H. E. aen. Phys. Rev. 8~ (1953, 1387. I61 Gunin, M. E.. A&. Rational hkh. AruL 44 (1972). 387.
AND *3acRo.sccPIc
Rzlvmsam
n?e%uaPatiotl~patioIl thcom is one of the fundamrntai LhcCWcmsin lirsar thamodynvnics of proccsra. A derivation oa the basis of sta&ical m&a&s has baa given by C&n and Welton [I]; it was ntbscqucntly simpli%cd by Kubo [IO& A thamodynamic derivation of this theorem has been presented by Callen and Grcme (m. [SD. This theorem rdates the comlation funaions of the %uctuations of cxrensiw varia& to the admittance matrix which uxmats the intensive vaziabIcs with their tha-modynamical& coojuga!e extcssi~e variabk. If, in addition, microscopic revcrsib12ity is invoked, then the comhtioa matrix has certain symmetfy pro@cs which, through the %uctuafioadissipation theorrm, arc rc%cctcd in corxsponding symmetry propcrtia of the admittaae lIm&k
properdcs of the adrniaanoc matrix are also obtained ift description variabies is used, if the OnsagerZasimir resipmcd rdatioas (PJ, (tl3. [3D are applied and if, &tally, the internal variabics are eliminated (11). The importance of the CaUen-Granc theorem, supplemented by microscopic rcvtibilty, lits in the faa that it can be fonnrJa& witboot considering cxplidtly internal variables as parametar of state As an example we mention the admittan= matrix of an daxr4 RCf-nerwork which is symmetric incspcc&ivc of the detaiIs of the structure of the nerworlr and of tbt charges and the currents through the various elements of the nczwo& Thuz tins theintcxsting question whethcrthesymme~properiia oftheadmittancz matrix occurring in the relation between extensive and intensive time varying variabkr oLa be derived from a principle which is puniy macroscopic and in which - hopcfuIly some kind of invariance a* time reversal, which should be of a macroscopic charact~, Nap a pan. llL first aswer to this problem has been given by Day (41 and by Gunin 163. who r+ These
symmetry
with intcmd
38
1. Mm
thmrskts to the constitutive equationsof linear viscoei&c materials wioh therma1 aspects being included by Gutin. We s&ail give in the scqncl a gencrakatkm of their msuits to other contitutive equations of the afmefkt type and we hail also make full ust of the powehli passivity property of suchcojstitutive equationsIn partiallar ‘h following points wiil be emphashd and treated: I. Appticatioas are possible amoag o&en in Linear elastic, tknnal ektricJ magnetic rehtioa with various moss eEciS Z The occurring constitutive equations are linear passive transformations; therefore the rich mathematical theory of linear passive transfbrmations (or Syncms) is applicable. 3. In addition to the time reversed procemes introduced by Day we shaJ3 also consider cyck proctsxt and the time reversed cyclic procmses. rnbothcasesthistimerevcSalis of a macroscopic nature and wiU therd~re be termed macrowpic time rcvusaL s&ml
4. Tjrme reversal win be properly defined for both types of procases and the oo~urzc~tcc of even as well as of odd parameters of state and of a st-pcrposcd constant megnetic 8dd wiil k taken into muot in *&isd&&ion. The distinction betuxen even and odd paxameters of state is impaative if one’wishcs to introduce the concept of macroscopic revem= ibilityatall. 5. Al~ough physically of minor imporran~ complex iinut pzJrive transformarioas will be inciuded. 7&y zqairc an intercsthg rukbition of time wusal. 6. It win k shoan that the postufate that a certain functional is unchanged under properly d&nod time reversal leads to the same symmetry prop&es of the admiw matrix of the amstitutive quations as the &xnutiow5issipation thccmm plus microscopic time reversal revcrsibiiity. Thus the conncctioa between microscopic and -pit mauoscopic reversibility are obtained from microwiU be cstablirhui and two kinds of scupic revcrsiity.
7. The results obtained for Iinear constitutive quations can easiiy be generaikd to Iiear structures built up from materials with linear ecnstitutive equations; thus such strucnms aiso exhibit macroscopic revcrsiiility as do *&e constitutive quatiom, 8. A dose relatioa between macroscopic revexsiiiiity and a certain raiprocity tbctnun canberzstaMished. Whrsk rmacmcopic revmibl2ity an be designated as an actual priaciplc remah stiU an open question. AU the results obtained in this paper refer to the l&r domah For noa-bar constitutivc equations we have so far no check by a simpie examptc, much less a microscopic theory, which would prove macroscopic revSWiry. lkrefon scopic revemibihty should be regarded within the mentioned Iimitations, but not yet as a geneal prkipik
A rather general set of Sinear constitutivc equations will be given which describes various kinds of rhxation phenomena: mechanical rclaxazioa, temperature daxatiou,
MAcItosa3Pfc
and the mpame
tedor
AND MICROSCOPlC RmERslBILITY
or me peeror
having sixtan compoaam ea& The fundamental inqualiiy of tbcnnod~mics (p3b [Ia of dectro-magnaic fields 1151.has now the simpk form both vedors
gcncntized to tbc prcsma
where the dot between f atrd g indicates the scalar product. The Iincar coostitutive equations express the fate vector g(r) as a fuactionaJ of &e
J. MEIfCNEiZ
40
islinear, imuiant~against any timeuandation and which sati+ the pppirirywnditiun (23) for all f(r) from a rusaabiy chomz class of fuoeriom; sncb axastitlnive CquaLiolu f(f) - g(t) arc called IineLv ~assin! ?n&wTar~~ or ~~;rorsirrjpsrrmx stimulus vector f(r) which
fia
sectionwe shall givethe d&tion v== (I% [91, VD and we shall
and some important prom of linear also introduce tile two w-Tunction?t we parsive adopt a more general point of view by admitting complex-vaiued stimu5 f(z) md xcIa this
spa== ,rfrr DBIXIIOX I. roc&sC,
A vector valued function f(t) with n componcats /I(t) is mid to bJors (m=O.1,2..,) if its compoaass
l.ared&eloa--cOst
.
A functional transformation f(z) - g(r) is said to em
D IInrrp
passire~anX 1, it assigns to cvuy f(f) e C, a vcaor-valued function g(r) E C, of the w with I! ampoaalts k(t); 2 the sapsporition principle holds; 3. the u2nsformatioa is k&ant under aXl t&c shifts; 4. for ail stimuli in C, ;b: passivity propeq Re _i, ,gI ~O)s&?
3 0
golds. The star * denotes tb conjuga? compla TEEOSW 1.
Emy
g(r) - Af(r)+@lp)+B
linear passk
MCXI
WI rtal3 value
qszem f(z) -+ g(r) has a repmwuatim
j f(r)&+ -. +
i P(t-r)f(s)&+ -9
where A, B are non-nega!ibc d&tie
Remitian
P(t) -
j [P(o)-P(t-s)]~(s)f&, -.
m!ricrs,
7 Pm d@(m) -0
u ir a HcrmiiliM mafAx, md
(331
with a mrri.r e(u). For epery 9ecror x, the /imction ji’P(r)x is a positive dq%&ejanction wifh mm raiuerao and the fktaion f g4(~)s is a real, boundc& non-drcreusing fun&w in - m < w e co which is conrinuol~ta: w = 0 and has fke the value zero. l 2% matrix @(a)
MACROSCOPXC AND MICXZOSCOPXC REVERSIBhzry
41
1.
2 3.
A rtid~ f(tj in ct r
I’(p) -pA+ht+~B+R(p), W9
(33) (3.10)
- M(O) + (17hARl -
d ~~~trix R(p) will be called the cbe propcrtks R(1) - R(r)*,
residual part
limpR(p)
(3.1 q
Rep > 0.
_i,~d@(t&
* 0,
of the odmit:ace
Iim~J?cp)
msttix Y(p); it hy
- 0.
(3.12)
1. MElxNER
42 The
ammu
matrixY(p) isa positive maeix vector I holomorphic
the individual tanu
in (34,
which acans that the function ji* Y(p)x is for tvciy iu Rep > 0 aud has that a non-negative real pan Also
PA, io, $ B, R(p) arc positive maties.
The kfr manher in the inequaIity (3.1) wii be dcsiguatcd as the W-/Mczio~Z because in t.hc PM sumemin reMon it it, apart from a factor I/P, the work trausf’ by the surf&z sznzzm to unit volume of the rzfcrcna state during the intcrml -QP c I ( 7. Ftoa TImrun I the foPowing result m be acrivad
We ag’x~e that f(-r) is also in &; then in (4.1) and (42) we can choose 7 - CDand obtain, now omitting the parameter r, the FV,-famajoa?i W, (f(t)) :-
Re j f(r)*. g(z)& -Q
j&VBF(O?+f j (1 +mz)$(m)’ &P(m)F(o),
(43
r-‘-f(r)&.
(4.4)
-0
WkC
F(m)
-
j
-a
No mtricdcms on the linear passive systems must he made, in order tit the W,-fune tional k d&nai For cxampic, it i! not ncceary that for the admitted s&nuG f(r) the response g(r) approaches a hitc ihit as I 4 a~, It is dikat with the IV,-functioml we are now going to define and which pert&s to qdk pA cyclicprocas is d&cd & such a way that the szimulus and the respcmse are periodic. Butas~utw~alwayrtostan.atsometimc~tmayalsostartatr---aPbutiD~~ a manner &at (3.19 holds); it can be periodic at hcst after some time r,, which we may take to - 0. We the&on choose a stimulus in c. which mm not be spaSed < 0 but which, in 0 c I < CI),is f(r)= C(qP, k
m+o.
' '. .
in - co e I (4.5)
MAcRoscoP
c
AND
MIcRoscoPxc
REvERsIB1m
The set of numbers k may also be inbite; but then appropriate conditions on the as& totic behavior of the ol as k - f ca must te imposed in order to acre approptiatt CCQ. vcrgence properties. But it s&m to consider the tite cast With such a cboiee of the stimulus the response b in gcncral not p&o& for 2 )r 0. But for some subclass of the IineZr passive SyStemSit IIWy become aqmptotic3Ily pdodc sf-+ 0o.Wbca we speak ofaqck proce wc always mean the asymptotic pehdic process fkom t to t+22+ with f -, a, ic after t5t transients have died 0uL Sufadent conditions for the cxktena: of a qdic prw with a szhuSu~ of the propary (45) &we call them ColtditionSs-arc ([la: 1. #(a) is absohatcly corJtinuous; 2 Q(m) exists waywhere, is continuog and in every 5nite interval of W W&tion; is. as a function of e, for all real 0 absolutdy iotcgabft 3. [@‘(a+&-ol(o-e)],” ia-c(eCL, when r>Q Under thex conditions the tit Y(iw) exists for all rcaJ w f 0 ;md the rnponse to the stimolIxs (49 is g(r) - B[ j f(S)&- z (Lkw)“&j + & Y(hJ)~c-+o(r), -m where 0(l) 4 0 as t + Q). This is a simple gawrakation of Thxem 52 in (la Atcnnwithk-Ointhespmin(4~hadtokomined,bec;lure~g(r)mi~ contain a tam which is aqmptotically proportionaI to t uniat B I 0.
(4.6)
We intmdwc now the Fyrfonctionalby #+%I-
W, (f(r)} : = Iim Rc W
I W’ - P(s) &. (4.7) I. In the special case of the purr strewstrain relation W, {f(r)) is apart from a factor l/T+ the work mnsf&cd by the surfaoe stresses to unit VOIUWof the refcmxce statr during oue puiod 2+0 afla the trans&ts have died ou& With f(t) from (4.5) and g( 2) from (4.6) the W&xnai~d assumesthevalue (4.g) QearIy, the vdua of f(t) in -co
(43)
The l&t result can be derived from IlSj, Theorem 43.
1. MpxNEIl
44
Atfintwcmnnaplainwhataruo~dbyatimercverscdnimnlurLcetfCr) be a stimulus in cs 50 tbzt f( - t) is also in E2. For tkc time revc& stimafur we cmnot jaas take the proczs f( -r). We snn~ also bear in mind that in physical systems there are vahbics and panmetus which arc odd under time remsaL & a rcprcscatative for odd ;Llld this is the most important casc-+a const2nt salpcrpo#l pamnetm we cIl_ maptic inductioa The proper d:finitioo of zicmcopic rmmi-ty is as foJ.Ious: D-ON. The the termed sttnuhcr to a pmccss f(t) taking place in the Iinar pusive _qstan with constant sqeqmed ~~~gncticinduction B+ is the stinmiau d(-t)’ taking pb in the iiucar passive a%lzna with the opposite value -B+ of the mag%tic inductioa The matrix z is a diagosai mwix with diagoaal dcmcW,s + 1 or - 1 asordhg as the rcspstive c.cmpooctn off is CISZor odd with rcspcct to time revrml; f(t) aad f(- f) aro both to be &osco in cz. Tllc transition to the uxljngatc ampIcx value in the time rot t& format f&o0 that o&y such simple rsUIt#saE rcvcn&stimuiusisrrca=ssary obtained (3 is we9 known that th time reversal opcr;itor in wave zrechks also changes
the wave function into its coajugatc compkx vztru). It is important to note that the mpow td the stimubxs f(t) is not simply rcfattd to the response to the time revencd srimotus Ir(-t)-. From (43) vx obraia W, (f(t), B-jW, (d(-t)‘, -B’) = ;~(O)‘[&B+)- rB(-B+)c]F(O)+ + $ _i, (1 +G)f(w)’ We 2sc ilow in a pcsiZio0 to cnunti
~Q(uJ, B+)- t&m,
tbc WphanwL
-B+)r]F(o).
(5.X)
a
Wl-T~~ Lrt borS f(t) and f(-t) be in 2;. 37xen the WI-$mctiona& for the process f(t) and its time rercxsedproccs “i( - t)* me &h&f nicK two funuioMt arc eq4 fir LG pomirtld f(t) if md o@ Q B(B+) -
&-B+)L,
#d)(m,B+) = r&w,
-B+)i
(-m
<.o < ca).
(52)
Proo/: The *-~-assertion foIIons hmdiatiy from (5.1). For the proof of the *oniy iT ass&on we chmsc a szimdus f(r) so that-51~ (4.4)-
MACROSCOPlC AND M.KROSCOP!C REVERSW
with
2tt
aad f(4)
2r’bi~
axmalltvectbt&Tba
arc dcariyin
ea.we d&e
t%(a) =
now the fuction x(a) by
(1+w~* u~@(w,B+)- r&u,
-B+)e]a
(SJ)
withthcnormati2atonX(O)-Oandwith x(o) - f[X(w+O)+X(ueO)),
(S-6)
Siacc @(a~, fB+) is continuous at pt = 0.90 is X(W). The egunliry of the W&Duiol& in the left lzlcmber of (5.1) yiads with (43)
ur X(0+) - X(0) - 0 ._ j &cr, 0 -0 0 for~points~,toCcontinuity?nd~~~~for~cr~inO((p,~aJ~virayoC (S&L We d&te now a new ftanczioo F(W) as the F(-co) in (53). and obtain in the same - Oia-as
We restrict BOWour attention to linear passive systems whicharcsubjat to the con&as d&n& in scctioo 3. tions S, both for B+ and -B+ and turn to the cycfic prThe first problem is now how to define the mverscd c/de. Given 2 szimulus f(r) in cx which in 0 g I e ca has the special form (43 with some w f 0, we introduce the function f1(r) - f(2x/o-
f) 0 7
.4@-
(Odtsi2z/cm).
(6.1)
Hcretoforr: we have coasiderrd q&al definitions of time reven& szimuli which 14 to the W, and IV,-thcormrr BCCAUXthe passivity property (3.1) of Iinur passive systcxm
MACROSCOPIC
AID
MXCXOSCOPXCREVERSIBILITY
47
symm&ic in the stimulus and reqow components, it is ‘also psiiie to consi& g(r) as the stimdus and f(r) as the respond. If f(t) is time reversed according to one of the @en two dchitions, g(z) is in gcnemI not time reversed. Also, if g(r) is takai as the stimu!us and is time rcvcmcd according to one of the given two dhitions, then the rcspoav f(r) is in gamzI not time reversed. That is to say, if we postulate equality of the respahe HMmctioaaIs for this stimulus g(r) and its time reversed stimdy we may obtain additioaai properties of the admittance matrix Y@, B+). In both casts we obtain fomdy the same result (4.5). but in the second GUC with @(w. B+) being the spscval matxix of . the impdance ma&ix Y(o,B+)-‘. we can proceed CVUImore gendy by clloosillg some compotits of tbc vcdot f(r) aad the remaining componcnt~ of the vector g(r) as the stimuhas, and the other v ua~~ of the vccton f(r) aad g(r) as the raponsc More precisely, we introda diagonal projatioa matrix 1 with A2 = dandd&casstimuhsandrcqoase
is
f(r; 1) - JJO)+ Cl- L)g(r),
(7.1)
LO; It) - (1 -MO+
c12)
telr).
lk &armation f( t; 1) - *( 2; A) exists for aU f(r; I) in T2 if (7.X), (72) and (33) can be solved for g(r, A) as a linear functionaI of f( r, A). The neczssaq and safE&nt condition for that is easily obtained if a eouatiom are written in Laph * dct[l+(l-%)qplrpll + 0. is Iincas and tims imaria! and suisda
This zmls5ocmation
Re j fir, l)‘.g(t, -*
A)& = Re 5 fv)*gtOdr --
transforms and WI
the passivity comiitioo 3r 0
(an rral XI.
(7.9
For the proof (7-I) and (72) are introduced and use is made of P - A; tbe rest foIlowr from (3.1). For stimuii f(r, 1) in c2 which vanish ia z < 0 oat obtains, in analogy to (3-Q, in terms of Lapfase aa&orms ie; wte& the inrrnirtmce
nutt*
2) - 0;
&,
1). (
04
I(p; A) is gjve~ by
I@; A) -
[X- f+ mp)J[l+(l~
i)Y(p)j-’
[A+ Y(p)(f-
1+ Y(jY)i).
I)]-‘[r-
(7.8
For A- 1 it rcdum to the admittana mat& for f = 0 to the ixtpcdana matrix. In all diagonal projectors f it is a positive mafrk The reprcscntatioa (3.9) to (3.12) of a positive ma&x also applies to the immittaaa m;luix I&; A, B+). We ootc the relations .
I@; 1. B’) - pA(1, B+)+iu(rt, B*)+ +(A.
B+)+.R(p: A, B’),
(7.7)
1. MPXNER
48
R(1; 1, P)
-
J@;A, B+)*;
SipR(y; ta
E&R@:1, B+)
1, B+) : 0,
I
0.
cI%;
FromtbcrrsuloiaScaionsSanddwc:obtainimmediatcty~foUowingi 1. If rhr W,-#tmctioMkfor allpwmssef f(z, ~)andttipropalytLntrcnrsrdpwacncs are qui-it is un&mmd that f( ft. 1) E c2 and that L is aqy diagonal pwjeuor+an (7.3),, (7-U) Md (7.13) hold n&h rhese PIajcaorr 2 r the W&m&mats fat all &ic processes f(r, A) and zh& pwHy rime rzIcIsd cycfic pwcesses me eqd-it is mdcrttood thar f(t, L) E c* ad rhur 1 is any &izgwd prajccroF-then (7.12) und (7.13) hp’d m% r&e ptojccrort. The problem is LIOWto invcstigata to what extcot (7.10) can be proved if V.12) ad (7.13) are assumed to hold for all diagonal pmjazton i. No use will be made of (52). It will aIs suf6ct to exploit tba consequences of ~.u) for the special projaton & - da- d-d,
(no summation
oyct a)
cf.14
which have only one zero on the diagonal WC assume that the diagonal efemenk I?of Y are invertible, irz other words that none of r&n is identically zero. If Y, %w zro in Rep > 0, then the Pti and the Y, (i I 1,2, . . . . n) would be constank satis@ing
?dKlCOSCOPlc
AND bfICJWScOPfC
?Y+Pm = 0. Wecho&nowinparh!ara
rqmseatations
rt’rl@; B+)-‘,
4&;
4 S+) -
I,&;
1, B+) - -Y&;
L(P;
4 B+) -
$15)
B+) ~n(p; B+)“,
m6)
K&P; B+)Y,,@;B+)-‘, -
-%a
(7.11)
h*
03)
(3‘9) and (7.7) will be writtea RP; B*) - $x@;
0;
49
- 1 and obtain from (7.6) G~!I k = 2,3; ..‘, m
d9d4 The
REVERSIBEJlY
1, B+) - $k(p;
.
B+) + R(p; B+),
cts)
1, B+)+R(p;
clro)
1, B+)
~I(P;l,B+)~Y(P;B+~etcTbeX@;%,B+)arepoiynomiasinpofdcgra~Z We take now the qwion (7.!s) in tile form fX&; 1
A B+)+%,(p;
and sttbtracz f’tu it the equation for B+. Denoting
1, B’)
II+X1,@; 1 t7w B+)+&(P;
B+)
- I
wIti& ii obtained from (721) w&n -B+ is s&&&d
XO@P;1, B+)- LY(&,(~X&
f, -B+)
- JXo@; 5 B’),
c122)
we obtain by virtue of (7.X2) and afta eliminating the R(p; 1, ET) by Cr.!91 asd using mr).
+
$dX,,(p;d, B+)dX,,@;
B+)Y,,(p;
B*)+c9X*,@; B-) - 0.
033)
?fdx,,@, B+) I 0, then Y,,(p;B+)ismninB*. Letmoow assomt that dX,, (P, B+) et 0; thea 6X”,(p, A, P) f 0 and Yxr@, B*) is a solution of (723) and therefoe has avayspahlform. Without proof WT note: Unakr fhc aaZitiona1 assumption tkt rk A(1, B+), a(l, B+). B(t, B+) are ho~omu@k in B+ tn a connected domein which with B+ ah c0ntah.s -B+, r/u fMaion Y,, @p;B+) ir glen in B+ with tk only exception that Yll(p, B+) * AlIp, or tit1 tw B,Jp wz’tk -41 3 4 4,
3 4 al1 =md.
Now we turn to the equations (7.16) and (7:17’) and assume hat Yll(p, B+) is &a in B+. We write (7.17) witb -BC in piace of B’, multiply this equation by rll(L) t&A) and subtract it from (7.16). Due to (720), (7.12) and (7.18) WCobtain with k - 2.3, ..l, I
dX,,(p; I, B+)Y,,(p; B+) - 4X,&;
B+).
(72s)
1. MEtXNE%
50
So we have either dX,,@; i, B+) = 0
and SX,,@; 8‘) - 0 or Y&p; B*) is tk ratio of two poiynomials in p of degree < 2 Only the SccoDdcase needs furtim mnsidaation. Tk equatioll LobuinedinaJimilarwayas~24).It~~tosctthatthezcrosandpobofY~’,,(p;B+) anocclxonlyon*iInaginarypaxk
Either aU d-ccxfBckws vatsisttor P%%(p,B+) takes auc of the fdowing
TEXBEM4.
Ginn an admittance mutrix Y(p; B+)
whirh is mui for
forrrrr
psi&e
p. hdb-
B’ ad -B’, anti has IW&god &nzaxr of tk forms (727). Zf the imnrittmrcl matrices I(p; I, B+) d&red /mm sub an a&nir~ctnce mopjlc by (7.6) exist /or al.t projeczors (7.14) and their residwl parts R(p; 1, B+) Ame the prom (‘?.U), then (7.10) and (7.~1) hofc. r Y(p; B+) is scalar, hoZomorphic in u domain W&&J cunnects B* wtd -B+, md has a residuaIpart which b an even fmction of B+. tkn I’@; B+) is M evenfmction of B+ U&S Y(p, B’) is prvpoMor& to p or lo pda or is an w morphic ia a &main
OoIutQnt,
which cmneets
..
It is assumed in tbc analysis of (7Z) that B+ 8 0, or thaw Ysr(pr B+) is not iadqa& at of B+. Otbmvisc (723) is triviqlly satisfied without any fi@cr consequence; kcsw of ax,,(p) = 0 and dX,,@, 1) = 0. The analysis of the dX,, and of the dXkI, rrrnainr valid and the first part of Theorem 4 remains tnte while the secondpast beama an aapty
statemen% The excepted admittance u3untcr-exampIcs. Che h
matrices nay aIso have the pmpcrq
(7.10). but thee arc
MACR~PIC
n
AND MIC%O5CDPlC REVEkStBILIW
reversibility when appkl to the stimuli f(i, A) with any 1 Bothtypcsof epic fnm (7.I4)--~ then speak of full woscopic revusibiiity-therefore lead to t& symmetry pro(7.10) except for a Smau &class of admktance ma* &Redprodtyaod
me
-* while w far m which kad from an quiliirium state to au quiliiriuat aatc and eycticphave been considered for ma& elements only, we now s&n wntemplate the colTMpooding situation in elastic uzatakl bodia and by dctitioa, unstmkd bomoWecbooxasarekuastatetheunnrrsud, gmeomstat+attnnperatracrofamatcrialbodyandpermitonly~d~ from this state which may, however, vaty with the position vector s and with time 1. Ektromagnetic fields are exdudcd Thar only the 5~3 ten compoaen6 in (23) aud (24)~~icvantWeintcgiatc(ZS)ovathc~~~u~ofthebodyaMiobtaio
Rx, 1) -
(t(x, 0. Q@. 01,
‘3~. r) - (b,
0. &,
0).
(82)
W&E x deaotcs a surfacz point of the body, T rT)/r-l,thef,&ilc~alx~ tioas and r,& = a,&, with t.bc arta demcot do; the do, are the compoamts of the ivc~ element vector, aad us(x, I) is the surfice stress tensor. The q[s. t) denotes the velocity Sdd of the surfpoints and 4,(x, t) is the ou~ard normal component of the beat ff~c vustor through the surface la the derivation of (8.1) from (23) use is made of the e Inwa
The summation amventioa is adopted. We expu that the koowlaige of the Sdd F(x. I), with the specifkation that the proa starts at r = -a~ from the qtibrium state 7 = 0, tl = 0, determines tbc motion and the heat flux in the bdy and on its surface In paxxicuiar, we see &a& due to (8.1). the relation F(x, t) -, G(x. r) coostituta a passive system with the surfaoc integral in (8.1) defining the scalar product of the fields F(x. t) and G(x, t). It is also linear if we rcstrkt ourscivcs to small departures from the original state, and time iovarian= is evident. if there is no other action oa the body. 73ercfore there exists a representation (Hackenbroth [7D which, when expressed in Laplace transforms (3.1). reads 6 (I, p) - j Y(x, x’; p) t(x’. p) do’.
(8.4)
s -
J. MUXNER
on x and i.
me admittane operator Y(x, it p) is a 4 x 4 matrix which also dcpds Itisa’positivegcntofw&ichmunsthp
Rcfdufdo’ &x)*Y(x:,
Y;p)&i> > 0
WI
for all vector fields 4%)wtich are absolutely squareintegrable over cbe body.Thisadmittanceo~torgcncraIirestheco~ofapositive~ From the theory of
is Id for positive p.
linear pvsivcsystems one dduao,
qtp)
=pA+$B+a+Rt,p).
condduing
sprf?a of
t,b
that Y(x,d;p)
(8-W
In~cumpi!:ancompoaentsofFandofOarrev~withscspeabtime~~ Mamoswpic memimty yidcb mx, i;p) Cmsidc+g
(84,
- R(i.
x:&S
R(p) * k(p).
ml)
(8.7), (S-II), (S-12), (8.13) and (8.14), UT obtain m,~;p)-d~,i)
- W*%3)+tiv% m-a
-
(8.18)
xP)+a.
Frh macroscopic revm%Zty yidds, hoi+zver, a(x, i) - 3 and a - 0 apar! from admittance amrices q(p) of one of the forms (727) and apart from admittmoz operatom macroscopic revcrsiiiby indodcs alI vandonnod stimuii p(l;i;p)ofsimiJarfonrnFulI =drapo= 3(x, r; L(x)) = I(x)F(x, r)+ [Z- Z(x:)lG(x. 2). (g-19) Gfx. r, 3x))
= II-Vx)W(x,
0+W)G(x,&
(839
with r(x) being a diagonal matrb who55 diagoaai ekn4mts A&x) m the valac 1 en thtpartDIofthebody~~aodnoootbtwmplemeotaypartDPofitfhc~ fonrredstimrrli~amixedboundayvalue~vtblcminthescnscthatachcompo~ i of F(x, r) is a givea fanaion of position and time on the mrfact domsinc L$ and tbc mqceive ampmcm of G(x, I) are given furaions of Htioa and time OLIthe wmp&mcntary pm @, wbiic the compooam of F(x, t) on fl and tbc wmpo~eoo of G(x, t) 00 LJ, (r’- 1,2,-) waszimtc t!KrcqwmsC lbe statement a(x, i) = 0 as a cmsqTJa~ pie lVvm%~ is made of full mby analogy to ot = 0, bat is. lacking a direct proof, given as a wajeurc Hmrva, a(x, 9) -Ocankdcrivedfromq(p)~ ij@)asenrraciataIia~~!. Weamsidcr aow two wmpkx stimuli P”(x, I) and F*‘(x, r), They prodncc response Gc*)(x, z) aud Gt*‘(x, r), aad a ~WCCSSinside the body aitb &r, I), #@(r. f), o(I)@. 2). $“(r, f), +(r. t) (r’ - 1.2 and r is tbe position v-or to points inside th body% By USCof the bafance oquatioag the apmsion L - f [W(x,
t)Fr”(X, t)-&“(X,
r)Tc*‘(x, z)] ch
(83)
caObCtaDdonwdinto
With the assumption of mail dcpanuxes from the original state I& hst inttgal caa be nglcctal and ,(” and @*’ cm be replaced by their values e* in the ori@iaji StatC, thus aegkring terms which are small of third order. We asstme now thaz Fc*‘(x, I) ad P*‘(x, t) WC two wmpkx stimuli 4th the same time depcndeoce @‘, Rep > 0. Due to the Iinevity of all LIB%under the precnt asrumptioa,
s
1. hEDU4ER
G(‘Vx. 1). p(‘)(r, tX r<“(r, t), r”“(r, ?), @(r, I) will have the same time dependen= bcCawcor+“-J+,(822)~IxoSto L =
$[y’)(r, f)cp(“(r, t)-
Bazausc of the assumed tbc ad (833) t = $d~$drr+(~‘,
y”(r,
f)cp<“(r, r)jg+ by.
and,
(823)
dcpendcncc we can use (8.4) and (8.5) to obtain from (82ij
I)[~(x,
x’;P)-
Y(x’, x;p~jF~(x,
2)
= S~p”‘(r,f)t~(P)-9~)1~(~)(~II,t)e+
N.
(824)
n¶is~&siIxun~tbc THEOREM5. I/ rk adnittw ma&ix of rk constitutive eqzwtia~ (8.10) it not of ON of zhr zprciol fomzs in (727). thenfun mucroscopicreversibiiily qpiieti to tkse aw sztii~eeqwtion.s of u materid elmem enwaim q(p) = i(p) and L = 0. Beaua tk nfmdi P*(x,z) (i - 1,Z) me arbitrary in ikir &p&cl on 5 it fwtkr /O&WS ihm fix,i;p)
-
Y(r’,%P)-
@u)
Gwuerseiy, if L - 0 for df SZirndi F(X.
f, then t) (i - 1,2) with time &pt&nce (825) iroldr and i/L - 0 for ail &bodies, howmr, smdl, thea 9(p) = $@) and thr cons25 Wi9e eqmZims of a materiaf element suotwy fUn ~WOSWpiC revs. The apation (821) with L 7 0 is cakd the reciproeirr ihoran. It takes thir mmc from the following property: Let a(x - x co? be a disttiiutioa over the surface of thebody
with d(x-r’0)
= 0 if xf P(s,
x@ and jb(x-x@) I) - d(x-x(~TP
do = I. Assume the’ spcdrl (i A 1.2;
stimuli
Rep > 0).
(SW
fbea &*‘(I(“, Rccipruity ekUrodynami~
r)(Q - &‘(ti”,
r)nt.
theorems of a related type have been givg previousiy ‘m ekstidry for materials with dispersion and losses (11% f1IQ.
9. The f%lnatioadbipation
theorem and miauscopic
(827) and in
rcvcrribill~
We now tntn back to the general stimulus vector f(r) in (2.3) and its response vator g(l) in (24). We multiply g(r) by eV*whcrr Y is M arbitrary volume and intrgratc from -Q up to time r. For small dcparturcs from the rcfcrcnce state we cm replace e(r) by e+ and obtain
If the stimulus f(t) vanishes at all tires, then in macroscopic terms uz have e(r) - 0. In microscopic terms we have, however, statistical fluctuations of the components of &(I) about their values in the reference state. According to the fluctuation-dissipation
MAcRoscoPIc
&f&-yare lekL&
tk#vwt
f(z)
4
g(r&
We d&c
AND
M1cR0scoP1C REvERsmnrrY
to the admittana the wrrU.ioa
mafrixY(p) of thc’p&Yc
.
55
linear tral&omIation
matrix
(93 whercg,,Q) is now the time depcndcncc of the statistkil fktuatioo of the variable g,, about the rtfdna state and ( > denotes au ensemble avuagr Thedactuation dissipation S(r)
-
c4rYl(OBYIO+m
-=-(12oD (93) zssxiatcdwit.bthelioearpoJsive Jystclnf(r)~~z)ret (3.9) aad (3.11). In the quantum mechanical formulatioa of the tluctuation tbeorcm tkrcis adigbxm0di6cati0a of (93),tbe oaumna of a scaiar factor (see [lb [IQ which b of no iaffrmc on our ES&S on the symmetry propdia of the residual part of tbc . odrmttaac= ma&i% It should also be noted that the ductuatio~ of the time iate@ of the heat 4oa q(s) may not be bounded, or that the intcgzd in (93) may be divcrgcnt. This izsmdmm compficatiozs which, however, m be 1~110ycd ‘I5 &x&sarion dissipation theorem as fonnuiatai in (93) has nothing to do with Ininoscopic m, 5&y. Rather, this is a separate and indepmdeat statanmt TO fmmrjarc microscopic revasibility we consider our material in a magnetic 5dd with the magartic induction 13. of -B+,thatbtoscry,~considaa~car~e~ withtbepnrnmctcrBwhicf,~~~thcvaluaB+or-B+w~rrfcftotwodiITarnt stats. ?beo rP quantities and functions that ctie linear payi= ryffcrn =Y dqald Gn w. Mcmswpk mwrsibiIiy is aprcsscci by Z!ICpzopaty
~whcre4i(~)isthespccBalnwix
AF(t, B+) - tR(r, of t&e am&rion
mat&
-B*)t
j-co
< t < co)
(PI, [211, @2D. By (93) this properry is refk%i
br(w, B’) = cb(w , -B+)r and from tbc representation
(-co
in the rr;iation
<(D 4 cc)
(3.11) of the residual adnakancz
Iz(p, B’) - rR(p, -B+)r
(9.4)
(9.3
matrix WCarrive at
(RIP ) a)-
(9.6)
lbpr the Pucruation-dissipation theorem comb&d with mi~pic m-9’ does not yield any syrnn?etry properties for A, u, B in (3.9) beyond those &a kloh which were that A, 8, B axz Hermitian mauias with A, B bein& morrova, non-I3QN.h deiinite and P hich hold for all linear passive systems, but which arc in genera! not of a type similar to (9.6). Anyhow, the result (9.6) is quivaknt to the consequence of ma=Pic reverw’bility (SZ?)~or (65) according to what type of macroscopic revcrsiiility is considered The Buctuatioodissipation theorem combined with macroscopic revenibility is e*haustcd with the result (K& or (6.4, but does not lead to full macroscopic mciu’biiity. Some further results QO be deduced for special cw uith additional argurncnt~.
JUEIXNER . .
56
Coos&r the special linear passive system f(1) - g(z) d&cd by physical point of view the system is said to have the prom ifforallf(~)e~~whichzrrcanstaatint,O,wilhVf .
by (2.3) and @A). Guided of approach to ctquihbriurn
- 0 ia I 3 0, the rcspciosc g(r)
approaches tcfo as 1 - 00 and if, moreover, the time integral of g(t) approaches a constant value as f 4 03 (but possibly except for the time integrat of the heat flow). Then one Qw prove that B - 0 which yie!ds (52), as a trivial consqucne Furthermore, one has P(t) - 0 as z -+ CDand that the principal minor of the matrix 4; which does not n&ate a ampormt of q or a component of Vf propaty Ia
WI2
quiliirium
or
to any of the components of f(2) or g(r), rcs~cclively, has the
.
(W.
the components of 5 g(r)&, with the cxcrptioo of the heat flow illtegraf -0
arr linear functions of the comp&ats
of f(O), Icaving out the component
of V+.
K one
rquim that these linear laws are in~t Under the mcr5& the0 the property (54a of (6.3) ~a k verified for the matrix components of A u%ch occur the&n. The full symmetry (7.10) of the admittauce matrix can k derived ftom the Oasag~Casimir reciprocal rciatiolls. They arc applicable if the state of the material ekmcnt at time I is spccikd ‘by the vector f(r) in (3) and by a complete set of even and odd internal variables whose vaks refer to the same time r. The quations of tkmn~odynvnio of imdblc proocsscJ can. for small departures from an qui&ium rcfautcr: Sate, be linearized and &mstitu:e a Joallcd linear thermodynamic formal&n. An cxampk is given in [zo]. The a~pticjtion of the Onsagtr-Casimir reciprocal rciations is straightforward. The eiirnination of the internal vtidzks then leads to a functional rdatioa f(r) - g(r) (see (2.2) and (2.4)) with an adrnitcncc matrix of the property (7.10). (For such an e!iminatioa see p-al. p. 484 and [19b This demonstrates that the Onsager-Cruimir reciprocal mkuions yield more iaforxnation than the fluctuation-dissipjtion t&mm combined aith microscopic rcvcnibiSity. That the Iattcr yields restricted results is due to the fact that only the fluctuatioos of cxtcb sivt quantities arc really mcaniagful. This prohibits the cxchztge of variables which has been discussed at the beginning of Section 7. The principk of full macroscopic reversibility is therefore a consquence of the Ousagcr-Casimir reciprocal rciati01~ REFERENCB (11Gkn. H. B.. and T. A. Wefton. Phyr RC-C. 83 (19s1). 34. [2) Cdkn. H. B., and R. F. Greene, ihid. 86 (1952). 702 131 Casimir.H. 8. G.. Rev. Mod. Php. 17 (1945). 343. (4 DRY. W. A.. Ad. RahuJ hi&. A&. JO (1971). (3 Gm, R. F.. and H. E. aen. Phys. Rev. 8~ (1953, 1387. I61 Gunin, M. E.. A&. Rational hkh. AruL 44 (1972). 387.