Robertson has derived from the Liouville equation an exact equation for the maxent
distribution which depends on a set of moments. The exact equations for these moments
verify predictions of Grad for the Maxwell and Cattaneo relaxation equations in a dilute
gas. Nonlinear reciprocity is applied to estimate contributions quadratic in heat flux Q,
to thermal conductivity and to second-order effects in Q, diffusion fIux , and traceless
pressure P cd in the reaction rate in a dilute gas mixture. All non-linear effects are too
small to see· readily

Nettleton, R. E.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

PERIODICA POLYTECHNICA SER. CHEM. ENG. VOL. 41. NO. 2, PP. 175-184 (1997) NONLINEAR RECIPROCITY: STATISTICAL FOUNDATIONS AND APPLICATIONS TO NONLINEAR EFFECTS IN HEAT TRANSPORT AND CHEMICAL REACTIONS R. E. NETTLETON Department of Physics C niversity of the Witwatersrand Johannesburg 20.50, South Africa Received: March 1, 1997 Abstract Robertson has derived from the Liouville equation an exact equation for the maxent distribution which depends on a set of moments. The exact equations for these moments verify predictions of Grad for the ?vlaxwell and Cattaneo relaxation equations in a dilute gas. I\onlinear reciprocity is applied to estimate contributions quadratic in heat flux Q, to thermal conductivity and to second-order effects in Q, diffusion fI ux k, and traceless ., pressure P cd in the reaction rate in a dilute gas mixture. All non-linear effects are too small to see· readily. Keywords: nonlinear reciprocity, nonlinear effects. 1. Introduction The maximum entropy formalism of JAY:\ES [1] (maxent) maximizes an entropy functional to obtain a distribution 0"(:1:, t) in the space of phase co-ordinates .1: which yields the best estimate (A) = Tr(AO") at time t of a dynamical function .4 (.l:), subject to specification of a set of values ({.4;}) (1 ::; i ::; I)) representing the available measured information at t. From 0" we calculate 5 = -f{. Tr (0" In 0") (1 ) which yields the information-theoretic model of entropy. Since 0" depends on only a finite number of moments, it is not a solution of the Liouville equation. Jaynes constructed a solution p(t) of the latter at t' > t by setting p(t') = exp[-ii (t' - t)]O"(t) (2) where L is the Liouville operator. To obtain a statistical derivation of the evolution equations of extended thermodynamics (ElT). we need to express p(t) as a functional of values of 176 R.E.NETTLETON the (rL) at t or over a range of times. \Ve shall proceed in the next section to consider two much-used ways of using (J" to construct approximate solutions of the Liouville equation. To prove that non-linear reciprocity does or does not exist and investi-gate whether terms introduced from symmetry [2,3] into the rate equations really belong there, we need an exact equation for iT of the type introduced by ROBERTSO:\ [4]. We study the latter in the third section. Specific exam-ples for a dilute gas are discussed in the fourth section. In the fifth section. \ve use the phenomenology to estimate the quad-ratic term in (3) for dilute-gas thermal conductivity. with Q = heat flux. The 0(Q2) will be found to be very small. The sixth section uses maxent to calculate chemical potentials {~l;} in a reacting gas mixture in which there is a heat flux. a diffusion flow. or a a traceless pressure. P =!= O. The reaction rate IS proportional to ~l;ii;. \\·here the {i';} are stoichiometric coefficients. 2. Derivation of ElT from Approximate Solutions to the Liouville Equation If we haw an approximate solution p(t} of the Liouyille equation which depends on a finite set of moments. we can substitute this expreSSiOn into the right-hand member of /j=-iLp and tai~e moments. This yields self-consistent moment equations which. together with the ansatz for p(t). soh'e Eg. (-lo) when the number of moments becomes infinite. -:;'he simplest such approach is the GRAD ansatz used in the clilute-gas Boltzmann equation. The GRAD expression linearizes a function which maximizes the entropy functional. A similar approach lllay be made to the LioU\'ille equation. Let {Ai )} (1 :S i :S z;) be a set of phase functions which are even under momentum reversal. Then {Ai(:r)} == {if } are odd. Defining g( G. c.') .I pU a d.l' .1 P:3 L'a d.l' . .I po(.r) exp(ii t)L'a d.l' .' c') . (.5) (6) (7) SO;';LISEAR RECIPROCITY 177 with PS the equilibrium canonical distribution, we set up the probability amplitude 9 for the numerical values of {--id, {.-i;}. \Ve assume equilibrium with a heat bath at temperature T. H. GRABERT [I] introduces a projection operator P into an identity for exp(iit). This leads to an eyolution equation for g(a,v): og at t (8) +! ds L ~! Dij(a, a', t - s)( -;) [g(a', s)/P3(a' 1] da' dv' . .. 00) at')' o I) I -1 " . Pg pgL'aA.i d:r . (9) qi ! PSI L'a (ii)2 AiP3 ch . (10 ) D· I) !d.rp3L'a'( )2Aj(1 - P)exp [ii(l The maxent expression for g(a. cl is [8]: g(a, v, t) == ! O'(.r)L'a('l') d.l' = ZcP3(a. v) exp [3F(0.. 1/)+ + ~{(j);(o..rl)(o.i - Cli) + cI>;(o..II)(lli - t' i )}]. ( 12) where 0'(.1') maximizes the functional in (1) subject to the concli tion~: (.4.;(.1')) = o.i . ( ... 1.;(.r ) ) 'Ii (1 ::; ! ::; 1/) . (13) Zc is the canonical partition function and F( 0.. 'I) the Helmholtz function. 0'(.1') is consistent with the Gibbs equation provided.3 (hT)-l and (j)i. cI>i are thermodynamic forces. If we substitute (1:2) into (8) and calculate first moments. these rep .. resent a self .. consistent approximation in the sense of Gracl. If we know the moments at time i. 9 in (1:2) is the distribution at t. Substinnillg this into the right hcmd member of we can calculate g(t) + (og/ot)6t which is a sufficient approximation if :'::"'i is short. a fraction of the relaxation time of fast variables. The moment equations resulting from use of gin (8) have the form (\i 'h """ (2)::. 'Ii = LLij 'l'j. j """ L ( :3) ,", . ..L """ L ( -±) ,f.. . L 0~) I L 0~) j j (l-.±) (15) 178 R. E. NETTLETOS 1>i and cI>j are nonlinear in deviations of the variables from equilibrium. and (k) - (k) the Lij are nonlinear 1>- and 1>-expansions to all orders. The Lij are not unique and satisfy reciprocity to arbitrary order in the expansions: (2) -L·· Jl L(4) Jl . (16) (1 I) Eqs. (16), (17) purport to justify nonlinear reciprocity under very general as-sumptions. A better approximate solution of the Liouville equation has been given by ZUBAREV [9]. One assumes a set {9i} of state variables which are classical or quantum mechanical operator averages (f;). The corresp onding maxent distribution is [10. ll] 0"( t, 0) = exp [-In Z (t) J(t)H + "L ~La(t)j\-a - "L !31>;(t)fij a I ( IS) The zero argument in 0" indicates that the Na and f; operators are time-in-dependent. Number operators 'Ya provide for particle non-conservation. The Zubarev approximate solution has the form: [ c_j~_' PE (t) = exp c ~~ 11nO(t',t't l dt'] (19) where the operators in (IS) are here taken to be Heisenberg operators e,'al-uated at t l - t. \-Vith PE' one calculates phenomenological equations in the form 0; (20) 3. The Exact Robertson Approach to Derivation of ElT \Ve generalize the foregoing results to a non-uniform system where the ther-modynamic variables dep end on position 1", (1 < I < v) . (21 The maxent distribution IS O"(t) = Z-l exp [- J dr t An(r. t)Fn(r .. r) - JHj. (22) 71=1 SOSLISEAR RECIPROCITY 179 where the {An} are Lagrange multipliers determined to satisfy Eq. (21) identically. The {in} may be quantum mechanical. We assume equilibrium with a heat bath at T = 1/ K.3. although \ve can relax this assumption later. Jaynesian statistical inference [1] predicts that CT(t) is the phase-space distribution provided we can derive exact equations for the {oi(r. t)}. The moment equations will be exact if CT plus these moment equations provide an exact solution for RobertsoIl's equation. The latter is derived from the Liouville equation by introducing a non-Hermitian operator PR with the property that er = PRfj . where er soh'es Eg. (4). \Ye define FR(th(:r) == t f dr[OCT(t)/OOn(r.t)] Tr [tn(i:·,.rlt] (23) 71=1 ' Here 00/00 71 is a functional derivati\·e. Operating with PR on the Liouville equation. ROBERTSO:\ cleriws [4]: er -iPrdt (t)CT(t) t ./ elf' F R ( t) i (t )i (t , t') i 1 o where i(t. t') 1" Cl solution of aT '""' '1 p' 't,JL~",t",I. 1,<'/ Il- - RI IJ fi) 1 ' \--=1 (25) :\lultiplying (24) by Fi(i".,r) and taking tile nace. we obtain equations for the 0i. COlltributions from the first term on the right in (24) yield ami-reciprocal relations to all orders in the {/\,,}, ::\oll-Henniticity of PR has frustrated the search for non-linear reciprocity from ,·he second term in (24). ROBEIlTSO:\ derives it only for the linear case. Hcm·ever. V'-C' call learn much from the first term. Let (r .. r) be Cl classical phase fUllction. e\'f'n under mOmeI11:um re-H.'rsal. "'e have. on multiplying (24) by and integrating O\'er phase space: CJ.li(r. t) \',7. ,Z',)'/\ = 1(1,7 '\' L(2) (,7 ,7) \ \',7 ") , L in" /n .(. , nEO (26) where the sum i~ O\'er forces /\, / j odd uncleI' time re\'ersal. If Fj IS odd under momentum re\'ersal. ,i f /-1 '\' L(3)- -I I (-I ' 'l'j = CT' L jll (r.T' )/lnJ .t) • n'::;:c + ... (2, ) 180 R. E. -,ETTLETO-, where the sum is over even forces. and the ellipsis refers to Onsager symmet-ric terms, mainly from the i term in (24). The detailed derivation shuws that L (2) (-; -;<) = _L(3) (-;< 7) IJ 7.7 nl 7 .7 (28) to all orders in the Pn}. 4. Examples: Cattaneo-Vernotte and Maxwell Stress Relaxation Equations \Ve now derive from (24) the linear evolution equations for heat flux Q(T'", t) o and traceless pressure P a 3(r. t) in a dilute gas. The operators are: Q(r, x) H(r, .r) Y L [UlT (2771) ;=] .Y 7Jl -] L Pi a Pi 3 is( 17; - '-'") . ;=] jJ; il(r;). .Y L(p;L /2ln )S(r; n. ;=1 The maxent distribution (221 for this choice of variables lS :-<PQir.t). Q(;'""c) + ~ <Po;!.Pii,".tlPOJii:-"C)}] Cl: ,j In linear approximation. which suffices here. the COllditioIl~ yield 110 Q + ... POP o .3+··· where (29 ) (30) (31 ) (32) (33) (34 ) (35) (3G) PO -2m3 2 V 5;Y -1/ 181 (37) (38) If we substitute 17(t) from (33) into (24), multiply by Q(r,.r) and Fof3(r,x). and then integrate over phase space. we obtain the time-evolution equations: o· Po:] EQ o (;;,T/m)'Vp - EQ('VT/T) .(Cattaneo) (39) o _ -(1/Tp)P o 3 - (4j.5)('VQ)5 - Ep('Vii)S, (?-.Iaxwell) (40) 1 - - T --2[(VQ) + ('VQ)] V· Qr5. (41) 5N(;;T)2 2Fm 2..'Y;;T \' (42) (43) The terms invoh'ing IQ. 'I' stem from the second term on the right in (24) and the remaining terms from the other term in (24). The coefficients of o _ vp and (VQ)$ agree exactly with Grad theory. This answers questions previously raised [12. 13] about the need for these terms. o The structure of (39). (40) shows that the fiuxes of Q. p can be ex-pressed in terms of these variables. To take these fiuxes as members of a hierarchy of internal variables. thus modifying the exact moment equa-tions. is equinllent to postulating an entropy model inconsistent with the information-theoretic one. _ _ S 0 If we write the identities Q = Q and p = p in the form: o Pn3 (44) (45) then. to linear terms. we esta blish an anti-symmetric Onsager coupling be-tween the - T 1 vT term in (39) and the ~O term in (44) and between the (VU)$ term in (40) and the <I>o3.p term in (4.5). Such a coupling should be valid to all orders since it is a necessary condition for positive definiteness of irreversible entropy production. 5. Non-Linear Heat Conduction The anti-reciprocal coupling in (39) and 44) can be applied to estimate /\2 in Eq. (3) for a dilute gas. \\'e extend 53) to the non-linear regllne by 182 R. E. ,,'ETTLETO,,' writing: Q LiPQ - EQT-1YT , (46) L ') 4 Lo + L2Q- + O(Q ), (47) EQ !\O + E2Q2 + 0(Q4), (48) ~q - 2 - -± -I/O Q + //2 Q Q + 0 (Q Q), (49) ° c If P is proportional to (v U)5, "'hen YU 7'= 0, we can reasonably take P = 0 ° -in steady heat conduction if u = 0, In 0-( t) we take P = 0 and Q as yariables and keep iPQ and ~a3,p' \Ve find that the presence of ~a3,p modifies //2 but not I/O, Then 19 1/51{T 2 F 2 m ' (50) /10 and /12 differ from (37) by a factor F, since here we take a small. homo-geneous system and do not integrate oyer r, Fi'om (44) and (49), we get [14]: -!\ 0 l/O -E 0/12 - E2J/0 F, O. (51 ) (52) Setting Q o in (46) and comparing the result with Fourier's law. we get: -1 !\O -T --10 I/O (53) E·) !\O _T- 1 _-_ + T- 1 ----::-10 1/(1 ( I 0 I/O (54) \Ve estimate 12 by supposing that TQ = (Iz'ul-1 1/1' \\'here r = mean free path and /' is nns speed. calculated from maxent. Finally, we obtaill: (0:) I For the case of A.r at 1O-2 atm. 0 °C. Q=1.209·10 6 ';/m 2 5 corresponding to I ~ 7.3 ' lO'1\:/m. we find 1'~2Q2 //\ul ~ 6.6 .10-4 . Large IVT! yields a second-order effect probably below the threshold of obserntbility. 6. Second-Order Perturbations in Chemical Gas Kinetics The quasi-steady reaction rate J in a dilute gas mixture is proportional to a chemical force (56) 183 VVe have for the Helmholtz function differential. dF. when {N;} are numbers of molecules and ~ D the thermodynamic force associated with a binary diffusion flux. k: L if?o.:3.pdP 0.:3 + L ilidNi - if?dJ . 0.3 The choice of variables is inspired by GARCL-\-COLI" et a1. [15]. The two vector forces in (51) are. in linear approximation: -Z/qqQ - I/qDJ~ , -Z'DqQ - l/DD·JD . (5T) (.58) (59 ) These can be calculated from maxent. using (29). (30). For a four-component mixture with _Y3 . _Y1 negligible. we get I/qD = 0 = l/Dq and l/DD 2\'2 5(hT)2 Li(-Yi /111 i) \-2 m1-Y1 o (60) (61) Contributions quadratic in Q. JD and p to the ~li can be calculated from the integrability condition and substituted into _-1 gi\'en by (56). For dissociation of :'\02 at 1125 E and Po =1 atm. we find the fractional change ':':::'J /J = (1/3)(P.cy /PO)2 produced by shear stress p.ry . \\-e estimate that a shear ;Cl 1;Cl lOG-I .. I 1 'J'J -0-') F .) rateVI1.r/U.I!'" S Isreqellrec tOlna.;:e.:...\./. "'1 -. or an equlInoar mixture of D2 and HCI at 600 E at the start of the reaction D:2 HC) -i- DH -;- DCI (62) we estimatE" that. to haw ':':::'J/J ",10-2 we need IvTi = 2.;) .10.5 Elm when the concentratioll gradient vanishes and IVPI i = 9.0 kg/m-i when vT = O. Very large gradients are needed for observability. References .L\Y:\ES. E. T.: Pity." Review. \'o!' 10(j. p. (j:20. 195, . . ]Ol'. D. C.·\5.·\5- \· .. \ZQl·EZ .. J. LEBO:\. G.: Rep!.". Progr. in Physics. \'o!. 51. p. l10·'5. 19:3:3. [:3] .]Ol'. D. :, C.-\. 5.-\. 5- V .. \ZQl'EZ . .J. LEBO:\. G.: Extended Irre\'ersible ThermodynaIll-ics. Springer. Berlin. 199:3. ROBERTSO:\. B.: Phys. Review. Vo!. 1-1-1. p. 1·51. 19(j(j. GR.-\D. H.: COmmlL7l. Pure Appl. Ala/h .. \'o!. :2. p. :3:31. [9-19. 184 R. E. i';ETTLETO;\' [6] NETTLETON. R. E.: J. Chem. Phys., Vol. 93, p. 824., 1990. [7] GRABERT. H.: Projection Operator Techniques in :-.ionequilibrium Statistical \le-chanics; chapter 4. Springer, Berlin, 1982. [8] NETTLETON, R. E. FREIDKIN, E. S.: Physica. Vol. A 1.58, p. 6.2, 1989. [9] Zc.;BAREV, D. );.: );onequilibrium Statistical Thermodynamics. Consultants Bureau, );ew 'York, 1974. [10] Lc.;ZZI, R. V"'SCONCELLOS, A. R.: Fortschr. Phys., Vol. 38. p. 887, 1990. [11] VASCONCELLOS, A. R. ALGARTE. A. C. - Luzl, R.: Physica. Vol. A 1.56, p. ·51 •. 1990. [12] :.iETTLEToN, R. E.: J. Chem. Phys. Vol. 97, p. 881.5,1992. [1:3] NETTLETor;, R. E.: Ann. Physik, Vol. 2, p. 490. 199:3. [14] :.iETTLEToN, R. E.: Phys. Review, Vol. E .54, p. 214 •. 1996. [1.5] GARcIA-CoLlN, L. S. DE LA SELVA. S. \1. T. - PI:\;A. E.: Phys. Letters. Vol. A 110. p. 36:3, 198.5. [16] :-.iETTLETON, R. E.: J. Phys. Chem., Vol. 100. p. 1100.5. 1996. 

NONLINEAR RECIPROCITY: STATISTICAL FOUNDATIONS AND APPLICATIONS TO NONLINEAR EFFECTS IN HEAT TRANSPORT AND CHEMICAL REACTIONS

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NONLINEAR RECIPROCITY: STATISTICAL FOUNDATIONS AND APPLICATIONS TO NONLINEAR EFFECTS IN HEAT TRANSPORT AND CHEMICAL REACTIONS

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