Nonequilibrium entropy of a gas

PHYSICAL REVIEW AVOLUME 45, NUMBER 12 15 JUNE 1992 None iuilib inm en opy o agas J.Ja ie B ey Fssica Teo ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, Sec o Su , E-41080Se illa, Spain And es San os Depa amen o de Fisica, Uni e sidad de Ex emadu a, E-06071 Badaj'oz, Spain (Recei ed 12 No embe 1991; e ised manusc ip ecei ed 27 Janua y 1992) The Bol zmann en opy o adilu e gas unde uni o m shea How is analyzed. The en opy a ia ion associa ed wi h iscous hea ing is e alua ed and compa ed wi h he local equilib ium exp ession. Fo in- e ac ion po en ials o he han he Maxwell po en ial, signi ican disc epancies a e ound ha can be e- la ed o he di e ence be ween he modynamic quan i ies, de ined om he en opy, and kine ic quan i- ies, de ined by means o local equilib ium. The disc epancies change, bu emain ele an , when a i icial ex e nal o ces a e in oduced in o de o c ea e an ideal s a iona y s a e. PACS numbe (s): 05.20.Dd, 05.60.+w, 51.10.+y, 05.70.Ln I. INTRODUCTION The ex ension o he modynamic ideas o a - om- equilib ium sys ems appea s as a undamen al and neces- sa y s ep owa d he de elopmen o agene al body o heo y o hose sys ems. F om a o mal poin o iew, i may be expec ed ha his would open he possibili y o looking o gene al ela ions simila o he ones exis ing a and nea equilib ium. These ela ions would apply o a wide ange o s a es, o ins ance s a iona y s a es. Bu , beyond he abo e possibili y, he e is abasic p oblem ha mus be sol ed o any heo y in o de o connec wi h wha is ac ually obse ed and measu ed in he eal wo ld: he empe a u e o a - om-equilib ium s a es mus be de ined in some way. I mus be no iced ha , concep ual- ly, one could a oid he use o he empe a u e, bo h in heo y and expe imen s, by employing he ene gy densi y ins ead. Ne e heless, expe imen alis s ha e ound i ui ul o cha ac e ize and classi y hei esul s by means o he empe a u e, wha e e i s meaning may be. In mos o he exis ing heo ies, he empe a u e is in- oduced by assuming some kind o local equilib ium, bu his assump ion is qui e dubious in a - om-equilib ium si ua ions. In ac , i is known ha in s ic local equilib- ium he e is no anspo . Amo e-consis en de ini ion o nonequilib ium empe a u e could be gi en i he de ini ion o some he modynamic po en ial, o ins ance he en opy, had been p e iously ex ended. The abo e commen s can also be applied o he p essu e. In spi e o he g ea deal o wo k de o ed o i , no gen- e al mic oscopic o mula ion o he en opy, ha ing he minimal equi emen s o dese e such aname, has been ound o nonequilib ium s a es. An excep ion e e s o dilu e gases obeying he Bol zmann equa ion (BE). In his case, anonequilib ium en opy S( ) can be de ined in e ms o he one-pa icle dis ibu ion unc ion, ( , , ), as S( )=— k&H ( )+cons , whe e kz is he Bol zmann cons an , H( ) is he quan i y H( )= d d ( , , )ln ( , , ), (1.2) and he cons an is simply p opo ional o he numbe o pa icles in he sys em. By using he symme y p ope - ies o his equa ion, Bol zmann himsel was able o p o e he H heo em, s a ing ha any ini ial dis ibu ion ap- p oaches equilib ium. Besides, he en opy S( ) g ows mono onically in ime, eaching i s maximum alue in he equilib ium si ua ion [1]. The heo em holds o cons an -ex e nal-po en ial ields, including he wall in- e ac ions, which a e eloci y independen [2]. A e y in- e es ing s onge e sion o he heo em has been gi en by Do man and an Beije en [9]. Fo bounda y condi- ions sa is ying a he mos a condi ion, hey p o ed a gene aliza ion o Clausius's o mula ela ing he change o en opy and he hea in e change o he sys em h ough he walls. F om knowledge o S( }i is possible o s udy nea - equilib ium s a es [4],bu o he bes o ou knowledge no use ul connec ion has been es ablished be ween Bol zmann's en opy and he quan i ies cha ac e izing a - om-equilib ium s a es. He e i will be seen ha his is no a i ial ask, e en o e y simple si ua ions. The e alua ion o S( } om Eqs. (1.1) and (1.2) o a gi en s a e equi es he knowledge o he dis ibu ion unc ion o he s a e; i.e.,one has o sol e he nonlinea BE. The only exac solu ions we a e awa e o co espond o homogeneous sys ems [5] o o dila ional lows no di ec ly ela ed o anspo p oblems [6]. The e a e o h- e cases whe e he dis ibu ion unc ion is no known, bu pa ial in o ma ion has been ob ained by compu ing a ini e numbe o i s momen s. They a e es ic ed o Maxwell*s in e ac ion and co espond o uni o m shea low [7],s eady hea low [8],and colo conduc i i y [9]. In he las yea s, anumbe o exac solu ions o he Bha naga -G oss-Kook (BGK) model kine ic equa ion [1] desc ibing a a ie y o in e es ing physical si ua ions ha e been de i ed [10,11]. The BGK equa ion can be 45 8566 1992 The Ame ican Physical Socie y NONEQUILIBRIUM ENTROPY OF AGAS 8567 conside ed as amodel o he BE wi h he collision e m eplaced by asingle- ime elaxa ion owa ds local equilib- ium. I keeps some o he main physical p ope ies o he BE,namely he conse a ion laws and he H heo em. Ade ini ion o en opy o nonequilib ium s eady s a es has ecen ly been p oposed by E ans [12]. In he low-densi y limi i educes o Bol zmann's en opy. Us- ing molecula -dynamics simula ion, E ans compu ed he en opy o alow-densi y gas o so disks unde uni o m shea low. In p inciple, his s a e is no s eady due o iscous hea ing [13],bu a he mos a o ce is in oduced in Re . [12] in o de o keep he ene gy cons an . F om he en opy da a, E ans e alua ed he modynamic em- pe a u es and p essu es, inding signi ican disc epancies wi h he alues o he co esponding kine ic quan i ies, he la e being de ined om he equipa i ion o ene gy and he p essu e enso , espec i ely. In his pape , we will analyze some p ope ies o he Bol zmann en opy o asys em unde uni o m shea low, using bo h he BEand he BGK equa ion. Because he solu ion o he BE o his s a e is only known o Maxwell molecules, ou esul s will be much mo e limi ed in he case o he BE. Ne e heless, i has been shown by compu e simula ion ha he BGK equa ion is aqui e good app oxima ion o he BE o he uni o m shea low, e en a aquan i a i e le el [14]. The dis ibu ion unc ion o he idealized s eady s a e o uni o m shea low has been de i ed in he BGK app oxima ion [15]. He e we ob ain he i s ew e ms o he expansion o he en opy in powe s o he ( educed) shea a e and com- pa e, a aquali a i e le el, wi h he esul s epo ed by E ans. As is o en he case, Maxwell molecules lead o a peculia beha io . In pa icula , he he modynamic and kine ic empe a u es a e he same o ha in e ac ion. This p ope y is also ue when he exac BEis used. To a oid misunde s andings, i is wo h men ioning ha he ole played by he he mos a o ces is no neu al, in he sense ha he ela ionship be ween esul s ob ained om he BEwi h and wi hou he he mos a is no simple o molecules o he han Maxwell molecules. The plan o he pape is as ollows. In he nex sec ion, he exis ence o an en opy unc ion o he uni o m shea low is pos ula ed and he modynamic empe a u e and p essu e a e de ined. Fo adilu e gas, hey can be easily w i en in e ms o he dis ibu ion unc ion i he Bol zmann de ini ion o en opy is adop ed. The case o he BE o Maxwell molecules is exp1ici ly conside ed. Using apowe s-se ies expansion in he shea a e, he BGK equa ion is sol ed in Sec. III, and he he mo- dynamic quan i ies a e ela ed o he kine ic ones. Also, i is shown ha he local equilib ium assump ion o he en opy a ia ion is no e i ied in gene al. The he mo- s a ed low is discussed in Sec. IV, while he inal sec ion is de o ed o some commen s. II. NONKQUILIBRIUM ENTROPY IN THE UNIFORM SHEAR FLOW Mac oscopically, he uni o m shea low (USF) is cha ac e ized by acons an densi y n, and by he uni o - mi y o all he hyd odynamic ields excep one o he componen s o he local eloci y u ha has alinea p o ile along adi ec ion pe pendicula o i . We will ake =c (2.1) whe e ais he shea a e. In he absence o an ex e nal he mos a o ce, wo k is done on he sys em so ha he s a e is ime dependen . Le us assume ha he e exis s o his s a e anone- quilib ium en opy ha is an ex ensi e unc ion o he numbe o pa icles, he olume, and he in e nal ene gy, and pa ame ically depends on he shea a e. The in e - nal ene gy is independen o u, and, he e o e, he en o- py densi y will be uni o m in he sys em. We can w i e s=s(n, e,a), (2.2) whe e sand ea e he en opy and in e nal ene gy pe pa icle, espec i ely. Now, anonequilib ium he mo- dynamic empe a u e T,hand anonequilib ium he mo- dynamic p essu e p,ha e de ined as T h Bs Bs 7 h= "~ e,a (2.3) i.e.,using he same ela ions as in equilib ium. O cou se, one can also in oduce aspeci ic nonequilib ium quan i y conjuga ed o he shea a e Bs ,h=— T,hBa n, e (2.4) In his way, we a i e a he gene alized Gibbs ela ion [12,16] de =T,hds+n p,hdn + hda .(2.5) The alidi y o he abo e scheme lies on he exis ence o an en opy unc ion sa is ying some minimal equi e- men s. In pa icula , i mus educe o he equilib ium alue o a=0. Besides, i we wan he de ini ions gi en in Eqs. (2.3) o be use ul, hei ela ion wi h mo e s an- da d de ini ions and wi h eal and compu e expe imen s has o be es ablished. Now we adop akine ic- heo y s andpoin and consid- e asys em desc ibed by he BE. The exis ence o he USF s a e is consis en wi h he BE [6,7,13,14] and also wi h he BGK equa ion [10,15]. The dis ibu ion unc- ion o agas unde USF is a unc ion o he o m (V, ), whe e V= — u, (2.6) i.e.,all he posi ion dependence occu s h ough he pecu- lia eloci y wi h espec o he local low eloci y [7,13]. The in e nal ene gy densi y is gi en by ne = dV—, 'mV , (2.7) and coincides wi h he kine ic ene gy densi y in he La- g angian ame. Fu he mo e, he kine ic empe a u e Tk is de ined as p opo ional o eby means o he local equilib ium ela ion (2.8) 8568 J.JAVIER BREY AND ANDRES SANTOS 45 Le us in oduce he dimensionless eloci y 2kB Tk V=V m and he co esponding dis ibu ion 3/2 2kB Tk '= nm (2.9) (2.10) is di e en om he kine ic p essu e pk. Mo e speci ically, p h na " h . Pk Pk (2.17) The abo e discussion can be pu in adi e en bu closely ela ed o m. The ime de i a i e o he Bol zmann en opy o he USF is om Eq. (2.11) (2k/ TI, Im) s=— =kB ln — h' +C, Nn(2.11) whe e Nis he numbe o pa icles, Cis acons an , and h'= JdV' 'ln ' .(2.12) In e ms o hese, he Bol zmann en opy, Eq. (1.1), o he USF eads d(ln Tk) d Td(ln Tk) dh* B ds d 2 — — kB — 3k BT h which o Maxwell molecules educes o ds, d(ln Tk) =-'kB (2.18) (2.19) As long as he e is ano mal solu ion o he BE o he USF, Eq. (2.11}p o ides an exp ession o he en opy ha has he dependence assumed in he he modynamic desc ip ion. No ice ha he use o Tk is ama e o con- enience, since i can always be elimina ed in a o o e by using Eq. (2.8}. F om Eqs. (2.3) and (2.11)we ha e '— 1 T,h=Tk 1', Tk ~T—— k(2.13) 1+ Bh '1+nBh '/Bn 1— — 'TkBh '/8Tk (2.14) (2.15) whe e Jis he Bol zmann collision ope a o and y=— ', A,nsinh [6cosh '[1+9(a/A, n) ]], (2.16) A, being acons an . Since e(o Tk) is he only ime- dependen pa ame e in he USF, h*canno depend on Tk. The e o e, o Maxwell molecules T,h=Tk. The dependence o hon he densi y is no known in de ail, bu Eq. (2.15) shows ha i is coupled o he dependence on a h ough he scaled a iable a/n. I ollows ha p,h whe e in he las equali y o Eq. (2.14) he kine ic p es- su e is de ined as one- hi d he ace o he p essu e en- so , i.e.,pk =nkBTk. I ollows ha he he modynamic empe a u e is equal o he kine ic empe a u e i and only i Bh'IdTk=0. I in addi ion i is Bh "IBn =0, he e is also ag eemen be ween he he modynamic p es- su e and he kine ic p essu e. F om ama hema ical poin o iew, p,h=pk, e en i T, AhT ik h' is a unc- ion o he scaled a iable n/TP . Fo Maxwell molecules, asolu ion o he BE co e- sponding o he USF s a e has been ound using he mo- men me hod [7,13]. Al hough he explici o m o ' has no been de e mined, i is known ha i does no de- pend on ime explici ly. In ac , ' obeys he equa ion [13] This is he exp ession assumed by he local equilib ium hypo hesis. The e o e, al hough he Bol zmann en opy does no ha e he unc ional o m assumed by local equi- lib ium, he la e co ec ly ep oduces i s ime a ia ion in he USF. Fo in e ac ion po en ials o he han Maxwell po en- ial, no solu ion o he BE has been ound o he USF. Fu he mo e, he ans o ma ion p ope ies o he Bol zmann collision ope a o show ha h'is expec ed o depend explici ly on ime [13]. Howe e , ealis ic es i- ma es can be ca ied ou using he BGK model kine ic equa ion. This is done in he nex sec ion. III. UNIFORM SHEAR FLOW FROM THE BGK MODEL The BGK equa ion o he USF is [15] d " 1dlnTk 8(V )Va ' B 2d BV' 8V„' whe e 0is he dimensionless Maxwellian dis ibu ion o =n ~exp( — V' )(3.2) a a(3.3) and Eq. (3.1) becomes and gis an e ec i e collision equency ha is linea in he densi y. The only o he dependence o gis on he empe a u e. He e we will ake g~ Tk wi h 0~a~ —, ', which co esponds o pu ely epulsi e powe -law po en- ials, including Maxwell molecules (a=0) and ha d sphe es (a=—, ') as limi cases. Taking in o accoun ha Tk inc eases mono onically due o iscous hea ing, he ime dependence o *can be accoun ed o h ough he educed shea a e 45 NONEQUILIBRIUM ENTROPY OF AGAS 8569 1d(ln Tk) () + I() aa' +—(V' ') Ba' 2BV' — a'V' = — '+ 0 .(3.4) "a * X We no ice ha he alue o aonly appea s a e he second-o de co ec ion in a o he local equilib ium dis ibu ion. Once he expansion o 'is known i is a ma e o simple algeb a o de i e he co esponding ex- pansion o h'de ined in Eq. (2.12). The esul is In o de o close his equa ion we need an exp ession o he e olu ion o Tk. This can be easily achie ed by ak- ing momen s in he equa ion i sel . Since he de ails ha e been al eady gi en elsewhe e [15], we me ely quo e he esul s. One inds h'=h *+a*2h ~+a~4h '+ 0247 wi h h* =— —, '— —, 'inn, 0 h2 — —, ', (3.13) (3.14) (3.15) 1d(ln Tk) =— a' }'(a'), d 3(3.5) h4 =— — '+ a. 43(3.16) wi h P„ i'(a ')=-nkg Tka (3.6) Upon de i ing he abo e exp essions use has been made o he p ope ies dV' „'=0 whe e P„„deno es he componen o he p essu e enso . The unc ion }'(a') is agene alized shea iscosi y ha e i ies aclosed nonlinea second-o de di e en ial equa- ion (see Eq. (4.1) in Re . [15]).The solu ion o his equa- ion co esponding o he hyd odynamic egime, i.e.,in he long- ime limi , can be cons uc ed nume ically o all alues o a' [15]. He e we will es ic ou sel es o he i s ew e ms o he expansion in powe s o a': and dV' kln o =0 (3.17) o k &1, which ollow di ec ly om he no maliza ion o 'and he exp ession o 0. The ac ha only e en powe s o a'a e p esen is aconsequence o he symme- y o he p oblem ha implies l'(a')= I— — , '(2— a)a' +— ', (7— 13a+4a )a' + (3.7) '( V„', V», V;;a'}= '( — V„*,V»', V;;— a') = '( V„',— V', V,';— a').(3.18) This se ies has been shown o be only asymp o ic o any alue o ao he han ze o [15]. In he case o Maxwell molecules (a =0),Eq. (3.4) educes o Taking in o accoun ha Bh' Ba' Bh' aBh' =— a BTk dTk Ba' Tk Ba' (3.19) ,(Vlll 0) aAVE 0— 4+ 0 gBV' BV„' (3.8) he exp ession o he he modynamic empe a u e, Eq. (2.13), can be ew i en as whe e y/g=y'= —, 'a' i'(a')= — ', sinh [—, 'cosh '(1+9a' )] .T,h(a )=Tx 1+—,&a ,Bh' Ba' (3.20) No ice he simila i y be ween Eqs. (2.15) and (3.8). Al- hough he solu ion o Eq. (3.8}is known [15],i will no be needed he e. In he ollowing, ou aim will be o ind an expansion o he Bol zmann en opy in powe s o he educed shea a e. The e o e, in he spi i o he Chapman- Enskog p ocedu e we w i e 4(V4 4) 4(V4)+a 4 (V )+ 42 8(Vo) +a' 3(V'}+.(3.9) He e we ha e al eady aken in o accoun ha o a'=0 he solu ion o Eq. (3.4) is gi en by Eq. (3.2). Subs i u- ion o Eqs. (3.7) and (3.9) in o Eq. (3.4) yields and subs i u ion o Eqs. (3.13)— (3.16) gi es Th(a')=TI F(a'), whe e (3.21) F(a")=1— —, 'aa' +—, '(1— 2a)aa'~+ (3.22) Fo a=0 (Maxwell molecules) i is F=1, and one e- co e s he esul ound in he p e ious sec ion o he BE. Al hough he expansion in Eq. (3.22) is only asymp o ic, i clea ly shows ha o a bi a y in e ac ion po en ials he e a e disc epancies be ween he kine ic and he mo- dynamic de ini ions o empe a u e. The p essu es a e s udied in asimila way. We ha e ;(V*)=— 2V„' V»* 0(V'), 2(V*)=[1— — 'V' — 2V' (1— 2V* )] 0(V'), 3(V*)=4V V[V (3— 2V )+— 'V* — —, '(5+a)] 0(V') . (3.10) (3.11) (3.12) Bh' a* Bh* Bn n and he e o e Eq. (2.14) eads 1— a'Bh */Ba ' 1+— 'aa *Oh "/Ba ' 3 (3.23) (3.24) 8570 J.JAVIER BREY AND ANDRES SANTOS 45 The i s ema k is ha , e en o Maxwell molecules, p,z is di e en om he kine ic p essu e pk =nk~Tk. Using he expansion gi en by Eq. (3.13) we ge leads o P 3nk~ Tk (4.3) p,h(a*)=p,M(a*), wi h M(a*)=l— (1+— ', a)a* +(1— 2a)(1+— ', a)a' + (3.25) (3.26) )s, d(ln Tk) — =— 'k~ G(a'), (3.27) Applying Eq. (2.18) we ind o he a e o change o he Bol zmann en opy densi y I can be shown [18] ha o any a bi a y ini ial dis i- bu ion *(V*,O), he solu ion o Eq. (4.2) app oaches a s a iona y o m ,*(V*) ha obeys Eq. (3.8). Thus, he educed dis ibu ion unc ion wi h he mos a o ce o a bi a y in e ac ion law is he same as ha o Maxwell molecules wi hou a he mos a when bo h a e w i en in e ms o he educed quan i ies V* and a*. This is a peculia p ope y o he BGK equa ion, and i is no held by he BE [13]. By making a=0in Eqs. (3.13)— (3.16) we now ha e G(a")=[F(a')] h*=— — '— — 'ln ~+— 'a* — — 'a*4+ S22 2 4(4.4) =1+— 'aa' — (— '— — "a)aa* + 3 3 9(3.28) Applying he same p ocedu e as in he p e ious sec ion, Eqs. (3.20}and (3.24), one ge s The e o e, he local equilib ium assump ion o he en- opy change is no e i ied o powe -law in e ac ion po- en ials o he han he Maxwell po en ial. As poin ed ou be o e, his is adi ec consequence o he di e ence be- ween T,&and Tk. Finally, he pa ame e ,&, de ined in Eq. (2.4), becomes wi h and T,„=T„F,(a'), F,(a*)=1— — 23aa* +— ', (1+— ', )a)ua" + (4.5) (4.6) k~ T h whe e k~ Tk a*R(a*), (3.29) p,„=pkM, (a *), wi h (4.7) F(a')— M(a') R(a*}= =1— (1— 2a)a" + 42 M, (a*)=1— (1+—, 'a)a* +(1+— 23a) a* + Also, Eq. (3.29) now becomes (4.8) (3.30) ka Tk h =a*R,(a*), (4.9) The se ies expansions ob ained in his sec ion show quali a i ely he in luence o bo h he po en ial pa ame e aand he shea a e a*on T,&, p,z, and ~,&. O cou se, a mo e ca e ul analysis would be needed in o de o e alu- a e F(a') and M(a') beyond he limi o small shea a es. wi h (4.10) R,(a*)=1— (1+— ', a)a* + IV. STATIONARY FLOW The USF is no as a iona y s a e due o he inc ease in ene gy associa ed wi h iscous hea ing. In o de o ge an isoene ge ic shea low, ex e nal d ag o ces mus be added o ex ac ene gy uni o mly om he gas [17]. Mo e p ecisely, ahomogeneous o ce I' p opo ional o he peculia eloci y Vo each pa icle is in oduced: (4.1) F=— myV . The BC+K equa ion o he USF including his noncon- se a i e o ce is 1d(ln T„} .(V* )aV* „ *— .(V* )= —P * o ).—— C} a * (4.2) The pa ame e yis de e mined om he condi ion ha he in e nal ene gy o he sys em emains cons an . This The s a iona y dis ibu ion unc ion gi en by Eq. (4.4) is analy ic a a' =0 [15]and, consequen ly, all he abo e se ies a e con e gen . Ne e heless, hei adius o con- e gence is no known, al hough i is p esumably he same as ha o g*(a*), namely ~a'~=&2/3. Fo a=0 (Maxwell molecules) i is F,=F=1, M, (a*)=M(a*),and R,(a )=R (a*), i.e., he ela ion- ship be ween he he modynamic and he kine ic quan i- ies is no a ec ed by he d ag o ce. Fo any o he in- e ac ion po en ial, he ela ions a e di e en wi h and wi hou he mos a o ces. This is amani es a ion o he non-neu al ole hey play [13]. Al hough he solu ion ,*(V*)o he BCiK equa ion is known o a bi a y shea a es [15],an explici exp es- sion o he co esponding unc ion h,*(a*)does no seem easible. Ne e heless, we can gain insigh in o i s main quali a i e ea u es by using in o ma ion heo y (o he maximum-en opy me hod [19)) o ge alowe es ima e. Mo e speci ically, we seek he dis ibu ion unc ion ,*, ha minimizes he unc ional h, ,subjec o he cons ain o ep oducing he ac ual p essu e enso . Asimple cal- cula ion yields 45 NONEQUILIBRIUM ENTROPY OF A GAS 8571 2,95 IIIIIIIIII I IIIII I IIIIII I I I 'IIIIIIIIII IIIIIIII I — 3.00 (3.29), he ac ha R,'mono onically dec eases does no mean ha so does ,h. In ac , in o ma ion heo y shows ha ,h eaches amaximum a a*=0.79, dec easing mono onically he ea e . V. COMMENTS AND DISCUSSION — 315 — 3.20 325 III I I I I I IIIIIIIIII I IIII I IIII I IIIIIIII I IIIII I I I II 000.20.40.60.8'1.0 a" FIG. 1. Reduced en opy unc ion o agas unde s a iona y uni o m shea low acco ding o in o ma ion heo y by using he esul s ob ained om he BGK equa ion o he p essu e enso . ; T(V )=m (de P ) Xexp[ (P ');— 1V; VJ'], h;,T(a')= — — ', — — ', Inn — —, 'ln(de PJ ), (4.11) (4.12) whe e P;*=P,J. /p.kis he educed-p essu e enso . I s de e minan is 1+3y* de P (I+2y') (4.13) 00.5 ~CA 0.3 0000 02 0.40.6 a" 08 1.0 FIG. 2. In o ma ion heo y es ima es o F,(a )(solid line), M,(a )(dashed line), and R,(a )(do ed line) o agas o ha d sphe es. wi h y' gi en below Eq. (3.8). The unc ion h;&T(a') is exac up o o de a', bu he coe icien o a* in he powe -se ies expansion is — —, ' a he han he exac alue Figu e 1shows h; T(a*) in he ange 0~a"~l. The cu e ep esen ing he ac ual unc ion h;(a*) would lie abo e he one plo ed in he igu e. F om h,*,T(a") one can ge decen es ima es o F,*,M,*,and R,'. These unc ions a e plo ed in Fig. 2 o ha d sphe es (a=—, '). We obse e ha p,hdec eases as he shea a e inc eases mo e apidly han T,hdoes. Due o he a* ac o in Eq. The Bol zmann de ini ion o en opy seems o be one o he mos sensible choices o adilu e gas ou o equilib i- um. The modynamic empe a u e and p essu e can hen be de ined in e ms o he Bol zmann en opy by ex end- ing he equilib ium ela ions. On he o he hand, akine - ic empe a u e is de ined as p opo ional o he in e nal ene gy, and akine ic p essu e, desc ibing he in e nal o ces in he luid, is de ined om he ace o he p es- su e enso . The esul s in his pape show ha he ela- ionship be ween he modynamic quan i ies, de ined in e ms o he Bol zmann en opy, and local equilib ium o kine ic quan i ies is no simple in a - om-equilib ium si ua ions. The complexi y is associa ed wi h he in i- ca e dependence o he dis ibu ion unc ion on he e- duced shea a e. Besides, he si ua ion does no imp o e when a i icial o ces a e in oduced o c ea e an ideal s a iona y s a e. On he con a y, one has o cope wi h he added p oblem o he ela ionship be ween quan i ies measu ed in sys ems wi h and wi hou a he mos a . E ans [12]pe o med amolecula -dynamics simula ion o asys em o so disks subjec o an isoene ge ic shea low. The densi y o he sys em was small and he com- pu ed he Bol zmann en opy a se e al shea a es, den- si ies, and ene gies. Using hese da a, he ob ained alues o he he modynamic empe a u e and p essu e, which he compa ed wi h he co esponding kine ic alues. The quali a i e beha io ound in Re . [12]is qui e simila o he one ob ained he e. In pa icula , T,hand p,hwe e smalle han Tk and pk, espec i ely, he disc epancy be- ing bigge in he case o he p essu e. Also, he en opy was ound o dec ease wi h he shea a e. Howe e , some quali a i e di e ences mus be men- ioned. Wi hin he accu acy o his da a, E ans go a quasilinea dependence o he en opy densi y as a unc- ion o he shea a e, which appa en ly ex ended o he limi o he shea a e going o ze o, while he analysis ca ied ou he e shows aquad a ic dependence in ha limi . As poin ed ou by E ans himsel , his simula ion alues o he shea a e a e p obably beyond he egion whe e he quad a ic beha io is dominan . This ac ex- plains also E ans's obse a ion ha ,„dec eases wi h he shea a e. E ans also conjec u ed ha he he mo- dynamic p essu e is equal o he minimum eigen alue o he p essu e enso . On he o he hand, ou analysis, based in he BGK equa ion, shows ha , in he he mo- s a ed case, he minimum eigen alue is p3=pk(1 — ~a ~+—, 'a' +), which is clea ly di e en om Eqs. (4.7) and (4.8). I mus be s essed ha he poin s add essed in his pa- pe a e no me ely o mal. The meaning o many o he calo ime ic measu es ca ied ou a om equilib ium is no clea , since hey a e based on equilib ium ela ions. I is also impo an o ealize ha ace ain deg ee o am- bigui y could exis in he de ini ion o nonequilib ium 8572 J.JAVIER BREY AND ANDRES SANTOS 45 he modynamic quan i ies. This ambigui y is ela ed o se e al possible choices o he nonequilib ium pa ame- e s (such as g adien s, ex e nal ields, e c.). In he con- ex o he uni o m shea low, i we had chosen o de ine T,hand p,hby Eq. (2.3), excep ha a* is kep cons an ins ead o a, hen we would ha e ob ained T,h=Tk, Pa =Pa. The e a e some p ope ies ha one would like he en- opy o ha e. Fo ins ance, one could expec ha none- quilib ium s a iona y s a es co espond o amaximum o he en opy when he app op ia e bounda y condi ions a e imposed. Also, i should be in e es ing i he en opy would inc ease uni o mly un il eaching s a iona i y. This would be ap oo o he s abili y o he s a iona y s a e. We ha e no been able o p o e any o he abo e p ope ies o he Bol zmann en opy o adilu e gas un- de uni o m shea low, e en in he BGK app oxima ion. Gi en he peculia i ies o he nonequilib ium s a es conside ed he e, especially he ideal s a iona y one, we plan o p esen in he nea u u e asimila analysis o he s eady hea low. ACKNOWLEDGMENTS Pa ial suppo om he Di eccion Gene al de In es igacion Cien i ica yTecnica (Spain) h ough G an Nos. PB 89-0618 (J.J.B.)and PS 89-0183 (A.S.)is g a e- ully acknowledged. [1]C. Ce cignani, The Bol zmann Equa ion and I s Applica ions (Sp inge -Ve lag, New Yo k, 1988). [2] G. E. Uhlenbeck and G. W. Fo d, Lec u es in S a is ical Mechanics (Ame ican Ma hema ical Socie y, P o idence, 1963). [3]J. R. Do man and H. an Beije en, in S a is ical Meehan ics, Pa B, edi ed by B. J. Be ne (Plenum, New Yo k, 1977),pp. 65-179. [4] M. Ma eschal, Phys. Re . A29, 926 (1984). [5]M. H. E ns , Phys. Rep. 78, 1(1981). [6]C. T uesdell and R. G. Muncas e , Fundamen als o Maxwell's Kine ic Theo y o aSimple Mona omic Gas (Academic, New Yo k, 1980). [7] E. Ikenbe y and C. T uesdell, J. Ra . Mech. Anal. 5, 55 (1956);5, 128 (1956). [8]E. S. Asmolo , N. K. Makashe , and V. I. Nosik, Dokl. Akad. Nauk SSSR 249, 577 (1979)[So . Phys. — Dokl. 24, 892 (1979)]. [9]V. Ga zo and A. San os, J.S a . Phys. 65, 747 (1991). [10]R. Zwanzig, J.Chem. Phys. 71, 4416 (1979). [11]A. San os, J. J. B ey, C. S. Kim, and J. W. Du y, Phys. Re . A39, 320 (1989);C. S. Kim, J.W. Du y, A. San os, and J. J. B ey, ibid. 40, 7165 (1989);J.J. B ey, ASan os, and J. W. Du y, ibid. 36, 2842 (1987);M. Alaoui and A. San os (unpublished). [12]D. J.E ans, J.S a . Phys. 57, 745 (1989). [13]J. W. Du y, A. San os, J. J. B ey, and R. F. Rod iguez, Phys. Re . A33, 459 (1986). [14]J. Gomez-O do ez, J. J. B ey, and A. San os, Phys. Re . A41, 810 (1990). [15]A. San os and J.J.B ey, Physica A174, 355 (1991). [16]D. J. E ans and H. J. M. Hanley, Phys. Le . A80, 175 (1980). [17]W. G. Hoo e , A. J. C. Ladd, and B. Mo an, Phys. Re . Le . 48, 1818 (1982); D. J. E ans and G. P. Mo iss, Compu . Phys. Rep. 1,299 (1984). [18]J.J. B ey and J.W. Du y (unpublished). [19]L. R. Mead and R. Papanicolau, J. Ma h. Phys. 25, 2404 (1984).