Nonlinear transport effects in mass separation by effusion

Transcription

1 Journal of Statistical Mchanics: Thory and Exprimnt () P34 ( pags) Nonlinar transport ffcts in mass sparation by ffusion Pirr Gaspard and David Andriux Cntr for Nonlinar Phnomna and Complx Systms, Univrsité Libr d Bruxlls, Cod Postal 3, Campus Plain, B-5 Brussls, Blgium Gnralizations of Onsagr rciprocity rlations ar stablishd for th nonlinar rspons cofficints of ballistic transport in th ffusion of gasous mixturs. Ths gnralizations, which hav bn stablishd on th basis of th fluctuation thorm for th currnts, ar hr considrd for mass sparation by ffusion. In this kintic procss, th man valus of th currnts dpnd nonlinarly on th affinitis or thrmodynamic forcs controlling th nonquilibrium constraints. Ths nonlinar transport ffcts ar shown to play an important rol in th procss of mass sparation. In particular, th ntropy fficincy turns out to b significantly largr than it would b th cas if th currnts wr supposd to dpnd linarly on th affinitis. I. INTRODUCTION For long, th undrstanding of nonquilibrium procsss has bn limitd to th linar rgim clos to th thrmodynamic quilibrium with rsults such as th Onsagr rciprocity rlations [. Nowadays, grat advancs hav bn carrid out sinc th discovry of tim-rvrsal symmtry rlations for larg-fluctuation proprtis on th basis of dynamical systms thory [ 4. Rcntly, ths rlations hav bn xtndd to stochastic procsss, as wll as to opn quantum systms [5 8. On of th most dtaild vrsions of such rlations is th fluctuation thorm for all th currnts flowing through a nonquilibrium systm [9,. Thanks to this fluctuation thorm, rlationships hav bn obtaind that gnraliz th Onsagr rciprocity rlations from linar to nonlinar rspons cofficints [ 3. Ths rlations hav alrady bn applid to linar chmical ractions [4, to Brownian sivs and molcular machins [5, and to lctronic quantum transport [6, 7. Th fact is that many nanosystms typically dvlop highly nonlinar rsponss to multipl nonquilibrium driving forcs, as it is th cas in lctronic transistors for instanc [. In such dvics, svral currnts ar coupld togthr so that diffrnt driving forcs may induc th rspons of on particular currnt. Sinc thy only hold in linar rgims, th Onsagr rciprocity rlations ar of limitd us to study such nonlinar dvics and w may xpct that thir rcntly discovrd gnralizations to nonlinar rgims would b valuabl to undrstand th coupling btwn transport proprtis in far-from-quilibrium systms [ 3. Th purpos of th prsnt papr is to addrss this issu for th procss of mass sparation by ffusion. In this procss, a mixtur of two or mor gass is forcd to flow from on rsrvoir to anothr through a small orific with a siz smallr than th man fr path [8. Effusion coupls th hat and particl flows and kintic thory has shown how ths flows dpnd on th mass of th particls [8. For a binary mixtur, thr currnts ar thus coupld togthr: th hat flow, th total mass flow, and th diffrntial flow of on spcis with rspct to th othr. Svral coupling ffcts ar possibl btwn ths thr flows and, in particular, th sparation btwn particls of diffrnt masss inducd by ithr th tmpratur or th prssur diffrnc btwn both rsrvoirs. Although a fluctuation thorm has rcntly bn obtaind for ffusion [ 4, nithr its xtnsion to mass sparation nor its consquncs on th nonlinar rspons cofficints hav bn rportd on. Ths cofficints ar worth to b studid bcaus thy oby rmarkabl rlationships that ar th consqunc of microrvrsibility [ 3, as th Onsagr rciprocity rlations ar in linar rgims [. Furthrmor, ths cofficints control th coupling btwn th diffrnt currnts far from quilibrium and thy influnc th fficincy of mass sparation. In this contxt, a concpt of ntropy fficincy was introducd as th ratio btwn th contribution to ntropy production causd by mixing of th two constitunts ovr th on causd by hat flow [5, 6. It turns out that th ntropy fficincy may b significantly highr in nonlinar rgims than in linar ons, as w show hr blow. Th papr is organizd as follows. In Sction II, w prsnt th fluctuation thorm for all th currnts in th ffusion of gas mixturs. Th gnrating function of th counting statistics of th diffrnt flows is obtaind from kintic thory. In Sction III, th man currnts and thir statistical cumulants ar calculatd from th gnrating function and w vrify that th nonlinar rspons cofficints indd oby rlationships prviously rportd in Rfs. [ 3 and which find thir origin in microrvrsibility. In Sction IV, th rsults ar applid to th mass sparation of two spcis. Th ntropy fficincy is shown to b significantly highr than possibl if linarity was assumd. Th conclusions ar drawn in Sction V.

2 II. FLUCTUATION THEOREM FOR CURRENTS IN THE EFFUSION OF GAS MIXTURES A. Thrmodynamic formulation W considr two gasous mixturs at diffrnt tmpraturs, prssurs and concntrations in two rsrvoirs R and R sparatd by a wall, through which a small por allows th flow of particls. Th rsrvoirs ar assumd to b so larg that th systm is maintaind out of quilibrium in a stady stat. Altrnativly, th far nds of both rsrvoirs can b closd by pistons moving in such a way that th thrmodynamic conditions of th gasous mixturs ar kpt stationary (s Fig. ). R T, P, {n i } V, E, S R ' T', P', {n' i } V', E', S' FIG. : Schmatic rprsntation of th stationary ffusion of gasous mixturs through a small por btwn two rsrvoirs R and R whr th thrmodynamic conditions ar kpt fixd by moving th pistons. Ths conditions ar dfind by th tmpratur T, th prssur P, and th particl dnsitis {n i} c. V, E, and S dnot th volum, th nrgy, and th ntropy. Th prim rfrs to th sam quantitis in th right rsrvoir. Th gass ar supposd to b idal mixturs of c monoatomic spcis so that th particl dnsitis {n i } c and {n i }c ar rlatd to th prssurs and tmpraturs by th prfct-gas quation of stat: P = n i kt and P = n ikt, () whr k is Boltzmann s constant. Th currnts of nrgy and particls from th lft rsrvoir to th right on ar dfind by J E Ė = +Ė, () J i Ṅi = +Ṅ i (i =,,..., c), (3) in trms of th nrgis E and E, and particl numbrs N i = n i V and N i = n i V in both rsrvoirs. Th stationary assumption that th prssurs, tmpraturs, and particl dnsitis rmain constant in both rsrvoirs implis that th volums of both rsrvoirs should satisfy th conditions V = c J i c n i c and V = + J i, (4) c n i so that P T V + P V T =. (5) Sinc th systm dos not xchang ntropy with its nvironmnt, d S/dt =, th tim drivativ of its total ntropy S tot = S + S givs th ntropy production of ffusion ds tot /dt = d S/dt + d i S/dt = d i S/dt. Using Gibbs rlation de = T ds P dv + c µ idn i for both gasous mixturs and Eq. (5), th ntropy production of ffusion is givn by d i S k dt = A E J E + in trms of th affinitis or thrmodynamic forcs dfind as [7 3 A i J i, (6) A E kt kt, (7) A i µ i kt µ i kt (i =,,..., c), (8)

3 whr {µ i } c and {µ i }c ar th chmical potntials of th diffrnt spcis composing th gasous mixturs in both rsrvoirs. Sinc th gass ar hr supposd to b idal and composd of monoatomic spcis, th affinitis of th particl flows ar givn by [ ( ) n i T 3/ A i = ln n (i =,,..., c). (9) i T If w tak th tmpratur and dnsitis of th rsrvoir R as th rfrnc conditions, th sam quantitis in th othr rsrvoir R can b xprssd in trms of th affinitis as kt = kt + kt A E, () n i = n i xp( A i ) ( + kt A E ) 3/ (i =,,..., c). () In this rgard, th affinitis ar th control paramtrs of th nonquilibrium constraints. Now, w still nd to dtrmin th hat and particl currnts ()-(3). For this purpos, kintic thory will allow us to tak into account th mchanistic aspcts of ffusion. 3 B. Kintic-thory formulation Effusion procds as a random squnc of particl passags through th por in both ways btwn th two rsrvoirs: R R. Ths random vnts ar statistically indpndnt of ach othr as long as th intraction btwn th particls is supposd to b ngligibl. In this rspct, th siz of th por should b smallr than th man fr path of th particls in th gasous mixtur. Thrfor, th ffcts of binary collisions btwn th particls can b ignord and th flow of particls across th por can b considrd to b purly ballistic. W furthr assum that th thicknss of th wall sparating both rsrvoirs is smallr than th diamtr of th por so that most particls ar crossing th por in straight trajctoris with ngligibl bouncing or intraction insid th por [3. Undr ths conditions, th flow of particls can b dtrmind following standard mthods from th kintic thory of gass [8. Th man numbr of particls passing th cross-sction ara σ of th por during som tim intrval can b calculatd in trms of th Maxwllian vlocity distribution and th particl dnsity in th rsrvoir, from which th particls arriv. Sinc ach particl of vlocity v carris th kintic nrgy ɛ = m i v /, th flow of particls also dtrmins th hat flow. Kintic thory dscribs th transport of nrgy and particls btwn th rsrvoirs as a Markovian stochastic procss [8. Th probability P t ( E, N) that an nrgy E and N = { N j } c j= particls ar transfrrd in th dirction R R during th tim intrval t is ruld by th mastr quation: d dt P t( E, N) = + P t ( E, N) dɛ W (+) i (ɛ) P t ( E ɛ, N i ) dɛ W ( ) i (ɛ) P t ( E + ɛ, N + i ) [ dɛ W (+) i (ɛ) + W ( ) i (ɛ), () whr w us th notation i = {δ ij } c j= with th Kronckr symbol δ ij and whr th transition rats ar givn by W (+) i (ɛ) = W ( ) i (ɛ) = σ n i πmi kt σ n i πmi kt ɛ ( kt xp ɛ kt ɛ kt xp ), (3) ( ɛ kt ), (4) for th particl flows R R and R R rspctivly [8. W notic that ths transition rats ar rlatd to ach othr and to th affinitis (7)-(8) by W (+) i (ɛ) W ( ) i (ɛ) = xp (ɛa E + A i ) (i =,,..., c), (5)

4 as th consqunc of th nonquilibrium constraints on th flow of particls btwn both rsrvoirs. At th quilibrium thrmodynamic stat whr th affinitis vanish, th two transition rats ar qual and th conditions of dtaild balancing ar rcovrd. Th rlations (5) ar th analogus for th prsnt systm of Schnaknbrg s conditions dfining th macroscopic affinitis driving gnral nonquilibrium stochastic procsss [9, 3. W introduc th gnrating function of th currnts and thir statistical cumulants as whr th statistical avrag is dfind by Q(λ E, λ) lim t t ln xp ( λ E E λ N) t, (6) xp ( λ E E λ N) t = E, N P t ( E, N) xp ( λ E E λ N), (7) in trms of th probability ruld by th mastr quation () [,. Taking th tim drivativ of this avrag and using th mastr quation, w find in th long-tim limit that th gnrating function is givn by Q(λ E, λ) = [ dɛ W (+) i (ɛ) ( λ Eɛ λ i ) + W ( ) i (ɛ) ( λ Eɛ+λ i ). (8) Using th transition rats (3) and (4), w gt th xplicit xprssion of th gnrating function: { [ [ } kt λi kt Q(λ E, λ) = σ n i πm i ( + kt λ E ) + σ n λi i πm i ( kt λ E ), (9) undr th condition that kt < λ E < kt. () Th xprssion (9), which was obtaind for a gas with a singl constitunt in Rf. [, is hr xtndd to gasous mixturs. Th rats of ffusion into vacuum ar known to b givn by whr r i σn i kt πm i = 4 σn i v i, () v i = v i = 8kT πm i () is th man valu of th spd of particls of mass m i [8. If w us ths rats and xprss th tmpratur T and th dnsitis {n i } in trms of th affinitis by using Eqs. ()-(), th gnrating function can b rwrittn as } Q(λ E, λ) = r i { Ai + ( + kt A E ) λi ( + kt λ E ) (Ai λi) [ + kt (A E λ E ). (3) Consquntly, w obsrv that th gnrating function satisfis th following symmtry rlation: Q(λ E, λ) = Q(A E λ E, A λ), (4) which is th fluctuation thorm for th currnts [9,. W notic that th gnrating function vanishs according to Q(, ) = Q(A E, A) =, (5) bcaus of its dfinition (6), th normalization condition t =, and th symmtry rlation (4). An quivalnt xprssion of th fluctuation thorm for th currnts is givn by P t ( E, N) P t ( E, N) xp(a E E + A N) for t. (6) 4

5 In this form, th fluctuation thorm compars th probabilitis of opposit fluctuations in th ffusiv transport of nrgy and particls across th por and shows that th ratio of ths probabilitis is controlld by th affinitis. At th thrmodynamic quilibrium whr th affinitis ar vanishing, th principl of dtaild balancing is rcovrd according to which th probabilitis of opposit fluctuations ar in balanc. Out of quilibrium, this balanc is brokn by th nonquilibrium probability distribution and th fluctuating currnts tnd to follow th dirctionality fixd by th affinitis. Th fluctuation thorm (6) is th consqunc of th rlations (5) satisfid by th transition rats of particls of nrgy ɛ and spcis i =,,..., c btwn both rsrvoirs. Th diffrnc btwn Eqs. (5) and (6) is that th rlations (5) concrn th rats of random transfrs of individual particls ovr an infinitsimal tim intrval, although th fluctuation thorm (6) holds for th cumulativ transport of an nrgy E and particl numbrs N ovr th long tim intrval t. Othrwis, th lattr rsults from th formr in much th sam way as th fluctuation thorm for currnts can b dducd from Schnaknbrg s dfinition of macroscopic affinitis in gnral nonquilibrium stochastic procsss [9, 3. Now, th avrag valus of th currnts can b dducd from th gnrating function (3) according to J E = Q (, ) = lim λ E t t E t, (7) J i = Q (, ) = lim λ i t t N i t (i =,,..., c). (8) Th nrgy and particl currnts ar thus givn in trms of th affinitis according to = kt r i [ Ai ( + kt A E ) 3, (9) J i = r i [ Ai ( + kt A E ) (i =,,..., c). (3) J E W rcovr th wll-known rsult that, if thr is th vacuum in th rsrvoir R, th nrgy and particl currnts bcom lim A J E = kt whr r i is th ffusion rat givn by Eq. () [8. Using th inquality X X [33, w hav that r i, (3) lim J i = r i (i =,,..., c) (3) A A E E A N t A E E t A N t. (33) 5 Thrfor, in th macroscopic stady stat rachd in th long-tim limit, w infr from th inquality (33) and Eq. (5) that d i S k dt = A E J E + A J = lim t t (A E E t + A N t ) lim t t ln A E E A N t = Q(A E, A) =, (34) which is th vrification that th scond law of thrmodynamics holds in th prsnt formulation. III. NONLINEAR RESPONSE COEFFICIENTS Th authors of th prsnt papr hav prviously provd that th symmtry rlation (4) has for consqunc not only th Onsagr rciprocity rlations which only hold in th linar rgims, but also gnralizations of ths rlations which xtnd to th nonlinar rspons cofficints [ 3, 36. Ths gnralizations of Onsagr rciprocity rlations ar rmarkabl, in particular, bcaus thy can b usd to xprss th rspons cofficints in trms of quantitis charactrizing th diffusivitis of th currnts and highr cumulants in th rgim of nonlinar transport by ffusion.

6 6 In this sction, w us th following notations: A = {A α } = {A E, A} = {A E, A, A,..., A c }, (35) λ = {λ α } = {λ E, λ} = {λ E, λ, λ,..., λ c }, (36) with th indics α {E,,,..., c}. Th rspons cofficints L α,β, M α,βγ, N α,βγδ, tc... ar dfind by xpanding th avrag valus of th currnts in powrs of th affinitis: J α (A) = Q λ α λ= = β L α,β A β + M α,βγ A β A γ + N α,βγδ A β A γ A δ + (37) 6 β,γ Hr, th comma sparats th indx of th currnt from th indics of th affinitis. W notic that th tnsors M α,βγ, N α,βγδ,... ar a priori only symmtric undr th xchang of th indics byond th comma. On th othr hand, th diffusivitis and th highr cumulants of th fluctuating currnts ar dfind by succssiv drivativs of th gnrating function with rspct to th paramtrs λ: β,γ,δ D αβ (A) Q λ=, (38) λ α λ β C αβγ (A) 3 Q λ α λ β λ γ λ=, (39) B αβγδ (A) 4 Q λ=, (4) λ α λ β λ γ λ δ so that all ths tnsors ar totally symmtric. Th fluctuation thorm for th currnts (4) has for consqunc th following raltionships starting from th Onsagr rciprocity rlations for th linar rspons cofficints and xtnding to th nonlinar rspons cofficints [ 3, 36. L α,β = D αβ () = L β,α, (4) ( Dαβ M α,βγ = + D ) αγ, (4) A γ A β A= ( D αβ N α,βγδ = + D αγ + D αδ ) A γ A δ A β A δ A β A γ B αβγδ, (43) A=. A gnralization of Onsagr rciprocity rlations is givn by th total symmtry of th following fourth-ordr tnsor: ( ) D αβ N α,βγδ + D αγ + D αδ, (44) A γ A δ A β A δ A β A γ which is th consqunc of Eq. (43) and th total symmtry of th fourth-cumulant tnsor (4) [ 3. Morovr, th fluctuation thorm for th currnts (4) has also for consqunc th following rlations [3 ( ) Cαβγ B αβγδ () =. (45) A δ Our purpos is now to apply ths rlationships to th ffusion procss charactrizd by th gnrating function (3). Bcaus th currnts ar fluctuating, th nrgy E and th particl numbrs N transfrrd ovr a tim intrval t undrgo random walks around thir man valus, which ar charactrizd by th diffusivitis or scond-ordr cumulants A= A=

7 7 (38): D EE = 3 (kt ) r i [ Ai + ( + kt A E ) 4, (46) D Ei = kt r i [ Ai + ( + kt A E ) 3, (47) D ii = [ r Ai i + ( + kt A E ), (48) whil D ij = for i j. Among th larg-dviation proprtis, w also hav th third-ordr cumulants (39): C EEE = 4 (kt ) 3 r i [ Ai ( + kt A E ) 5, (49) C EEi = 6 (kt ) r i [ Ai ( + kt A E ) 4, (5) C Eii = kt r i [ Ai ( + kt A E ) 3, (5) C iii = r i [ Ai ( + kt A E ), (5) whil th othr third-ordr cumulants ar vanishing. Th fourth-ordr cumulants can b obtaind by using thir dfinition (4) in trms of th gnrating function or thanks to th rlations (45): B EEEE () = B EEEi () = C EEE A i B EEii () = B Eiii () = B iiii () = C EEi C EEE A E () = (kt ) 4 r i, (53) () = C EEi A E () = 4 (kt ) 3 r i, (54) () = C Eii A E () = 6 (kt ) r i, (55) A i C Eii () = C iii A i () A E = kt r i, (56) C iii () A i = r i, (57) whil th othr cumulants ar vanishing. Th rmarkabl rlationships (4)-(43) for th rspons cofficints of th currnts (9) and (3) ar satisfid as wll: L E,E = D EE () = 6 (kt ) r i, (58) L E,i = D Ei () = kt r i, (59) L i,e = D Ei () = kt r i, (6) L i,i = D ii () = r i, (6)

8 M E,EE = D EE A E () = 4 (kt ) 3 8 r i, (6) M E,Ei = D EE () + D Ei () = 6 (kt ) r i, (63) A i A E M i,ee = M E,ii = M i,ie = M i,ii = D Ei A E () = 6 (kt ) r i, (64) D Ei () A i = kt r i, (65) D ii () + D Ei () A E A i = kt r i, (66) D ii A i () = r i, (67) N E,EEE = 3 D EE A E () B EEEE() = (kt ) 4 r i, (68) N E,EEi = D EE () + D Ei () A E A i B EEEi() = 4 (kt ) 3 r i, (69) A E N i,eee = 3 D Ei A () E B EEEi() = 4 (kt ) 3 r i, (7) N E,Eii = D EE A () + D Ei () i A E A i B EEii() = 6 (kt ) r i, (7) N i,iee = N E,iii = D ii A () + D Ei () E A E A i B EEii() = 6 (kt ) r i, (7) 3 D Ei A () i B Eiii() = kt r i, (73) N i,iie = D ii () + D Ei () A E A i B Eiii() = kt r i, (74) A i N iiii = 3 D ii A () i B iiii() = r i, (75) which shows that th rspons cofficints can indd b obtaind from th cumulants. Th quality btwn th cofficints (59) and (6) is th Onsagr rciprocity rlation L E,i = L i,e, which rsults from th total symmtry of th diffusivity tnsor (38). Howvr, w point out that, for instanc, th qualitis btwn th cofficints (63) and (64) [or btwn th cofficints (73) and (74) ar spcific to th ffusion procss and thy ar not th consquncs of Eqs. (4)-(45) w ar hr concrnd with. A systm whr such systm-spcific qualitis do not hold has bn dscribd in Rf. [4. Nvrthlss, th qualitis N E,EEi D EE () = N i,eee D Ei () =, (76) A E A i N E,Eii D EE A i A E () = N i,iee D ii A () = 3 (kt ) r i, (77) E N E,iii D Ei A () = N i,iie D ii () =, (78) i A E A i do rsult from th total symmtry of th fourth-ordr tnsor (44) and ar th mntiond gnralizations of Onsagr rciprocity rlations [ 3. Ths qualitis, as wll as Eqs. (53)-(57), ar th consquncs of th fluctuation thorm (4). W hav thus shown that th fluctuation thorm for currnts has fundamntal implications on th nonlinar rspons cofficints, which indd satisfy th rmarkabl rlationships (4)-(45) in th ffusion procss. Ths rsults hold as wll for othr ffusion procsss [3, 4.

9 9 IV. MASS SEPARATION A. Th coupling btwn th currnts Hr, w considr a binary mixtur of particls with diffrnt masss m m. Mass sparation is th procss in which th molar fraction of on of th spcis incrass as th gasous mixtur flows btwn th rsrvoirs. Sinc th mixing ntropy should b rducd, mass sparation rquirs nrgy supply. Th ffusion procss provids this nrgy by th coupling of th particl flows to th hat flow, as shown in prvious sctions. For th ffusion of a binary mixtur, thr xist thr coupld flows: th hat flow and th two flows of particls of masss m and m. In ordr to achiv mass sparation, th diffrnc btwn th two particl flows must b controlld by th othr flows. Thrfor, w introduc th currnt of th total numbr of particls and th diffrnc btwn th particl currnts w shall call th diffrntial currnt [37 Th conjugatd affinitis ar dfind as J N J + J, (79) J D J J. (8) A N (A + A ) = ln [ n n n n ( ) T 3, (8) T (A A ) = ln n /n n. (8) /n Th affinity associatd with th diffrntial currnt is rlatd by = ln(/ f) to th sparation factor: f n /n n /n, (83) for th nrichmnt of th binary mixtur by spcis during th transfr R R. Th currnts ar givn in trms of th affinitis by [ J E = kt r A N [ ( + kt A E ) 3 + kt r A N + [ J N = r A N [ ( + kt A E ) + r A N + J D = r [ A N ( + kt A E ) ( + kt A E ) 3, (84) ( + kt A E ), (85). (86) r [ A N + ( + kt A E ) If w introduc th molar fractions, also calld th molar concntrations, n i c i (i =, ), (87) n + n th currnts can b xprssd as follows [( c J E = σ + c ) P ( ) c kt πm + c P kt, (88) πm πm πm [( c J N = σ + c ) ( ) P c πm + c P, (89) πm πm kt πm kt J D = σ [( c c ) πm πm ( ) P c πm c P, (9) kt πm kt in trms of th prssurs and tmpraturs in both rsrvoirs. Ths xprssions show that all th currnts {J E, J N, J D } ar coupld togthr so that th diffrntial currnt J D can b controlld by both th currnts J E and J N and, thrfor, by th diffrncs of prssurs or tmpraturs. Th thr currnts (84)-(86) ar dpictd in Fig. as a function of th affinity associatd with th sparation factor (83). Thy ar compard with thir linar approximations, J α (lin) = β L α,βa β, givn in trms of th linar rspons cofficints (58)-(6). W obsrv that th currnts manifst important nonlinar bhavior and that thir linar approximation may soon bcom vry crud away from quilibrium. Th diffrntial currnt J D is obsrvd in Fig. c to b positiv alrady in a small intrval at ngativ valus of th corrsponding affinity. It is in this small intrval that mass sparation occurs bcaus th diffrntial currnt J D > is thr opposit to th affinity <.

10 (a) (b) (c).5 J E J N J D.5 FIG. : Th avrag currnts vrsus th affinity in th ffusion of particls of masss m = and m = composing a binary mixtur at th tmpratur kt = and dnsitis n = and n = through a small por of ara σ = from th rsrvoir R to th rsrvoir R at th affinitis A E =. and A N =.3 with rspct to th formr: (a) th nrgy currnt J E; (b) th total particl currnt J N ; (c) th diffrntial currnt J D. Th dashd lins dpict th linar approximations of th currnts givn by J α (lin) = β L α,βa β with th linar rspons cofficints (58)-(6). B. Fluctuation thorm and ntropy production Bcaus of microrvrsibility, th fluctuating transfrs of nrgy E and particls oby th following fluctuation thorm: N N + N, (9) D N N, (9) P t ( E, N, D) P t ( E, N, D) xp(a E E + A N N + D) for t, (93) and th rspons cofficints ar rlatd to th statistical cumulants by Eqs. (4)-(45). Th thrmodynamic ntropy production (34) is hr givn by d i S k dt = A E J E + A N J N + J D, (94) and is always non ngativ as th consqunc of th fluctuation thorm. W notic that this non-ngativity is othrwis non trivial to stablish, givn th analytic xprssions of th currnts (84)-(86) and th affinitis (7), (8), and (8). Th positivity of th ntropy production undr nonquilibrium conditions is illustratd in Fig. 3 whr w s that it is ovrstimatd by its approximation basd on th linarizd currnts. Th non-ngativity of th ntropy production has implications in th organization of th domains whr th diffrnt currnts tak opposit dirctions in th thr-dimnsional spac of th affinitis. Figur 4 rprsnts two sctions in this spac: th sction at A N = in Fig. 4a and th sction at A E = in Fig. 4b. In th plan (, A E ) dpictd in Fig. 4a, th dirctions of th currnts J D and J E switch on th lins J D = and J E =. Both lins intrsct at th stat of thrmodynamic quilibrium locatd at th origin = A E =. In th vicinity of this point, th linar approximation of th currnts holds, but th lins J D = and J E = bcom curvd away from quilibrium bcaus of th nonlinar transport ffcts. Th scond law of thrmodynamics dtrmins th signs of th currnts in th diffrnt domains limitd by th lins J D = and J E =. Indd, th ntropy production (94) implis that J D on th lin J E = in th plan A N =, whrupon w find that J D > for >, as it is indd th cas on th right-hand sid of th origin in Fig. 4a. As a rlatd consqunc, th diffrntial currnt has th opposit sign J D < on th lft-hand sid whr <, again along th lin J E =. A similar rasoning dtrmins th sign of th currnt J E along th lin J D =, as sn in Fig. 4a. Thrfor, both currnts J D and J E hav th sam sign in th whit domains and thy hav opposit signs in th gry domains of Fig. 4a, in consistncy with th scond law. On th othr hand, th ntropy production (94) rducs to A N J N + J D in th plan A E = dpictd in Fig. 4b, whr th lins corrsponding to J D = and J N = ar rprsntd. Hr also, th diffrnt domains ar organizd around th thrmodynamic quilibrium stat at th origin according to th scond law of thrmodynamics.

11 .5 d i S/dt.5 FIG. 3: Th thrmodynamic ntropy production (94) vrsus th affiinity for ffusion in th sam conditions as in Fig.. Th dashd lin dpicts its approximation givn by th linarizd currnts J α (lin) = β L α,βa β with th linar rspons cofficints (58)-(6). 5 4 (a) J D > J E > 4 3 (b) J D > J N > 3 J D = A E J D = A N J N = J E = J D < J E < 3 3 J D < J N < 3 3 FIG. 4: Effusion of a binary mixtur of particls of masss m = and m = at th tmpratur kt = and dnsitis n = n = through a por of ara σ = : (a) th plan (, A E) at A N = ; (b) th plan (, A N ) at A E =. Th domains whr th currnts J D and J N hav th sam dirction ar in whit and thos whr th currnts hav opposit dirctions ar in gry. C. Entropy fficincy A concpt of ntropy fficincy has bn introducd by Onsagr [5, 6 as th fficincy of th thrmodiffusiv sparation procss to dcras Gibbs mixing ntropy at th xpns of incrasing th othr contributions of ntropy. In th cas of ffusion, th ntropy fficincy can b dfind as minus th ratio btwn th contribution to ntropy production causd by th mixing of th two spcis ovr th othr sourcs of dissipation: J D η. (95) A E J E + A N J N This quantity charactrizs th fficincy of mass sparation in trms of th powr of th diffrntial currnt dividd by th powr dissipatd by th nrgy and th total particl currnts. In th rgims whr mass sparation is ffctiv, th diffrntial currnt J D is opposit to th corrsponding affinity = ln(/ f) so that J D <. Thrfor, th fficincy (95) is positiv bcaus th non-ngativity of th ntropy production (94) implis that A E J E + A N J N J D >. Furthrmor, th ntropy fficincy (95) cannot xcd unity, which is its maximum valu allowd by th scond law of thrmodynamics (94): in th rgims of ffctiv mass sparation whr η >. η, (96)

12 With rgard to th xprssions (84)-(86) of th currnts, w notic that th ntropy fficincy (95) dpnds only on four variabls: η = η(kt A E, A N, ; q), (97) that ar th combination kt A E = (T/T ) of th tmpratur with th nrgy affinity, th affinitis A N and, and, finally, th ratio of th ffusion rats: q r = n m, (98) r n m which plays a cntral rol. This rsult shows that th fficincy of mass sparation by ffusion dpnds on both th mass ratio and th initial concntration ratio. In th linar rgim clos to th thrmodynamic quilibrium whr th currnts can b rplacd by thir linar approximation J α (lin) = β L α,βa β in trms of th linar rspons cofficints (58)-(6), th maximum valu of th ntropy fficincy is givn by ( ) q η max (lin) =, (99) q + which is rachd at A E = and = A N ( q )/( q +). Accordingly, th maximum fficincy can approach th unit valu if ithr q or q. Howvr, w should xpct dviations from th prdiction (99) in th nonlinar rgims furthr away from quilibrium, which is indd th cas...5 η FIG. 5: Th fficincy (95) vrsus th affinity undr th sam conditions as in Figs. and 3, for which q =.58. Th dashd lin dpicts th fficincy for th linar approximation of th currnts, which would rach th maximum valu η max (lin) =.3 if A E =...5 η FIG. 6: Th fficincy (95) vrsus th affinity for th ffusion at th affinitis A E =. and A N =.3 of a binary mixtur of particls of masss m = and m = at th tmpratur kt = and dnsitis n = and n = in th rsrvoir R Th por is of ara σ =. Th dashd lin dpicts th fficincy for th linar approximation of th currnts. Hr also, q =.58 and η max (lin) =.3 if A E =.

13 Figur 5 dpicts th fficincy (95) undr th sam conditions as in Figs. and 3. As sn in Fig. 5, th fficincy is positiv in a small intrval of valus of th affinity xtnding from th locus whr J D ( ) = to =. It is in this small intrval that mass sparation is powrd by th othr currnts J E and J N. In Fig. 5, th xact valu of th fficincy is compard with th prdiction of th linar approximation, which is obsrvd to undrstimat th xact valu. Howvr, if ngativ affinitis A E and A N ar takn, w obsrv in Fig. 6 that th linar approximation givs an ovrstimation of th actual fficincy. Sinc th conditions of Figs. 5 and 6 ar clos to quilibrium and sinc th ratio (98) is of th ordr of unity, th fficincy rmains vry small with rspct to its maximum possibl valu (96) η. 4 4 FIG. 7: Th fficincy (95) vrsus th affinity for mixturs of masss m = and m =. Th lft rsrvoir R is at th tmpratur kt = and dnsitis n = n = whil th right rsrvoir R is at th affinitis A E =.5 and A N = 3. Th ratio of th ffusion rats is hr givn by q = 3.6. Th dashd lin dpicts th fficincy for th linar approximation of th currnts, which would rach th maximum valu η max (lin) =.78 if A E =. Figur 7 shows th fficincy undr conditions furthr away from quilibrium whr th ffct of nonlinaritis is strongr. Th maximum valu of th fficincy is hr largr than in th conditions of Fig. 5. Morovr, th actual valu of th fficincy turns out to b much largr than th valu (99) prdictd by th linar approximation. W rmark that th actual valu would b smallr than its linar approximation if th affinitis A E and A N wr ngativ, as alrady obsrvd in Fig. 6. Accordingly, th largst valus of th fficincy ar typically obtaind for positiv rathr than ngativ affinitis A E and A N. Ths rsults clarly dmonstrat that th nonlinar dpndnc of th currnts on th affinitis plays an important rol in th procss of mass sparation by ffusion..5.4 η m /m FIG. 8: Th ntropy fficincy (95) for mass sparation by ffusion vrsus th affinity and th mass ratio m /m. Th lft rsrvoir R is at th tmpratur kt = and dnsitis n = and n =.5, whil th right rsrvoir R is at th affinitis A E = and A N =. Th ntropy fficincy vanishs at th mass ratio m /m = (n /n ) =.5 on th straight lin =. Th dpndnc of th fficincy (95) on th mass ratio m /m and th affinity is dpictd in Figs. 8- for diffrnt conditions. In ach plot, th fficincy is positiv btwn th straight lin = and th curvd lin whr J D =. According to Eq. (86) for th diffrntial currnt, th intrsction btwn th lins = and J D = happns if th ratio (98) of th ffusion rats taks th unit valu q = whil A N ln( + kt A E ), in which cas m m = ( n n ). ()

14 4 η log (m /m ) FIG. 9: Th ntropy fficincy (95) for mass sparation by ffusion vrsus th affinity and th logarithm log (m /m ) of th mass ratio, for th right rsrvoir R at th affinitis A E = and A N =, whil th lft rsrvoir R is at th tmpratur kt = and dnsitis n = and n =. Th ntropy fficincy vanishs at th mass ratio log (m /m ) = log (n /n ) = on th straight lin =. Not that th fficincy narly rachs its maximum possibl valu (96) allowd by th scond law undr ths conditions corrsponding to a larg diffrnc of prssur btwn both rsrvoirs η.. 5 log (m /m ) FIG. : Th ntropy fficincy (95) for mass sparation by ffusion vrsus th affinity and th logarithm log (m /m ) of th mass ratio, for th right rsrvoir R at th affinitis A E = and A N =, whil th lft rsrvoir R is at th tmpratur kt = and dnsitis n = and n =. Th ntropy fficincy vanishs at th mass ratio log (m /m ) = log (n /n ) = on th straight lin =. Not that th fficincy rmains low undr ths conditions corrsponding to a larg diffrnc of tmpratur btwn both rsrvoirs. Figur 8 dpicts th fficincy for a modrat tmpratur diffrnc in a situation whr th affinity A N is vanishing. W obsrv that th fficincy is positiv for < if q > and for > if q <. If q = so that th mass ratio is rlatd by Eq. () to th initial concntration ratio, mass sparation cannot b prformd by ffusion. Mass sparation thus bcoms ffctiv for q. Nvrthlss, th fficincy rmains low undr th conditions of Fig. 8, which ar clos to quilibrium. Figur 9 shows that th fficincy may approach its maximum possibl valu (96) allowd by th scond law if th affinity A N taks larg positiv valus, i.., for vry diffrnt prssurs in both rsrvoirs. Indd, using th xprssions (84)-(86) for th currnts and th dfinition (95), th maximum fficincy rachd in this limit can b stimatd to b givn by η max q q + for A N +, () whr q is th ratio dfind in Eq. (98). W notic that this maximum fficincy is always largr than th maximum valu (99) of th linar rgim. Thrfor, th fficincy can b much largr in ths nonlinar rgims than prdictd by th linar approximation. In contrast, th fficincy cannot rach high valus in th limit whr th tmpratur diffrnc is vry larg bcaus th fficincy dcrass as in this limit, as illustratd in Fig.. η max q q + ln(kt A E ) kt A E for A E +, ()

15 5 V. CONCLUSIONS In th prsnt papr, w hav rportd a study of nonlinar transport proprtis in th procss of mass sparation by ffusion in th light of th rcnt advancs in nonquilibrium statistical mchanics. Effusion is th ballistic transport of particls through a small por in a wall sparating two rsrvoirs containing gasous mixturs at diffrnt tmpraturs, prssurs, and concntrations. This is th classical analogu of ballistic transport in opn quantum systms whr tim-rvrsal symmtry rlations known undr th nam of fluctuation thorm hav bn rcntly provd [7, 8. For th ffusion of gasous mixturs, classical kintic thory basd on th Maxwllian vlocity distribution can b usd to obtain th counting statistics and its gnrating function in trms of th affinitis or thrmodynamic forcs, which ar th control paramtrs of th nonquilibrium constraints imposd by th particl rsrvoirs and driving th transport procss. Th fluctuation thorm for th currnts stablishd in Rfs. [9, can b considrd for th stochastic procsss of ffusion. This fluctuation thorm - which finds its origin in microrvrsibility - is th consqunc of th rlations (5) btwn th forward and backward transition rats and th affinitis. Th main point is that th fluctuation thorm for th currnts implis that th nonlinar rspons cofficints oby rmarkabl rlationships that ar th gnralizations of th Onsagr rciprocity rlations [ 3, as w hav hr dmonstratd for ffusion in gasous mixturs. On th on hand, ths rlationships rval that th nonlinar rspons cofficints can b obtaind from th statistical cumulants charactrizing th fluctuations of th currnts in nonquilibrium stady stats. On th othr hand, thy stablish symmtris among th cofficints as givn by Eqs. (53)-(57) and Eqs. (76)-(78). In sction IV, ths considrations ar applid to mass sparation by ffusion in binary mixturs whr th flows of two particl spcis ar coupld to th nrgy flow. Hr, w hav introducd th total particl currnt and th diffrntial currnt that ar coupld not only to th nrgy currnt but also togthr, allowing th diffrntial currnt to b controlld by both th nrgy and total particl currnts. Th nonlinar dpndnc of th thr currnts on th thr conjugatd affinitis is vidncd by comparing thir actual valus to thir linar approximations. Ths rsults show th importanc to go byond linar rspons thory in considring th nonquilibrium thrmodynamics of mass sparation by ffusion. Bcaus of th nonlinar transport ffcts, th ntropy fficincy may b significantly largr than it would b th cas in th linar approximation. Th scond law of thrmodynamics, which is hr provd thanks to th fluctuation thorm, imposs an uppr bound on th ntropy fficincy of mass sparation by ffusion. This fundamntal limit is not rachd undr modrat nonquilibrium conditions, but can nvrthlss b approachd for larg prssur diffrncs btwn both rsrvoirs. In conclusion, w s that our knowldg of transport phnomna can nowadays b xtndd from th linar to th nonlinar rgims thanks to th rcnt advancs in nonquilibrium statistical mchanics. Acknowldgmnts D. Andriux thanks th F.R.S.-FNRS Blgium for financial support. This rsarch is financially supportd by th Blgian Fdral Govrnmnt (IAP projct NOSY ). [ L. Onsagr, Phys. Rv. 37, 45 (93). [ D. J. Evans, E. G. D. Cohn, and G. P. Morriss, Phys. Rv. Ltt. 7, 4 (993). [3 G. Gallavotti and E. G. D. Cohn, Phys. Rv. Ltt. 74, 694 (995). [4 S. Tasaki and P. Gaspard, J. Stat. Phys. 8, 935 (995). [5 J. Kurchan, J. Phys. A: Math. Gn. 3, 379 (998). [6 J. L. Lbowitz and H. Spohn, J. Stat. Phys. 95, 333 (999). [7 M. Esposito, U. Harbola, and S. Mukaml, Rv. Mod. Phys. 8, 665 (9). [8 D. Andriux, P. Gaspard, T. Monnai, and S. Tasaki, Nw J. Phys., 434 (9). [9 D. Andriux and P. Gaspard, J. Stat. Phys. 7, 7 (7). [ David Andriux, Nonquilibrium Statistical Thrmodynamics at th Nanoscal (VDM Vrlag, Saarbrückn, 9) ISBN [ D. Andriux and P. Gaspard, J. Chm. Phys., 667 (4). [ D. Andriux and P. Gaspard, J. Stat. Mch.: Th. Exp. P (6). [3 D. Andriux and P. Gaspard, J. Stat. Mch.: Th. Exp. P6 (7). [4 D. Andriux and P. Gaspard, Phys. Rv. E 77, 337 (8). [5 R. D. Astumian, Phys. Rv. E 79, 9 (9). [6 K. Saito and Y. Utsumi, Phys. Rv. B 78, 549 (8).

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