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1 Physca A 392 (2013) Contents lsts avalable at ScVese ScenceDect Physca A jounal homepage: Themodynamcs n the lmt of evesble eactons A.N. Goban a,, E.M. Mkes b, G.S. Yablonsky c a Depatment of Mathematcs, Unvesty of Leceste, Leceste LE1 7RH, UK b Insttute of Space and Infomaton Technologes, Sbean Fedeal Unvesty, Kasnoyask, Russa c Paks College, Depatment of Chemsty, Sant Lous Unvesty, Sant Lous, MO 63103, USA a t c l e n f o a b s t a c t Atcle hstoy: Receved 8 August 2012 Receved n evsed fom 18 Septembe 2012 Avalable onlne 13 Octobe 2012 Keywods: Entopy Fee enegy Reacton netwok Detaled balance Ievesblty Fo many complex eal physcochemcal systems, the detaled mechansm ncludes both evesble and evesble eactons. Such systems ae typcal n homogeneous combuston and heteogeneous catalytc oxdaton. Most complex enzyme eactons nclude evesble steps. Classcal themodynamcs has no lmt fo evesble eactons, wheeas knetc equatons may have such a lmt. We epesent systems wth evesble eactons as the lmts of fully evesble systems when some of the equlbum concentatons tend to zeo. The stuctue of the lmt eacton system cucally depends on the elatve ates of ths tendency to zeo. We study the dynamcs of the lmt system and descbe ts lmt behavo as t. If the evesble systems obey the pncple of detaled balance then the lmt system wth some evesble eactons must satsfy the extended pncple of detaled balance. It s fomulated and poven n the fom of two condtons: () the evesble pat satsfes the pncple of detaled balance and () the convex hull of the stochometc vectos of the evesble eactons does not ntesect the lnea span of the stochometc vectos of the evesble eactons. These condtons mply the exstence of the global Lyapunov functonals and allow an algebac descpton of the lmt behavo. Themodynamc theoy of the evesble lmt of evesble eactons s llustated by the analyss of hydogen combuston Elseve B.V. All ghts eseved. 1. Intoducton 1.1. The poblem: non-exstence of themodynamc functons n the lmt of evesble eactons We consde a homogeneous chemcal system wth n components A. The concentaton of A s c 0, the amount of A n the system s N 0, V s the volume, N = Vc, and T s the tempeatue. The n-dmensonal vectos c = (c ) and N = (N ) belong to the closed postve othant R n + n Rn. (R n + s the set of all vectos x Rn such that x 0 fo all.) Classcal themodynamcs has no lmt fo evesble eactons, wheeas knetc equatons have. Fo example, consde a smple cycle k 1 k 2 k 3 A 1 A 2 A 3 A 1 k 1 k 2 k 3 wth the equlbum concentatons c eq = (c eq 1, ceq k c eq = k c eq +1, 2 ceq 3 ) and the detaled balance condtons Coespondng autho. Tel.: E-mal addess: ag153@le.ac.uk (A.N. Goban) /$ see font matte 2012 Elseve B.V. All ghts eseved. do: /j.physa

2 A.N. Goban et al. / Physca A 392 (2013) unde the standad cyclc conventon, whch s A 3+1 = A 1 and c 3+1 = c 1 hee. The pefect fee enegy has the fom F = c RTVc ln c eq 1 + const. Let the equlbum concentaton c eq 1 0 fo fxed values of c eq 2,3 > 0. Ths means that k 1 k 1 = c eq 1 c eq 0 2 and k 3 = c k 3 eq 1 c eq 3 0. If we take fxed values of the ate constants k 1, k ±2 and k 3 then the lmt knetc system exsts and has the fom A 1 k 1 A2 k 2 k 2 A 3 A 1. k 3 It s a outne task to wte a fst-ode knetc equaton fo ths scheme. At the same tme, the fee enegy functon F has no lmt: t tends to fo any postve vecto of concentatons because the tem c 1 ln(c 1 /c eq 1 ) nceases to. The fee enegy cannot be nomalzed by addng a constant tem because the vaaton of the tem c 1 ln(c 1 /c eq 1 ) on an nteval [0, c] wth fxed c also nceases to ; t vaes fom c eq 1 /e (fo the mnmze c 1 = c eq 1 /e) to a lage numbe c(ln c ln ceq ) 1 (fo c 1 = c). The logathmc sngulaty s athe soft and does not cause a eal physcal poblem, because even fo c eq 1 /c 1 = the coespondng lage tem n the fee enegy wll be just 23RT pe mole. Nevetheless, the absence of the lmt causes some mathematcal questons. Fo example, the densty f = F/(RTV) = c (ln(c /c eq ) 1) s a Lyapunov functon fo a system of chemcal knetcs fo a pefect mxtue wth detaled balance unde sochoc sothemal condtons. Hee, c s the concentaton of the th component and c eq s ts equlbum concentaton fo a selected value of the lnea consevaton laws, the so-called efeence equlbum. Ths functon has been used fo analyses of the stablty, exstence and unqueness of chemcal equlba snce the wok of Zeldovch n 1938 [1]. Shapo and Shapley pesented a detaled analyss of the connectons between detaled balance and the fee enegy functon [2]. The fst detaled poof that f s a Lyapunov functon fo chemcal knetcs of pefect systems wth detaled balance was publshed n 1975 [3]. Of couse ths does not dffe sgnfcantly fom the Boltzmann poof of hs H-theoem n 1873 [4]. Fo evesble systems obtaned as lmts of systems wth detaled balance, we should expect pesevaton of the stablty of the equlbum. Moeove, we can expect the exstence of Lyapunov functons, whch ae as unvesal as the themodynamc functons ae. Ths unvesalty means that these functons depend on the components pesent and on the equlbum concentatons but do not dectly depend on the eacton ate constants. The themodynamc potental of a component A cannot be defned n the evesble lmt when the equlbum concentaton of A tends to 0. Nevetheless, hee we constuct unvesal Lyapunov functons fo systems wth some evesble eactons. Instead of detaled balance, we use the weake assumpton that these systems can be obtaned fom systems wth detaled balance when some constants tend to zeo Extended fom of detaled balance condtons fo systems wth evesble eactons Consde a eacton mechansm n the fom of the followng system of stochometc equatons: α A β j A j ( = 1,..., m), j whee α 0 and β j 0 ae the stochometc coeffcents. The evese eactons wth postve ate constants ae ncluded n (2) sepaately (f they exst). The stochometc vecto γ of the elementay eacton s γ = (γ ), γ = β α. We always assume that thee exsts a stctly postve consevaton law, a vecto b = (b ), b > 0 and b γ = 0 fo all. Ths may be the consevaton of mass o of the total numbe of atoms, fo example. Accodng to the genealzed mass acton law, the eacton ate fo an elementay eacton (2) s [5] (cf. to [6, Eqs. (4), (7), and (14)] and [7, Eq. (4.10)]) n w = k a α, (3) =1 whee a 0 s the actvty of A, µ µ 0 a = exp. RT Hee, µ s the chemcal potental and µ 0 s the standad chemcal potental of component A. (1) (2) (4)

3 1320 A.N. Goban et al. / Physca A 392 (2013) Ths law has a long hstoy [8 10,7]. It was fomulated to meet the themodynamc estctons on knetcs. Fo ths pupose, accodng to the pncple of detaled balance, the ate of the evese eacton s defned by the same fomula and ts ate constant should be found fom the detaled balance condton at a gven equlbum. It s woth mentonng that the fee enegy has no lmt when some of the eacton equlbum constants tend to zeo. Fo example, fo the deal gas the chemcal potental s µ (c, T) = RT ln c + µ 0 (T). In the evesble lmt, some µ0. On the contay, the actvtes eman fnte (fo the deal gases a = c ) and the appoach based on the genealzed mass acton law and the detaled balance equatons w + = w can be appled to fnd the evesble lmt. The eacton mechansm (2) ncludes eactons wth eacton ate constants k > 0. Fo each, we defne k + = k, w + = w, k s the eacton ate constant fo the evese eacton f t s n (2) and 0 f t s not, and w s the eacton ate fo the evese eacton f t s n (2) and 0 f t s not. Fo a evesble eacton, K = k + /k. The pncple of detaled balance fo the genealzed mass acton law s as follows: fo gven values k, thee exsts a postve equlbum a eq > 0 wth detaled balance w + = w. We ecently fomulated the extended fom of the detaled balance condtons fo systems wth some evesble eactons [11]. Ths extended pncple of detaled balance s vald fo all systems that obey the genealzed mass acton law and ae the lmts of systems wth detaled balance when some of the eacton ate constants tend to zeo. It conssts of two pats: The algebac condton: The pncple of detaled balance s vald fo the evesble pat. (Ths means that fo the set of all evesble eactons thee exsts a postve equlbum n whch all the elementay eactons ae equlbated by the evese eactons.) The stuctual condton: The convex hull of the stochometc vectos of the evesble eactons has an empty ntesecton wth the lnea span of the stochometc vectos of the evesble eactons. (Physcally, ths means that the evesble eactons cannot be ncluded n oented cyclc pathways.) We ecall the fomal conventon: the lnea span of the empty set s {0} and the convex hull of the empty set s empty. We pevously studed systems wth some evesble eactons that ae the lmts of evesble systems wth detaled balance and we dentfed stuctual and algebac condtons [11]. Hee we focus on the dynamcal consequences of these condtons. We pove that attactos always consst of fxed ponts. These lmt ponts ( patal equlba ) ae stuated on the faces of the postve othant of concentatons. We descbe these faces and patal equlba. The emande of the pape s oganzed as follows. In Secton 2 we study systems wth detaled balance, the multscale lmts and lmt systems that satsfy the extended pncple of detaled balance. The classcal Wegschede denttes fo eacton ate constants ae pesented. The lmts when some of the equlba tend to zeo gve the extended pncple of detaled balance. We use the genealzed mass acton law fo the eacton ates. Fo analyss of the equlba fo geneal systems, the fomulas wth actvtes ae the same as fo deal systems and t s convenent to wok wth actvtes unless we need to study dynamcs. The dynamcal vaables ae amounts and concentatons. In Secton 2.3 we dscuss the elatons between concentatons and actvtes, fomulate the man assumptons and pesent fomulas fo the dsspaton ate. We nvestgate attactos of systems wth some evesble eactons n Secton 3 and pesent the cental esults of the pape. We fully chaacteze the faces of the postve othant that nclude ω-lmt sets. On such a face, the dynamcs s completely degeneate (zeo ates) o s dven by a smalle evesble system that obeys classcal themodynamcs. Hydogen combuston s a wdely studed and vey mpotant gas eacton. It seves as a benchmak example fo many studes of chemcal knetcs. Ths s aleady not a toy example but the complexty of ths system s not extemely hgh: n the usual models thee ae sx to eght components and elementay evesble eactons. Unde vaous condtons some of these eactons ae pactcally evesble. We use ths system as a benchmak n Secton 4 and gve an example of coect sepaaton of the eactons nto evesble and evesble pats. The lmt behavo of ths system n tme s descbed. In Secton 5 we befly dscuss the esults wth a focus on unsolved poblems. 2. Multscale lmt of a system wth detaled balance 2.1. Two classcal appoaches to the detaled balance condton Thee ae two tadtonal appoaches to the descpton of evesble systems wth detaled balance. Fst, we can stat fom the ndependent ate constants of the elementay eactons and consde the solvablty of the detaled balance equatons as an addtonal condton on the admssble values of the ate constants. Hee, fo m elementay eactons we have m constants (m should be an even numbe, m = 2l, l = m/2) and some equatons that descbe connectons between these constants. Ths appoach was ntoduced by Wegschede n 1901 [12] and developed futhe by many authos [13,14]. Second, we can select a fowad eacton n each pa of mutually evese elementay eactons. If a postve equlbum s known, then we can fnd the eacton ate constants fo the evese eacton fom the constants fo the fowad eacton and the detaled balance equatons. Theefoe, the fowad eacton ate constants and a set of the equlbum actvtes fom the complete descpton of the eacton. Hee we have l + n ndependent constants, l = m/2 ate constants fo fowad eactons and n (numbe of components) equlbum actvtes. Fo these l + n constants, the pncple of detaled

4 A.N. Goban et al. / Physca A 392 (2013) balance poduces no estctons. Ths second appoach s popula n appled chemcal themodynamcs and knetcs [15 17] because t s convenent to wok wth ndependent paametes fom scatch. The Wegschede condtons appea as the necessay and suffcent condtons of solvablty of the detaled balance equatons [9]. We jon the fowad and evese elementay eactons and wte α A β j A j ( = 1,..., l). (5) j The stochometc matx s Ɣ = (γ ), γ = β α (gan mnus loss). The stochometc vecto γ s the th ow of Ɣ wth coodnates γ = β α. Both sdes of the detaled balance equatons, w + = w, ae postve fo postve actvtes. The solvablty of ths system fo postve actvtes s equvalent to the solvablty of the followng system of lnea equatons: γ x = ln k + ln k = ln K ( = 1,... l) (6) fo x = ln a eq. Of couse, we assume that f k + > 0 then k > 0 (evesblty) and the equlbum constant K > 0 s defned fo all eactons fom (5). Poposton 1. The necessay and suffcent condton fo the exstence of the postve equlbum a eq s as follows. Fo any soluton λ = (λ ) of the system l λγ = 0.e. λ γ = 0 fo all =1 the Wegschede dentty holds: > 0 wth detaled balance (7) l l (k + )λ = (k )λ. =1 =1 (8) It s suffcent to use n (8) any bass of solutons of system (7): λ {λ 1,..., λ q } Multscale degeneaton of equlba We consde systems wth some evesble eactons as the lmts of fully evesble systems when some eacton ate constants tend to zeo. In evesble systems, the pncple of detaled balance mples the Wegschede denttes (8). Theefoe, the lmt system s not abtay. Some consequences of the Wegschede denttes pesst although some of the eacton ate constants n these denttes become zeo. We pevously compaed these consequences wth the Cheshe cats gn [11]: the whole cat (the evesble system wth detaled balance) vanshes but the gn pessts. We can postulate that some eacton ate constants go to zeo. Howeve, the eacton ate constants ae not ndependent. They ae connected by the Wegschede denttes. The ate constants should tend to zeo wth pesevaton of the elatons. Theefoe, the smple stategy of neglectng the ates of some of the eactons cannot be appled fo complex eactons. Nevetheless, we can change the vaables and use the ndependent set eacton ate constants fo the fowad eactons + equlbum actvtes [15 17,11]. Evey set of postve values of these vaables coesponds to a evesble system wth detaled balance and no addtonal estctons ae needed. If the evesble system degeneates to a system wth some evesble eactons, then some of the equlbum actvtes tend to zeo. We study ths pocess of degeneaton of evesble eactons nto evesble ones statng fom the coespondng degeneaton of equlbum actvtes to zeo. We consde a system wth detaled balance and send some of the equlbum actvtes to zeo: a eq 0 when I fo some set of ndexes I. Immedately we supsngly fnd that ths assumpton s not suffcent to fnd a lmtng evesble mechansm. It s necessay to take nto account the ates of the convegence to zeo of dffeent a eq. Consde a vey smple example, k 1 k 2 A 1 A 2 A 3 k 1 k 2 when a eq 1, aeq 2 0. If a eq, 1 aeq 2 0, a eq 1 /aeq 2 = const > 0 and a eq k 1 3 = const > 0, then the lmt system should be A 1 A 2 A 3 and we can k 1 keep k 1, 1,2 = const, wheeas k 2 0. If a eq, 1 aeq 2 k 1, , aeq 1 /aeq 2 0, then the lmt system should be A 1 A 2 A 3 and we can keep k 1,2 = const > 0, wheeas If a eq 1, aeq 2 0, a eq 2 /aeq 1 0, then n the lmt only one eacton suvves, A 2 A 3 (f we assume that all the eacton ate constants ae bounded).

5 1322 A.N. Goban et al. / Physca A 392 (2013) We study asymptotcs a eq = const ε δ, ε 0 fo vaous values of non-negatve exponents δ 0 ( = 1,..., n). At equlbum, each eacton ate n the genealzed mass acton law s popotonal to a powe of ε: w eq+ = k + const ε α δ, w eq Accodng to the pncple of detaled balance, w eq+ = k const ε β δ. = w eq and k + k = const ε (γ,δ), (9) whee δ s the vecto of exponents, δ = (δ ). Thee ae thee goups of eactons wth espect to the gven vecto δ: 1. (γ, δ) = 0; 2. (γ, δ) < 0; 3. (γ, δ) > 0. In the fst goup ((γ, δ) = 0), the ato k + /k emans constant and we can take k ± = const > 0. In the second goup ((γ, δ) < 0), the ato k /k+ 0 and we should take k 0, wheeas k + may eman constant and postve. In the thd goup ((γ, δ) > 0), the stuaton s nvese: k + /k 0 and we can take k = const > 0, wheeas k + 0. These thee goups depend on δ, but ths dependence s pecewse constant. Fo evey γ, thee sets of δ ae defned: () hypeplane (γ, δ) = 0, () hemspace (γ, δ) < 0, and () hemspace (γ, δ) > 0. The space of vectos δ s splt nto the subsets defned by the values of functons sgn(γ, δ) (±1 o 0). We consde bounded systems, and hence negatve values of δ should be fobdden. At least one equlbum actvty should not vansh. Theefoe, δ j = 0 fo some j. Below we assume that δ 0 and δ j = 0 fo a non-empty set of ndces J 0. Moeove, the atom balance n equlbum should be postve. Hee, ths means that fo the set of equlbum concentatons ( J 0 ) the coespondng values of all atomc concentatons ae stctly postve and sepaated fom zeo. Let the vecto of exponents δ = (δ ) be gven and let the thee goups of eactons be found. Fo the eactons of the thd goup (wth (γ, δ) > 0) the fowad eacton vanshes n the lmt ε 0. It s convenent to tanspose the stochometc equatons fo these eactons and swap the fowad eactons wth evese ones. Afte pefomng ths tansposton, α swaps wth β, γ tansfoms to γ, and the nequalty (γ, δ) > 0 tansfoms to (γ, δ) < 0. To summaze, we use the gven vecto of exponents δ and poduce a system wth some evesble eactons fom a c eq system of evesble eactons and detaled balance equlbum a eq 1. If δ > 0 then we assgn a eq = 0, and f δ = 0 then a eq 2. If (γ, δ) = 0 then k ± do not change. 3. If (γ, δ) < 0 then we assgn k = 0 and k + does not change. does not change. accodng to the followng ules: 4. If (γ, δ) > 0 then we assgn k + = 0 and k does not change. (In the last case, we tanspose the stochometc equaton and swap the fowad eacton wth the evese one; fo convenence, γ changes to γ and k becomes 0. Theefoe, ths case tansfoms to case 3.) Ths s a lmt system caused by the multscale degeneaton of equlbum. The multscale chaacte of the lmt a eq = const ε δ 0 (fo some ) s mpotant because eactons may have dffeent domnant dectons fo dffeent values of δ and the set of evesble eactons n the lmt may change. The geneal fom of the knetc equatons fo homogeneous systems s dn dt = V w γ, whee N s the amount of A, N s the vecto wth components N and V s the volume. Consde a lmt system fo the degeneaton of equlbum wth the vecto of exponents δ. Fo ths system (γ, δ) 0 fo all and, n patcula, (γ, δ) < 0 fo all evesble eactons and (γ, δ) = 0 fo all evesble eactons. (10) Poposton 2. A lnea functonal G δ (N) = (δ, N) deceases along the solutons of knetc equatons (10) fo ths lmt system: dg δ (N)/dt 0 and dg δ (N)dt = 0 f and only f all the eacton ates fo the evesble eactons ae zeo. Poof. In fact, dg δ (N) = V w (γ, δ) 0, dt (11) because (γ, δ) = 0 fo evesble eactons and w = w + 0 and (γ, δ) < 0 fo evesble eactons. All the tems n ths sum ae non-postve and hence t may be zeo f and only f each summand s zeo.

6 A.N. Goban et al. / Physca A 392 (2013) Ths Lyapunov functon may be used to pove that the ates of all evesble eactons n the system tend to 0 wth tme. Indeed, f they do not tend to zeo then on a soluton of (10) N(t) then G δ (N(t)) when t and N(t) s unbounded. Eq. (11) and Poposton 2 allow us to pove the extended pncple of detaled balance n the followng fom. Consde a eacton mechansm that ncludes evesble and evesble eactons. Assume that the eacton ates satsfy the genealzed mass acton law (3) and that the set of eacton ate constants s gven. We ask f t s possble to obtan ths eacton mechansm and eacton ate constants as a lmt n the multscale degeneaton of equlbum fom a fully evesble system wth the classcal detaled balance. The answe to ths queston gves the followng theoem about the extended pncple of detaled balance. Theoem 1. A system can be obtaned as a lmt n the multscale degeneaton of equlbum fom a evesble system wth detaled balance f and only f () the eacton ate constants of the evesble pat of the eacton mechansm satsfy the classcal pncple of detaled balance and () the convex hull of the stochometc vectos of the evesble eactons does not ntesect the lnea span of the stochometc vectos of evesble eactons. Poof. Let the gven system be a lmt of a evesble system wth detaled balance n the multscale degeneaton of equlbum wth the exponent vecto δ. Then (γ, δ) = 0 fo the evesble eactons and (γ, δ) < 0 fo the evesble eactons. Fo evey vecto x fom the convex hull of the stochometc vecto of the evesble eactons (x, δ) < 0 and fo any vecto y fom the lnea span of the stochometc vectos of the evesble eactons (y, δ) = 0. Theefoe, these sets do not ntesect. The eacton ate constants fo the evesble eactons satsfy the classcal pncple of detaled balance because they do not change n the equlbum degeneaton and keep ths popety of the ognal fully evese system wth detaled balance. Convesely, let a system satsfy the extended pncple of detaled balance: () the eacton ate constants of the evesble pat of the eacton mechansm satsfy the classcal pncple of detaled balance and () the convex hull of the stochometc vectos of the evesble eactons does not ntesect the lnea span of the stochometc vectos of evesble eactons. Accodng to the classcal theoems of convex geomety, thee exsts a lnea functonal that sepaates ths convex set fom the lnea subspace (stong sepaaton of closed and compact convex sets). Ths sepaatng functonal can be epesented n the fom (x, θ) fo some vecto θ. Then (γ, θ) = 0 fo the evesble eactons and (γ, θ) < 0 fo the evesble eactons. It s possble to fnd a vecto δ wth ths sepaaton popety and non-negatve coodnates. Accodng to the basc assumptons, thee exsts a lnea consevaton law wth stongly postve coodnates. Ths s a vecto b (b > 0) wth the popety (γ, b) = 0 fo all eactons. Fo any λ, the vecto θ + λb has the same sepaaton popety as the vecto θ has. We can select λ such that δ = θ + λb 0 and δ = θ + λb = 0 fo some. We take ths lnea combnaton δ as a vecto of exponents. We ceate a fully evesble system fom the ntal patally evesble one. We do not change the evesble eactons and the ate constants. Because the evesble eactons satsfy the classcal pncple of detaled balance, thee exsts a stongly postve vecto of equlbum actvtes a > 0 fo the evesble eactons. Fo each evesble eacton wth the stochometc vecto γ and eacton ate constant k = k + > 0, we add a evese eacton wth the eacton ate constant k = k + (a ) γ. Fo ths fully evesble system the actvtes a > 0 povde the pont of detaled balance. In the multscale degeneaton pocess, the equlbum actvtes depend on ε 0 as a eq = a εδ. Fo eactons wth (γ, δ) = 0, the eacton ate constants do not depend on ε and fo eactons wth (γ, δ) < 0, the ate constant k tends to zeo as ε (γ,δ) and k + does not change. We etun to the ntal system of eactons n the lmt ε 0. Ths s a patcula fom of the extended pncple of detaled balance [11]. Fg. 1 llustates the geometc sense of the extended detaled balance condton: the convex hull of the stochometc vectos of the evesble eactons does not ntesect wth the lnea span of the stochometc vectos of the evesble eactons. In ths llustaton, {γ J 0 } ae the stochometc vectos of the evesble eactons and {γ J 1 } ae the stochometc vectos of the evesble eactons Actvtes, concentatons and affntes To combne the lnea Lyapunov functons G δ (N) = (δ, N) (11) wth the classcal themodynamc potental and study the knetc equatons n the closed fom, we have to specfy the elatons between actvtes and concentatons. We accept the assumpton a = c g (c, T), whee g (c, T) > 0 s the actvty coeffcent. It s a contnuously dffeentable functon of c, T n the whole dapason of the values. In a bounded egon of concentatons and tempeatue we can always assume that g > g 0 > 0 fo some constant g 0. Ths assumpton s vald fo the non-deal gases and fo lqud solutons. It holds also fo the suface gas n knetcs of heteogeneous catalyss [9] and does not hold fo the sold eagents (e.g. analyss of cabon actvty n methane efomng [11]). The system of unts should also be noted. Tadtonally, a s assumed to be dmensonless and fo pefect systems a = c /c, whee c s an abtay standad concentaton. To avod the ntoducton of unnecessay quanttes, we always assume that c 1 n the system of unts selected.

7 1324 A.N. Goban et al. / Physca A 392 (2013) / / Fg. 1. Man opeatons n the applcaton of the extended detaled balance condtons. (a) In the concentaton space R n we should fnd the subspace spanned by all the stochometc vectos {γ = 1,..., l}. In ths subspace we have to select the ntenal coodnates. (b) In span {γ = 1,..., l} we have to select the subspace spanned by the stochometc vectos of the evesble eactons (the dashed vectos). The stochometc vectos of the evesble eactons n b, c and d ae sold bold lnes. Owng to the extended pncple of detaled balance, span{γ J 0 } should not ntesect conv{γ J 1 } (dotted tangle). Fo analyss of ths ntesecton, t s convenent to poceed to the quotaton space span {γ = 1,..., l}/span {γ J 0 }. In ths quotaton space, span { γ J 0 } s {0} and two stuatons ae possble: (c) 0 conv{ γ J 1 } (the dotted tangle ncludes the ogn) o (d) 0 conv{γ J 1 } (the dotted tangle does not nclude the ogn). In case (c) the extended detaled balance condton s volated. Case (d) satsfes ths condton. If the themodynamc potentals exst, then owng to the themodynamc defnton of actvty (4), ths hypothess s equvalent to the logathmc sngulaty of the chemcal potentals, µ = RT ln c +, whee... denotes fo a contnuous functon of c, T (all the concentatons and the tempeatue). In ths case, the fee enegy has the fom F(N, T, V) = RT N (ln c 1 + f 0 (c, T)), (12) whee the functons f 0 (c, T) ae contnuously dffeentable fo all possble values of the aguments. Functons f 0 on the ght-hand sde of (12) cannot be estoed unambguously fom the fee enegy functon F(N, T, V), but fo a small admxtue A t s possble to ntoduce the patal pessue p, whch satsfes the law p = RTc + o(c ). Ths s because of the tems N ln c n F. In fact, P = F(N, T, V)/ V = RTc + o(c ) + P c =0. Connectons between the equaton of state, the fee enegy and the knetcs ae dscussed n moe detal elsewhee [7,18,5]. Thee ae seveal smple algebac coollaes of the assumed connecton between actvtes and concentatons. Consde an elementay eacton α A β A wth α, β 0. Then, accodng to the genealzed mass acton law, fo any vecto of concentatons c (c 0) we have the followng: 1. If, fo some, c = 0, then γ w(c) If, fo some, c = 0 and γ < 0, then α > 0 and w(c) = 0. Smlaly, fo a evesble eacton α A β A we have the followng: 1. If, fo some, c = 0 and γ > 0, then β > 0 and w (c) = If, fo some, c = 0 and γ < 0, then α > 0 and w + (c) = 0. These statements, as well as Poposton 3 and Coollay 1 below, ae consequences of the genealzed mass acton law (3) and the connecton between actvtes and concentatons wthout any assumptons about the extended pncple of detaled balance. Each set of ndexes J = { 1,..., j } defnes a face of the postve polyhedon F J = {c c 0 fo all and c = 0 fo J}. By defnton, the elatve nteo of F J, (F J ), conssts of ponts wth c = 0 fo J and c > 0 fo J.

8 A.N. Goban et al. / Physca A 392 (2013) Poposton 3. Fo a pont c (F J ) and an ndex J, let γ w (c) = 0. Then ths dentty holds fo all c F J. Poof. Fo convenence, we wte all the fowad and evese eactons sepaately and epesent the eacton mechansm n the fom (2). All the tems n the sum γ w (c) ae non-negatve because c = 0. Theefoe, f the sum s zeo then all the tems ae zeo. The eacton ate w (3) wth a non-zeo ate constant takes a zeo value f and only f α j > 0 and a j = 0 fo some j. The equalty a = 0 s equvalent to c = 0. Theefoe, w (c) = 0 fo a pont c (F J ) f and only f thee exsts j J such that α j > 0. If α j > 0 fo an ndex j J, then w (c) = 0 fo all c F J because c j = 0 n F J. We call a face F J of the postve othant R n + nvaant wth espect to a set S of elementay eactons f S γ jw (c) = 0 fo all c F J and evey j J. We now consde a eacton mechansm n the fom (2) n whch all the fowad and evese eactons patcpate sepaately. Coollay 1. The followng statements ae equvalent: 1. γ S w (c) = 0 fo a pont c (F J ) and all ndexes J. 2. The face F J s nvaant wth espect to the set of eactons S. 3. The face F J s nvaant wth espect to evey elementay eacton n S. 4. Fo evey S ethe γ j = 0 fo all j J o α j > 0 fo some j J. We now analyze the asymptotc behavo of the knetc equatons n the multscale degeneaton of equlbum descbed n Secton 2.2. Fo ths pupose, we have to detemne how the elatons between actvtes a and concentatons c depend on the degeneaton paamete ε 0. We do not ty to fnd the maxmally geneal appopate answe to ths queston. Fo known applcatons, the elatons between a and c do not depend on ε 0. In patcula, ths s tvally tue fo deal systems. The smple genealzaton a = c g (c, T, ε), whee g (c, T, ε) > g 0 > 0 ae contnuous functons, s not a genealzaton at all, because we can use fo ε 0 the lmt case that does not depend on ε, g (c, T) = g (c, T, 0). Ths ndependence fom ε mples that the evesble pat of the eacton mechansm has the themodynamc Lyapunov functons such as fee enegy. If we just delete the evesble pat, then the classcal themodynamcs s applcable and the themodynamc potentals do not depend on ε. Fo the genealzed mass acton law, the tme devatve of the elevant themodynamc potentals have a vey nce geneal fom. Let the functon Φ(N,...) be gven, whee... denotes quanttes that do not change n tme unde the gven condtons. Ths s the themodynamcs potental f Φ(N,...)/ N = µ. Fo example, t s the fee Helmholtz enegy F fo V, T = const and the fee Gbbs enegy G fo P, V = const. We calculate the tme devatve of Φ(N,...) accodng to knetc equaton (10). The eacton ates ae gven by the genealzed mass acton law (3) wth defnton of the actvtes though chemcal potental (4). We assume that the pncple of detaled balance holds (t should hold fo the evesble pat of the eacton mechansm accodng to the extended detaled balance condtons). Moe pecsely, thee exsts an equlbum wth detaled balance fo any tempeatue T, a eq (T): fo all, w + (aeq ) = w (aeq ) = w eq (T). It s convenent to epesent the eacton ates usng the equlbum fluxes w eq (T): w + α = w eq (µ µ eq ) exp, w β = w eq (µ µ eq ) exp, RT RT whee µ eq = µ (a eq, T). These fomulas mmedately gve the followng epesentaton of the dsspaton ate: dφ dt = Φ(N,...) dn = dn µ N dt dt = VRT (ln w + ln w )(w+ w ) 0. (13) The nequalty holds because ln s a monotone functon and hence the expessons ln w + ln w and w + w always have the same sgn. Fomulas of ths type fo dsspaton ae well known snce the famous Boltzmann H-theoem [4,10]. The entopy ncease n solated systems has a smla fom: ds dt = VR (ln w + ln w )(w+ w ) 0. Note that ln w + ln w = 1 RT µ (α β ) = (γ, µ). RT

9 1326 A.N. Goban et al. / Physca A 392 (2013) The quantty (γ, µ) s one of the cental notons of physcal chemsty, affnty [19]. It s postve f the fowad eacton pevals ove the evese one and s negatve n the opposte case. It measues the enegetc advantage of the fowad eacton ove the evese one (fee enegy pe mole). The actvty dvded by RT shows the magntude of ths enegetc advantage compaed to the themal enegy. We call t the nomalzed affnty and use a specal notaton fo ths quantty: A = (γ, µ). RT We apply the elementay dentty exp a exp b = (exp a + exp b) tanh a b to the eacton ate, w = w + w : 2 w = (w + + w ) tanh A 2. (14) Ths epesentaton of the eacton ates mmedately gves the followng fo the dsspaton ate: dφ = VRT (w + + w dt )A tanh A 0. (15) 2 In ths fomula, the knetc nfomaton s collected n the non-negatve factos, the sums of eacton ates (w + + w ). The puely themodynamc multples A tanh(a /2) ae also non-negatve. Fo small A, the expesson A tanh(a /2) behaves lke A 2 /2 and fo lage A t behaves lke the absolute value A. Theefoe, we have two Lyapunov functons fo the two pats of the eacton mechansm. Fo the evesble pat, t s just a classcal themodynamc potental. Fo the evesble pat, t s the lnea functonal G δ (N) = (δ, N). Moe pecsely, the evesble eactons decease ths functonal, wheeas t s the consevaton law fo the evesble eactons. Theefoe, t deceases monotoncally n tme fo the whole system. 3. Attactos 3.1. Dynamc systems and lmt ponts Knetc equaton (10) does not gve a complete epesentaton of the dynamcs. The ght-hand sde ncludes the volume V and the eacton ates w, whch ae functons (3) of the concentatons c and tempeatue T, wheeas the left hand sde has Ṅ. To close ths system, we need to expess V, c and T though N and quanttes that do not change n tme. Ths closue depends on condtons. The smplest expessons appea fo sochoc sothemal condtons V, T = const, c = N/V. Fo othe classcal condtons (U, V = const, P, T = const o H, P = const) we have to use the equatons of state. Thee may be moe sophstcated closues that nclude models o extenal egulatos of the pessue and tempeatue, fo example. Poposton 2 s vald fo all possble closues. It s only mpotant that the extenal flux of the chemcal components s absent. Futhe n ths pape, we assume that the condtons ae selected, a closue s pefomed, the ght-hand sde of the esultng system s contnuously dffeentable and thee exsts a postve bounded soluton fo the ntal data n R n +, and V and T eman bounded and sepaated fom zeo. The natue of ths closue s not cucal. Fo some mpotant patcula closues, poofs of the exstence of postve and bounded solutons ae well known [3]. Stctly speakng, such a system s not a dynamc system n R n + but s sem-dynamc: the solutons may lose postvty and leave Rn + fo negatve values of tme. The theoy of the lmt behavo of sem-dynamc systems was developed fo applcatons to knetc systems [20]. We now descbe the lmt behavo of the systems as t. Unde the extended detaled balance condton, the lmt behavo s athe smple and the system wll appoach steady states, but to pove ths behavo we need the moe geneal noton of the ω-lmt ponts. By defnton, the ω-lmt ponts of a dynamc system ae the lmt ponts of the motons when tme t. We consde a knetc system n R n +. In patcula, fo each soluton N(t) of the knetc equatons, the set of the coespondng ω-lmt ponts s closed, connected and conssts of the whole tajectoes [20, Poposton 1.5]. Ths means that moton that stats fom an ω-lmt pont emans n R n + fo all tme moments, both postve and negatve. Poposton 4. Let N(t) be a postve soluton of the knetc equaton and let x be an ω-lmt pont of ths soluton and x = 0. Then ẋ x = 0 at ths pont. Poof. Let x(t) be a soluton of the knetc equatons wth the ntal state x(0) = x. All the ponts x(t) ( < t < ) belong to R n +. Thee exsts a sequence t j such that N(t j ) x. Fo any τ (, ), N(t j + τ) x(τ). Fo suffcently lage j, t j + τ > 0 and N(t j + τ) R n +. Theefoe, x(τ) Rn + ( < τ < ) and fo any τ the pont x(τ) s an ω-lmt pont of the soluton N(t). Let x = 0 and ẋ x = v 0. If v > 0 then fo small τ and τ < 0 the value of x becomes negatve, x (τ) < 0. Ths s mpossble because of the postvty. Smlaly, f v < 0 then fo small τ > 0 the value of x becomes negatve, x (τ) < 0. Ths s also mpossble because of the postvty. Theefoe, ẋ x = 0. We use Poposton 4 n the followng combnaton wth Poposton 3. (Below we wte the eacton mechansm n the fom of (2).)

10 A.N. Goban et al. / Physca A 392 (2013) Coollay 2. If an ω-lmt pont belongs to the elatve nteo F J of the face F J R n +, then the face F J s nvaant wth espect to the eacton mechansm and fo evey elementay eacton ethe γ j = 0 fo all j J o α j > 0 fo some j J. Poof. If an ω-lmt pont belongs to F J then all ċ j = 0 fo j J at ths pont accodng to Poposton 4. Theefoe, we can apply Coollay Steady states of evesble eactons Unde extended detaled balance condtons, all the eacton ates of evesble eactons ae zeo at evey lmt pont of knetc equaton (10) accodng to Poposton 2. In ths secton, we gve a smple combnatoal descpton of steady states fo the set of evesble eactons. Ths descpton s based on Poposton 2 and theefoe uses the extended detaled balance condtons. We contnue to study multscale degeneaton of a detaled balance equlbum. The vecto of exponents δ = (δ ) s gven, δ 0 fo all and δ = 0 fo some. Thee ae two sets of eactons. Fo one of them, (γ, δ) = 0 and n the lmt k ± > 0. Fo the second set, (γ, δ) < 0 and n the lmt we assgn k = 0 and k + s the same as n the ntal system (befoe equlbum degeneaton). If necessay, we tanspose the stochometc equatons and swap the fowad eactons wth the evese ones. Fo convenence, we change the notaton. Let γ be the stochometc vectos of evesble eactons wth (γ, δ) = 0 ( = 1,..., h), and let ν l be the stochometc vectos fo eactons fom the second set, (ν l, δ) < 0 (l = 1,..., s). Fo the eacton ates and constants fo the fst set, we keep the same notaton: w, w ±, k±. Fo the second set, we use q l = q + l fo the eacton ate constants and v l = v + l fo the eacton ates. They ae also calculated accodng to the genealzed mass acton law (3). The nput and output stochometc coeffcents eman α and β fo the fst set and fo the second set we use the notaton α ν l and βν l. Let the ates of all the evesble eacton be equal to zeo. Ths does not mean that all the concentatons a wth δ > 0 each zeo. A bmolecula eacton A + B C gves us a smple example: w = ka A a B and w = 0 f ethe a A = 0 o a B = 0. On the plane wth coodnates a A, a B and wth the postvty condton, a A, a B 0 and the set of zeos of w s a unon of two sem-axes, {a A = 0, a B 0} and {a A 0, a B = 0}. In a moe geneal stuaton, the set n the actvty space, whee all the evesble eactons have zeo ates, has a smla stuctue: t s the unon of some faces of the postve othant. Let us descbe the set of steady states of the evesble eactons. Accodng to Poposton 2, f v l lν l = 0 then all v l = 0. Theefoe, we should descbe the set of zeos of all v l n the postve othant of actvtes. Fo evey l = 1,..., s the set of zeos of v l n R n + s gven by the condtons αν l 0 and a = 0 fo at least fo one. It s convenent to epesent ths condton as a dsjuncton. Let J l = { α ν l 0}. Then the set of zeos of v l n the postve othant of actvtes s epesented by J l (a = 0). The set of zeos of all v l s epesented by the followng conjuncton fom: Jl (a = 0). (16) s l=1 To tansfom ths nto the unons of subspaces, we have to move to a dsjuncton fom and make some cancelatons. Fst, we epesent ths fomula as a dsjuncton of conjunctons: Jl (a = 0) = 1 J 1,..., s J s (a1 = 0) (a s = 0). (17) s l=1 Fo a cotege of ndexes { 1,..., s }, the coespondng set of values may be smalle because some values l may concde. Let ths set of values be S {1,..., s }. We can delete fom (17) a conjuncton (a 1 = 0)... (a s = 0) f thee exsts a cotege { 1,..., s } ( l J l ) wth a smalle set of values, S {1,..., s } S { 1,..., s}. We check the coteges n some ode and delete a conjuncton fom (17) f a tem wth smalle (o the same) set of ndex values emans n the fomula. We can also substtute the coteges n (17) by the sets of values. The esultng mnmzed fomula may be shote. Each elementay conjuncton epesents a coodnate subspace and afte cancelatons each subspace does not belong to a unon of othe subspaces. The fnal fom of (17) s j ( Sj (a = 0)), whee S j ae sets of ndexes, S j {1,..., n}, and fo evey two dffeent S j, S p, nethe of them ncludes the othe, S j S p. The elementay conjuncton Sj (a = 0) descbes a subspace. The steady states of the evesble pat of the eacton mechansm ae gven by the ntesecton of the unon of the coodnate subspaces (18) wth R n +. Fo applcatons of ths fomula, t s mpotant that the equaltes a = 0, c = 0 and N = 0 ae equvalent and the postve othants of the actvtes a, concentatons c o amounts N epesent the same sets of physcal states. Ths s also tue fo the faces of these othants: F J fo the actvtes, concentatons o amounts coespond to the same sets of states. (The same state may coespond to dffeent ponts of these cones, but the totaltes of the states ae the same.) (18) 3.3. Sets of steady states of evesble eactons nvaant wth espect to evesble eactons Hee we study the possble lmt behavo of systems that satsfy the extended detaled balance condtons and nclude some evesble eactons. All the ω-lmt ponts of such systems ae steady states of the evesble eactons accodng to

11 1328 A.N. Goban et al. / Physca A 392 (2013) Poposton 2, but not all these steady states may be the ω-lmt ponts of the system. A smple fomal example gves us the eacton couple A B, A + B C. Hee, we have one evesble and one evesble eacton. The condtons of extended detaled balance hold (tvally): the lnea span of the stochometc vecto of the evesble eacton, ( 1, 1, 0), does not nclude the stochometc vecto of the evesble eacton, ( 1, 1, 1). Fo a descpton of the multscale degeneaton of equlbum, we can take the exponents δ A = 1, δ B = 1, δ C = 0. The steady states of the evesble eacton ae gven n R n + by the dsjuncton (c A = 0) (c B = 0), but only the ponts (c A = c B = 0) may be the lmt ponts when t. In fact, f c A = 0 and c B > 0, then dc A /dt = k 1 c B > 0. Accodng to Poposton 4, ths s not an ω-lmt pont. Smlaly, the ponts wth c A > 0 and c B = 0 ae not ω-lmt ponts. We combne Popostons 2 and 4 and Coollay 2 n the followng statement. Theoem 2. Let the knetc system satsfy the extended detaled balance condtons and nclude some evesble eactons. Then an ω-lmt pont x F J exsts f and only f F J conssts of steady states of the evesble eactons and s nvaant wth espect to all evesble eactons. Poof. If an ω-lmt pont x F J exsts, then t s a steady state fo all evesble eactons (accodng to Poposton 2). Theefoe, the face F J conssts of steady states of the evesble eactons (Poposton 4) and s nvaant wth espect to all evesble eactons (Poposton 4 and Coollay 2). To pove the evese statement, assume that F J conssts of steady states of the evesble eactons and s nvaant wth espect to all evesble eactons. The evesble eactons that do not act on c j fo j J defne a sem-dynamcal system on F J. The postve consevaton law b defnes a postvely nvaant polyhedon n F J. The dynamcs n such a compact set always has ω-lmt ponts. We now detemne the faces F J that contan the ω-lmt ponts n the elatve nteo F J. Accodng to Theoem 2, these faces should consst of the steady states of the evesble eactons and should be nvaant wth espect to all evesble eactons. We look fo the maxmal faces wth ths popety. Fo ths pupose, we always mnmze the dsjunctve foms by cancelaton. We do not lst the faces that contan the ω-lmt ponts n the elatve nteo and ae the pope subsets of othe faces wth ths popety. All the ω-lmt ponts belong to the unon of these maxmal faces. We stat fom the mnmzed dsjunctve fom (18), whch epesents the set of steady states of the evesble pat of the eacton mechansm by a unon of the coodnate subspaces Sj (c = 0) n ntesecton wth R n +. Ths s the unon of the faces, j F Sj. If a face F J conssts of the steady states of the evesble eactons, then J S j fo some j. The followng fomula s tue on a face F J f t contans ω-lmt ponts n the elatve nteo F J (Theoem 2): (c = 0),γ >0 j,αj >0(c j = 0),γ <0 j,βj >0(c j = 0). (19) Hee, c = 0 n F J may be ead as J. Followng the pevous secton, we use hee the notaton γ, α and β fo the evesble eactons and eseve ν l, α ν l and β ν l fo the evesble eactons. The set of γ n ths fomula s the set of the stochometc vectos of the evesble eactons. The equed faces F J may be constucted n an teatve pocedue. Fst we ntoduce an opeaton that tansfoms a set of ndexes S {1, 2,..., n} n a famly of sets S(S) = {S,..., 1 S l }. We use (19) and fnd the set fo whch t s vald fo all S. Ths set s descbed by the followng fomula: S (c = 0),γ >0 j,αj >0(c j = 0),γ <0 j,βj >0(c j = 0). (20) We poduce a dsjunctve fom of ths fomula and mnmze t by cancelatons as descbed n Secton 3.2. Ths yelds j=1,...,k S (c = 0). (21) j Because of cancelatons, the sets S j do not nclude each anothe and they gve the esult S(S) = {S,..., 1 S l }. Each S j S(S) s a supeset of S, S S. We extend the opeaton S to the sets of sets S = {S 1,..., S p } wth the popety S S j fo j. We apply S to all S and take the unon of the esults: S 0 (S) = S(S ). Some sets fom ths famly may nclude othe sets fom t. We oganze cancelatons: f S, S S 0 (S) and S S, then we etan the smallest set, S, and delete the lagest one. We cay out cancelatons untl ths s possble and we call the fnal esult S(S). The esult does not depend on the ode of these opeatons. We stat fom any famly S and teate the opeaton S. Afte a fnte numbe of teatons, the sequence S d (S) stablzes, S d (S) = S d+1 (S) =, because fo any set S all sets fom S(S) nclude S. The poblems of popostonal logc that ase n ths and the pevous secton seem vey smla to elementay logcal puzzles [21]. In the soluton we just use the logcal dstbuton laws (dstbuton of conjuncton ove dsjuncton and dstbuton of dsjuncton ove conjuncton), commutatvty of dsjuncton and conjuncton, and elementay cancelaton ules such as (A A) A, (A A) A, [A (A B)] A and [A (A B)] A. We ae now n a poston to descbe the constucton of all F J that have the ω-lmt ponts n the elatve nteo and ae the maxmal faces wth ths popety. 1. We follow Secton 3.2 and constuct the mnmzed dsjunctve fom (18) fo a descpton of the steady states of the evesble eactons. 2. We calculate the famles of sets S d ({S j }) fo the famly of sets {S j } fom (18) and d = 1, 2,... up to stablzaton.

12 A.N. Goban et al. / Physca A 392 (2013) Let S d ({S j }) = S d+1 ({S j }) = {J 1, J 2,..., J p }. Then the famly of faces F J ( = 1, 2,..., p) gves the answe: the ω-lmt ponts ae stuated n F J and fo each thee ae ω-lmt ponts n F J Smple examples In ths secton we pesent two smple and fomal examples of the calculatons descbed n the pevous sectons. 1. A 1 + A 2 A 3 + A 4, γ = ( 1, 1, 1, 1, 0); A 1 + A 2 A 5, ν = ( 1, 1, 0, 0, 1). The extended pncple of detaled balance holds: the convex hull of the stochometc vectos of the evesble eactons conssts of one vecto γ 2 and t s lnealy ndependent of γ 1. The nput vecto α fo the evesble eacton A 1 + A 2 A 5 s ( 1, 1, 0, 0, 0). The set J = J l fom conjuncton fom (16) s defned by the non-zeo coodnates of ths α ν : J = {1, 2}. The conjuncton fom n ths smple case (one evesble eacton) loses ts fst conjuncton opeaton and s just (c 1 = 0) (c 2 = 0). It s, at the same tme, the mnmzed dsjuncton fom (18) and does not eque addtonal tansfomatons. Ths fomula descbes the steady states of the evesble eacton n the postve othant R n +. Fo ths dsjuncton fom, the famly of sets S = {S j} conssts of two sets, S 1 = {1} and S 2 = {2}. We now calculate S(S 1,2 ). Fo both = 1, 2 thee ae no evesble eactons wth γ = 0. Theefoe, one expesson n paentheses vanshes n (20). Fo S = {1} ths fomula gves (c 1 = 0) ((c 3 = 0) (c 4 = 0)) and fo S = {2} t gves (c 2 = 0) ((c 3 = 0) (c 4 = 0)). The elementay tansfomatons gve the dsjunctve foms [(c 1 = 0) ((c 3 = 0) (c 4 = 0))] [((c 1 = 0) (c 3 = 0)) ((c 1 = 0) (c 4 = 0))], [(c 2 = 0) ((c 3 = 0) (c 4 = 0))] [((c 2 = 0) (c 3 = 0)) ((c 2 = 0) (c 4 = 0))]. Theefoe, S(S 1 ) = {{1, 3}, {1, 4}}, S(S 2 ) = {{2, 3}, {2, 4}} and S({S 1, S 2 }) = {{1, 3}, {1, 4}, {2, 3}, {2, 4}}. No cancelatons ae needed. The teatons of S do not poduce new sets fom {{1, 3}, {1, 4}, {2, 3}, {2, 4}}. In fact, f c 1 = c 3 = 0 o c 1 = c 4 = 0 o c 2 = c 3 = 0 o c 2 = c 4 = 0, then all the eacton ates ae zeo. Moe fomally, fo S({1, 3}), fo example, (20) gves [(c 1 = 0) ((c 3 = 0) (c 4 = 0))] [(c 3 = 0) ((c 1 = 0) (c 2 = 0))]. Ths s equvalent to (c 1 = 0) (c 3 = 0). Theefoe, S({1, 3}) = {1, 3}. The same esult s tue fo {1, 4}, {2, 3} and {2, 4}. All the ω-lmt ponts (steady states) belong to the faces F {1,3} = {c, c 1 = c 3 = 0}, F {1,4} = {c, c 1 = c 4 = 0}, F {2,3} = {c, c 2 = c 3 = 0} o F {2,4} = {c, c 2 = c 4 = 0}. The poston of the ω-lmt pont fo a soluton N(t) depends on the ntal state. Moe specfcally, ths system of eactons has thee ndependent lnea consevaton laws: b 1 = N 1 + N 2 + N 3 + N 4 + 2N 5, b 2 = N 1 N 2 and b 3 = N 3 N 4. Fo gven values of these b 1,2,3, vecto N belongs to the 2D plane n R 5. The ntesecton of ths plane wth the selected faces depends on the sgns of b 2,3 : If b 2 < 0 and b 3 < 0, then t ntesects F {1,3} at only one pont, N = (0, b 2, 0, b 3, b 1 + b 2 + b 3 ) (N 5 should be non-negatve, b 1 + b 2 + b 3 0). If b 2 = 0 and b 3 < 0 then t ntesects both F {1,3} and F {2,3} at one pont, N = (0, 0, 0, b 3, b 1 + b 3 ) (N 5 should be non-negatve, b 1 + b 3 0). If b 2 < 0 and b 3 = 0, then t ntesects both F {1,3} and F {1,4} at one pont, N = (0, b 2, 0, 0, b 1 + b 2 ) (N 5 should be non-negatve, b 1 + b 2 0). If b 2 > 0 and b 3 < 0, then t ntesects F {2,3} at only one pont, N = (b 2, 0, 0, b 3, b 1 + b 2 + b 3 ) (N 5 should be nonnegatve, b 1 + b 2 + b 3 0). If b 2 > 0 and b 3 = 0, then t ntesects F {2,3} and F {2,4} at the pont N = (b 2, 0, 0, 0, b 1 + b 2 ) (N 5 s non-negatve because b 1 + b 2 + b 3 0). If b 2 < 0 and b 3 > 0, then t ntesects F {1,4} at only one pont, N = (0, b 2, b 3, 0, b 1 + b 2 + b 3 ) (N 5 should be nonnegatve, b 1 + b 2 + b 3 0). If b 2 = 0 and b 3 > 0, then t ntesects F {1,4} and F {2,4} at one pont, N = (0, 0, b 3, 0, b 1 + b 3 ) (N 5 s non-negatve because b 1 + b 3 0). If b 2 > 0 and b 3 > 0, then t ntesects F {2,4} at only one pont N = (b 2, 0, b 3, 0, b 1 + b 2 + b 3 ) (N 5 s non-negatve because b 1 + b 2 + b 3 0). As we can see, the system has exactly one ω-lmt pont fo any admssble combnaton of the values of the consevaton laws. These ponts ae the lsted ponts of ntesecton. Fo the second smple example, we change the decton of the evesble eacton. 2. A 1 + A 2 A 3 + A 4, γ 1 = ( 1, 1, 1, 1, 0), A 5 A 1 + A 2, ν = (1, 1, 0, 0, 1). The extended pncple of detaled balance holds. The steady states of the evesble eactons ae gven by one equaton, c 5 = 0. Fomula (20) gves just (c 5 = 0) fo S({5}). Face F {5} ncludes ω-lmt ponts n F {5}. The dynamcs on ths face s defned by the fully evesble eacton system and tends to the equlbum of the eacton A 1 + A 2 A 3 + A 4 unde the gven consevaton laws. On ths

13 1330 A.N. Goban et al. / Physca A 392 (2013) Table 1 H 2 combuston mechansm [22]. No. Reacton Stochometc vecto 1 H 2 + O 2 2OH ( 1, 1, 2, 0, 0, 0, 0, 0) 2 H 2 + OH H 2 O + H ( 1, 0, 1, 1, 1, 0, 0, 0) 3 OH + O O 2 + H (0, 1, 1, 0, 1, 1, 0, 0) 4 H 2 + O OH + H ( 1, 0, 1, 0, 1, 1, 0, 0) 5 O 2 + H + M HO 2 + M (0, 1, 0, 0, 1, 0, 1, 0) 6 OH + HO 2 O 2 + H 2 O (0, 1, 1, 1, 0, 0, 1, 0) 7 H + HO 2 2OH (0, 0, 2, 0, 1, 0, 1, 0) 8 O + HO 2 O 2 + OH (0, 1, 1, 0, 0, 1, 1, 0) 9 2OH H 2 O + O (0, 0, 2, 1, 0, 1, 0, 0) 10 2H + M H 2 + M (1, 0, 0, 0, 2, 0, 0, 0) 11 2H + H 2 H 2 + H 2 (1, 0, 0, 0, 2, 0, 0, 0) 12 2H + H 2 O H 2 + H 2 O (1, 0, 0, 0, 2, 0, 0, 0) 13 OH + H + M H 2 O + M (0, 0, 1, 1, 1, 0, 0, 0) 14 H + O + M OH + M (0, 0, 1, 0, 1, 1, 0, 0) 15 2O + M O 2 + M (0, 1, 0, 0, 0, 2, 0, 0) 16 H + HO 2 H 2 + O 2 (1, 1, 0, 0, 1, 0, 1, 0) 17 2HO 2 O 2 + H 2 O 2 (0, 1, 0, 0, 0, 0, 2, 1) 18 H 2 O 2 + M 2OH + M (0, 0, 2, 0, 0, 0, 0, 1) 19 H + H 2 O 2 H 2 + HO 2 (1, 0, 0, 0, 1, 0, 1, 1) 20 OH + H 2 O 2 H 2 O + HO 2 (0, 0, 1, 1, 0, 0, 1, 1) face, thee exst bode equlba, whee c 1 = c 3 = 0 o c 1 = c 4 = 0 o c 2 = c 3 = 0 o c 2 = c 4 = 0, but they do not attact the postve solutons. 4. Example: H 2 + O 2 system Fo the case study, we selected the H 2 + O 2 system. Ths s one of the man model systems of gas knetcs. Hydogen combuston gves us an example of medum complexty wth eght components (A 1 = H 2, A 2 = O 2, A 3 = OH, A 4 = H 2 O, A 5 = H, A 6 = O, A 7 = HO 2 and A 8 = H 2 O 2 ) and two atomc balances (H and O). Fo the example, we selected the eacton mechansm descbed by Vlachos [22]. The lteatue on hydogen combuston mechansms s huge and ncludes ecent dscussons [23,24]. We do not am to compae the dffeent schemes of ths eacton but use ths eacton mechansm as an example and a benchmak. The symbol M denotes a thd body that may be any molecule and povdes the enegy balance. The effcency of dffeent molecules n ths pocess dffes, and theefoe the concentaton of the thd body s a weghted sum of the concentatons of the components wth postve weghts. The thd body does not affect the equlbum constants and does not change the zeos of the fowad and evese eacton ates, but modfes the non-zeo values of the eacton ates. Theefoe, fo ou analyss we can omt these tems. Elementay eactons 10, 11 and 12 can be consdeed as a sngle eacton, 2H H 2, afte cancelaton of the thd bodes, and thus we analyze the mechansm of 18 eactons. Unde vaous condtons, some of the eactons ae (almost) evesble and some of them should be consdeed as evesble. Fo example, consde the H 2 + O 2 system at o nea atmosphec pessue and n the tempeatue nteval K. Reactons 1, 2, 4, 18, 19, and 20 ae supposed to be evesble (accodng to the eacton ate constants povded by Vlachos [22]). The fst queston s: f these eactons ae evesble, then whch eactons may be evesble? Accodng to the geneal cteon, the convex hull of the stochometc vectos of the evesble eactons does not ntesect wth the lnea span of the stochometc vectos of the evesble eactons. Theefoe, f the stochometc vecto of a eacton belongs to the lnea span of the stochometc vectos of the evesble eactons, then ths eacton s evesble. Smple lnea algeba gves that γ 3,5,9 span{γ 1, γ 2, γ 4, γ 18, γ 19, γ 20 }. In patcula, γ 3 = γ 1 + γ 4, γ 5 = γ 1 γ 18 + γ 19 and γ 9 = γ 2 γ 4. Theefoe, the lst of evesble eactons should nclude eactons 1, 2, 3, 4, 5, 9, 18, 19, and 20. Reactons 6, 7, 8, 10, 11, 12, 13, 14, 15, and 17 may be evesble. Fomally, thee ae 2 8 = 256 possble combnatons of the dectons of these eght eactons (eactons 10, 11 and 12 have the same stochometc vecto and n ths sense should be consdeed as one eacton). The geneal cteon and smple lnea algeba gve that thee ae only two admssble combnatons of the dectons of evesble eactons: ethe k = 0 fo all of them o k + = 0 fo all of them. Hee, the fowad and evese eactons and the notaton k ± ae selected accodng to Table 1. It s mmedately evdent that the nvese decton fo all eactons s vey fa fom the ealty unde the gven condtons; fo example, t ncludes the evesble dssocaton H 2 2H. We now demonstate n detal how the geneal cteon poduces ths educton fom the 256 possble combnatons of dectons of evesble eactons to just two admssble combnatons. We assume that the ntal set of eactons s splt n two: evesble eactons wth numbes J 0 and evesble eactons wth J 1, ank{γ 1, γ 2,..., γ l } = d, ank{γ J 0 } = d 0. The ank of all vectos γ, d, must exceed the ank of the stochometc vectos of the evesble eactons, d > d 0, because f d = d 0 then all the eactons must be evesble and the poblem becomes tval.