Random activation energy model: from disordered kinetics, to population genetics. and genomics

Transcription

1 ando activation energy odel: fro disordered inetics, to population genetics and genoics Marcel Ovidiu Vlad #,*, Gianfranco Cerofolini + and Peter Oefner # Institute of Applied Matheatics and Matheatical Statistics, Casa Acadeiei oane, Calea Septebrie 3 Bucharest, oania * Departent of Cheistry, Stanford University, Stanford CA ST Microelectronics, 24 Agrate MI, Italy Institute of Functional Genoics, University of egensburg, Josef-Engert Strasse 9, 9353 egensburg, Gerany. Introduction The rando activation energy odel (AE) is widely used for describing the inetics of rate processes in disordered systes -4. The ain assuption of this odel is that the activation energies of the rate coefficients have rando coponents selected fro certain probability laws, typically froen Maxwell-Boltann distributions. This odel has been successfully applied to any physical, cheical and biological probles, such dielectric and other olecular relaxation processes, the cheical transforations of active interediates in radiation cheistry, the reaction inetics on heterogeneous surfaces, the charge or ass transport in disordered systes, and the ligand protein interactions in olecular biology -4. In its siple for the AE assues that a fluctuation of the activation energy barrier, once it occurs, lasts forever, that is the syste displays static disorder. This is a reasonable assuption for low and oderate teperature systes. As the teperature increases the disorder becoes dynaic 5 and the rando coponents of the activation energy barriers becoe rando functions of tie. Siilar situations occur in single-olecule inetics 6-7 Although various odels of rando activation barriers with dynaic disorder have been suggested in the literature -4,8-, they are based on various ad hoc assuptions and lac unity. The purpose of this chapter is the developent of a unified approach for the AE odels, which describes any type of stochastic behavior of the rando coponents of the rando activation energy barriers, ranging fro static disorder with infinite eory to long range and short range dynaic disorder, and finally extree dynaic disorder with rapid, independent fluctuations. The structure Produced with Applixware for Linux V. 5.

2 2 of the chapter is the following. In Section 2, we give a general forulation of the proble by starting fro the classical for of AE with static disorder with infinite eory. In Section 3 we develop general ethods for static and dynaic averaging based on the use of a generalied Zubarev-McLennan nonequilibriu enseble approach -2. In Section 4 we apply the general theory for coputing the average survival function, which is an experiental observable, for various types of static and dynaic disorder. In Section 5 we illustrate our approach by considering a single type of dynaic disorder of the renewal type. In Sections 6 and 7 we generalie the nonequilibriu enseble approach to reversible reactions and apply the results to a desorption experient reported in the literature In Section 8 we discuss the possible iplications of our theory for interpreting experiental data. Finally in Section 9 we discuss the iplications of our approach in olecular biology and genetics. 2. Forulation of the proble In this section we forulate the objective of this chapter by starting out fro the siplest version of the rando activation energy odel, which in the language of cheical inetics, corresponds to a first order process with a rando activation energy barrier with static disorder. Both the classical AE odel as well as the dynaic generaliations introduced in this chapter can be easily extended to nonlinear inetics. For first order inetics the evolution equation of the process is: l t = l, l= t. () Here is a rate coefficient and l is an instantaneous, fluctuating survival function. In olecular relaxation, l is the instantaneous, fluctuating probability that the relaxation process has not occurred fro the initial oent t until the current tie t. In cheical inetics l is the ratio of the concentration of a cheical species at tie t and the concentration of the species at the initial tie t. The rate coefficient obeys the Arrhenius equation, that is, L N M O Q P E = Aexp, (2) B T where T is the teperature of the syste, B is Boltann s constant, E is the activation energy of the process and A is a pre-exponential aplitude factor which is (alost) independent of teperature. The Produced with Applixware for Linux V. 5.

3 3 activation energy of the process is assued to be ade up of two different coponents: a constant ter E and a rando coponent which is selected fro a given probability law, E = E + E. (3) The constant ter E is assued to be non-negative, E, and the variable ter is assued to tae any rando variable between ero and infinity. If we denote by η d coponent has a value between and + d, we have: b g the probability that the rando η d= b g. (4) The experiental observable of the odel is the average survival function, l t, which can be expressed as a static average: b g l t l t η d E. (5) = We solve Eq.(), insert the result into Eq.(5) and ae used Eqs.(2)-(3), resulting in: b g b gr b g l t = exp T χ t η d E, (6) where L N M O Q P E = T Aexp, (7) B T is the constant, non-rando coponent of the rate coefficient corresponding to = and b g b B g χ E = exp E T (8) is a rando transparence factor. The siplest version of AEM assues that the probability density η d b g is a Maxwell- Boltann distribution froen at a teperature T, which is higher than the current teperature T of the syste a f c B h c B h η d= T exp T d, with T > T. (9) The froen Maxwell-Boltan distribution was introduced a long tie ago in heterogeneous catalysis; subsequently its use has expanded in other applications of the AE odel. By inserting Eq.(9) into Eq.(6) we coe to: Produced with Applixware for Linux V. 5.

4 l t b g b b g αγ α, T t t αγ α = ~ α T t t T x b g g αb t t 4 g α b g, for t >> t, () where γ ax x a a, = exp xdxax,, > and Γ ax, = x exp xdxa, > are the incoplete and the coplete gaa functions, respectively, and α is a fractal scaling exponent given by α = T T. We note that for a Maxwell-Boltann distribution the AE odel leads to a fractal scaling law of the negative power law type; the experiental occurrence of this fractal scaling law is well docuented in the literature. For soe systes, for exaple in the case of ligand protein interactions 3, the distribution of the rando energy barriers is not always of the Maxwell-Boltann type. In this case the probability density b g of the rando energy barrier can be evaluated fro experiental data. It is convenient to introduce η the probability density pd with b g pd =, () of the rate coefficient. We have: p d = d δ b T χ Eg rb η Egd E = S L NM F HG = T T T B η T B ln and Eq.(6) becoes: b g a fi K JO d for QP T af af for T l t = p exp t d, (3) that is, the observed average survival function l t is the Laplace transfor of probability density p of rate coefficient; it follows that p can be obtained fro the observed survival function by eans of an (2) inverse Laplace transforation. After p is coputed fro the experiental data the distribution of energy barriers can be evaluated fro Eq.(2). We obtain: b g F I L HG KJ NM η T T T p T E = exp exp. (4) T B B B F HG IO KJ QP Produced with Applixware for Linux V. 5.

5 5 Now we can introduce a generalied AE odel which is valid both for static as well as dynaic disorder. In general the rando coponent of the energy barrier is a stochastic function of tie, =, t t t t ; its stochastic properties can be described by a probability density functional: 3 t $ t with 3 t $ t =, (5) is a suitable integration easure over the space of functions where $ E t E =, t t t t ; and the sybol denotes the operation of path integration. Eq.(6) for the average survival function is replaced by the ore general forula: ST t b g c b g t h UVW l t = exp T E t dt χ, (6) with... =... 3 t $ t, (7) which includes Eq.() as a particular case. In order to show that this is true, we discretie the tie interval fro t into tie intervals liited by the ties tu, u=,,... which have the lengths t = t t, u=,,... By recalling that for static disorder a rando selection of is ade only once, u u u 2 at the beginning of the process and then that selection lasts forever, we obtain: static static 3 t $ t = S T c U h c b gh V. (8) u> W = li η t dt δ t t dt u u u t u By inserting Eq.(8) into Eq.(7) we coe to Eq.(6), as expected. It follows that, for a unified approach to the AE odel, which includes both static and dynaic disorder as particular cases, we have to solve the following probles:. Find a suitable ethod for evaluating the probability density functional (5) for the rando energy barriers. 2. Develop ethods for coputing the average survival function of the process fro Eqs.(6)-(7). 3. Use the theory for evaluating the inetic paraeters of the process fro experiental data. Produced with Applixware for Linux V. 5.

6 6 These three probles are investigated in the reainder of this chapter. 3. Nonequilibriu enseble approach to rando energy barriers The first proble to be solved is the introduction of a suitable representation for the probability density functional of the rando energy barriers. Different approaches can be used for solving this proble. Our suggestion is to use a generaliation of the ethod of nonequilibriu statistical ensebles introduced by Zubarev and MacLennan 2. Copared to other ethods this approach has the advantage that it leads to the Maxwell-Boltann density as a particular case, which aes it possible to discuss the transition fro static disorder to dynaic disorder by using a unified approach. By using the Zubarev-Khalashniov version of the nonequilibriu enseble theory a nonequilibriu probability density functional for the energy barriers can derived fro the axiu inforation entropy approach by requiring nowledge on the whole previous histories of the average height of the rando energy barriers = t, t t t. (9) By using Eq.(9) as an isoperietric condition we arrive at a nonequilibriu enseble which is a dynaic generaliation of the static Maxwell-Boltann distribution of the rando energy barriers. We shall show later that this dynaical Maxwell-Boltann enseble can only describe the particular case of extreely fast dynaic disorder without eory. In order to consider the transition fro static to dynaic disorder we need to use a ore general type of non-equilibriu statistical enseble. We use a generaliation of the Zubarev-Khalashniov ethod suggested by Vlad and Macey 4. This ethod consists of requiring nowledge not only of the previous history of the average values of the rando variables but also of the history of various correlation functions. Applied to the proble of rando energy barriers this approach leads to the isoperietric conditions:, = 2... = t... t, t t,... t t,,... (2) The isoperietric conditions (2) can be deterined either fro theoretical considerations or fro experiental data. The siplest choice is to assue that the oents (2) have constant values for all ties. By using the axiu inforation entropy principle we arrive at the following nonequilibriu enseble: Produced with Applixware for Linux V. 5.

7 where 7 b g 3 t $ t = ] λ t,..., λ t,..., t,.. t t U exp S... λ bt,..., t gt b gdt... t b gdt V $ t b g, (2) = T t t W b g ] λ t,..., λ t,..., t,.. = t t U = exp S... λ bt,..., t gt b gdt... t b gdt V $ t b g, (22) = T t t W b g is a partition functional and the Lagrange ultipliers λ t,..., λ t,..., t,.. are deterined by the functional equations δ t... t = ln ] λ t t t bt t g,..., λ b,..., g,... (23) δλ,..., The characteristic functional of the nonequilibriu enseble (22) is: S t U & σ t = exp T i σ t t dt V 3 t $ t, (24) t W where σ t is a test function conjugated to. t The derivation of Eqs.(2)-(24) is long but standard; it is a straightforward functional generaliation of the inforation theory approach to statistical echanics.for a siilar coputation see Vlad and Macey. 4 This characteristic functional can be easily evaluated in ters of the partition functional (22). By inserting Eq.(2) into Eq.(24) and using Eq.(22) we coe to & b g b g ] λ t iσ t, λ2 t, t2,..., λ t,..., t,.. σ t = ] λ t,..., λ t,..., t,.. b g. (25) Now we can try to evaluate the dynaic average in Eq.(7). We notice that for dynaic disorder the rate coefficient is a rando function of tie. It is convenient to introduce a probability density functional for the rate coefficient : Produced with Applixware for Linux V. 5.

8 3 t $ t with 8 3 t $ t =, (26) and its characteristic functional S t U & ζ t = exp T i ζ t t dt V 3 t $ t t W, (27) where $ t is a suitable integration easure over the space of functions =, t t t t ; and ζ t is a test function conjugated to t. The probability functional 3 t $ t derived fro 3 t $ t We obtain: c h and & t and its characteristic functional & t σ by eans of a functional transforation: ζ can be t = Tχ Et. (28) δ χc h 3 t $ t = t T Et $ t = Sδ χ T c h U V E W t T t $ t 3 t $ t, (29) and S t & ζ t = exp i ζ t t dt T t U V W δ c h t Tχ t $ t 3 t $ t = S t U = exp i ζ t Tχct hdt V 3 t $ t. (3) T t W By coparing Eqs.(6), (27) and (3) we notice that the average survival function l t can be expressed in ters of the characteristic functional & o b g b gt ζ t. We have l t = & i ϑ t t ϑ t t, (3) Produced with Applixware for Linux V. 5.

9 9 where δ is a functional delta sybol and ϑ x is Heaviside s step function. Eq.(3) aes it possible to evaluate the average survival function in ters of the characteristic functional & ζ t ; in a few cases the characteristic functional & ζ t can be evaluated exactly; otherwise, if the cuulants of the rate coefficient exist and are finite, & investigated in detail in the following section. ζ t can be expressed as cuulant expansion. These probles are 4. Average survival functions for different types of disorder 4. Extree dynaic disorder. We start out by investigating the case of rapidly changing, independent fluctuating energy barriers. This is the siplest possible type of dynaic AE odel and corresponds to extree, very strong dynaic disorder. Other types of disorder are interediates between this extree case of independent, dynaic disorder without eory and the classic AE odel with static disorder and infinite eory discussed in the literature and suaried in Section. For rapid, independent fluctuations the nonequilibriu enseble is deterined copletely by the tie evolution of the average value, t, t t t of the rando coponent of the activation energy barrier. By applying the general theory developed in Section 3 we coe to the following expression of the probability density functional of : t ST UVW t t 3 t $ t = ] exp λ t t dt $ t. (32) In this case the Lagrange ultiplier λ t can be easily evaluated in ters of t, t t t. We have λ t = E t. (33) We notice that in this case the nonequilibriu enseble can be expressed as the continuous liit of a product of independent Maxwell-Boltann probability densities: 3 t $ t = li t exp t t dt = u u u u tu u B u u B u u { } = li T t exp t T t d E t, (34) tu u { } Produced with Applixware for Linux V. 5.

10 where T t = E t, (35) u u B are characteristic teperatures corresponding to different interediate ties t u. The nonequilibriu enseble is the continuous liit of a product of Maxwell-Boltann distributions froen at various characteristic teperatures T t, T t2,..., T t u,... This is a feature siilar to the one displayed by the static AE odel discussed in Section 2. The average survival function, l t, can be expressed as: t Et l t btg L = dt t N M O U exps exp T B Q P V T W = t L t = S N M O U exp af a f T exp Q P dt V 3 t af $ t af. (36) T t T B W In Appendix A we show that Eqs.(34)-(36) leads to the following expression for the average survival function t laf t = exps atf T t L NM a f B O QP U V W T B Et + T dt. (37) Eq.(37) connects the average survival function, which is an experiental observable, to the teperature of the syste, T, and to the inetic paraeters of the odel, the non-rando coponent of the rate coefficient, btg, and the previous average values of the rando coponent t of the activation energy barrier. The inetic paraeters in Eq.(37), btg and t, can be evaluated by coparing Eq.(37) with experiental data. In order to clarify the physical significance of Eq.(37) we consider a slight generaliation of the independent Maxwell-Boltann nonequilibriu enseble (34). We replace in Eq.(34) the Maxwell- Boltann probability densities corresponding to different ties by arbitrary, independent probability densities ηb; t u g: Produced with Applixware for Linux V. 5.

11 o c h t 3 t $ t = li tu tu dtu η ;. (38) tu u In Appendix B we show that the probability density functional (38) leads to the following expression for the average survival function t l t = S exp bt gdt t T U V W. (39) In the particular case of Maxwell-Boltann distribution Eq.(39) reduces to Eq.(37). By differentiating Eq.(39) we coe to: t, (4) l t = t l t that is, for rapid dynaic fluctuations the average survival function l t obeys the sae evolution equation as the fluctuating survival function l t (Eq.()), with the difference that the fluctuating rate coefficient t is replaced by its average value t. The physical significance of Eqs. (37)-(39) is clear: they are the integral inetic law derived fro an averaged differential inetic equation (Eq.(53)) characteried by an averaged rate coefficient t. In particular, for a Zubarev-McLennan enseble we have c h T BT t = T Et Et Et Et d Et = χ exp, (4) t + T and Eq.(39) reduces to Eq.(37) General treatent of dynaic disorder. The generaliation introduced above and developed in Appendix B also provides a clue about a possible way of generaliing Eq.(39) for an arbitrary type of dynaic disorder; it suggests to express the average inetic curves in ters of the cuulants t... t, = 2,,... of the rate coefficient. By assuing that these cuulants exist and are finite, the characteristic functional & expressed as: & t t i t exps =! T t t ζ =... t... t ζ t... ζ t dt... dt U V W B ζ t can be, (42) Produced with Applixware for Linux V. 5.

12 2 and Eq.(3) leads to: l t S T t t... t... t dt... dt t t bg = exp =! U V W. (43) Eq. (3) and (43) are exact equations for the average survival function. Unfortunalely, in order to apply these equations we need to evaluate the characteristic function & ζ t of the total rate coefficient or the cuulants t... t, = 2,,... in ters of the generalied nonequilibriu enseble (2) or the partition functional (22). These quantities can be evaluated, at least in principle, by using perturbation approaches siilar those used in quantu field theory, by starting out fro the independent Zubarev- McLennan enseble (32) and then evaluating corrections induced by the correlation functions of the energy barriers. In this chapter we do not attept to carry out such coputations; instead we try to analye the general physico-cheical iplications of Eqs.(3) and (43). We start out by discussing the proble of the tie-dependence of the average of the rando coponent of the energy barrier, t. Fro the theoretical point of view, such a dependence arises naturally fro the application of the Zubarev-McLennan approach. However, for ost experiental systes described by the AE odel the disorder is stationary and thus such a tie dependence can be ruled out. Tie-dependent averages t ay be needed for describing the relaxation under the influence of external fields with a tie dependent average. The tie dependence of t ay coplicate the analysis of experiental data. For exaple the condition that is constant applied to rapid, independent fluctuations of the Zubarev-McLennan type leads to exponential relaxation, with a b g B + B. If is linearly increasing in tie: constant rate T T E T b g b g t = t + btt, (44) the inetic curve displays a long tail with a scaling exponent btg BT b: l t = L NM T B + t T+ t + btt B O b g QP Siilarly, if is a periodic function of tie, T T b B. (45) Produced with Applixware for Linux V. 5.

13 3 t = t b g + acos ωbt tg+ ϕ, t > a (46) then the relaxation function is given by T T L N 2 T BT c c t t l - F ϕ b g t = I a c M c HG K J P ω + ϕ exps S arctg M 2 ω + tg - arctg 2 c + HG KJ tg 2 O Q L NM F IOUU QP V V W W, (47) where c= t + BT a >. (48) In this chapter we do not discuss these cases in further detail. In the following, for siplicity, we assue that the stochastic process which describes the fluctuations of the energy barrier is stationary and therefore is constant. Produced with Applixware for Linux V. 5.

14 4 4.3 elations between static and dynaic disorder Now we discuss the relations between static and dynaic disorder for AE odels. For siplicity we assue the case where the probability density of energy barriers η b g has an arbitrary shape but is tie independent; this includes the case of Maxwell-Boltan profiles as a particular case. In Appendix C we show that, under these circustances, for static disorder the cuulants and oents of the total rate coefficient are constant. t = t... t, t t,..., t t, (49) t = t... t, t t,..., t t. (5) The physical interpretation of Eqs.(49)-(5) is clear: for the static AE odel the oents and the cuulants of the rate coefficient do not decay; for any ties they have the sae value as at the initial oent t. Under these circustances a fluctuation of the energy barrier and of the rate coefficient, once it occurs, lasts forever. It follows that the AE odel with static disorder has infinite eory. Eqs.(49)-(5) ae it possible to discuss the transition fro static to dynaic disorder. Based on these equations, in Appendix C we show that, b g l t = G t = p exp t d, (5) static which is equivalent to Eq.(3), derived in Section for systes with static disorder. The transition fro static to dynaic disorder can be understood in ters of two tie scales, the inetic (relaxation) tie scale, which has the agnitude order τ inetic ~, T and the tie scale characteristic for the decay of the fluctuations of the rate coefficient, τ decay. If τ decay >> τ, the cuulants of the rate coefficients decay very slowly and the transition fro static to dynaic disorder is also very slow. Fro Eqs.(43), (5) and (5) it follows that S T l bg t = exp b t t tg ~ l t! for τ = decay U V W inetic static, >> τ, τ decay >> t t. (52) inetic Produced with Applixware for Linux V. 5.

15 5 That is, for sall to interediate ties, the decay of fluctuations of the rate coefficient has little influence on the inetics of the process and the average survival function is alost the sae as the average survival function for a syste with static disorder. For large ties, however, the inetics of the process is no longer doinated by the inetic tie scale, τ inetic, and the decay of the fluctuations of the process play a doinant role. If the two tie scales have the sae order of agnitude, τ decay ~ τ, then there is a strong interaction between the inetics of the process and the decay of the fluctuations. For a better understanding of this interediate regie we consider a special type of cuulants of the rate coefficient which is capable of describing a wide range of types of disorder. Since we discuss only the case of stationary processes, it follows that the cuulants ust obey the condition of tie invariance: d i d i t... t = t t... t t b inetic. (53) where t = in t,..., t g. Furtherore we assue that decay of the cuulants of a given order can be described in ters of a single daping function g t t d i d i d i d i which depends on the cuulant order : t t... t t = κ g t u t. (54) u Siple shot noise odels for dynaic disorder produce 5 cuulants of the type (54). After soe algebraic anipulations involving transforations of integration variables and doains, Eqs.(43) and (54) lead to the following expression for the survival function: tt t t l b g L t = S d M θ exp κ... θ... g =! T b g We consider two types of eory. (a) Long eory. If the daping functions gt N d O Q P U V W τ dτ. (55) t i have long tails of the negative power law type, then the fluctuations of the rate coefficients have long eory. The siplest case of long eory corresponds to H g ~ τ τ,> > H, (56) Produced with Applixware for Linux V. 5.

16 6 where H are fractal exponents attached to the different cuulants. By inserting Eq.(56) into Eq.(55) we obtain: T + bgbhg t t l bg κ b g t = exps! H + H = b gb g b gb g c U h V W, (57) that is, the average survival function is given by a generalied stretched exponential. (b) Short eory. If the daping functions gt d t i decay fast and have short tails of the exponential type, then the fluctuations of the rate coefficients have short eory. The siplest case of long eory corresponds to: g τ b ν τg ~exp, (58) where ν are decay frequencies which depend on the cuulant order. In this case the aplitude factors κ are equal to the cuulants of the rate coefficient for static disorder: κ = t. (59) Eq.(59) can be obtained fro Eqs. (54) and (58) applied for t,..., t = t. The average survival function becoes: l t = exp In this case τ exponentially = exp S T = S T = bg b bg b! ν b decay ~ax btg g! btg gb g S T tt Lexp ν bt t... dθ NM ν go QP U V W dθ (6a) UU W V W u νbt tg expcν bt tgh V. (6b) u u= ν g. According to Eq.(6b) for large ties the average survival function decays b g l t = exp t t Ω, for t t >> (6) where: b g = b g b t g Ω=! ν, (62) is a characteristic frequency. Produced with Applixware for Linux V. 5.

17 7 expected, for τ for τ decay Fro Eqs.(6a-b) we can also recover the two extree types of disorder as particular cases. As decay >> τ, Eq. (6a) reduces to Eq.(5) valid for static disorder. In the other extree, inetic << τ, we have ν and Eqs.(6a-b) reduce to inetic b g l t = exp t tt, (63) which is a particular case of Eqs.(37) and (39) valid for rapid, independent fluctuations described by the Zubarev-MacLennan enseble: Eqs.(37) and (39) reduce to Eq.(63) for constant. A popular approach for disordered inetics is based on the use of ultiplicative Gaussian, white noise 6 for describing the fluctuations of rate coefficients. Applied to the AE odel such an approach is based on the assuption that the first cuulant of the rate coefficient,, which is identical to the average value is constant, the cuulant of second order is delta correlated and all other cuulants are equal to ero: b g = constant, t t2 = σδ 2 tt2,, t... t > 2 =. (64) Despite its popularity, the Gaussian approxiation (64) leads to unphysical results for the AE odel, due to the fact that Gaussian probability laws are syetric and ae it possible for the rate coefficient to tae negative values, which is incorrect. Although the probability of taing negative values is usually very sall, its effect is increased by the ultiplicative nature of the noise. This shortcoing can be easily overcoe by introducing a generalied white noise for which all cuulants of order bigger than one are delta correlated : b u u+ g u= = constant, t... t = σ δ t t, = 23,,... (65) In the literature, inetic systes obeying Eqs.(65) are usually described in ters of Langevin equations. However, in this chapter we study the iplications of Eqs.(65) by using the nonequilibriu enseble approach, which leads to results copatible with the ethod of Langevin equations. We intend to investigate the connections between these two approaches in a separate publication. The approach developed in this section is based on the assuption that the cuulants of the rate coefficient exist and are finite. It follows that, strictly speaing, a syste with white noise of type (65) Produced with Applixware for Linux V. 5.

18 8 cannot be used; the noral approach for such systes is to use stochastic calculus of the Stratonovich or Ito type. In this chapter, however we use an alternative approach suggested by Van Kapen and West 7 for other probles; such an approach has the advantage that we can use partial results developed in this section. We consider cuulants which lead to the delta correlated expressions (65) for certain given values of soe paraeters, copute the average survival function for arbitrary values of the paraeters and then pass to the liit which reproduces the delta correlated cuulants (65). Such an approach can be easily ipleented by considering a slight odification of the daping functions (58) b g g = τ ν exp ν τ. (66) By introducing the preexponential frequency factor ν, the daping factor g τ has the foral structure of a probability density for the tie difference τ ; in the liit ν, the daping factor g τ tends towards a delta function. In this case Eq.(59) is replaced by: κ = b t νg, (67) and the cuulants described Eqs.(54) reduce to the delta correlated cuulants described by Eqs.(65) in the liit, ν with the constraint li κ ν = σ ν b g. (68) For arbitrary ν the average survival function can be easily evaluated fro Eq.(55): tt l bg κbνg U t = exps... dθ t t d b g o exp ν b gt θ V. (69) =! T W In the liit ν with the constraint (68), Eq.(69) leads to an exponential law for the average survival function b g l t = exp t t Ω, (7) where: Ω = b g b = σ! g. (7) It is interesting to copare the Zubarev-MacLennan fluctuations with the case of white, non-gaussian noise with delta correlated cuulants. In both cases the fluctuations of the energy barrier and rate Produced with Applixware for Linux V. 5.

19 9 coefficient are fast and statistically independent. Both types of fluctuations correspond to the liit case of very strong dynaic disorder. There is only one difference between the two: for Zubarev-MacLennan fluctuations the cuulants of the rate coefficient exist and are finite, whereas for white noise the cuulants display delta type singularities. In both cases the average survival function decays exponentially; although the decay rates are different in the two cases, experients based on easureent of average concentrations cannot ae a distinction between these two types of dynaic disorder. The typical pattern which eerges fro the exaination of the different types of dynaic disorder investigated in this section is that for long ties the dynaics activation energy fluctuation decays prevails and outweighs the contribution of the reaction inetics. If the fluctuations are slowly decaying the contribution of cheical inetics can be seen in the shape of the average survival function for sall to interediate ties; otherwise, if the decay of fluctuations is fast, the overall inetics of the process is doinated by fluctuation dynaics. In the following section we illustrate the general results derived in this section by considering a siple exaple of AE odel with dynaic disorder. 5. Exaple: AE odels with renewal-type dynaic disorder We introduce a siple AE odel with dynaic disorder, based on the following assuptions. () The rando coponent of the energy barrier is randoly selected fro a tie invariant probability density ηbg. Different values are selected at rando ties fro the probability density ηbg; the selected value at a given tie is independent of the other values of selected at previous ties. (2) The tie interval τ between two selection events is selected fro two probability laws: an initial b g for the tie interval that elapses fro the beginning of the process to the first probability density ψ τ selection event and a second probability density ψ τ for the tie interval between two selection events. The AE odel defined by these two assuptions is a odel of the continuous tie rando wal 8 type (CTW) which is also referred to as a renewal process. CTW (renewal) processes are coonly used for describing transport processes in disordered systes -4. By using the general theory developed in this chapter, in Appendix D. we show that the Laplace transfor of the average survival function is given by: Produced with Applixware for Linux V. 5.

20 L d p s d p s la f b g NM b gol + QP NM b go Ψ + QP ψ dpafψ as + f + t t = d p Ψ + s + 2, (72) where: τ ψ τ d τ, Ψ τ = ψ τ d τ, (73) τ Ψ = τ the overbar and + t denotes the Laplace transforation and s is the Laplace variable conjugated to the tie variable. Fro Eq.(72) we notice that the tie dependent behavior of the average survival function is deterined by the eigenvalues of the secular equation Is = dp b ψ s+ g=. (74) The eigenvalues of Eq. (74) depend only on the probability density ψ τ of the tie interval between two arbitrary selection events and are independent of the probability density ψ bτg attached to the first selection event. It follows that ψ bτg ay influence the values of aplitude factors present in the expression of the average survival function but does not influence the tie dependence of this function. For this reason in the following we assue that b g ψ τ = ψ τ. (75) The approxiation (75) is coonly used in the CTW literature and in ost cases it does not lead to serious liitations. b g in ters of ψb s b g b g b g b g We express the Laplace transfors Ψ + s + g. We have: s Ψ + s = dt + s t t dt = ψ exp ψ +. (76) t + s We also notice that the Laplace transfor l s static l t static, can be expressed as: of the average survival function for static disorder, b g c b g h p l s = l t exp st dt = p dexp s+ t dt = s d. (77) static static + Produced with Applixware for Linux V. 5.

21 2 By using Eqs.(75)-(77) Eq.(72) becoes: ψ s+ l s dp static s + t l b + g t = dpb ψ s + g b g. (78) Eq.(78) can be used for evaluating the average survival function for different types of disorder. The static disorder corresponds to the particular case where the tie interval between two selection events tends towards infinity. We assue that the probability density ψ τ depends on the average value τ. Diensional analysis requires ψ τ to be of the for c h ψτ = τ ϕτ τ, (79) where ϕ x is a diensionless probability density and x = τ τ is a diensionless tie interval. We have: ψ s = exp c τ xshϕ x dx. (8) Fro Eq.(8) it follows that in the liit τ we have ψ s and fro Eq.(78): li τ as expected. l t = l t static, (8) In the other extree of fast, independent fluctuations we have τ. We notice that x = τ τ =, that is, the average value of the diensionless tie interval x is equal to one and thus, according to Eq.(8) we have = x = xϕ x dx = ϕ, (82) where b g ϕ = x ϕ x dx exp, (83) is the Laplace transfor of ϕ x and is the Laplace variable conjugated to the diensionless tie interval x. Fro Eqs. (78), (8) and (82)-(83) we obtain: Produced with Applixware for Linux V. 5.

22 22 li + t t τ l dp ϕ = = s + dpb s + g ϕ, (84) fro which, by coing bac to the tie variable we coe to: = l t exp t, (85) which is the sae as Eq.(63) for the average survival function for independent, rapidly fluctuating MacLennan-Zubarev fluctuations. For studying the interediate types of dynaic disorder we need to now the solutions of the secular equation (74). We notice that dpbg bs g dpbg ψ + = exp tbs + gψ t dt = χ s, (86) where b g χ s = exp st ψ t l t dt, (87) static and thus the secular equation (74) can be put in a for siilar to Lota s secular equation fro atheatical deography, which has been extensively studied in the literature 9 : b g χ s = exp st ψ t l t dt. (88) static It has been shown that if χ s is analytic in s then equations of type (88) have a single real root s, which is nonpositive 9. We have: s < if χ< and s = if χ=. (89) Eq.(88) can also have at ost a countable nuber of coplex roots sq = uq ± ivq, q = 2,,... with real parts u q saller or at least equal to the real root s (ref) u q s, q = 2,,... (9) For siplicity we assue that the coplex eigenvalues sq = uq ± ivq are siple and evaluate the inverse Laplace transfor of Eq. (78). After lengthy algebraic anipulations we coe to: Produced with Applixware for Linux V. 5.

23 were! and ) q ± q t l t =! expc t s h+ 2 expetuqj, (9) q ) + )! = are constants d qi d qi dp dθo exp θbs + gtψ θ s +, (92) dp dθθ exp θ s + ψ θ b g e js T cos w θ θ ± q ) q = d p dθθ exp θ u q sin wq q t are periodic functions of q t = dd dθθ d dxθθ pb p g d d iu iv W ψθ, (93) e qj e qj d qb gi d q i b g. (94) exp xθ u θ u cos w t xθ cos w θ ψ θ ψ θ For large ties the decay of the average survival function is exponential l t ~! expct s h for t >>, (95) a result which is consistent with the general analysis in Section 4. An interesting situation which was not considered in the general discussion in Section 4 corresponds to the case where the probability density ψ τ of the tie interval between two selections is also described by a rate process with a rando activation energy barrier. More precisely, we assue that ψτ is described by a superposition of exponential distributions: µ µ d µ i d µ i d µ i ψτ = d η µ exp τµ, (96) where µ dµ i is an activated rate coefficient for which the energy barrier has a rando coponent µ d i L N M O Q P µ Eµ µ µ = µ T exp with µ = T Aµ exp, (97) B T B T and η µ d µ i is the probability density of the rando coponent µ of the energy barrier of the selection process. In this particular case the odel is siilar to a aster equation approach developed in a L N M O Q P Produced with Applixware for Linux V. 5.

24 24 different physical context in ref. The inetics of the process is represented by the superposition of two different static AE odels, one for the cheical process itself and the second for the selection process of the energy barriers. By cobining Eqs.(78) and (97), after soe algebraic anipulations we coe to + t t l L p p µ p p b s g = dd N M O µ + dd s+ + Q P µ µ µ µ s+ + µ where p is given by Eq.() and p µ µ is given by: S F I U HG d i T B KJ V W L F µ IO NM HG KJ S T dµ T B ηµ T B ln for µ µ T µ QP µ T for µ µ T µ pµ µ dµ = d δ µ T exp µ η µ µ d E µ = T = µ, (98) Although not identical, Eq.(98) has a siilar structure with Eq.(8.4) fro ref. This siilar structure can be used for evaluating the inverse Laplace transfor in Eq.(98) for different types of distribution functions. By using this ethod we can identify different types of asyptotic regies which can be used for the analysis of experiental data. We distinguish three different cases: ) We assue that the rate process is described by a Maxwell-Boltann distribution of activation energies and the selection process by a constant activation energy E ϑ T pd= α T α α d = E. We have: µ µ (99), () pµ µ dµ = δ µ T µ dµ, () and the large tie behavior of the average survival function is given by l t α Γb + αg btg ~ α αµ T T B α, α, T T + µ T c h t α exp µ T t, for t x q b g b g >> T (2) p where Bpqx,, = x x dxpqx,,, > is the incoplete beta function. According to Eq.(2) the beginning of the tail of the survival function, for µ T >> t >> T, is described by a negative power Produced with Applixware for Linux V. 5.

25 25 law, whereas the end of the tail, for t >> T is described by an exponential functions. Thus, for large ties, the decay of the fluctuations deterines the overall inetics of the process, a result which is copatible with the general approach in Section 4. 2) The rate process is described by a Maxwell-Boltann distribution of activation energies and the selection process by a positive Gopert law with a characteristic variation of the activation energy µ, d i d i o d i η µ µ = µ µ µ µ µ t exp exp. (3) We get the following asyptotic for expression the large tie behavior of the average survival function: where b g α α ε l t ~ Γ + α T µ T t exp ξt, (4) ε = T B dµ + T B i <, (5) is a second fractal exponent and L NM O QP + µ BT T B T B ξ = µ T M + P, (6) µ µ is a characteristic frequency. In this case the beginning of the tail is given by a power law and the end of the tail by a stretched exponential. 3) We assue that both the rate process and the dynaic fluctuations of the energy barriers are described by Maxwell-Boltann distributions. The probability density η d η µ µ i is given by a siilar exponential law b g is given by Eq.(9) and d i= exp, (7) η µ µ µ µ µ where µ is the average value of the rando coponent of the energy barrier attached to the selection process. The probability density p of the reaction rate is given by the power law () and the probability density p µ µ of the rate µ is given by a siilar power law Produced with Applixware for Linux V. 5.

26 26 S L O U N d i T B Q V W µ pµ µ dµ = dµ δ µ T expm P µ η µ µ d E µ = T ϑµ µ T = µ µ T α µ α µ α µ dµ, (8) and α µ = T B E µ, (9) is a third scaling exponent between ero and unity. Fro Eqs.(98), (8) and (9) we get the following expression for the large tie behavior of the average survival function: α dα+ α i b g d i µ l t ~ Γ + α Γ + α µ µ T T µ T t for T >> µ T, t >> µ T. () In this case there are two AE odels, one for the rate process itself, and the second one for the selection of the energy barriers which describes the decay of the dynaic fluctuations of the activation energy. In the long run these two AE odels interfere, resulting in a fractal exponent α + which is the su of the fractal exponents attached to the two processes. In conclusion, in this section we have shown that continuous tie rando wal (renewal) odels for dynaic disorder lead to results which are generally copatible with the general analysis in Section 4: for sall to oderately large ties the process can be approxiately described by a static AE odels, whereas for large ties the decay of dynaic fluctuations of the energy barrier prevail. There is however one exception, corresponding to long eory of the decay of fluctuations of the energy barrier: in this case the two AE odels interfere in the long run and the total fractal exponent of the resulting power tail is the su of the fractal exponents corresponding to the two AE processes. 6. Nonequilibriu enseble approach to reversible cheical processes Until now we focused on the siplest disordered rate process, a first-order irreversible reaction with static or dynaic disorder. In this section we consider a ore coplicated type of process, a reversible (quasi) reaction, involving a set of forward and bacward steps: α η A+ B C, () Produced with Applixware for Linux V. 5.

27 27 eaction () describes different types of experiental systes, for exaple adsorption inetics on a heterogeneous surface, where A is a cheical species in the gas or liquid phase, B is a free adsorption site on a surface and C is an occupied site on the surface. In the following we liit ourselves to the study of the adsorption proble; in the next section, an adsorption experient reported in the literature is analyed based on the theoretical results derived in this section. We assue that the surface is energetically heterogeneous, the adsorption energy of the species A varies fro place to place. By following the hoottatic patch approxiation, we assue that the surface can be divided into patches which are characteried by the sae adsorption energy and that these patches are big enough so that a (quasi) acroscopic description of the adsorption is possible. For each patch we have a set of local adsorption and desorption rate coefficients where c h L N M O exp B Q P, (2) ± ± ± ± E local E = Alocal T T ± ± local are the adsorption and desporption rate coefficients at teperature, T, respectively A local are preexponential factors, and E ± are adsorption and desorption activation energies, with are rando functions of tie and position. For a given patch, the ratio between the two rate coefficients: a f c h c h L exp N M E E K = E E = A A T P = K + local local local local local local B is a local equilibriu constant and O Q L N M T B O Q P, (3) + = E E = H, (4) is the adsorption energy. In general the activation energies E ± are both rando functions of tie. Instead of carrying out dynaic averages over two rando functions, E + and E, it is ore convenient to use Eq.(4) and tae an average over one of the two activation energies, say the activation energy for desorption, E and the adsorption energy. This type of averaging is easier because the adsorption energy is a rando nuber not a rando function and the corresponding average is static. We denote by A = ρ, the concentration of species in the bul phase (liquid or gas), by B = b d the surface concentration of free sites with an adsorption energy between and + d Produced with Applixware for Linux V. 5.

28 28 and by C ε = c ε dε the corresponding surface concentration of occupied sites. We have the balance equation:, (5) b d + c d = g d where g d is the total concentration of adsorption sites, free and occupied, which are characteried by an adsorption energy between and + d. The local coverage corresponding to an adsorption energy between and + d is given by: θ local = c g. (6) The inetic equation for this local coverage is given by: c h + θlocal = local t ρ θlocal local t θlocal, (8) t In typical adsorption experients the variation of the bul concentration ρ is negligible and Eq.(8) can be easily solved, resulting in: where: locala f local a f localb g local af localaf t local t L NM c h af equil. equil. θ, t = θ + θ, t θ exp + K ρ t dt. (9) θ equil. local a f a f a f Klocal ρ =, (2) + K ρ local is the local equilibriu coverage corresponding to an absorption energy between and + d. The experiental observable is the total coverage θ, t which can be obtained fro Eq.(9) by eans of a double averaging procedure: ) over all possible rando functions local t and 2) over all possible equilibriu constants K local a f, corresponding to different adsorption energies. We rewrite Eq.(9) in the following for. The dynaic averaging over all possible rando functions local t can be easily carried out by using the nonequilibriu enseble approach developed in Section 3. Eq.(9) can be expressed in ters of the odified survival function: where af expl κ NM af χc a fh l t t O QP t = E t dt, (2) O QP Produced with Applixware for Linux V. 5.

29 29 = E E, (22) is the rando coponent of the desorption activation energy, κ a f c hc a f h = local E + Klocal ρ, (23) is an effective rate coefficient, and d i χ t = exp t T, (24) is a rando transparence factor siilar to the one defined by Eq.(8). We have: a f a f b g a f a f B equil. equil. θ local, t = θ local + θ local, t θ local l t (25) By exaining Eq.(25) we notice that the only ter dependent of the fluctuations of the energy barrier E t is l t and that all ters in this equation depend on the adsorption energy. The total coverage θ t can be expressed as the double average: θ t = θ b tg E = local, t b g b g b g equil. equil. = θlocal, T + θlocal, t t θ local, T t l l where the dynaic average l t t t t, (25) can be coputed by using the nonequilibriu enseble techniques developed in Section 3 and the static average... can be expressed in ters of the probability dendsity p d of the adsorption energy:... = p d. (26) As pointed out by a referee, for coplex processes, the average concentrations or survival functions do not contain enough inforation for the deterination of the inetic paraeters of the process. In soe cases the evaluation of the statistical properties of the lifetie of a species provides the necessary inforation for deterining the inetics and echanis of the process. In the following we copute the probability density of the lifetie τ of the species on the surface. We introduce a joint density function for the adsorption heat and the trapping tie: στ b, gdd τ with c d = d στ b, gdτ. (27) Produced with Applixware for Linux V. 5.

30 3 Here στ b, gdd τ is the surface concentration of occupied sites with an adsorption energy between and + d and which have trapped a olecule for a tie interval between τ and τ + d τ. The density function can be coputed fro the balance equations: F + I HG K J = t t τ στ b, g στ b, g b g c b gh local, (28) στ= +, = ρ t θ g local local. (29) In ters of the density function σbτ, gdd τ we can copute the conditional probability density στ, dτ ϕτ e ; E t j b g dτ=, with ϕτ e, E t jdτ=, (3) στ, dτ b g of the trapping tie corresponding for a given value of the adsorption energy. Finally, the unconditional probability density of the trapping ties can be evaluated by averaging over all possible values of the adsorption energy and of the activation energy E t of the desorption process ϕτ dτ= dτ ϕτ e ; E t j p d = ϕτ e ; E t j with E t E ϕτ dτ =.(3) t In conclusion, in this section we have shown that the nonequilibriu enseble approach can be easily extended to reversible cheical reactions involving a forward and bacward step. For illustration we considered the case of adsorption inetics on a heterogeneous surface. The obtained results are used in the following section for analying an experient on desorption inetics Interpretation of desorption experients by Draer and Zanette In their experients Draer and Zanette 2 used nonconsolidated pacings of relatively unifor, spherical, activated carbon grains obtained fro apricot pits, with an average radius of d = (. 3±. ) c. The carbon grains were paced in a 3c high, 2.54c inner diaeter cylinder. In the experients, the porous ediu is initially filled up with aqueous.m NaI solution tagged with I 3. These authors carried out desorption experients where the untagged NaI solution was replaced by distilled water and showed that for large ties the concentration profiles of I 3 decrease slowly to ero according to a negative power law, µ ρ~ t t, t >>, µ = 63., (32) Produced with Applixware for Linux V. 5.

31 3 characteried by a fractal exponent µ = 63.. Draer and Zanette have also shown that the adsorption equilibriu at low concentrations is described by a Freundlich adsorption isother θ ~ ρ ν, ρ sall, ν = µ = 63., (33) where θ is the surface coverage and ν = µ = 63.. The experiental results of the authors suggest that the displaceent of the radioactive isotope I 3 involves a very slow, dispersive (Montroll) diffusion processs 8. The qualitative physical picture suggested by the authors 2 is the following: the otion of an ato of the radioactive isotope along the colun can be represented by a hopping echanis involving a succession of desorption and readsoption processes, which is basically a rando wals in continuous tie 8. According to Montroll s theory of dispersive difffusion 8, such a continuous tie rando wal ay lead to concentration profiles with long tie tails of the negative power law type if the probability density of the trapping tie of the radioactive isotope in the adsorbed state on the surface, has a long tie tail. In a previous article 2 we cae up with a theoretical interpretation of Draer and Zanette s experient, based on the rando activation energy odel and on the assuption that the adsorption and desorption activation energies are related to each other through a linear free energy relation. In the following we shall show that, within the fraewor of the nonequilibriu enseble approach developed in this chapter, the existence of a linear free energy relation between the adsorption and desorption activation energies, is not necessary for explaining the experiental results of Draer and Zanette; however the linear free energy relation is copatible with their results. µ By assuing that the generation of a concentration profile of the type ρ~ t t, t >>, is due to a CTW otion, it can be shown that the probability density of the lifetie of a olecule on the surface also has a long tail of the type 22 : µ ϕτ ~ τ b + g, τ >>. (34) Now we can apply the results derived in Section 6 to the experients of Draer and Zanette. Although the whole colun is in a nonequilibriu state, it is reasonable to assue that at each section of the colun there is a local stationary state. By assuing static disorder, the stationary solution of Eqs.(28)-(29) is given by: a f localafb locala fg af localc h c h + στ, = ρ t θ g exp τ E χ E, (35) Produced with Applixware for Linux V. 5.