A new approach to the derivation of binary non-Markovian kinetic equations

Physica A 268 (1999) 567–606

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A new approach to the derivation of binary non-Markovian kinetic equations A.A. Kipriyanova , O.A. Igoshina , A.B. Doktorovb; ∗ b Institute

a Novosibirsk State University, Novosibirsk, 630090, Russia of Chemical Kinetics and Combustion, The Russian Academy of Sciences, Novosibirsk 630090, Russia

Received 21 March 1998; received in revised form 26 November 1998

Abstract A universal method of derivation of in nite hierarchies for partial distribution functions and correlation forms in the thermodynamic limit has been developed. It is based on the consideration of reacting systems in the Fock space. Hierarchy closure methods available in the literature are shown to give incorrect binary kinetic equations of the reaction A+B → B in some critical cases. A new approach to hierarchy closure has been proposed. It consists in neglecting contributions from four-particle correlations and in adapting the Faddeev method of the three-body theory to the extraction of a binary part of three-particle evolution. For the model of the reaction A + B → B the proposed method gives correct kinetic equations obtained earlier on the basis of diagram summation. It gives the theoretical basis for derivation of binary kinetic equations for c 1999 Elsevier Science B.V. All rights reserved. realistic reacting systems.

1. Introduction The presence of a cage eect [1,2] in chemical reactions proceeding at binary encounters of reactants in liquid solutions leads to the necessity of describing adequately the non-Markovian stage of binary kinetics, where the rate constant is time dependent [3– 6]. Thus the development of the binary non-Markovian theory of bimolecular reactions calls for taking proper account of corrections to the density parameter of reactants, and may be realized solely on the basis of the many-particle consideration of reacting systems [4,7–10]. At present many papers dealing with such a consideration are available. We recognize two main approaches to the derivation of binary kinetic equations. The rst ∗

Corresponding author. Fax: +7-3832-342-350. E-mail: [email protected] (A.B. Doktorov)

c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 0 2 0 - 5

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one is based on the closure of in nite hierarchies, which are similar to the BBGKY hierarchies of the non-equilibrium statistical mechanics [11–13]. The most commonly used closure is based on the Waite superpositional decoupling [7,14 –17] (adaptation of the Kirkwood superpositional decoupling [18,13] to reacting systems). In describing irreversible reactions in spatially uniform systems, its simplest version results in the dierential theory [7,14,15]. The simplicity of the derivation of non-Markovian kinetic equations in this theory is an obvious advantage of the rst approach. It does not call for the use of diagram technique of the many-body theory. However, the construction of hierarchies employed in it has only been done for spatially uniform reacting systems on the basis of semi-intuitive considerations. This, however, is justi ed by the level of model description. Nevertheless, consideration of nonuniform systems, or abandoning of at least some simplifying assumptions in the description of reactions calls for the development of a universal procedure of hierarchy derivation. It should be based on the consideration of reacting systems in the Fock space. Besides, the method of closure by the Waite superpositional decoupling is not quite satisfactory. If applied to the description of reversible reactions, it gives kinetic equations which are rather dicult to interpret physically [19,20]. However, its principal disadvantage is the impossibility to properly control the accuracy. Therefore, there is no sense in further development of the rst approach without referring to the second one. The second approach is based on the construction of irreducible evolution operator – “mass” operator – and its subsequent simpli cation by an expansion in concentration of reactants [21–24]. The expansion to the rst non-vanishing terms leads to the integro-dierential encounter theory [25]. This is the theory able to study the reactions that cannot be described by rate constant [26]. Now this theory is frequently used for studying kinetics of multistage reactions of charge and energy transfer in solutions [27,28]. However, the encounter theory has been shown [29] to have a rather narrow time interval of applicability. It means that the linear term=in concentration=of the mass operator does not account properly for the binary kinetics, and thus the encounter theory needs modi cation [30]. Recently for irreversible reactions A + B → B, this modi cation has been performed [10] on the basis of diagrammatic summation methods. Owing to simplifying assumptions in the description of reactant structure (point reactants, no force interaction) and conservation of number of Bs, calculation of many-particle=in both As and Bs=kinetics has been reduced to the calculation of survival probability of one A reactant in the ensemble of B reactants. This has made it possible to obtain the master kinetic equation which is the analogue of the Prigozhine–Resibois [31] equation by a rather simple diagram method. Selection of the necessary binary diagrams using scaling principles allowed us to obtain correct binary non-Markovian kinetic equations. However, abandoning any of the simplifying assumptions or considering the reactions in which both species decay – e.g. A + B → C – makes the above reducibility impossible. Generalization of the described derivation to this case is really complicated. On the other hand, binary non-Markovian kinetic equations of the reaction A+B → B obtained in [10] may be very useful as a test for judging closure approximations and

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nding correct binary closure method. The simplicity of deriving kinetic equations by closure methods may allow to nd a rather simple but universal way for a many-particle derivation of binary kinetic equations. The goal of the paper is to develop such a method. It will be done by the development of a universal procedure of hierarchy derivation and its closure. The latter is based on the adaptation of the Faddeev reduction in the quantum-mechanical theory of three-bodies [32] to reacting systems. The outline of the paper is as follows. In Section 2 the simplifying assumptions made in paper [10] are formulated, and the basic results from the paper are given. In particular, two equivalent forms of the binary non-Markovian kinetic equations, integro-dierential and dierential, are demonstrated; and two ways of describing the evolution of the reacting pair by the propagator or t-matrix are shown. Section 3 consists of three subsections wherein a universal method for the derivation of in nite hierarchies for partial distribution functions (PDFs) in the thermodynamic limit for nonuniform systems is given. The rst subsection describes the evolution of a many-particle reacting system in the Fock space by the Liouville equations. In the second subsection a microscopic point density is introduced, and the in nite set of equations for the average – over the Fock space – point densities is derived from the Liouville equations. In the third subsection the passage to the thermodynamic limit is performed in the obtained set of equations, and PDFs are introduced. Based on the analysis of the hierarchy derived for the PDFs, the correlation loss condition in the reacting systems is established, and the rules for the transitions between the PDFs are formulated. Those rules set up a correspondence between PDFs of dierent order and are similar to those in non-equilibrium statistical mechanics. Section 4 analyzes the Waite superpositional decoupling, with the systems spatially nonuniform in A species distribution and uniform Bs as an example. It is shown that this procedure is incompatible with the internal symmetry of the reacting system under study, and thus can hardly be a universal method for obtaining binary kinetic equations. In Section 5 correlation forms for the reacting system and their diagram representation are introduced. This allows one to give a graphic representation for any “correlation dynamics” equation. In Section 6 a closure of hierarchies for correlation forms is performed on the basis of physical argumentation used in non-equilibrium statistical mechanics [11]. Inconsistency of kinetic equations thus obtained is shown. Modi cation of the closure is shown to lead to integro-dierential equations of the encounter theory. In Section 7 we neglect four-particle correlations and use the ideas that form the basis of the Faddeev reduction in the quantum-mechanical theory of three-bodies [32]. This made it possible to perform the closure of hierarchies for the correlation forms, and to obtain the eective pair approximation found in [10] by diagram technique methods. This approximation serves as the rst stage in the development of a binary approximation. With a system spatially nonuniform in A species and uniform in B species as an example, Section 8 demonstrates the reduction of the eective pair approximation to the binary approximation. The compatibility of the closure procedure found with internal symmetry of the reacting system is also shown there. The basic results are summarized in Section 9.

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2. Statement of the problem To describe the reaction A + B → B, we use the following generally accepted simpli cations [4,33]. The irreversible event of chemical conversion in a reaction pair will be described by the ordinary two-center rate of elementary event w(r) dependent on a relative position vector r of reactants. Thus the in uence of the reaction event on the position of the reactants – rebound eect – is completely neglected. It means that B reactant does not change its coordinate during the elementary event. Translational motion of reactants in solution is described by a stationary Markovian random process over the collection of coordinates of all reactants. Since further reactants are considered point, and their force interaction is ignored, their space coordinates change independently. Accordingly, the multidimensional Markovian random process breaks into a collection of independent Markovian processes (random walks) over space coordinates of each reactant. The above simpli cations refer to the description of the “mechanics” of a pair encounter of reactants in solution. As for many-particle statistical hypothesis, we completely neglect the initial correlations in the position of reactants. The possibility of considering both spatially uniform and spatially nonuniform reacting systems remains. In our examination the systems spatially uniform in B species and nonuniform in A species are of particular importance. Due to internal symmetry, their evolution is described by a closed equation for macroscopic kinetics [10]. The many-particle kinetics P(t) is de ned as a ratio of the mean concentration of As at the instant t to its initial value. Under the simpli cations made, this kinetics coincides [10] with the survival probability of one A particle in the ensemble of B particles. Owing to this reducibility, one can construct the exact kinetic equation of the integro-dierential type for P(t) by diagram methods ˆ + (t) ; @t P(t) = −P(t)

(2.1)

where the Dirac delta-function (t) takes account of the initial conditions, while the kernel – the memory function – of the memory operator ˆ has a shift symmetry in time (t|t0 ) = (t − t0 |0)

(2.2)

even for the systems nonuniform in A species. It is de ned by a diagram series given in Ref. [10]. In the binary approximation the memory function depends on relative mobility only [30], and is approximated by the memory function of the modi ed encounter theory [10,30] Z (t|0) ∼ m (t|0) = −[B]exp(−[B]kt)

drdr0 t(r; t|r0 ; 0) ;

(2.3)

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where [B] is the concentration of B particles, and k is the stationary value of rate constant that can be calculated via t-matrix of the reaction pair [34,35] Z ∞ Z dt drdr0 t(r; t|r0 ; 0) : (2.4) k =− 0

The t-matrix t(r; t|r0 ; 0) may be found from the solution of any of the two conjugate operator equations tˆ = vˆ + vˆgˆ0 tˆ = vˆ + tˆgˆ0 vˆ :

(2.5)

These equations describing the evolution of the reaction pair in relative coordinates are analogous to the Lippmann–Schwinger equations for the t-matrix of quantum scattering theory [36]. The integral reactivity operator vˆ de nes the reaction of reactants in such a pair. Its kernel is commonly considered local in space and time [4] v(r; t|r0 ; t0 ) = −w(r)(r − r0 )(t − t0 ) ;

(2.6)

where w(r) is the elementary event rate in irreversible reaction depending on the relative position vector r of reactants. As is seen from Eq. (2.6), a time variable is not a parameter but is considered on equal terms with space coordinates. This consideration will be used throughout the paper. The free propagator gˆ0 describes translational motion of reactants by stochastic jumps in relative coordinates. Its kernel obeys the equation ˆ 0 (r; t|r0 ; t0 ) = (r − r0 )(t − t0 ) ; (@t − L)g

(2.7)

where the integral operator Lˆ de nes the process of stochastic jumps, and has the kernel [37] L(r; t|r0 ; t0 ) = −−1 [(r − r0 ) − f(r|r0 )](t − t0 ) ;

(2.8)

where  is the mean time between the jumps, and f(r|r0 ) is the density of conditional probability that the reactant will nd itself at point r as a result of a jump from point r0 . By virtue of shift symmetry (2.2), the integro-dierential equation for the nonMarkovian binary kinetics P(t) @t P(t) = −ˆ m P(t) + (t)

(2.9)

is the convolution type equation. So, by identical transformations, it can be brought into equivalent form @t P(t) = −[B]K(t)P(t) + (t)

(2.10)

with time dependent rate constant K(t) [30]. In the limits of binary approximation this constant is related to the t-matrix of the reaction pair as [34,35] Z t Z d drdr0 t(r; |r0 ; 0) : (2.11) K(t) = − 0

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Further, another representation, which is used more frequently in the theory of elementary reactions will be needed for the constant K(t) [4]. To obtain it, we introduce the propagator gˆ de ning the evolution of the pair in relative coordinates [34,35] gˆ = gˆ0 + gˆ0 tˆgˆ0 :

(2.12)

Its kernel obeys the equation ˆ t|r0 ; t0 ) = vg(r; ˆ t|r0 ; t0 ) + (r − r0 )(t − t0 ) (@t − L)g(r;

(2.13)

that generalizes Eq. (2.7). The operation of the propagator on the initial distribution density of reactants in a pair (which is equal to unity for all r for uncorrelated initial conditions) gives the value of the above density at the instant of time t n(r; t) = g|(t)1(r)i ˆ :

(2.14)

Using this de nition in Eq. (2.13), we obtain the equation for n(r; t) ˆ t) = vn(r; ˆ t) + (t) : (@t − L)n(r;

(2.15)

To express the rate constant in terms of the density introduced, we operate on the initial density of reactants in a pair by the operator identity [34,35] vˆgˆ = tˆgˆ0 : As a result, we have vn ˆ = tˆ|(t)1(r)i =

(2.16) Z 0

t

d dr0 t(r; |r0 ; 0) :

(2.17)

With this equation in Eq. (2.11), we have the desired traditional [4] representation for the rate constant Z Z (2.18) K(t) = − dr vn ˆ = d3 rw(r)n(r; t) : The dierential form of the binary non-Markovian kinetic equation is considerably more simple than the integro-dierential one. That is why it is widely used in applications [4]. Though mathematically rigorous, the derivation of non-Markovian equation (2.10) employs essentially the reducibility of consideration of the problem many-particle in A species to that of the decay of one A particle in the ensemble of B particles. Abandoning any of the above simpli cations in the description of the reaction A+B → B makes this reducibility impossible. Generalization of the derivation obtained to this case is rather complicated. An alternative derivation of binary non-Markovian kinetic equations not requiring essential development of many-particle diagram technique is the Waite superpositional decoupling of hierarchies for distribution functions. In the theory of irreversible reactions for spatially uniform systems – in both A and B species – this approach has resulted in the binary non-Markovian kinetic equations of the dierential form [7]. This success has become the grounds for the use of the superpositional decoupling in the theory of reversible reactions [16,17]. However, further we shall show that the superpositional decoupling for the nonuniform reaction system does not produce the closed

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kinetic equation for the binary kinetics P(t). Therefore, the decoupling procedure is incompatible with fundamental=internal=symmetry of the reaction system manifesting itself in the existence of the closed equation for the exact many-particle kinetics P(t) (see Eq. (2.1)). Thus the superpositional decoupling can hardly serve as a general method of deriving the binary non-Markovian kinetic equations. The development of a universal method for binary closure of hierarchies is an urgent problem. Its solution will allow one to study non-model reaction systems avoiding diculties of diagram description. 3. Hierarchies The analog of BBGKY hierarchies for the reacting systems was rst suggested by Waite [7] for spatially uniform irreversible reacting systems. The hierarchies were obtained from qualitative physical considerations. Later the same approach was used in the consideration of reversible reactions [16,17] in uniform systems. The goal of this section is to develop a universal technique of deriving similar hierarchies. This should be important in the description of spatially nonuniform systems, or in taking account of force interactions between reactants. 3.1. The Fock space The most general description of many-particle reaction systems is realized in the Fock space on the basis of the conception of identical ‘indistinguishable’ particles [12,38]. Just in such a representation, the Liouville equations describing the reacting system evolution are easy to interpret physically, and, therefore, may be readily formulated for dierent levels of the model description of real systems. In the reaction A + B → B the number of A particles varies, while the number of B particles remains the same. If in the initial state the number of A particles varies in accordance with the Grand canonical ensemble, then the reacting system is described by the in nite set of distribution functions (DF) {(0) (rBM ; t); (1) (rA1 ; rBM ; t) : : : (N ) (rAN ; rBM ; t) : : :} :

(3.1)

It describes all possible outcomes generated by the reaction from the initial state by time t. Each function from the set is the density of the probability that at time t N identical As and M identical Bs will be found in the macroscopic volume at B } respectively. Every function is points rAN ≡ {r1A ; r2A ; : : : ; rNA } and rBM ≡ {r1B ; r2B ; : : : ; rM normalized to the probability pN of nding the system in the state with a de nite number N of A particles. Z drAN drBM (N ) N M (3.2)  (rA ; rB ; t) = pN (t) : N! M! This probability varies in time due to the reaction. Integration in Eq. (3.2) is performed in in nite limits. Convergence of the integral is provided by a fast decay of functions

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 N outside the volume . The completeness condition is expressed in terms of the following normalization of probabilities pN (t) (t ¿ 0) ∞ X

pN (t) = 1 :

(3.3)

N =0

It indicates that at any time a many-particle system resides in the above space of the elementary outcomes. The set of distribution functions (3.1) carries complete information on the reacting system. In particular, the average number of A particles remaining in the system at the instant of time t is de ned by the expression hN i(t) =

∞ X

NpN (t) :

(3.4)

N =0

By time t the survival probability P(t) of A particles is related to N as hN i(t) : hN i(0)

P(t) =

(3.5)

According to simpli cations made in Section 2, the irreversible reaction A + B → B is described by the following set of Liouville equations: ! N M X X Lˆ Ai − Lˆ B (N ) (rAN ; rBM ; t) @t − i=1

=

N; M X

=1

Vˆ i;  (N ) (rAN ; rBM ; t)

i; =1

−

M Z X =1

drNA +1 Vˆ N +1;  (N +1) (rAN ; rNA +1 ; rBM ; t) + (t)0(N ) (rAN ; rBM ) :

(3.6)

The operator operating on (N ) in the left-hand side de nes a free – i.e., in the absence of reaction – evolution of the reacting system. Apart from the time derivative, it involves operators Lˆ Ai and Lˆ B specifying the process of random walks of A and B reactants, respectively [33]. The structure of these operators is similar to Eq. (2.8). The rst term in the right-hand side describes the escape from the state with N A particles caused by the reaction between A and B. The reactivity operator Vˆ i;  de nes the reaction between A and B reactants located at points riA and rB , respectively. As usual, we assume its kernel to be local in space and time [4,33] (compare to Eq. (2.6)) B B ; t0 ) = −w(riA − rB )(riA − r0iA )(rB − r0 )(t − t0 ) ; Vˆ i;  (riA ; rB ; t|r0iA ; r0

(3.7)

where w(riA −rB )) is the elementary event rate in the irreversible reaction introduced in Eq. (2.6). The second term describes the income from the state with (N +1) A particle. Summation and integration exhaust all possibilities in (N + 1)-particle state at which the decay of one A particle results in the N -particle state with the con guration rAN . The third term takes account of the initial conditions speci ed by the set of functions {0(N ) }.

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It can easily be seen that the Liouville equations are compatible with normalization condition (3.3). For this purpose, we obtain the equation for probabilities pN using de nition (3.2). Since in the absence of reaction the number of reactants remains the same, we have [39,33] Z Z (3.8) driA Lˆ Ai = drB Lˆ B = 0 : Now, using the identity of particles, we have from Eq. (3.6) Z drAN drBM ˆ V 1; 1 (N ) (rAN ; rBM ; t) @t pN (t) = MN N! M! Z drAN +1 drBM ˆ (3.9) V 1; 1 (N +1) (rAN +1 ; rBM ; t) + (t)pN (0) : −M N! M! Summing Eq. (3.9) over N , one should take into account that the right-hand side of the equation for p0 (t) involves no escape term as well as the normalization of the set {pN (0)}. As a result, we have ∞ X pN (t) = (t) : (3.10) @t N =0

At t ¿ 0 Eq. (3.3) is the solution of this equation. Thus a mathematically closed problem of calculating the kinetics P(t) of the manyparticle reaction system is formulated in the Fock space. Before we start deriving a hierarchy from it, let us introduce contracted notation of the reactant coordinates riA ≡ Ai ;

rB ≡ B ;

rAN ≡ A N ;

rBM ≡ B M ;

(3.11)

to be used below along with the accepted designations. 3.2. Microscopic point density As in the non-equilibrium statistical mechanics [13,11] for the N -particle state, we consider a microscopic point density ) p q n(N p; q (A ; B ; t)=

N; M X

(A1 − A0i1 ) · · · (Ap − A0ip )

i1 ···ip ;1 ···q =1

i1 6=···6=ip ; 1 6=···6=q

×(B1 − B01 ) · · · (Bq − B0q ) :

(3.12)

This is a microscopic random quantity parametrically dependent on the set of observation points Ap and B q . Its nonzero values are realized when the points are occupied by reactants. The case where two reactants occupy the same point is improbable. That is why the average value of the point density over N -particle state Z dA0N dB0M (N ) (N ) N M (N ) p q np; q (A ; B ; t) = n  (A0 ; B0 ; t) N !M ! p; q Z dAp+1 : : : dAN dBq+1 : : : dBM (N ) N M (3.13) =  (A ; B ; t) (N − p)!(M − q)!

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is the average density of the number of situations when p A reactants and q B reactants are found at the observation points at a time. Exhaustion of all possible positions of the observation points is performed by integration Z M! N! dAp dB q (N ) p q q pN (t) = CNp CM np; q (A ; B ; t) = pN (t) : p!q! p!(N − p)! q!(M − q)! (3.14) It gives the product of the realization probability of this state and the number of all possible samples over p A particles and q B particles from the ensemble of reactants involved in the N -particle state. From Eq. (3.13) it follows that the mean value of the highest order point density (p = N; q = M ) coincides with DF ) N M (N ) n(N (A N ; B M ; t) : N; M (A ; B ; t) = 

(3.15)

This equality provides a new physical interpretation of (N ) , and reveals obvious advantages of the conception of identical particles over distinguishable ones. The higher the order of the average density, the more statistical information on the reaction system it carries. Thus it can serve to reconstruct the average density of the lower order. For r ¡ p and s ¡ q, we have from Eq. (3.13) Z (N − p)!(M − q)! (N ) r ) s p q dAr+1 : : : dAp dBs+1 : : : dBq n(N nr; s (A ; B ; t) = p; q (A ; B ; t) : (N − r)!(M − s)! (3.16) Just as in the non-equilibrium statistical mechanics, for a xed N -particle state the Liouville equation gives the hierarchy of equations for the average densities of dierent orders that is equivalent to the Liouville equation employed. In view of de nition (3.16), properties (3.8) and the identity of reactants from Eq. (3.6), we have ! p q p; q p Z X X X X (N ) (N ) ) ˆ ˆ ˆ V i;  np; q + dBq+1 Vˆ i; q+1 n(N LAi − LB np; q = @t − p; q+1 i=1

=1

i; =1

+

i=1

q Z X =1

Z + −

) dAp+1 dBq+1 Vˆ p+1; q+1 n(N p+1; q+1

q Z X =1

Z −

) dAp+1 Vˆ p+1;  n(N p+1; q

+1) dAp+1 Vˆ p+1;  n(N p+1; q

+1) dAp+1 dBq+1 Vˆ p+1; q+1 n(N p+1; q+1

) + (t))n(N p; q (t = 0) :

(3.17)

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Unlike the non-equilibrium statistical mechanics problems, in the hierarchy derived the ) evolution of (p+q)-particle density n(N p; q is explicitly related to both (p+q+1)-particle density and to the densities of concurrent location of (p + q + 2) particles. The average number of A particles residing in the reacting system under consideration at time t can also be expressed in terms of the average densities. In view of normalization (3.14), de nition (3.4) may be represented as Z ∞ Z X (N ) (3.18) dA n1; 0 (A; t) ≡ dA hn1; 0 i(A; t) ; hN i(t) = N =0

where the completely average single-particle density is introduced hn1; 0 i(A; t) =

∞ X N =0

) n(N 1; 0 (A; t) :

(3.19)

Being a microscopic single-particle density (see Eqs. (3.12)) averaged over the Fock space, it falls under the category of macroscopically observable quantities [13]. Generalizing de nition (3.19), we introduce the completely average (p + q)-particle density hnp; q i(Ap ; B q ; t) =

∞ X

) p q n(N p; q (A ; B ; t)

(3.20)

N =0

which is the average of microscopic density (3.12) over the Fock space. Note that in the language of completely average densities normalization condition (3.3) takes the form hn0; 0 i = 1 :

(3.21)

Since the number of reactants in the Fock space states does not appear in hierarchic equations (3.17) explicitly, we can obtain the closed hierarchy for the completely average densities. Summing up equations of hierarchy (3.17), in accordance with Eq. (3.20), we get ! p q p; q p Z X X X X ˆ ˆ ˆ V i;  hnp; q i + dBq+1 Vˆ i; q+1 hnp; q+1 i LAi − LB hnp; q i= @t − i=1

=1

i; =1

i=1

+ (t)hnp; q i (t = 0) :

(3.22)

Thus the hierarchy for the complete average densities is much more simple than hierarchy (3.17). Similarly to the non-equilibrium statistical mechanics hierarchies, in the above hierarchy the evolution of (p + q)-particle density is explicitly related solely to the density evolution of concurrent location of (p + q + 1) particles. Despite formal similarity, general properties of solutions of hierarchy (3.22) dier essentially from those of the non-equilibrium statistical mechanics. First of all, it is impossible to introduce an order for the completely average density. In other words, with r ¡ p and s ¡ q it is impossible to reconstruct hnr; s i by the known density hnp; q i.

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This follows from the fact that a recurrent relation for the completely average densities cannot be obtained from recurrent relation (3.16) due to coecients explicitly involving N . However, the situation changes radically, if we restrict ourselves to the reacting system in the thermodynamic limit [13]. 3.3. The thermodynamic limit For the systems in the thermodynamic limit, the observable is not the average number of particles hN i but their concentration averaged over the macroscopic volume Z hN i(t) dA (3.23) = lim ’1; 0 (A; t) : [A]t = T − lim v→∞ v v

The operator T − lim denotes the thermodynamic limit [13]. The second equality is obtained using Eq. (3.18) with the subsequent passing to the thermodynamic limit. The passage has been performed by the well-known non-equilibrium statistical mechanics procedure [12,13]. It consists in the separation of an extensive part of the integral in Eq. (3.18) by partitioning the macroscopic volume into cells of the volume v followed by the passage to the limit v → ∞. As a result, the integrand in Eq. (3.23) is expressed in terms of the completely average density in the thermodynamic limit ’1; 0 (A; t) = T − limhn1; 0 i(A; t) :

(3.24)

Symbol v under the integral sign indicates that integration in Eq. (3.23) is performed over the cell of the volume v. As concentration [A]t is a nite ‘nonzero’ quantity, the function ’1; 0 (A; t) is a non-normalizable function, unlike hn1; 0 i(A; t). The latter describes the reaction in the system with a nite average number of particles thus (see Eq. (3.18)) being a normalizable function. Note that, according to the de nition of the density hn1; 0 i, the function ’1; 0 coincides with local concentration of A reactants which is a widely used object in chemical kinetics [4,40]. The many-particle kinetics P(t) can also be expressed in terms of the concentration. With Eq. (3.23) in Eq. (3.5), we have P(t) =

[A]t ; [A]0

(3.25)

where [A]0 is the concentration of reactants at the initial instant of time. Generalizing Eq. (3.24), we introduce the functions ’p; q (Ap ; B q ; t) = T − limhnp; q i(Ap ; B q ; t) ;

(3.26)

which are also non-normalizable. Further the functions introduced will be called partial distribution functions (PDFs), since: (1) they have all typical properties of partial distribution functions of the non-equilibrium statistical mechanics, (2) they are the generalization of these functions to the reacting systems. This will be shown below.

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The hierarchy for the PDFs is obtained from hierarchy (3.22) for the completely average densities by passing to the thermodynamic limit ! p q p;q p Z X X X X Vˆ i;  ’p; q + dBq+1 Vˆ i; q+1 ’p; q+1 Lˆ Ai − Lˆ B ’p; q = @t − i=1

=1

i; =1

i=1

+ (t)’p; q (t = 0) :

(3.27)

Despite formal coincidence of hierarchies (3.22) and(3.27), operators Lˆ Ai , Lˆ B and Vˆ i;  , appearing in both hierarchies (3.27) are dierent. Though their physical meaning remains the same, in Eq. (3.27) they are the operators in the thermodynamic limit. It means that they describe translational motion of reactants and the reaction between them in in nite three-dimensional space. In spite of these changes, for simplicity of notation, their designations are left the same. Misunderstanding is not likely, because all calculations will be done for the reacting system in the thermodynamic limit. The initial completely uncorrelated state of the reacting system spatially nonuniform in A and B species corresponds to the following initial conditions for hierarchy (3.27) ’p; q (t = 0) =

p Y

A (Ai ) ×

i=1

q Y

B (B ) :

(3.28)

=1

The functions A and B are the initial local concentrations of A and B reactants, respectively. Therefore, they must satisfy the relations analogous to Eq. (3.23), i.e., Z Z dA dB A (A); [B] = lim B (B) ; (3.29) [A]0 = lim v→∞ v v v→∞ v v where we introduce the concentration [B] of B reactants averaged over macroscopic volume. It is time independent in the reaction in question. Together with the requirement for non-negativeness of the functions A and B , i.e., A ¿0 and B ¿0, requirement Eq. (3.29) completely de nes the class of permissible initial distributions. A fundamental property of the PDFs forming the basis of the introduction of correlation forms is the correlation loss condition [12]. The essence of this condition is as follows. Consider two groups of points {Ap ; B q } and {Ar0 ; B0s }. If the distance between the two groups is rather large, the observation of particles at the points of each group will be an independent event. Thus the PDF ’p+r; q+s (Ap ; Ar0 ; B q ; B0s ; t) is factorized ’p+r; q+s (Ap ; Ar0 ; B q ; B0s ; t)

∼

Ar0 ; B0s →∞

’p; q (Ap ; B q ; t)’r; s (Ar0 ; B0s ; t) :

(3.30)

To prove Eq. (3.30), it is sucient to show that the equations for its right- and left-hand sides coincide. These equations are easily obtained from hierarchy (3.27). They do coincide, if we take into account that the reaction between the reactants located at the points of dierent groups can be neglected. The correlation loss conditions make it possible to prove the rules of transition between the PDFs which coincide with those in the non-equilibrium statistical mechanics Z dAp ’p; q (Ap ; B q ; t) = [A]t ’p−1; q (Ap−1 ; B q ; t) ; (3.31) lim v→∞ v v

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Z lim

v→∞

v

dBq ’p; q (Ap; B q; t) = [B]’p; q−1 (Ap; Bq−1; t) : v

(3.32)

Thus introducing an order for PDFs becomes possible. The proof of Eq. (3.31) is nontrivial, since the number of A reactants varies in the course of the reaction. As for Eq. (3.32), it is easily derived from Eq. (3.16) using de nitions (3.20) and (3.26), because the number of B reactants remains unchanged. To prove Eq. (3.31) for the lowest order function ’1; 0 , note that transformation (3.31) coincides with (3.23) by virtue of equality ’0;0 = 1 following from Eqs. (3.21) and (3.26). In the general case, the idea of the proof is based on the establishment whether transformation (3.31) of the equation for ’p; q from hierarchy (3.27) yields the equation coinciding with that for ’p−1; q . The above transformation does yield ! p−1 q X X ˆ ˆ LAi − LB [A]t ’p−1; q @t − i=1

=1

p−1; q

= [A]t

X

q Z X dAp ˆ V p;  ’p; q v→∞ v v

Vˆ i;  ’p−1; q + lim

i; =1 p−1 Z

+ [A]t

X

=1

dBq+1 Vˆ i; q+1 ’p−1; q+1 + lim

v→∞

i=1

+ (t)[A]0 ’p−1; q

(t = 0) :

Z v

dAp dBq+1 ˆ V p; q+1 ’p; q+1 v (3.33)

First of all, note that the second term in the right-hand side of this equality is equal to zero, since the integrand is normalizable. The fourth term can be transformed using correlation loss condition (3.30) Z dAp dBq+1 ˆ V p; q+1 ’p; q+1 lim v→∞ v v Z dAp dBq+1 ˆ V p; q+1 ’1; 1 (Ap ; Bq+1 ; t) : =’p−1; q lim (3.34) v→∞ v v This is possible because the main contribution into the integral of this term is made by the region wherein the observation points {Ap ; Bq+1 } are located asymptotically far from points {Ap−1; B q }. Further one can determine – as it was done, for example, in Appendix A – how a time derivative operates on the product of two functions in the left-hand side of Eq. (3.33), and use the equation Z dA1 dB1 ˆ V 1; 1 ’1; 1 (A1 ; B1 ; t) + (t)[A]0 : (3.35) @t [A]t = lim v→∞ v v It follows from Eq. (3.23) and the equation for ’1; 0 from hierarchy (3.27), in view of Eq. (3.29). Collecting the like terms in Eq. (3.33) and reducing it by [A]t , we obtain the equation that is the hierarchical equation for ’p−1; q . In conclusion note that hierarchy (3.27) for PDFs for spatially uniform systems coincides with the hierarchy obtained in [7] from qualitative physical considerations.

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4. The superpositional decoupling The Kirkwood decoupling of hierarchies for PDFs [18] is widely used in the nonequilibrium statistical mechanics. Its analog in the theory of elementary reactions was applied [7,14 –17] to derive binary kinetic equations. In this section we show that this decoupling is a rather non- exible procedure not allowing for possible symmetries in the evolution of the reacting systems. The validity of this statement will be proved using the example of the reacting system the initial state of which is de ned by the distribution q

’p; q (t = 0) = [B]

p Y i=1

A (Ai ) ×

q Y

1(B ) ;

(4.1)

=1

where 1(B ) is a unit function introduced in Eq. (2.14). We shall show that the Waite simple superpositional decoupling for the reacting system under study [7] does not lead to kinetic equation (2.10) of the dierential theory with the rate constant de ned by Eqs. (2.18) and (2.15). The rst four equations of in nite hierarchy (3.27) for initial distribution (4.1) are of the form Z   (4.2) @t − Lˆ A1 ’1; 0 (A1 ; t) = dB1 Vˆ 1; 1 ’1; 1 (A1 ; B1 ; t) + (t)A (A1 ) ; 

 @t − Lˆ B1 ’0; 1 (B1 ; t) = (t)[B] ;



(4.3)

Z  @t − Lˆ A1 − Lˆ B1 − Vˆ 1; 1 ’1; 1 (A1 ; B1 ; t) = dB2 Vˆ 1; 2 ’1; 2 (A1 ; B1 ; B2 ; t) + (t)[B]A (A1 ) ;



 @t − Lˆ B1 − Lˆ B2 ’0; 2 (B1 ; B2 ; t) = (t)[B]2 :

(4.4) (4.5)

Eqs. (4.3) and (4.5) show that, in full agreement with physical intuition, local concentration ’0; 1 of B particles always coincides with concentration [B] at all times, and the PDF ’0; 2 is also expressed in terms of concentration [B] ’0; 1 (B; t) = [B];

’0; 2 (B1 ; B2 ; t) = [B]2 :

(4.6)

Thus space uniformity in B species and uncorrelatedness of B particles persist in time. In the approach being described the other two equations will serve as a basis for the development of a binary approximation. They form a closed set, if we allow for the superpositional decoupling [18,7], that in view of Eq. (4.6) takes the form ’1; 2 (A1 ; B1 ; B2 ; t) =

’1; 1 (A1 ; B1 ; t)’1; 1 (A1 ; B2 ; t) : ’1; 0 (A1 ; t)

(4.7)

Following the derivation of the kinetic equation in paper [7], we introduce a twoparticle density (A; B; t) and single-particle functions f(A; t) and f0 (A) by the

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equalities ’1; 1 (A; B; t) = (A; B; t)’1; 0 (A; t)[B];

’1; 0 (A; t) = [A]0 f(A; t);

A (A) = [A]0 f0 (A) :

(4.8)

In view of Eqs. (3.7), Eqs. (4.2), (4.4) and (4.7) give the closed set for functions  and f that takes the following form for the frequently considered model with immobile traps ‘Lˆ B = 0’ Z ˆ (@t − LA )f(A; t) = −[B]f(A; t) dBw(A − B)(A; B; t) + (t)f0 (A) ; (@t − Lˆ A + w(A − B))f(A; t)(A; B; t) Z = − [B](A; B; t)f(A; t) dB2 w(A − B2 )(A; B2 ; t) + (t)f0 (A) :

(4.9)

It is easily seen from Eqs. (3.35) and (4.8) that in the framework of the binary approximation the kinetics P(t) of the system in question is de ned by an expression Z Z dA f(A; t) dBw(A − B)(A; B; t) + (t) @t P(t) = −[B] lim v→∞ v v Z Z dr B f(r B + r; t)(r B + r; r B; t) + (t) (4.10) drw(r) = −[B] lim v→∞ v v thus being easily expressible in terms of functions  and f. If the binary approximation could lead to the kinetic equation of the dierential theory, as it happens for spatially uniform systems [7], then Eqs. (4.9) and (4.10) would be equivalent to Eqs. (2.10), (2.18) and (2.15). In particular, the comparison between Eq. (4.10) and Eq. (2.10) (with the rate constant from Eq. (2.18)) suggests the introduction of the density of reactants ne (r; t) in some eective pair Z dr B (4.11) f(r B + r; t)(r B + r; r B ; t) = P(t)ne (r; t) : lim v→∞ v v Then Eq. (4.10) takes the form of Eq. (2.10). Now all we need to do is to obtain an equation for ne that should lead to the possibility of identifying ne and n from Eq. (2.15). Such an equation is derived in Appendix A. It is of the form (r; t) + (t) ; (@t − Lˆ r + w(r))ne (r; t) = 2 P (t)

(4.12)

where the function (r; t) speci es spatial position of the source in the eective pair, and is de ned by the expression Z dr B dr0B dr0 w(r0 )f(r B + r0 ; t)f(r0B + r0 ; t) (r; t) ≡ −[B] lim v→∞ v v2 ×(r B + r0 ; r B + r0 − r; t)[(r B + r0 ; r B ; t) − (r0B + r0 ; r0B ; t)] :

(4.13)

It is shown in Appendix A that if f0 (r A → ∞) → const:, then  = 0, and, therefore, ne (r; t) = n(r; t). By analogy, it can be proved that in the systems with two mobile

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reactants wherein non-uniformity is local the superpositional decoupling gives a correct binary approximation obtained earlier on the basis of many-particle consideration by diagram technique [10]. However, in the systems wherein space non-uniformity is not local=i.e., there exists no limit f0 (r A → ∞)=, the function (r; t) 6= 0, and, therefore, ne (r; t) 6= n(r; t). In other words, the decoupling procedure proves to be incompatible with internal symmetry of the system. The study of the incompatibility value calls for rather cumbersome calculations, and will be conducted elsewhere. In the present paper we restrict ourselves to the proof of the fact  6= 0 in the region where the source value is obviously small, and may be calculated by the perturbation theory methods. Assume that the required smallness is provided by the reactivity weakness. Then all the necessary functions are representable as a series in powers of its intensity, i.e.,  = (0) + (1) + (2) + · · · ; f = f(0) + f(1) + f(2) + · · · ;

(4.14)

ne = n(0) + n(1) + n(2) + · · · : Substituting Eqs. (4.14) into Eq. (4.13) yields (Appendix B) the following expression for the leading term of the expansion of the function (r; t): (r; t) ∼ 3 (r; t) Z dr B dr0B dr0 w(r0 )fA(0) (r B + r0 ; t)fA(0) (r0B + r0 ; t) ≡ −[B] lim v→∞ v v2 ×(1) (r B + r0 ; r B + r0 − r; t)[(1) (r B + r0 ; r B ; t) − (1) (r0B + r0 ; r0B ; t)] : (4.15) Thus the value of the source in the eective pair is of the third order in the reactivity intensity. In accordance with this, for the rst time the existence of the source manifests itself just in the calculation of n(3) (@t − Lˆ r )n(3) (r; t) = −w(r)n(2) (r; t) + 3 (r; t)

(4.16)

and, therefore, in that of the rate constant in the fourth order of the perturbation theory (see Eq. (2.18)). To see that 3 6= 0, the reaction system should be made more speci c. Assume that the elementary event rate is of the form w(r) = kr

(r − a) ; 4ra

(4.17)

where kr is the reaction constant (intrinsic constant). The motion of A reactants is a continual diusion with the constant D, and their initial distribution is de ned by the sum of a uniform distribution and spherical wave of the amplitude C f0 (r A ) = 1 + C cos(rA =b) :

(4.18)

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We assume that the deviation from a uniform distribution is small, and the characteristic size of non-uniformities b exceeds considerably the reaction zone size, i.e., C 1;

a b;

r b :

(4.19)

We are interested in the instants of time for which diusion displacement of A reactant is large as compared to the reaction zone but small in comparison with the non-uniformity scale max(a2 ; r 2 ) Dt . b max(a; r) :

(4.20)

Under the assumptions made, the value of the source 3 calculated in Appendix C is of rather simple form   1     ; r¿a [B]kr3 C 2 t 2 r × : (4.21) 3 (r; t) = −  162 ab4  1 ; r6a   a The absence of the dependence on A reactant mobility stands out. Thus, generally speaking, the binary kinetic equation obtained by the superpositional decoupling cannot be reduced to Eq. (2.10) derived on the basis of diagram summation. That is why it is necessary to examine another hierarchy closure methods based on more advanced approaches. 5. Correlation forms The concept of correlation forms introduced due to the existence of correlation loss condition (analogous to (3.30)) for PDFs is widely used in the non-equilibrium statistical mechanics. The way of introducing correlation forms relies on a group expansion of PDF that may be brought into the following form for the system studied: X p; q (Ap ; B q ; t; s ) (5.1) ’p; q (Ap ; B q ; t) = s

where s de nes a given partitioning of the collection of (p + q) particles. In a more detailed notation, partitioning is explicitly de ned by vertical lines [13]. For the lowest order PDF expansion (5.1) has the form ’1; 0 (A1 ) = 1; 0 (A1 );

’0; 1 (B1 ) = 0; 1 (B1 ) ;

’2; 0 (A1 ; A2 ) = 2; 0 (A1 |A2 ) + 2; 0 (A1 ; A2 ) ; ’0; 2 (B1 ; B2 ) = 0; 2 (B1 |B2 ) + 0; 2 (B1 ; B2 ) ; ’1; 1 (A1 ; B1 ) = 1; 1 (A1 |B1 ) + 1; 1 (A1 ; B1 ) ; ’1; 2 (A1 ; B1 ; B2 ) = 1; 2 (A1 |B1 |B2 ) + 1; 2 (A1 ; B1 |B2 ) + 1; 2 (A1 ; B2 |B1 ) + 1; 2 (A1 |B1 ; B2 ) + 1; 2 (A1 ; B1 ; B2 ) :

(5.2)

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For brevity, the time coordinate is omitted. Each correlation form involving the partitioning into several groups is factorized [13]. For example, 2; 0 (A1 |A2 ) = 1; 0 (A1 )1; 0 (A2 );

1; 1 (A1 |B1 ) = 1; 0 (A1 )0; 1 (B1 ) ;

1; 2 (A1 ; B1 |B2 ) = 1; 1 (A1 ; B1 )0; 1 (B2 ) ; 1; 2 (A1 |B1 ; B2 ) = 1; 0 (A1 )0; 2 (B1 ; B2 ) :

(5.3)

This allows one to determine the completely correlated forms from Eqs. (5.2), i.e., the forms that involve the partitioning consisting of a single group, e.g., 2;0 (A1 ; A2 ), 1; 1 (A1 ; B1 ), 1; 2 (A1 ; B1 ; B2 ) etc. Transition rules for PDFs (see Eqs. (3.31) and (3.32)) in terms of the correlation forms are as follows: Z Z dA dA (5.4) 1; 0 (A) = [A]t ; lim 0; 1 (B) = [B] ; lim v→∞ v v v→∞ v v Z lim

v→∞

v

dA 1; 1 (A; B) = lim v→∞ v = lim

v→∞

Z v

dB 1; 1 (A; B) v

v

dA 1; 2 (A; B1 ; B2 ) = · · · = 0 : v

Z

(5.5)

Condition (5.5) provides the decay of the completely correlated forms for widely separated observation points. A hierarchy for the correlation forms responsible for their evolution may be derived from de nitions (5.2), (5.3) and hierarchy (3.27) by conventional technique. Since in the reacting system under study the reaction does not aect the motion of B reactants, therefore, the hierarchy for the correlation forms involving solely B reactants is the simplest and closed one. Initial distribution (3.28) includes the completely uncorrelated forms only, thus at any time correlations in the ensemble q of B particles are described solely by such forms. They obey the equation ! q X ˆ LB 0; q (B1 |B2 | : : : |Bq ) = (t)0;0 q (B1 |B2 | : : : |Bq ) ; (5.6) @t − =1

where according to Eq. (3.28), the initial state of the completely uncorrelated form, is 0;0 q (B1 |B2 | : : : |Bq )

=

q Y

B (B ) :

(5.7)

=1

Eq. (5.6) may be brought into the form integral in time 0 0; q (B1 |B2 | : : : |Bq ) = Gˆ 0; q |(t)0;0 q (B1 |B2 | : : : |Bq )i

(5.8)

0 if a free propagator Gˆ 0; q de ning translational motion of B reactants is included into 0 consideration. It is a speci c case of a free propagator Gˆ p; q of a more general form.

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Fig. 1. Graphic representation of the simplest free propagators.

Its kernel obeys the equation @t −

p X

Lˆ Ai −

q X

i=1

= (t − t0 )

Lˆ B

=1 p Y

! p q 0 p q Gp; q (A ; B ; t|A0 ; B0 ; t0 )

(Ai − A0i ) ×

i=1

q Y

(B − B0 )

(5.9)

=1

and, hence, is expressed in terms of the kernels of individual free propagators p q 0 p q Gp; q (A ; B ; t|A0 ; B0 ; t0 ) =

p Y i=1

UA0 (Ai ; t|A0i ; t0 ) ×

q Y

UB0 (B ; t|B0 ; t0 ) :

(5.10)

=1

The explicit form of kernels UA0 and UB0 is determined by the form of operators LˆAi and LˆB de ning the Markovian motion of A and B reactants, respectively. For example, if the motion of B species is continual diusion – with the coecient DB – then   (t − t0 ) (r B − r0B )2 0 B B : (5.11) exp − UB (r ; t|r0 ; t0 ) = (4DB (t − t0 ))3=2 4DB (t − t0 ) The general analytical form of other hierarchical equations correlation forms is rather complex. Just as in the non-equilibrium statistical mechanics [13], its mathematical structure is conveniently appreciated by diagram technique. First of all, construct a diagram representation for Eq. (5.2). Though mathematically the time coordinate is considered on equal terms with the space one, physically it is natural to show the directed time axis in the diagrams explicitly. If the necessity arises, the time axis for each reactant will be denoted by a dashed line, with the direction from the right to 0 0 the left being considered positive. Since individual free propagators Uˆ A and Uˆ B are nonlocal (see Eq. (5.11)), graphically, they will be represented by sections on the time 0 axes of the corresponding reactants. Fig. 1 illustrates some lowest operators Gˆ p; q . The reactants are given on the right of the time axes. They are commonly numbered from bottom to top. For clarity, the instants of times that relate the kernels of propagators 0 0 Uˆ A and Uˆ B to one another are given under the arrows. This is conveniently done by passing from graphic representation to the analytical one. Not to clutter the gure, the time axes of other reactants (not involved explicitly in free propagators represented) are

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587

Fig. 2. Graphic representation of the simplest correlation forms of B reactants.

not given, though implied. Diagram representations for completely uncorrelated forms 0; q (B1 |B2 | : : : |Bq ) are introduced by analogy. Fig. 2 exempli es several relations from Eq. (5.8). The reaction aects the evolution of correlation forms containing A reactants. For example, for the simplest three forms from Eqs. (3.27), we have Z (@t − Lˆ A1 )1; 0 (A1 ) = (t)A (A1 ) + dB1 Vˆ 1; 1 1; 1 (A1 |B1 ) Z +

dB1 Vˆ 1; 1 1; 1 (A1 ; B1 ) ;

(@t − Lˆ A1 − Lˆ B1 )1; 1 (A1 |B1 ) = (t)A (A1 )B (B1 ) + Z +

(5.12) Z

dB2 Vˆ 1; 2 1; 2 (A1 |B1 |B2 )

dB2 Vˆ 1; 2 1; 2 (A1 ; B2 |B1 ) ;

(5.13)

(@t − Lˆ A1 − Lˆ B1 )1; 1 (A1 ; B1 ) = Vˆ 1; 1 1; 1 (A1 |B1 ) + Vˆ 1; 1 1; 1 (A1 ; B1 ) Z + dB2 Vˆ 1; 2 1; 2 (A1 |B1 ; B2 ) Z + Z +

dB2 Vˆ 1; 2 1; 2 (A1 ; B1 |B2 ) dB2 Vˆ 1; 2 1; 2 (A1 ; B1 ; B2 ) :

(5.14)

The right-hand side of Eq. (5.14) includes the terms relating the evolution of the completely correlated form 1; 1 (A; B) to that of two-particle and three-particle correlation forms. The operators ahead of the forms are most easily expressed in terms of the reactivity operator (3.7). They are local in time, i.e., proportional to the Dirac delta function of time. Accordingly, their graphic representation is a point on the time axis

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Fig. 3. Graphic representation of the reactivity operator.

Fig. 4. Graphic representation of the simplest correlation forms.

of A reactants (see Fig. 3). “Tails” of the point indicate B reactants related by the operator. This diagram representation is similar to that introduced earlier [10]. Its obvious asymmetry relative to A and B species is an important distinction of the diagram technique under discussion from the similar Balescu method of the non-equilibrium statistical mechanics [13]. For correlation forms of a general form we introduce a diagram representation by generalizing the introduced representation of completely uncorrelated forms and diagram representation of non-equilibrium statistical mechanics correlation forms. For illustration, Fig. 4 gives diagram representations for some correlation forms of a lower order. It is easily seen that they completely agree with Eqs. (5.3), and are in essence their graphic representation. The introduced graphic elements make it possible to formulate the equivalent diagram representation for the integral form of Eqs. (5.12) – (5.14). Representations are given in Fig. 5. The analysis shows that the right-hand side of any equation involves all ways of obtaining the desired correlation form – the left-hand side of the equation – from all possible correlation forms of permissible symmetries. It is done by joining on the left the graphic elements that describe a single reaction event followed by the free evolution of reactants included in the desired correlation form. Using this general rule, one can easily construct a diagram representation for the equation for the completely correlated form 1; 2 (A1 ; B1 ; B2 ) that we shall need further to develop the binary approximation. The equation is given in Fig. 6. Its right-hand side involves eleven terms. However, as already mentioned, by virtue of initial conditions and the absence of the reaction in uence on the motion of B reactants, there will be no correlations in the evolution of the group containing reactants B only. Thus the terms with numbers 5,6,8,9,10 are

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589

Fig. 5. Graphic representation of evolution of the simplest correlation forms.

Fig. 6. Graphic representation of evolution of 1; 2 (A1 ; B1 ; B2 ).

equal to zero at all times. Note that the third term in Eq. (5.14) is equal to zero for the same reason. The obtained set of equations for correlation forms is equivalent to the initial hierarchy (3.27). In other words, de nition of correlation forms completely characterizes

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the state of the system in accordance with Eq. (5.1). However, some correlation forms obey dierent equations of motion, even if they correspond to the same number of reactants. That is why the description of the system evolution in terms of the correlation forms is more detailed. In many aspects, passing from the PDFs to the correlation forms is similar to the well-known quantum mechanical transition in the description of degenerate state in the language of irreducible representations of the Hamiltonian symmetry groups. 6. The Klimontovich pair collisions approximation and the encounter theory Treating the time evolution of the reacting system as a correlation dynamics poses the problem of obtaining the binary approximation in terms of correlation forms. In the non-equilibrium statistical mechanics such a binary approximation is well-known [11]. The extension of its physical argumentation to the system under study leads to neglecting the third and the fth terms in the right-hand side of Eq. (5.14) (involving correlation forms 1; 2 (A1 |B1 ; B2 ) and 1; 2 (A1 ; B1 ; B2 ), respectively). For the system in question this requirement seems to be quite acceptable, since, as mentioned earlier, 1; 2 (A1 |B1 ; B2 ) = 0. As for the completely correlated form 1; 2 (A1 ; B1 ; B2 ), its presence is intuitively associated with taking account of triple collisions of reactants. However, below it will be shown that the ful llment of this requirement results in the kinetic equation which describes inadequately both the non-Markovian and Markovian stages of the kinetics in migration control: |V | → ∞. The Klimontovich pair collisions approximation will be analyzed using the example of spatially nonuniform reacting system with the initial state de ned by distribution (4.1). Since 0; 1 (B) = [B] for it, a single-particle operator of the averaged reactivity may be introduced Z Z Vˆ ≡ dB1 Vˆ 1; 1 1(B1 ) = − dBw(A − B)Iˆ ≡ V Iˆ; (6.1) where the unit operator Iˆ is de ned by the kernel I (A; t|A0 ; t0 ) = (A − A0 )(t − t0 ). In view of this de nition and Eqs. (5.3), in the framework of this binary approximation Eqs. (5.12) and (5.14) take the form Z (@t − Lˆ A1 )1; 0 (A1 ) = (t)A (A1 ) + [B]V 1; 0 (A1 ) + dB1 Vˆ 1; 1 1; 1 (A1 ; B1 ) ; (6.2) (@t − Lˆ A1 − Lˆ B1 − Vˆ 1; 1 − [B]V )1; 1 (A1 ; B1 ) = [B]Vˆ 1; 1 [1; 0 (A1 )1(B1 )] :

(6.3)

The above equations form the closed set. To solve it, we introduce the propagator Gˆ 1; 1 of the reaction pair whose the kernel obeys the equation (@t − Lˆ A1 − Lˆ B1 − Vˆ 1; 1 )G1; 1 (A1 ; B1 ; t|A01 ; B01 ; t0 ) =(t − t0 )(A1 − A01 )(B1 − B01 ) :

(6.4)

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This equation is a generalization of Eq. (5.9) for the pair. In view of Eq. (6.4), Eq. (6.3) yields [ (t − t ))]Vˆ [ (A )1(B )] ; (6.5) 1; 1 (A1 ; B1 ) = [B][G1; 1 exp([B]V 0 1; 1 1; 0 1 1 where [d : : : ] is the operator the kernel of which is given in square brackets. Substituting Eq. (6.5) into Eq. (6.2) yields (@t − Lˆ A1 )1; 0 (A1 ) = (t)A (A1 ) Z [ (t − t ))]Vˆ } +[B] dB1 {Vˆ 1; 1 + Vˆ 1; 1 [G1; 1 exp([B]V 0 1; 1 ×[1; 0 (A1 )1(B1 )] :

(6.6)

Introduce a pair T -operator [10] of the reaction pair Tˆ 1; 1 = Vˆ 1; 1 + Vˆ 1; 1 Gˆ 1; 1 Vˆ 1; 1 :

(6.7)

Then Eq. (6.6) with allowance for Eq. (3.7) gives the closed kinetic equation for local concentration Z [ (t − t ))] (@t − Lˆ A1 )1; 0 (A1 ) = (t)A (A1 ) + [B] dB1 [T1; 1 exp([B]V 0 ×[1; 0 (A1 )1(B1 )] : Using the expression

(6.8)

Z

dA (6.9) 1; 0 (A) v v that follows from Eqs. (3.25), (3.23) and (5.2), and a shift space symmetry of the T -operator lim P(t) = [A]−1 0

v→∞

T1; 1 (A; B; t|A0 ; B0 ; t0 ) = T1; 1 (A − A0 ; B − A0 ; t|0; B0 − A0 ; t0 )

(6.10)

we obtain a closed integro-dierential equation for the binary kinetics from Eq. (6.8) @t P(t) = ˆ p P(t) + (t) :

(6.11)

Its kernel – the memory function – has a shift time symmetry (see Eq. (2.2)), and is expressed in terms of the T -operator kernel as follows: Z (6.12) p (t|0) = −[B]exp([B]V t) dA dB dB0 T1; 1 (A; B; t|A0 ; B0 ; 0) : This expression may be simpli ed, if in the description of the reacting pair we replace the individual coordinates of reactants by relative coordinates and the pair’s center ones [10]. Z (6.13) p (t|0) = −[B]exp([B]V t) dr dr0 t(r; t|r0 ; 0) : Comparison between the result obtained and Eq. (2.3) shows that they coincide at the kinetic stage – V → 0 – of the reaction when [41,4] Z (6.14) k = −V = dr w(r) :

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In the reverse limiting case of migration control – |V | → ∞ – p → 0, and no reaction takes place. That is why not only does Eq. (6.13) dier from Eq. (2.3), but it also roughly disagrees with qualitative physical considerations. Though the above method of extracting pair collisions results in incorrect binary kinetic equation, it has an important advantage. Within the framework of this approach it is possible to obtain the closed equation for the kinetics P(t) for nonuniform system under study. Thus, unlike the Waite decoupling, it is compatible with the fundamental ‘internal’ symmetry of the reacting system. In conclusion of this section note that the developed formalism of the correlation forms allows one to obtain the encounter theory approximation [23–25,29]. It is also the analog of the binary equation derivation method of non-equilibrium statistical mechanics [42]. This approach is based on the diagram construction and subsequent simpli cation of the mass operator by expansion in powers of reactant concentrations. For the system in question, retention of only the rst non-vanishing terms in concentration yields the integro-dierential equation for the kinetics P(t), with the memory operator being proportional to concentration [B] [42]. The analysis has shown that this equation describes correctly the Markovian stage of the reaction course, but provides no correct description of the non-Markovian one [29]. The main reason is that the conception of pair encounters of reactants in liquid solutions is more complicated than that in gases. Thus derivation of the encounter theory equation on the basis of the correlation form formalism is the rst important stage in the development of the desired binary approximation corresponding to the complicated conception of pair encounters. Now we turn to the diagram representation of Eq. (5.14) (see Fig. 5) neglecting all contributions of three-particle correlation forms into the evolution of two-particle completely correlated form 1; 1 (A; B). Eq. (6.3) is thus simpli ed: (@t − Lˆ A1 − Lˆ B1 − Vˆ 1; 1 )1; 1 (A1 ; B1 ) = [B]Vˆ 1; 1 [1; 0 (A1 )1(B1 )] :

(6.15)

Then, proceeding as described above, we easily obtain the integro-dierential kinetic equation of Eq. (6.11) type but with the memory function Z e (t|0) = −[B]

Z dA dB dB0 T1; 1 (A; B; t|A0 ; B0 ; 0) = −[B]

dr dr0 t(r; t|r0 ; 0) : (6.16)

It coincides with the memory function of the encounter theory equation [10]. As the encounter theory describes correctly the Markovian stage of the reaction over the entire range of micro-parameters, it is preferable to the theory based on the Klimontovich approach. Accordingly, a regular procedure in the construction of expansions in density parameter is neglecting all contributions from three-particle forms in the equation for two-particle forms rather than ignoring selected ones as in the Klimontovich approach.

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7. The eective pair approximation The diagram analysis of the reacting system carried out in [10] has shown that partial summation performed by a conventional procedure with allowance for diagrams necessary from the standpoint of scaling procedure leads to the eective pair approximation. It is the rst stage in the derivation of the binary non-Markovian kinetic equations. In this section the eective pair approximation will be obtained on the basis of the correlation form formalism for the system spatially nonuniform both in A and B species (initial distribution (3.28)). First of all, note that intuitive physical considerations that relate the completely correlated form 1; 2 (A1 ; B1 ; B2 ) to triple encounters of reactants are erroneous. Really, the Waite decoupling is a correct binary procedure for the uniform reacting system. In this case the correlation form 1; 2 (A1 ; B1 ; B2 ) diers from zero, and is de ned by the equality 1; 1 (A1 ; B1 )1; 1 (A1 ; B2 ) (7.1) 1; 2 (A1 ; B1 ; B2 ) = 1; 0 (A1 ) following from Eqs. (4.7) and (5.2). Thus, constructing correct binary expansion, it is necessary to consider the time equation for this form graphically shown in Fig. 6. Following the idea that a proper expansion in density parameter calls for neglecting contributions of all more complicated forms into the right-hand side of the equation for the form under study, we retain solely three-particle forms in the equation for 1; 2 (A1 ; B1 ; B2 ). As a result, we have (@t − Lˆ A1 − Lˆ B1 − Lˆ B2 − Vˆ 1; 1 − Vˆ 1; 2 )1; 2 (A1 ; B1 ; B2 ) = Vˆ 1; 1 1; 2 (A1 ; B2 |B1 ) + Vˆ 1; 2 1; 2 (A1 ; B1 |B2 ) :

(7.2)

This equation is solved using the propagator Gˆ 1; 2 de ning the evolution in the ensemble of three reactants: one A reactant and two B reactants (@t − Lˆ A1 − Lˆ B1 − Lˆ B2 − Vˆ 1; 1 − Vˆ 1; 2 )G1; 2 (A1 ; B1 ; B2 ; t|A01 ; B01 ; B02 ; t0 ) = (t − t0 )(A1 − A01 )(B1 − B01 )(B2 − B02 ) :

(7.3)

Thus one can obtain the following expression for the local rate of the reaction determined by the form 1; 2 (A1 ; B1 ; B2 ), and appearing in the integrand in the last term in the right-hand side of Eq. (5.14): Vˆ 1; 2 1; 2 (A1 ; B1 ; B2 ) = Vˆ 1; 2 Gˆ 1; 2 (Vˆ 1; 1 1; 2 (A1 ; B2 |B1 ) + Vˆ 1; 2 1; 2 (A1 ; B1 |B2 )) : (7.4) Let us emphasize that it is this value – not completely correlated form 1; 2 (A1 ; B1 ; B2 ) – that should be calculated for the derivation of the binary kinetic equation. It is the value that admits physically clear extraction of the contribution from pair encounters of reactants. This can be done by the three-body theory methods [32]. We introduce a three-particle T -operator which is a formal analogue of the two-particle T -operator (6.7) 0 0 0 (7.5) Tˆ = Vˆ + Vˆ Gˆ 1; 2 Vˆ ; Gˆ 1; 2 = Gˆ 1; 2 + Gˆ 1; 2 Tˆ Gˆ 1; 2 ;

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where a three-particle reactivity operator Vˆ is de ned as follows: [ [ ˆ ˆ Vˆ = [V1; 1 (B 2 − B02 )] + [V1; 2 (B1 − B01 )] ≡ V 1 + V 2 :

(7.6)

It describes a chemical reaction of A reactant with the rst (Vˆ 1 ) and the second (Vˆ 2 )B reactants. Two conjugate operator equations for the three-particle T -operator are derived from de nitions (7.5) by conventional methods [34] (compare with Eq. (2.5)) 0 0 Tˆ = Vˆ + Vˆ Gˆ 1; 2 Tˆ = Vˆ + Tˆ Gˆ 1; 2 Vˆ :

(7.7)

Further we perform the reduction similar to the Faddeev one in the quantum mechanical three-body problem [32]. For this purpose, we introduce “channel” T -operators 0 Tˆ 1 = Vˆ 1 + Vˆ 1 Gˆ 1; 2 Tˆ ;

0 Tˆ 2 = Vˆ 2 + Vˆ 2 Gˆ 1; 2 Tˆ :

(7.8)

From their de nition and Eqs. (7.7) it follows that the three-particle T -operator is their sum (7.9) Tˆ = Tˆ 1 + Tˆ 2 : Physical meaning of the “channel” T -operators becomes absolutely clear, if we note that the following relations take place 0 Vˆ 1 Gˆ 1; 2 = Tˆ 1 Gˆ 1; 2 ;

0 Vˆ 2 Gˆ 1; 2 = Tˆ 2 Gˆ 1; 2 :

(7.10)

The rst relation follows from the equalities 0 0 0 0 Tˆ 1 Gˆ 1; 2 = Vˆ 1 (Gˆ 1; 2 + Gˆ 1; 2 Tˆ Gˆ 1; 2 ) = Vˆ 1 Gˆ 1; 2

(7.11)

wherein the rst equation from Eqs. (7.8) and the second one from Eqs. (7.5) are used. The second relation is proved by analogy. The use of the second relation from Eqs. (7.10) in the right-hand side of Eq. (7.4) shows that the desired extraction of the contribution of pair encounters must be performed in the calculation of the “channel” operator Tˆ 2 . To do this, we obtain the equation for the “channel” operators. First of all note that, in view of Eq. (7.9), Eqs. (7.8) give 0 0 Tˆ 1 − Vˆ 1 Gˆ 1; 2 Tˆ 1 = Vˆ 1 + Vˆ 1 Gˆ 1; 2 Tˆ 2 ;

0 0 Tˆ 2 − Vˆ 2 Gˆ 1; 2 Tˆ 2 = Vˆ 2 + Vˆ 2 Gˆ 1; 2 Tˆ 1 :

(7.12) The above equations may be brought into a less singular form [32]. With this aim, we ˆ 2 as the solutions of operator equations ˆ 1 and T introduce three-particle operators T ˆ 1; ˆ 1 = Vˆ 1 + Vˆ 1 Gˆ 01; 2 T T

ˆ2: ˆ 2 = Vˆ 2 + Vˆ 2 Gˆ 01; 2 T T

(7.13)

These equations are similar in structure to those for the two-particle T -operators (see, ˆ 2 are T -operators of a three-particle ˆ 1 and T e.g. Eq. (7.7)), therefore, the operators T problem wherein A reactant reacts only with one of the two B reactants. This is most clearly seen from the equation for the propagator of the problem expressed in terms of the T -operator in a standard fashion analogous to Eqs. (7.5). For example, for the ˆ 2 the corresponding propagator is operator T (2) 0 0 ˆ 2 Gˆ 01; 2 ; Gˆ = Gˆ 1; 2 + Gˆ 1; 2 T

ˆ 2 = Vˆ 2 + Vˆ 2 Gˆ (2) Vˆ 2 : T

(7.14)

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Its kernel obeys the equation (@t − Lˆ A1 − Lˆ B1 − Lˆ B2 − Vˆ 1; 2 )G (2) (A1 ; B1 ; B2 ; t|A01 ; B01 ; B02 ; t0 ) = (t − t0 )(A1 − A01 )(B1 − B01 )(B2 − B02 ) :

(7.15)

(1) It is seen that the rst B reactant does not react and moves freely. The propagator Gˆ ˆ 1 is introduced similarly. for the operator T Note that Eqs. (7.13) have formal solutions 0 T1 = (1 − Vˆ 1 Gˆ 1; 2 )−1 Vˆ 1 ;

0 T2 = (1 − Vˆ 2 Gˆ 1; 2 )−1 Vˆ 2 :

(7.16)

In view of these solutions, Eqs. (7.12) may easily be brought into the desired form ˆ1+T ˆ 1 Gˆ 01; 2 Tˆ 2 ; Tˆ 1 = T

ˆ2+T ˆ 2 Gˆ 01; 2 Tˆ 1 : Tˆ 2 = T

(7.17)

The derived equations for “channel” T -operators are analogous to Faddeev equations in the quantum mechanical three-body theory [32]. For large characteristic distances ˆ 1 and Tˆ 2 ' T ˆ 2 . So one can between reactants in a three-particle problem Tˆ 1 ' T neglect the second terms in the right-hand sides of Eqs. (7.17). The existence of them is associated with the presence of essentially three-particle part in triple encounters of reactants. Thus in the frame of the binary approximation we have 0 ˆ 2 Gˆ 01; 2 = Vˆ 2 Gˆ (2) : Vˆ 1; 2 Gˆ 1; 2 = Tˆ 2 Gˆ 1; 2 ' T

(7.18)

In the last equality the relation following from Eq. (7.10) for a speci c case of a three-particle problem de ned by Eq. (7.15) is used. Substituting Eq. (7.18) into the right-hand side of Eq. (7.4), it is necessary to neglect the rst term, since it is of the same structure as the second terms in Eqs. (7.17) ignored in the derivation of Eq. (7.18). So local reaction rate (7.4), with only pair encounters of reactants taken into account, is de ned by the expression (2) Vˆ 1; 2 1; 2 (A1 ; B1 ; B2 ) ' Vˆ 2 Gˆ Vˆ 2 1; 2 (A1 ; B1 |B2 ) :

(7.19)

In view of the above expression, Eq. (5.14) for a two-particle completely correlated form is as follows: (@t − Lˆ A1 − Lˆ B1 − Vˆ 1; 1 )1; 1 (A1 ; B1 ) Z (2) = Vˆ 1; 1 1; 1 (A1 |B1 ) + dB2 (Vˆ 2 + Vˆ 2 Gˆ Vˆ 2 )1; 2 (A1 ; B1 |B2 ) :

(7.20)

(2) Now we take into account that the kernel of the three-particle propagator Gˆ satisfying Eq. (7.15) is of the form

G (2) (A1 ; B1 ; B2 ; t|A01 ; B01 ; B02 ; t0 ) = G1; 1 (A1 ; B2 ; t|A01 ; B02 ; t0 )UB0 (B1 ; t|B01 ; t0 ) (7.21) that agrees completely with physical considerations. So in view of de nition (6.7) of the two-particle T -operator, the last term in the right-hand side of Eq. (7.20) may be

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represented as Z Z (2) ˆ ˆ ˆ ˆ UB0 (1)](1; 1 (A1 ; B1 )0; 1 (B2 )) : dB2 (V 2 + V 2 G V 2 )1; 2 (A1 ; B1 |B2 ) = dB2 [T1; 2[ (7.22) UB0

we give the number In round brackets after the kernel of the individual propagator of B reactant whose coordinates are the arguments of this kernel. Using Eq. (7.22), Eq. (7.20) is put into the form UB0 (1)])1; 1 (A1 ; B1 ) = Vˆ 1; 1 1; 1 (A1 |B1 ) ; (@t − Lˆ A1 − Lˆ B1 − Vˆ 1; 1 − [M1[ where we introduce the kernel Z M1 (A1 ; t|A01 ; t0 ) = dB2 dB02 T1; 2 (A1 ; B2 ; t|A01 ; B02 ; t0 )0; 1 (B02 )

(7.23)

(7.24)

of the “mass” operator with allowance for pair encounters of reactants only [10]. The form of Eq. (7.23) makes it possible to consider the eective pair the propagator of which satis es the equation UB0 (1)])Ge (A1 ; B1 ; t|A01 ; B01 ; t0 ) (@t − Lˆ A1 − Lˆ B1 − Vˆ 1; 1 − [M1[ = (t − t0 )(A1 − A01 )(B1 − B01 ) :

(7.25)

Using this propagator, the solution of Eq. (7.23) has the form 1; 1 (A1 ; B1 ) = Gˆ e Vˆ 1; 1 1; 1 (A1 |B1 ) : as

(7.26)

Thus the kinetic Eq. (5.12) for local concentration of A reactants may be represented (@t − Lˆ A1 )1; 0 (A1 ) = (t)A (A1 ) +

Z

dB1 (Vˆ 1; 1 + Vˆ 1; 1 Gˆ e Vˆ 1; 1 )1; 1 (A1 |B1 ) : (7.27)

De ne the two-particle T -operator of the eective pair of the problem by the equality analogous to the rst equation from Eqs. (7.5) or Eq. (6.7) Tˆ e = Vˆ 1; 1 + Vˆ 1; 1 Gˆ e Vˆ 1; 1 :

(7.28)

This gives the desired kinetic equation for the local concentration of A reactants in the eective pair approximation [10] Z ˆ (7.29) (@t − LA1 )1; 0 (A1 ) = (t)A (A1 ) + dB1 Tˆ e (1; 0 (A1 )0; 1 (B1 )) : It is more convenient to calculate the eective pair T -operator by deriving the closed equation for Tˆ e similar to Eqs. (7.13) or Eqs. (7.7). For this purpose, we introduce 0 a “free” propagator Gˆ e of the eective pair the kernel of which is de ned by the equation 0 UB0 (1)])Ge (A1 ; B1 ; t|A01 ; B01 ; t0 ) (@t − Lˆ A1 − Lˆ B1 − [M1[

=(t − t0 )(A1 − A01 )(B1 − B01 ) :

(7.30)

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This equation shows that the reaction aects a “free” motion of A reactant in the eective pair by the averaged ‘background’ action of all B reactants. Thus it is not 0 surprising that the “free” propagator Gˆ e does not preserve the number of A reactants. Nevertheless, it has an important property of free propagators of the system under study. Its kernel is representable as a product of kernels of individual propagators, i.e., it describes independent evolution in the reaction pair 0 (A1 ; B1 ; t|A01 ; B01 ; t0 ) = UAe (A1 ; t|A01 ; t0 )UB0 (B1 ; t|B01 ; t0 ) ; Ge

(7.31)

where UAe is the kernel of the so-called encounter theory propagator [10]. It obeys a single-particle equation (@t − Lˆ A1 − Mˆ 1 )UAe (A1 ; t|A01 ; t0 ) = (t − t0 )(A1 − A01 ) :

(7.32)

The validity of representation (7.31) is veri ed by substituting it immediately into Eq. (7.30) with allowance for the Chapman–Kolmogorov relation [39] that has the following form in the frame of the terminology employed: Z (7.33) UB0 (B; t|B0 ; t0 ) = UB0 (B; t|B1 ; t1 ) dB1 UB0 (B1 ; t1 |B0 ; t0 ) : 0 The introduction of the “free” propagator Gˆ e allows one to use the conventional procedure of deriving the closed equation for the pair T -operator [5,34,36]. As a result, Tˆ e satis es either of the two operator equations e 0 ˆ e 0 ˆ [ ˆ ˆ Tˆ e = Vˆ 1; 1 + Vˆ 1; 1 [U[ A UB ]T e = V 1; 1 + T e [UA UB ]V 1; 1 :

(7.34)

These equations have been obtained in Ref. [10] by performing partial summation of the “mass” operator diagram expansions. Thus Eqs. (7.29) and (7.34) are the essence of the eective pair approximation. Rigorously speaking, they are not binary, since in the case of uniform systems the irreversible reaction kinetics established on their basis is not expressed solely in terms of reactivity parameters and the relative motion in a pair [10], as it must be in the binary approximation. A change to a binary approximation calls for further simpli cations. 8. The binary approximation For the model of the reacting system under discussion, the eective pair approximation is the rst stage in the development of the binary approximation in the nonMarkovian kinetic equation [10]. In the general case of spatially nonuniform systems the details of further simpli cations of Eqs. (7.29) and (7.34) are given in [10]. Principal points of the derivation of the binary non-Markovian kinetic equations from the equations of the eective pair approximation will be described below using the systems spatially uniform in B species. As is shown in Section 2, for such systems there exists the closed kinetic equation for the kinetics P(t) (see Eq. (2.9)). Recall that examination of exactly these systems has been of fundamental importance in revealing

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the advantages and disadvantages of the non-equilibrium statistical mechanics binary approaches to in nite hierarchies closure (see Sections 4 and 6). For the systems uniform in B species 0; 1 (B) = [B], and so the “mass operator” has a shift symmetry in space and time M (A; t|A0 ; t0 ) = M (A − A0 ; t − t0 |0; 0) :

(8.1) e Uˆ A

By virtue of Eq. (7.32), the encounter theory propagator has the same symmetry. Taking this fact into account in Eq. (7.34), we conclude that a two-particle T -operator of the eective pair has a shift symmetry as well Te (A; B; t|A0 ; B0 ; t0 ) = Te (A − A0 ; B − A0 ; t − t0 |0; B0 − A0 ; 0) :

(8.2)

Thus the operator Tˆ e exhibits an important property of the operator Tˆ 1; 1 that has made it possible to obtain the closed equation for the kinetics in Section 6. The integro-dierential equation for the kinetics Pe in the eective pair approximation may be derived similarly @t Pe (t) = −ˆ e Pe + (t) :

(8.3)

The memory function of this equation has a shift symmetry in time, and is expressed in terms of the kernel of the operator Tˆ e Z e (t|0) = −[B] dA dB dB0 Te (A; B; t|A0 ; B0 ; 0) : (8.4) So the eective pair approximation for nonuniform system (4.1) leads to the closed equation for the kinetics, and, therefore, is compatible with the fundamental /internal/ system of the reacting system. Calculation of the eective pair T -operator in the binary approximation calls for the use of point encounter approximation for the “mass” operator in Eq. (7.32) [10] M (A; t|A0 ; 0) ' −[B]k(A − A0 )(t) ;

(8.5)

where k is a stationary rate constant de ned in Eq. (2.4). Then Eq. (7.32) takes a form (@t − Lˆ A1 + [B]k)UAe (A1 ; t|A01 ; 0) = (t)(A1 − A01 ) :

(8.6)

Its solution is as follows: UAe (A; t|A0 ; 0) = exp(−[B]kt)UA0 (A; t|A0 ; 0) :

(8.7)

Solving Eq. (7.34) with the kernel UA0 of Eq. (8.7) type gives the desired T -operator of the eective pair in the frame of the binary approximation [10] Te (A; B; t|A0 ; B0 ; 0) ' exp(−[B]kt)T1; 1 (A; B; t|A0 ; B0 ; 0) :

(8.8)

Substitution of this result into Eq. (8.4) yields a binary approximation for the memory function of integro-dierential kinetic equation (8.3) of the form Z (8.9) e (t|0) ' −[B]exp(−[B]kt) dA dB dB0 T1; 1 (A; B; t|A0 ; B0 ; 0) : The identity of this result with Eq. (2.3) is established by changing to relative coordinates and the pair’s center ones in the description of the reacting pair [10]. The same expedient was used earlier in going from Eq. (6.12) to Eq. (6.13).

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9. Summary A universal method of derivation of in nite hierarchies for partial distribution functions and correlation forms in the thermodynamic limit has been developed by considering the reacting systems in the Fock space. They describe the irreversible reaction A + B → B in spatially nonuniform systems. These hierarchies are the analogues of BBGKY ones of the non-equilibrium statistical mechanics. In the theory of elementary reactions they were introduced earlier on the basis of semi-intuitive considerations only for spatially uniform systems. The extension of physical argumentation of the approaches existing in the nonequilibrium statistical mechanics to the closure of the obtained hierarchies has led to wrong binary non-Markovian kinetic equations in some critical cases. We suggest the following closure method: (a) neglecting the contributions from the four-particle correlation forms in the description of the evolution of the three-particle completely correlated form, (b) adapting the method for channel extraction by the Faddeev reduction – known in the quantum mechanical three-body theory and based on the formalism of T -operators – in extracting a binary part of the evolution of three-particle systems. Together with the point approximation, in the eective pair evolution the above closure method yields correct binary non-Markovian kinetic equations of the reaction A+B → B derived earlier on the basis of diagram summation. Thus the developed new method for closing in nite hierarchies employs solely general physical principles. Therefore, it admits the necessary generalizations to the case of the abandonment of simplifying assumptions in the description of the structure of reactants and to a wider class of reactions. Some of the papers performing such generalizations will be published soon. Acknowledgements The authors are grateful to the Russian Foundation of Basic Research (RFBR) for nancial support under the project 96-03-32917. Appendix A In view of de nition (4.11), integration of both sides of the second equality from Eq. (4.9) yields (@t − Lˆ + w(r))ne (r; t − 0)P(t) = (t − 0) = − [B] lim

v→∞

Z

Z v

dr B f(r B + r; t)(r B + r; r B ; t) v

dr2B w(r B + r − r2B )(r B + r; r2B ; t) ; (A.1)

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where, to de ne a time derivative of the product ne P unambiguously, we separate the discontinuity points of these functions by in nitesimal value. Let us introduce a new variable r0 = r B + r − r0B , and change to new independent variables (r0 ; r2B ) in the integral in Eq. (A.1). So Eq. (A.1) may be represented as P(t)(@t − Lˆ + w(r))ne (r; t − 0) + n(r; t − 0)@t P(t) Z Z dr2B f(r2B + r0 ; t)(r2B + r0 ; r2B ) dr0 w(r0 ) = (t − 0) − [B] lim v→∞ v v ×(r2B + r0 ; r2B + r0 − r; t) :

(A.2)

Using the separation of discontinuities of functions ne and P and in view of Eqs. (4.10) and (4.11), the second term in the left-hand side of Eq. (A.2) is reduced to the form Z ne (r; t − 0)@t P(t) = −[B]ne (r; t)P(t) dr0 w(r0 )ne (r0 ; t) Z = −[B] lim

v→∞

Z ×

v

dr B f(r B + r; t)(r B + r; r B ; t) v

dr0 w(r0 )ne (r0 ; t) :

(A.3)

Substituting (r0 ; r2B ) for (r0 ; r B ) in the integral in Eq. (A.3) and subtracting Eq. (A.3) from Eq. (A.2), we get the equation for ne (r; t) Z dr B ˆ dr0 2 w(r0 )f(r2B + r0 ; t) P(t)(@t − L + w(r))ne (r; t) = (t − 0) − lim v→∞ v v ×(r2B + r0 ; r2B + r0 − r; t) ×[(r2B + r0 ; r2B ; t) − ne (r0 ; t)] : Apply the relation ne (r0 ; t) =

R

dr0B B B B v f(r0 + r0 ; t)(r0 + r0 ; r0 ; t) R dr0B limv→∞ v v f(r0B + r0 ; t)

limv→∞

(A.4)

v

(A.5)

following from de nition (4.11) in the right-hand side of Eq. (A.4). As a result, we have P 2 (t)(@t − Lˆ + w(r))ne (r; t) Z dr B dr B dr0 2 2 0 w(r0 )f(r2B + r0 ; t)f(r0B + r0 ; t) = − [B] lim v→∞ v v   ×(r2B + r0 ; r2B + r0 − r; t) (r2B + r0 ; r B ; t) − (r0B + r0 ; r0B ; t) + (t − 0) : (A.6) Since the function P 2 (t) is positive, Eq. (A.6) may be represented as Eq. (4.12).

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Let us show that the source (r; t) is equal to zero in the case where non-uniformity is local. It follows from the set (4.9) (@t − Lˆ A + w(A − B))f(A; t)(A; B; t) = (A; B; t)(@t − Lˆ A )f(A; t) :

(A.7)

Calculation of the time derivative of the product f requires that the discontinuity points of these functions be separated, just as in Eq. (A.1). However, for brevity this procedure will not be given explicitly in further calculations. If at A → ∞, then the function f(A; t) is independent of the coordinate A, therefore, Lˆ A f(A; t) → 0. Thus Eq. (A.7) gives the equation for  (@t − Lˆ A + w(A − B))(A; B; t) = (t) :

(A.8)

In relative coordinates the above equation takes the form (@t − Lˆ + w(r))(r B + r; r B ; t) = (t) :

(A.9)

As in this equation all operators operating on  do not depend on r B , hence, (r B + r; r B ; t) is independent of r B , and the integrand in Eq. (A.6) is equal to zero. If both reactants are mobile, one should change to a relative coordinate r and a pair’s center coordinate Rc in Eq. (A.1) [10] r = A − B;

Rc = A + B;

(A.10)

where  = DB =(DA + DB ) and = DA =(DA + DB ) are expressed in terms of macroscopic diusion of A and B reactants. The function (r;  Rc ; t) = (Rc + r; Rc − r; t) is a B natural generalization of the function (r + r; r B ; t). Further calculations are similar to those given above.

Appendix B Substitute expansion (4.14) into the set of Eqs. (4.9). In the zero approximation, we have (@t − Lˆ A )f(0) (A; t) = (t)f0 (A) ; (@t − Lˆ A )f(0) (A; t)(0) (A; B; t) = (t)f0 (A) :

(B.1)

Hence 0 f(0) (A; t) = Uˆ A [f0 (A)(t)] ;

(0) (A; B; t) = 1 : Thus in the zero approximation in reactivity no correlations take place. Introducing the designation Z w = dB w(A − B)

(B.2)

(B.3)

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we have from Eq. (4.9) in the rst order in w (@t − Lˆ A )f(1) (A; t) = −[B]f(0) (A; t)w ; (@t − Lˆ A )[f(0) (A; t)(1) (A; B; t) + f(1) (A; t)] = − w(A − B)f(0) (A; t) − [B]wf(0) (A; t) :

(B.4)

This system is equivalent to the following: (@t − Lˆ A )f(1) (A; t) = −[B]f(0) (A; t)w ; (@t − Lˆ A )f(0) (A; t)(1) (A; B; t) = −w(A − B)f(0) (A; t) :

(B.5)

In view of Eqs. (B.2), its solution is of the form 0 0 f(1) (A; t) = −[B]wUˆ A Uˆ A [f0 (A)(t)] = −[B]f(0) (A; t)wt ; 0 g0 [w(r)f(B + r; t)] : f(0) (A; t)(1) (A; B; t) = −Uˆ A [w(A − B)f(0) (A; t)] = −ˆ

(B.6)

Substitute expansion (4.14) into Eq. (4.13). It follows from Eqs. (B.2) that the leading term in the expression in square brackets is of zero order, and the contribution from the leading term of the expansion (r B + r0 ; r B + r0 − r; t) (i.e., from (0) = 1) is equal to zero by virtue of the integrand antisymmetry in integration variables r B and r0B . Appendix C Let the non-uniformity of the initial distribution be nonlocal (see Eq. (4.18)). Since the diusion Green function   1 |r A − r0A |2 ; exp − UA0 (r A; t|r0A ; t0 ) = 4D(t − t0 ) [4D(t − t0 )]3=2   1 |r − r0 |2 : (C.1) exp − g0 (r; t|r0 ; t0 ) = 4D(t − t0 ) [4D(t − t0 )]3=2 Therefore, we have from Eqs. (B.2) and (C.1) for f(0) Z  f(0) (r A; t) = Re dr0A G(r A; t|r0A ; 0)(1 + C exp(ib−1 r0A )) (r A )2 Z   ∞ (r0A )2 2Dt 2Ce− 4Dt A A −1 A dr r exp ib r − =1+ 0 0 0 (4Dt)3=2 0 4Dt rA  A A    A A r r r r0 − exp − 0 × exp 2Dt 2Dt   A 2   exp − (r4Dt) 2Dt @ Z ∞ (r0A )2 A −1 A dr0 exp ib r0 − =1+C (4Dt)1=2 r A @r A 0 4Dt  A A    A A r r r r0 + exp − 0 : × exp 2Dt 2Dt

(C.2)

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Here integration over the angles of the vector r0A is performed, and the result is expressed in terms of the derivative of the sum of two integrals with respect to the parameter r A . Calculation of the rst integral yields    Z ∞ (r A )2 rA dr0A exp − 0 + r0A ib−1 + 4Dt 2Dt 0     A Z ∞ (r A + r A + 2ib−1 Dt)2 (r + 2ib−1 Dt)2 dr0A exp − 0 = exp 4Dt 4Dt 0   A 1√ (r + 2ib−1 Dt)2 : (C.3) = 4Dt exp 2 4Dt The second integral is given by substitution of −r A for r A , or the equivalent substitution of −i for i. Eventually, we have     (r A + 2ib−1 Dt)2 − (r A )2 r A + 2ib−1 Dt C (0) A exp − + c:c: : f (r ; t) = 1 + 2 4Dt rA (C.4) If we pass to the limit in Eq. (4.15), the nonzero contribution into the integral can only be made by the integrands not tending to zero with r B → ∞ and r0B → ∞. Thus Eq. (C.4) may be simpli ed f(0) (r A; t) ∼ 1 + C exp(−b−2 Dt) cos(b−1 r A ) + o(1) :

(C.5)

rA →∞

Similar reasoning holds for the function (1) , and, therefore, for the function h(r B + r; r B ; t) ≡ f(0) (r B + r; t)(1) (r B + r; r B ; t) :

(C.6)

From Eqs. (B.6) and (C.4), we have h(r B + r; r B ; t) ∼ −gˆ0 w(r) − C gˆ0 [w(r)e−b rB →∞

−2

Dt0

cos(b−1 |r B + r|)] + o(1) : (C.7)

The rst term is expressed in terms of the Green function g0 (r; t|a; t0 ) averaged over the angles, as (see (4.17)) Z t Z Z t dt0 d 0r g0 (r; t|r0 ; t0 )|r0 =a = kr dt0 g0 (r; t|a; t0 ) : (r; t) ≡ gˆ0 w(r) = kr 0

0

(C.8)

To calculate the second term in Eq. (C.7), we restrict ourselves to the leading term of the expansion in 1=r B and a=b (i.e., replace cos(b−1 |r B + r|) by cos(b−1 r B )). This gives gˆ0 [w(r)exp(−b−2 Dt0 ) cos(b−1 |r B + r|)]   Z t Z |r − r0 |2 rB exp(−b−2 Dt0 ) + o(1) Re dt w(r ) exp − + i dr ∼ 0 0 0 r B →∞ 4D(t − t0 ) b [4D(t − t0 )]3=2 0 a=b 1 =C cos(b−1 r B )kr

Z 0

t

dt0 exp(−b−2 Dt0 )g(r; t|a; t0 ) + o(1) :

(C.9)

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Thus it follows from Eqs. (C.7) – (C.9) that h(r B + r; r B ; t) B ∼ −(r; t) − C(r; t) cos(b−1 r B ) + o(1) ; r →∞

where

Z (r; t) = kr

t

0

dt0 exp(−b−2 Dt0 )g0 (r; t|a; t0 ) :

(C.10)

(C.11)

Express 3 (r; t) from Eq. (4.15) in terms of the function h from Eq. (C.7) Z dr B dr B dr0 0 2 w(r0 )h(r B + r0 ; r B + r0 − r; t) 3 (r; t) = [B] lim v→∞ v v   f(0) (r B + r0 ; t) B h(r + r0 ; r B; t) : (C.12) × h(r0B + r0 ; r0B ; t) − (0) 0B f (r + r0 ; t) On substitution of Eqs. (C.5) and (C.10) into Eq. (C.12), one can explicitly perform integration over the variable r0B using the identity Z lim

v→∞

v

rB dr B cos = 0 : v b

(C.13)

In Eq. (C.12) the functions h(r B + r0 ; r B + r0 − r; t) may be replaced by h(r B + r; r B; t), and f(0) (r B + r0; t) – by f(0) (r B; t) up to the leading terms in a=b, r=b. Then, in view of Eq. (C.13), we obtain Z dr B w(r0 )h(r B + r; r B; t) dr0 3 (r; t) = [B] lim v→∞ v v   h(r B + r0 ; r B; t) : (C.14) × −(r0 ; t) − 1 + C cos(b−1 r B ) Now we use the smallness of C (see Eq. (4.19)), and expand 3 (r; t) in a power series of C so as to nd the leading term. In view of Eq. (C.10), the zero-order term of the integrand expansion is equal to zero, and the rst-order term of the source 3 (r; t) expansion is equal to zero because of Eq. (C.13). Thus the leading term of the 3 expansion is of the second order in the parameter C 3 (r; t) ∼ −

C 2 kr [(r; t) − (r; t)][(a; t) − (a; t)] : 2

Substituting the de nitions of  and  from Eqs. (C.8) and (C.11) gives Z [B]kr3 C 2 t dt0 [1 − exp(b−2 Dt0 )]g0 (a; t|a; t0 ) 3 (r; t) = − 2 0 Z t dt1 [1 − exp(b−2 Dt1 )]g0 (r; t|a; t1 ) : × 0

(C.15)

(C.16)

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For the time interval (4.20) with allowance for Eq. (4.19), the exponent in Eq. (C.16) is small; and the exponents may be expanded. This gives    Z [B]kr3 C 2 D2 t a2 t0 dt0 p 1 − exp − 3 (r; t) = − 32b4 3 ra3 0 D(t − t0 ) 4D(t − t0 )      Z t (r + a)2 (r − a)2 t1 dt1 p − exp − : exp − × D(t − t1 ) D(t − t1 ) 4D(t − t1 ) 0 (C.17) By virtue of the condition Dtmax(a2 ; r 2 ), the integrals in Eq. (C.17) may be estimated asymptotically. For example,    Z t    Z t a2 a2 t dt (t − )d p 0 0 √ 1 − exp − = 1 − exp − D(t − t0 ) D 4D 4D(t − t0 ) 0 0    √ Z ∞ a2 d  √ 1 − exp − = t|a| : (C.18) ∼ t 2 D D 4D 0 t aD Similar calculation of the asymptotics of the second integral in Eq. (C.17) gives      Z t (r + a)2 t1 dt1 (r − a)2 p − exp − exp − D(t − t1 ) D(t − t1 ) 4D(t − t1 ) 0 √ t  (r + a − |r − a|) : (C.19) ∼ 2 +r 2 D t a D Substituting asymptotics (C.19) and (C.18) in (C.17) yields the desired result (4.21). References [1] R.M. Noyes, J. Amer. Chem. Soc. 78 (1956) 5486. [2] S.G. Entelis, R.P. Tiger, Reaction Kinetics in Liquid Phase (Quantitative Account of the Medium In uence), Khimiya, Moscow, 1973 (in Russian). [3] A.I. Burshtein, Yu.D. Tsvetkov, Proc. USSR Acad. Sci. 214 (1974) 369 (in Russian). [4] A.A. Ovchinnikov, S.F. Timashev, A.A. Belyi, Kinetics of Diusion Controlled Chemical Processes, Nova Science, Commack, NY, 1989. [5] A.B. Doktorov, A.A. Kipriyanov, Mol. Phys. 88 (1996) 453. [6] A.A. Kipriyanov, I.V. Gopich, A.B. Doktorov, Chem. Phys. 187 (1994) 251. [7] T.R. Waite, Phys. Rev. 107 (1957) 463. [8] M. Doi, J. Phys. A 9 (1976) 1479. [9] I.V. Gopich, A.B. Doktorov, J. Chem. Phys. 105 (1996) 2320. [10] A.A. Kipriyanov, I.V. Gopich, A.B. Doktorov, Physica A 255 (1998) 347. [11] Yu.L. Klimontovich, Statistical Physics, Harwood Academic, New York, 1986. [12] N.N. Bogolubov, N.N. Bogolubov Jr., Introduction to Quantum Statistic Mechanics, World Scienti c, Singapore, 1982. [13] R. Balescu, Equilibrium and Non-equilibrium Statistical Mechanics, Wiley, New York, 1975. [14] V.N. Kuzovkov, E.A. Kotomin, Phys. State Sol.(b) 108 (1981) 37. [15] E. Kotomin, V. Kuzovkov, Chem. Phys. 76 (1983) 489. [16] S. Lee, M. Karplus, J. Chem. Phys. 86 (1987) 1883. [17] S. Lee, J.J. Lee, K.J. Shin, Bull. Korean Chem. Soc. 15 (1994) 311.

606 [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]

A.A. Kipriyanov et al. / Physica A 268 (1999) 567–606 J.G.J. Kirkwood, Chem. Phys. 3 (1935) 300. A. Szabo, R. Zwanzig, N. Agmon, Phys. Rev. Lett. 61 (1988) 2496. A. Szabo, J. Phys. Chem. 93 (1989) 6929. H. Mori, Prog. Theor. Phys. 33 (1965) 423. R. Kapral, Adv. Chem. 48 (1988) 1479. V.P. Sakun, Physica A 80 (1975) 128. A.B. Doktorov, Physica A 90 (1978) 109. A.A. Kipriyanov, A.B. Doktorov, A.I. Burshtein, Chem. Phys. 76 (1983) 149. N.N. Lukzen, A.B. Doktorov, A.I. Burshtein, Chem. Phys. 102 (1986) 289. A.I. Burshtein, P.A. Frantsuzov, J. Chem. Phys. 106 (1997) 3948. A.I. Burshtein, N.N. Lukzen, J. Chem. Phys. 103 (1995) 9631. A.A. Kipriyanov, I.V. Gopich, A.B. Doktorov, Chem. Phys. 187 (1994) 241. A.A. Kipriyanov, I.V. Gopich, A.B. Doktorov, Chem. Phys. 191 (1995) 101. P. Resibois, M. de Leener, Classical Kinetic Theory of Fluids, Wiley, New York, 1980. E. Schmid, H. Ziegelmann, The Quantum Mechanical Three-body Problem, Pergamon Press, Braunschweig, 1978. A.A. Kipriyanov, I.V. Gopich, A.B. Doktorov, Physica A 205 (1994) 585. A.A. Kipriyanov, A.B. Doktorov, Physica A 230 (1996) 75. A.A. Kipriyanov, A.B. Doktorov, Chem. Phys. Lett. 246 (1995) 359. J. Taylor, The Scattering Theory, the Quantum Theory of Non-relativistic Collisions, Wiley, New York, 1972. A.A. Kipriyanov, I.V. Gopich, A.B. Doktorov, Physica A 205 (1994) 585. M. Doi, J. Phys. 9 (1976) 1465. A.B. Doktorov, E.A. Kotomin, Phys. Stat. Sol.(b) 114 (1982) 9. H. Eyring, S.H. Lin, S.M. Lin, Basic Chemical Kinetics, Wiley, New York, 1980. M. Yokota, O. Tatimoto, J. Phys. Soc. Japan 22 (1967) 779. S. Fujita, Introduction to Non-equilibrium Quantum Statistical Mechanics, Saunders, Philadelphia, 1966.