Joint correlation function

How could we take into account the fluctuations of the order parameter Let us return to the well-studied example of the gas-liquid system. A general equation of the state of gases and liquids proved in statistical physics [9] has a form p = nk T - n2G(x) where G(x) is some integral containing the interaction potential of particles and the joint correlation function x(r). Therefore, the equation for the long-range order parameter n contains in itself the functional of the intermediate-order parameter x r)-... 

The only problem necessary for developing the condensation theory is to add to the above-mentioned equation of the state the equation defining the function x(r)- Unfortunately, it turns out that the exact equation for the joint correlation function, derived by means of basic equations of statistical physics, contains f/iree-particle correlation function x 3), which relates the correlations of the density fluctuations in three points of the reaction volume. The equation for this three-particle correlations contains four-particle correlation functions and so on, and so on [9], This situation is quite understandable, since the use of the joint correlation functions only for description of the fluctuation spectrum of a system is obviously not complete. At the same time, it is quite natural to take into account the density fluctuations in some approximate way, e.g., treating correlation functions in a spirit of the mean-field theory (i.e., assuming, in particular, that three-particle correlations could be expanded in two-particle ones). 

On this way we arrive at Bom-Green-Ivon, Percus-Yevick and hyperchain equations [5, 9], all having a general form (x,Vx,n,T) = 0. These non-linear integro-differential equations are close with respect to the joint correlation function, and Percus-Yevick equation gives the best approximation amongst known at present. An important point is that the accuracy of... 

However, a question arises - could similar approach be applied to chemical reactions At the first stage the general principles of the system s description in terms of the fundamental kinetic equation should be formulated, which incorporates not only macroscopic variables - particle densities, but also their fluctuational characteristics - the correlation functions. A simplified treatment of the fluctuation spectrum, done at the second stage and restricted to the joint correlation functions, leads to the closed set of non-linear integro-differential equations for the order parameter n and the set of joint functions x(r, t). To a full extent such an approach has been realized for the first time by the authors of this book starting from [28], Following an analogy with the gas-liquid systems, we would like to stress that treatment of chemical reactions do not copy that for the condensed state in statistics. The basic equations of these two theories differ considerably in their form and particular techniques used for simplified treatment of the fluctuation spectrum as a rule could not be transferred from one theory to another. 

It is convenient to divide a set of fluctuation-controlled kinetic equations into two basic components equations for time development of the order parameter n (concentration dynamics) and the complementary set of the partial differential equations for the joint correlation functions x(r, t) (correlation dynamics). Many-particle effects under study arise due to interplay of these two kinds of dynamics. It is important to note that equations for the concentration dynamics coincide formally with those known in the standard kinetics... 

The said allows us to understand the importance of the kinetic approach developed for the first time by Waite and Leibfried [21, 22]. In essence, as is seen from Fig. 1.15 and Fig. 1.26, their approach to the simplest A + B —0 reaction does not differ from the Smoluchowski one However, coincidence of the two mathematical formalisms in this particular case does not mean that theories are basically identical. Indeed, the Waite-Leibfried equations are derived as some approximation of the exact kinetic equations due to a simplified treatment of the fluctuational spectrum a complete set of the joint correlation functions x(rJ) for kinds of particles is replaced by the only function xab (a t) describing the correlation of chemically reacting dissimilar particles. Second, the equation defining the correlation function X = Xab(aO is linearized in the function x(rJ)- This is analogous to the... 

Omstein-Zemike theory of the critical opalescence which operates also with a linear equation for the joint correlation function. 

A new principal element of the Waite-Leibfried theory compared to the Smoluchowski approach is the relation between the effective reaction rate K(t) and the intermediate order parameter x = Xab (r,i). In its turn, the Smoluchowski approach is just an heuristic attempt to describe the simplest irreversible bimolecular reactions A + B- B,A + B- B and A + B -> 0 and cannot be extended for more complicated reactions. The Waite-Leibfried approach is not limited by these simple reactions only it could be applied to the reversible reactions and reaction chains. However, in the latter case the particular linearity in the joint correlation function x = Xab (r, ) does not always mean linearity of equations since additional non-linearity caused by particle densities can arise. 

Neglecting third-order momenta, the joint correlation functions (2.1.85), (2.1.86) become [36, 37]... 

Therefore P2(oo) = n2. It is convenient to eliminate dimension-dependent concentration co-factor n2, defining the joint correlation function... 

Fig. 2.17. A pattern of joint correlation function in a homogeneous condensed system.

The joint correlation function x (r) characterizes short and intermediate orders in particle spatial relative distribution. However, what is observed in diffraction experiments is not the joint correlation function but the corresponding structure factor [77]... 

These joint correlation functions have transparent physical interpretation [78] mean density of a number of i/-kind particles at a distance r from a given /z-kind particle (placed at origin of coordinates) is nothing but... 

Therefore, the joint correlation functions Xvjl (a t)> being at least potentially observable, are more a theoretical than an experimental tool for the description of interacting particles in condensed media. Both these joint functions and macroscopic concentrations nv t) determine the lowest level to characterize the spatio-temporal structure of a system. 

Since these characteristics are time-dependent, let us assume particle birth-death and migration to be the Markov stochastic processes. Note that making use of the stochastic models, we discuss below in detail, does not contradict the deterministic equations employed for these processes. Say, the equations for nv t), Xu(r,t), Y(r,t) given in Section 2.3.1 are deterministic since both the concentrations and joint correlation functions are defined by equations (2.3.2), (2.3.4) just as ensemble average quantities. Note that the... 

A structure of the obtained set of equations derived by us in [81, 86] is very close to the famous BBGKI set of equations widely used in the statistical physics of dense gases and liquids [76]. Therefore, we presented the master equation of the Markov process in a form of the infinite set of deterministic coupled equations for averages (equation (2.3.34)). Practical use of these equations requires us to reduce them, retaining the joint correlation functions only. 

Taking into account (2.3.27)-(2.3.29), two-particle densities could be expressed easily through the joint correlation functions... 

A procedure similar to the condensate separation in the imperfect Bose gas was employed by Lifshitz and Pitaevski [78]. The diagrammatic technique allows us to calculate the reaction rate and steady-state joint correlation functions. A separation of a condensate from terms with k = 0 cannot be done without particle production (p = 0), in which case nA, tiq —> 0 as t —> oo. In this respect the formalism presented by Lushnikov [111] for the non-stationary processes is of certain interest. 

In this Section we consider again the kinetics of bimolecular A + B -A 0 recombination but instead of the linearized approximation discussed above, the complete Kirkwood superposition approximation, equation (2.3.62) is used which results in emergence of two new joint correlation functions for similar particles, Xu(r,t), v = A,B. The extended set of the correlation functions, nA(f),nB(f),Xfi,(r,t),Xa(r,t) and Y(r,t) is believed to be able now to describe the intermediate order in the particle spatial distribution. 

Substitution of equation (2.3.62) into a set of equations (4.1.13) to (4.1.16) for noncharged (neutral) particles (Uvil r) = 0) does not affect equations (4.1.18) and (4.1.19) whereas the linear equation (4.1.23) describing the correlation dynamics splits now into three integro-differential equations. Main stages of the passage from general equations (4.1.14)—(4.1.16) for the joint densities to those for the joint correlation functions have been demonstrated earlier, see (4.1.20) and (4.1.21). Therefore let us consider only those terms which are affected by the use of superposition approximation. Hereafter we use the relative coordinates f=f — f(, f = r 2 — r[ and... 

The point is that this approach ignores the distinctive feature of the bi-molecular process - its non-equilibrium character. The fundamental result known in the theory of non-equilibrium systems [2, 3] is that they tend to become self-organised to a degree which could be characterised by the joint correlation functions, Xv(r, t) and Y(r, t). The idea to use n t)r as a small parameter were right, unless there are no other distinctive parameters of the same dimension as tq. 

For the joint correlation functions of similar particles the native boundary condition is absence of the flux over the coordinate origin... 

Numerical solution of a set of the kinetic equations (6.1.45) and (6.1.63) to (6.1.66) for the joint correlation functions is presented in Figs 6.11 and 6.12. (To make them clear, double logarithmic scale is used.) The auto-model variable 77 is plotted along abscissa axis in Fig. 6.11 showing the correlation function X(r, R). 

Behaviour of the joint correlation functions (see Figs 6.15 to 6.17 as typical examples of the black sphere model) resembles strongly those demonstrated above for immobile particles at the scale r < = Id the similar particle function exceeds its asymptotic value Xv(r,t) 2> 1. As r Id. both the correlation functions strive for their asymptotics Y(r,t), Xu(r,t) 1. The only peculiarity is that for mobile particles the boundary condition (5.1.40) tends to smooth similar particle correlation near the point r = 0. On the other hand, if one of diffusion coefficients, say Da, is zero, the corresponding... 

Fig. 6.16. A logarithmic plot of the joint correlation functions for similar, Xv r, t), and dissimilar, Y(r, t), particles for d = 2 and symmetric (a), Da = Db, and asymmetric (b), Da = 0, cases, respectively. Full curves are Y(r,t), broken and dotted lines XA(r,t) and Xs(r, t). The initial concentration n(t) — 1.0. The dimensionless time Dtjr is 10 (curve 1) ...

The origin of this unusual behaviour is partly clarified from Fig. 6.34(a) where the relevant curves 2 demonstrate the same kind of the non-monotonous behaviour as the critical exponents above. Since, according to its definition, equation (4.1.19), the reaction rate is a functional of the joint correlation function, this non-monotonicity of curve 2 arises due to the spatial re-arrangements in defect structure. It is confirmed by the correlation functions shown in Fig. 6.34(a). The distribution of BB pairs is quasi-stationary, XB(r,t) X°(r) = exp[(re/r)3], which describes their dynamic aggregation. (The only curve is plotted for XB in Fig. 6.35(a) for t = 102 (the dotted line) since for other time values XB changes not more than by 10 per cent.) This quasi-steady spatial particle distribution is formed quite rapidly already at t 10° it reaches the maximum value of XB(r, t) 103. The effect of the statistical aggregation practically is not observed here, probably, due to the diffusion separation of mobile B particles. 

Fig. 6.35. The joint correlation functions for dissimilar particles Y(r,t) (full curves), immobile similar particles Xa r,t) (broken curve) and mobile particles Xb(r,t) (dotted curve). Parameters used are given. The dimensionless time (in units rg/D) is (a) t = 101 (1) 102 (2) 103 (3) 104 (4) (b) Da = De, t = 102 (1) 104 (curve 2).

Of greater interest is the behaviour of the joint correlation functions presented in Fig. 6.35(b). At any reaction time X (r, f) > X (r,t) holds now an increase of the maximum of X (r,t) in time is very slow. According to the above-given estimates for neutral non-interacting particles, it has a logarithmic character ... 

The deviation of the joint correlation function X from Xa arises due to the additional effect of the dynamic aggregation, which is observed mainly at the relative distances r < re. It follows from equation (6.3.6), that the... 

Fig. 6.38. The joint correlation function for the asymmetric (a) and symmetric (b) cases. Full curves are Y r,t) broken curves XAdotted curves Xb(t, t). Dimensionless time is 101 (1) 103 (2). Note that UAA = 0.

Before discussing mathematical formalism we should stress here that the Kirkwood approximation cannot be used for the modification of the drift terms in the kinetics equations, like it was done in Section 6.3 for elastic interaction of particles, since it is too rough for the Coulomb systems to allow us the correct treatment of the charge screening [75], Therefore, the cut-off of the hierarchy of equations in these terms requires the use of some principally new approach, keeping also in mind that it should be consistent with the level at which the fluctuation spectrum is treated. In the case of joint correlation functions we use here it means that the only acceptable for us is the Debye-Htickel approximation [75], equations (5.1.54), (5.1.55), (5.1.57). 

The equation for the time development of macroscopic concentrations formally coincides with the law of mass action but with dimensionless reaction rate K(t) = K(t)/ AnDr ) which is, generally speaking, time-dependent and defined by the flux of the dissimilar particles via the recombination sphere of the radius tq, equation (5.1.51). Using dimensionless units n(t) = 4nrln(t), r = t/tq, t = Dt/r, and the condition of the reflection of similar particles upon collisions, equation (5.1.40) (zero flux through origin), we obtain for the joint correlation functions the equations (6.3.2), (6.3.3). Note that we use the dimensionless diffusion coefficients, a = 2k, IDb = 2(1 — k), k = Da/ Da + Dq) entering equation (6.3.2). 

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