Many-point densities

The scope of this book is as follows. Chapter 2 gives a general review of different theoretical techniques and methods used for description the chemical reactions in condensed media. We focus attention on three principally different levels of the theory macroscopic, mesoscopic and microscopic the corresponding ways of the transition from deterministic description of the many-particle system to the stochastic one which is necessary for the treatment of density fluctuations are analyzed. In particular, Section 2.3 presents the method of many-point densities of a number of particles which serves us as the basic formalism for the study numerous fluctuation-controlled processes analyzed in this book. 

To conform to Section 5.1 let us use again the approach of the many-point densities. The fact that the quantity of interest is the survival probability of a single particle A in terms of mathematics means that from the complete set of equations for many-point densities pm, m, equation (2.3.38), we can restrict ourselves to those with only the first index equal to one m1 = 1 m = 0,1,..., oo, that is... 

In this Section we consider several approaches which differ from the many-point density formalism discussed above. Szabo et al. [45] have introduced a novel method based on the density expansion for the survival probability, u>(t). Consider a system containing walkers (particles A) and N traps (quenchers B) in volume V in d-dimensional space. We assume that the particles have a finite size but the traps can be idealized as points and hence are ignorant of each other. When the concentration of the walkers is sufficiently low so that excluded volume interactions between them are negligible, one might focus on a single walker. 

Let us start with first-principles approach in the particular case of immobile particles (low temperatures). An infinite set of equations for many-point densities pm,m = Pm,m r m r m> t), where f m = n,..., rm (below a symbol f lm denotes the vector fj omitted in a set r m) could be derived by summing recombination and generation contributions which yields ... 

The distinctive feature of equation (7.1.1) is that all many-point densities Pm,m become zero in the limiting case of instant annihilation, op —> oo,... 

Taking into account preceding discussion of the virtual configurations for many-point densities pmp or p >mi let us define new functions... 

Equation (7.1.16) is asymptotically (cto — oo) exact. It shows that the accumulation kinetics is defined by (i) a fraction of AB pairs, 1 — u>, created at relative distances r > r0, (ii) recombination of defects created inside the recombination volume of another-kind defects. The co-factor (1 - <5a - <5b ) in equation (7.1.16) gives just a fraction of free folume available for new defect creation. Two quantities 5a and <5b characterizing, in their turn, the whole volume fraction forbidden for creation of another kind defects are defined entirely by quite specific many-point densities pmfl and po,m > he., by the relative distribution of similar defects only (see equation (7.1.17)). 

Its structure is self-evident each coordinate entering the many-point density Pmtm is associated with the relevant concentration co-factor n (t) (if the coordinate is from a set f m or nB(t) if it is from a set r m ). Each pair of coordinates corresponds to the particular joint correlation function, all these functions are multiplied. An analog of the substitution (7.1.18) in the case of many-point densities is an expression... 

Paper [109] determined the value of Uo upon approach to the steady state from above. One-dimensional crystals were simulated of length from 8 x 103 to 2 x 104ao (ao is a lattice constant the spatial correlation in genetic pairs is neglected). The limiting values Uo = 3.5-3.6 for 500 and 700 sites in the recombination sphere (Table 7.3, third column) are close to the value 3.43 obtained in the continuum approximation by an approximate method [22] and considerably exceed the estimate 1.36 implied by the approach based on many-point densities in the linear approximation [31, 111] remember that... 

Staying within a class of mono- and bimolecular reactions, we thus can apply to them safely the technique of many-point densities developed in Chapter 5. To establish a new criterion insuring the self-organisation, we consider below the autowave processes (if any) occurring in the simplest systems -the Lotka and Lotka-Volterra models [22-24] (Section 2.1.1). It should be reminded only that standard chemical kinetics denies their ability to selforganisation either due to the absence of undamped oscillations (the Lotka model) or since these oscillations are unstable (the Lotka-Volterra model). 

Since the many-point density formalism in its practical applications assumes macroscopically homogeneous system, we will restrict ourselves to a particular class of microscopically self-organized autowave processes. Without investigating in Chapter 8 all possible kinds of autowave processes, we are aimed to answer a principal question - whether these two models under question could be attributed to the basic models useful for the study of autowave processes. 

To formulate this stochastic model in terms of concentrations and joint correlation functions only, i.e., in a manner we used earlier in Chapters 2, 4 and 5, it is convenient to write down a master equation of the Markov process under study in a form of the infinite set of coupled equations for many-point densities. Let us write down the first equations for indices (m + m ) = 1 ... 

The considerable progress made in the studies of simple bimolecular reactions (which has led to such fundamental conclusions) was achieved by a more rigorous mathematical treatment of the problem, avoiding the use of the simplest approximations which linearize the kinetic equations. We focus main attention on the many-point density formalism developed in [26, 28, 49] since in our opinion it seems at present to be the only general approach permitting treatment of all the above-mentioned problems, whereas other theoretical methods so far developed, e.g., those of secondary quantization [19, 29-32], and of multiple scattering [72, 73], as well based on... 

Its structure is self-evident each coordinate entering the many-point density Pm,m is associated with the relevant concentration co-factor (if the... 

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