Correlation functions for diffusion-limited annihilation,A+A→0

Abstract

The full hierarchy of multiple-point correlation functions for diffusion-limited annihilation, A + A → 0, is obtained analytically and explicitly, following the method of intervals. In the long time asymptotic limit, the correlation functions of annihilation are identical to those of coalescence, A + A → A, despite differences between the two models in other statistical measures, such as the interparticle distribution function.02.50. Ey, 05.50.+q, 05.70.Ln, The kinetics of nonequilibrium processes, in particular diffusion-limited reactions, have attracted much recent interest [1][2][3][4][5][6][7][8]. Because of the lack of a comprehensive approach for the study of such systems, models that yield to an exact analysis are of prime importance. In this respect, none have been studied more than diffusion-limited annihilation, A + A → 0 [9-28], and coalescence, A + A → A [9, [13][14][15]17,21,23,26,[28][29][30][31][32][33]. Known exact results include the time dependence of the particle concentration and the two-point correlation function (for finding two particles at two different points, simultaneously). It has also been shown that the full hierarchy of n-point correlation functions for the two processes is identical [21,23,34,35], but explicit expressions for n > 3 are unavailable.Here we attack the problem of correlation functions for annihilation, using the method of parity intervals (or even/odd intervals) [20,26,27,36]. We recover the identity relation of the n-point correlation functions for annihilation and coalescence, and we derive explicit expressions, valid in the long time asymptotic limit, for all n.Consider the annihilation model, defined on the line −∞ < x < ∞. Particles A are represented by points which perform unbiased diffusion with a diffusion constant D. When two particles meet they annihilate instantly. Since the reaction step is infinitely fast, the system models the diffusion-limited annihilation process A + A → 0.An exact treatment of the problem is possible through the method of parity intervals [20,26,27,36]. The key parameter is G(x, y; t)-the probability that the interval [x, y] contains an even number of particles at time t [37]. Particles near the edges of an interval may diffuse into or out of the interval, affecting the probability G. (On the other hand, reactions inside the interval cannot affect its parity.) With this observation in mind, one derives a rate equation for the probability G(x, y; t) [26]:

Lattice kinetics of diffusion-limited coalescence and annihilation with sources

“…In the small h limit, the expressions for ρ s ≈ h 1/2 and τ R agree with the continuum limit [26]. These results have also been obtained by Rácz by mapping the above model into the kinetic Ising model; to do so, one chooses Γ = 2D/a 2 , δ = (1 − h)/(1 + h), and w (2) i ({σ}) = 0 in Eq.…”

Multiscaling of correlation functions in single species reaction-diffusion systems

“…Large fluctuation effects are accounted for in binary * Electronic address: ranm@maths.warwick.ac.uk † Electronic address: rrajesh@brandeis.edu ‡ Electronic address: olegz@maths.warwick.ac.uk reaction-diffusion models using Empty Interval methods (EIM) and its generalizations [6,7,8,9]. This approach is restricted to d = 1 and does not extend to higher dimensions.…”

“…Most studies have concentrated on computing the average density of particles (N = 1) [4,12]. To the best of our knowledge, the computation of multi-particle probabilities are only considered in [6,7,8], with the analysis restricted to one dimension. Dynamical RG method allows us to obtain answers for the large time limit in the form of an ε-expansion (ε = 2 − d) for d < 2 and logarithmic corrections to the MF scaling for d = 2:…”

Numerical study of A+A→ and A+B→ reactions with inertia

“…The A+A → 0 and A+A → A reactions have also been extensively studied [2,[14][15][16][17][18][19][20][21][22][23]. In the diffusion-limited regime these reactions in one dimension have provided a wealth of information because they can be solved exactly for the concentrations as a function of time.…”