Fluctuation effects in the pair-annihilation process with Lévy dynamics

We investigate the density decay in the pair-annihilation process $A+A\to \varnothing$ in the case when the particles perform anomalous diffusion on a cubic lattice. The anomalous diffusion is realized via Lévy flights, which are characterized by long-range jumps and lead to superdiffusive behavior. As a consequence, the critical dimension depends continuously on the control parameter of the Lévy flight distribution. This instance is used to study the system close to the critical dimension by means of the nonperturbative renormalization group theory. Close to the critical dimension, the assumption of well-stirred reactants is violated by anticorrelations between the particles, and the law of mass action breaks down. The breakdown of the law of mass action is known to be caused by long-range fluctuations. We identify three interrelated consequences of these fluctuations. First, despite being a nonuniversal quantity and thus depending on the microscopic details, the renormalized reaction rate ${\lambda }_0$ can be approximated by a universal law close to the critical dimension. The emergence of universality relies on the fact that long-range fluctuations suppress the influence of the underlying microscopic details. Second, as criticality is approached, the macroscopic reaction rate decreases such that the law of mass action loses its significance. And third, additional nonanalytic power law corrections complement the analytic law of mass action term. An increasing number of those corrections accumulate and give an essential contribution as the critical dimension is approached. We test our findings for two implementations of Lévy flights that differ in the way they cross over to the normal diffusion in the limit $\sigma \to 2$.

Figure 1

The RG flow from microscopics ($k=\infty$) to macroscopics ($k=0$) can be divided into three characteristic regimes. The degrees of freedom being integrated out for $k>k^{*}$ are very short ranged and are thus restricted to the lattice sites. For $\tilde{k}<k<k^{*}$ the fluctuations propagate through the lattice and thereby perceive the lattice structure. In the continuum regime $k<\tilde{k}$, the flow is driven by the very long-ranged modes, which cannot resolve the underlying lattice. The schematic plots below show the typical partition of the two-dimensional Brillouin zone into low (black) and high (white) energy modes for the three characteristic regimes.

Figure 2

The macroscopic reaction rate (solid red) is plotted against $(d-\sigma )$ for the microscopic rate $\lambda =\infty$ in one spatial dimension. The dotted black line represents the power law Eq. (20) which is approached in the limit $(d-\sigma )\to 0$. The numerical integration of $\int 1/\epsilon {}_1$ is carried out by expanding the dispersion function in $\mathbf{p}$ [see Eq. (A1) of Appendix ]. The lack of an analogous expansion in two dimensions makes the integration numerically unfaithful for $d=2$.

Figure 3

The flow of the running reaction rate ${\lambda }_k$ (solid red) is divided into the three separate regimes $k<\tilde{k}$, $\tilde{k}<k<k^{*}$, and $k^{*}<k$ (compare Fig. 1). Before the macroscopic limit ${\lambda }_{k=0}$ is reached (i.e., for $k<\tilde{k}$), the flow is dominated by long-range fluctuations and can be approximated by the continuum limit Eq. (24) (dotted blue). These long-range fluctuations generate nonanalytic corrections to the law of mass action. As the long-range fluctuations cannot resolve the microscopic details, the nonanalytic terms can only depend on the microscopic model indirectly through the macroscopic reaction rate ${\lambda }_0$. Hence, the correction terms are universal functions of the nonuniversal rate ${\lambda }_0$.

Figure 4

(a) According to the law of mass action, the steady state is determined by $J=2{\lambda }_0{\rho }^2$ (dotted black line). As the critical dimension $\sigma$ approaches the spatial dimension $d=1$ from below, the macroscopic reaction rate ${\lambda }_0$ vanishes according to Eq. (20) and the law of mass action term loses its significance. This explains the discrepancy between the simulated data (red crosses) and the theoretical law of mass action (manifest in the last two plots). Additional nonanalytic correction terms have to be included (blue solid line) in order to compensate the loss of significance of the law of mass action term and to ensure good agreement with the simulations. The red squares in the lower right plot show the simulation results for finite microscopic reaction rate $\lambda =1$ as opposed to the instantaneous annihilation ($\lambda =\infty$). Due to the universal behavior of the renormalized rate ${\lambda }_0$ close to the critical dimension [compare the discussion around Eq. (20)], both data sets collapse on the same curve. For this plot, the number of nonanalytic corrections was chosen such that the exponent $2+n\epsilon$ remains below 3. The Lévy flights were implemented according to the pure power law $p_1$. (b) Same plot as in (a), except that the Lévy flights are now implemented via $p_2$. The qualitative results are unchanged.

Figure 5

Analogous to Fig. 4 for $d=2$. (a) The Lévy flights are implemented via the jump length probability distribution $p_1$ and the nonanalytic corrections to the law of mass action term $2{\lambda }_0{\rho }^2$ are marginal. (b) If the Lévy flights are implemented via $p_2$, the correction terms give significant contributions close to the critical dimension. Mathematically, the difference between (a) and (b) follows from the different behavior of ${\lambda }_0$ as the critical dimension is approached; see Eqs. (21a) and (21b).

Figure 6

Some of the one-loop diagrams with one outgoing line are shown. The most general diagram, which can be built out of vertices with only two ingoing legs is shown in (c). Diagrams (a), giving rise to the law of mass action term, and (b) are special cases of (c). Diagram (d) is obtained from (c) by contracting two of the vertices to one vertex with three incoming legs. As argued in the text, the contribution of (d) is negligible compared to (c).

Figure 7

The particle input $J$ is plotted against the average density $\rho$ of the stationary state, similar to the plot of Fig. 4a. The different point styles correspond to different choices of the system size $N$. As all data points clearly lie on the same line, we can exclude finite size effects.

Figure 8

The result of a numerical simulation of the pair-annihilation process with particle input $J$ using the Gillespie algorithm on a finite ($N={10}^6$) lattice. Initially, at $t=0$, the lattice is empty. After a short transient regime the stationary state is reached and the average particle density (dotted blue line) can be measured.