Long-range and many-body effects in coagulation processes

We study the problem of diffusing particles which coalesce upon contact. With the aid of a nonperturbative renormalization group, we first analyze the dynamics emerging below the critical dimension two, where strong fluctuations imply anomalously slow decay. Above two dimensions, the long-time, low-density behavior is known to conform with the law of mass action. For this case, we establish an exact mapping between the physics at the microscopic scale (lattice structure, particle shape and size) and the macroscopic decay rate in the law of mass action. In addition, we identify a term violating this classical law. It originates in long-range and many-particle fluctuations and is a simple, universal function of the macroscopic decay rate.

Figure 1

Important one-loop diagrams determining the flow of the vertex functions ${\Gamma }_k^{(m,n)}$, where $m$ and $n$ are the number of outgoing and incoming legs, respectively. The fact the number of legs can only decrease along the time arrow puts significant restrictions on the possible diagrams. Diagram (a) is not created in the flow equation because the (2,1)-vertex function is always zero. It follows that the propagator is not renormalized. Diagram (b) implies a term linear in ${\Gamma }_k^{(1,3)}$ to the flow of ${\Gamma }_k^{(1,3)}$. Similarly there arise linear contributions to the flow of general ${\Gamma }_k^{(m,n)}$ (except for $m=n=2$, where there is a quadratic term in ${\Gamma }_k^{(2,2)}$). Since diagram (c) is composed only of (2,2) vertices and propagators, the flow for ${\Gamma }_k^{(2,2)}$ can be solved without knowledge of further vertex functions.

Figure 2

Illustration of the order for calculating the nonzero fixed-point coefficients ${\tilde{g}}^{☆(m,n)}$. In each step the result is independent of the coefficients to follow. This can be shown by looking at all possible one-loop diagrams which determine the flow of the vertex function ${\Gamma }_k^{(m,n)}$ with $n$ incoming and $m$ outgoing legs. For the physically relevant line with $m=1$ all the coefficients ${\tilde{g}}^{☆(m^{\prime },n^{\prime })}$ with $n^{\prime }\le n$ must be known. In contrast, the diagonal elements $g^{☆(n,n)}$ only depend on $g^{☆(n^{\prime },n^{\prime })}$ with $n^{\prime }<n$.

Figure 3

Double logarithmic plot of the rescaled nonequilibrium force $f_{\tau }=f^{☆}$ (solid red line) at the fixed point for a one-dimensional system. It was obtained by calculating its expansion up to order 125 in $\chi$ within the local potential approximation. For small $\chi$, $f^{☆}\left(\chi \right)=\frac{\pi }{2}{\chi }^2+\frac{{\pi }^2}{6}{\chi }^3+\cdots$ is dominated by the power law $\frac{\pi }{2}{\chi }^2$ (flat dashed line). When $\chi$ is increased, one reaches a regime where $f^{☆}$ is described well by $4.2{\chi }^3$ (steep dashed line), before the Padé approximation—utilized to extend the regime of convergence—breaks down. As expected [cf. Eq. (20)] for large enough $\chi$, in addition to the third-order term $\frac{{\pi }^2}{6}{\chi }^3$, the rest of the expansion of $f^{☆}$ combines to another term scaling as ${\chi }^3$.

Figure 4

Results for the amplitude ${\mathcal{A}}_d$ of the long-time decay $\rho \sim {\mathcal{A}}_d{\rho }^{-\frac{d}{2}}$ ($d<2$). The circles are our estimates from the nonperturbative renormalization group calculations within the local potential approximation. They are compared to the exact result ${\mathcal{A}}_1=\frac{1}{\sqrt{2\pi }}$ in one dimension [43] and to the perturbative result $\frac{1}{2\pi \epsilon }+\frac{2\ln \left(8\pi \right)-5}{8\pi }$ of [16] (solid line). The latter is an expansion in the deviation $\epsilon$ from the upper critical dimension, $\epsilon =d_c-d$. It is doubtful whether $\epsilon =1$ (corresponding to one dimension) can be considered small. Indeed, despite the relatively crude truncation, the nonperturbative approach provides a substantially better result.

Figure 5

In (a) and (b) the one-loop Feynman diagrams for the calculation of the flow of the decay rate ${\lambda }_k=\frac{1}{2}{g}_k^{(1,2)}=\frac{1}{2}{\Gamma }_{k(\mathbf{0},0;\mathbf{0},0;\mathbf{0},0)}^{(1,2)}{\left(2\pi \right)}^{12}{\left(VT\right)}^{-1}$ are shown. Diagram (a) determines the flow of the (1,2)-vertex function ${\Gamma }_{k(\mathbf{0},0;\mathbf{p},\omega ;-\mathbf{p},-\omega )}^{(1,2)}$ and involves a (2,2)-vertex function ${\Gamma }_{k({\mathbf{p}}^{\prime },{\omega }^{\prime },-{\mathbf{p}}^{\prime },-{\omega }^{\prime };\mathbf{p},\omega ;-\mathbf{p},-\omega )}^{(2,2)}$, whose flow [deduced from diagram (b)] provides a closed and exact formula for the (2,2)-vertex function. Diagram (c) determines the flow of the coefficient $g_k^{(1,3)}$. The three internal propagators imply a factor ${\left(\frac{1}{k^2}\right)}^3$ for the behavior of $g_k^{(1,3)}$ in the limit of small $k$. Integration over momentum and frequency space in the Wetterich equation abates this singularity by a factor $k^{d+2}$. Hence, when $2<d<4$ the coefficient $g_k^{(1,3)}$ diverges as $k^{d-4}$. Above four dimensions it approaches a finite value as $k\to 0$.

Figure 6

Rescaled data for the universal correction to the nonequilibrium force $F$. In the stochastic simulations, $F$ was determined directly by introducing homogeneous particle input and considering stationary states. On this double logarithmic plot, we show the rescaled data for $(F\left(\psi \right)-\mu {\rho }^2)/{\mu }^{\frac{5}{2}}$ for a range of models. We predict this term to be of the universal form ${\rho }^{\frac{5}{2}}/\left(2\sqrt{2}\pi \right)$ (solid black line), independent of the model [cf. Eq. (33)]. The data evidently corroborate our theoretical results.

Figure 7

Two-dimensional versions of the reaction kernels of extended objects (solid red). In three dimensions, for extended object 1, the kernel $\lambda \left(\mathbf{z}\right)={\tilde{\lambda }}_{\infty }$ if $\mathbf{z}\in S=\left\{\right(\pm 1,0,0),(0,\pm 1,0),(0,0,\pm 1\left)\right\}$. Otherwise it is zero. Notice that for instantaneous reactions, the striped square can be regarded as part of extended object 1, which is then a discretization of the sphere. The support $S$ [with $\lambda \left(\mathbf{z}\right)={\tilde{\lambda }}_{\infty }$ if $\mathbf{z}\in S$] of the three-dimensional reaction kernel of extended object 2 is crated by the union of the set $\left\{\right(1,1,2),(1,2,1),(2,1,1\left)\right\}$ with its mirror images in each octant. Since the flow conserves the support of the reaction kernel, their shape remains the same. For instantaneous coagulation reactions we find ${\mu }^{-1}=0.086064343192\left(3\right)$ (extended object 1) and ${\mu }^{-1}=0.036287603611\left(2\right)$ (extended object 2).