Validity of the Law of Mass Action in Three-Dimensional Coagulation Processes

Diffusion-limited reactions are studied in detail on the classical coalescing process. We demonstrate how, with the aid of a recent renormalization group approach, fluctuations can be integrated systematically. We thereby obtain an exact relation between the microscopic physics (lattice structure and particle shape and size) and the macroscopic decay rate in the law of mass action. Moreover, we find a strong violation of the law of mass action. The corresponding term in the kinetic equations originates in long-wavelength fluctuations and is a universal function of the macroscopic decay rate.

Figure 1

Important one-loop Feynman diagrams. Because of causality, the propagator only connects earlier $\overline{\psi }$ (“creation”) to later $\psi$ (“annihilation”). Since for coagulation processes the number of legs can only decrease as time passes, the set of possible diagrams is substantially restricted. Indeed, there is no such diagram as in (a) and, therefore, no renormalization of the dispersion $ϵ$ and of the field amplitude $Z$. The diagrams depicted in (b) and (c) both stand for the renormalization of the reaction kernel ${\lambda }_k$. Finally, diagram (d) exemplifies the divergence of the RG flow contribution of one-loop diagrams which contain only $(2,1)$ and $(2,2)$ vertices: Each of the three propagators gives rise to a factor $\sim 1/k^2$. For small $k$, the volume of reciprocal space and time that is integrated is $\sim k^5$. Thus, $g_k^{(1,3)}\sim k^{-1}$.

Figure 2

Relaxation of the density for one-site objects. On double logarithmic plots, data of stochastic simulations (solid red line) are compared to the theoretical prediction for the density decay $\rho (t)\sim {\mu }^{-1}{t}^{-1}$ (dashed line), with the macroscopic decay rate ${\mu }^{-1}={\lambda }^{-1}+0.253\dots$; cf. Eq. (5). The initial states in both plots were randomly distributed with $\rho (0)=0.2$.

Figure 3

Numerical solution to the flow equation, Eq. (4), for spheroids of volume $\frac{4}{3}\pi$ as a function of their eccentricity. We observe that the macroscopic decay rate $\mu$ increases with the eccentricity and is larger for prolates. In the images, as yellow (light gray) turns to orange and red (dark gray), the $\mathbf{x}$-dependent macroscopic rate ${\lambda }_0(\mathbf{x})$ grows. As a rule of thumb, the more jagged the reaction kernel, the higher $\mu$ becomes.

Figure 4

Rescaled data for the deviation from the LMA. We ran simulations for a range of models and compared the results with the predicted universal correction ${\rho }^{5/2}/(2\sqrt{2}\pi )$ (solid black line); cf. Eq. (6). The nonequilibrium force $F$ can be determined directly by introducing homogeneous particle input with rate $J$. This gives rise to an additional term $J$ in the kinetic equation, such that for the stationary state $F(\rho )=J$. Extended object 1 is a discretization of the sphere, made up of seven sites. Together with extended object 2, it is discussed in 23, where also the data for extended object 2 are provided (not shown here for clarity but in excellent agreement with the theory as well).