Finite Size Effects in Addition and Chipping Processes

TL;DR

The study analyzes finite-size effects in addition and chipping (AC) processes, where monomer-cluster collisions yield either monomer addition or monomer chipping with probability $p$ and $1-p$. In infinite systems, three regimes emerge: a jammed phase for $p>1/2$, a critical regime at $p=1/2$ with algebraic decay and a non-Gaussian jammed state, and a quasi-stationary steady state for $p<1/2$. For finite systems, jammed states are inevitable; lifetimes scale as $T\sim \ln N$ in the jammed regime, linearly with $N$ at criticality, and exponentially $T\sim e^{A(p)N}$ in the quasi-stationary regime, with intricate fluctuations and sub-extensive final island structures (supercluster states) in the critical case. The paper also extends to AC processes with rates proportional to cluster mass, revealing a rich phase diagram with a critical line $\lambda_c=1$ and scaling laws $T\sim N^{(5-2a)/(5-a)}$, $k_{typ}\sim N^{1/(5-a)}$ for $a<2$. Overall, finite-size effects induce giant fluctuations and nontrivial jammed states, offering a comprehensive scaling picture across regimes and rate choices.

Abstract

We investigate analytically and numerically a system of clusters evolving via collisions with clusters of minimal mass (monomers). Each collision either leads to the addition of the monomer to the cluster or the chipping of a monomer from the cluster, and emerging behaviors depend on which of the two processes is more probable. If addition prevails, monomers disappear in a time that scales as $\ln N$ with the total mass $N\gg 1$, and the system reaches a jammed state. When chipping prevails, the system remains in a quasi-stationary state for a time that scales exponentially with $N$, but eventually, a giant fluctuation leads to the disappearance of monomers. In the marginal case, monomers disappear in a time that scales linearly with $N$, and the final supercluster state is a peculiar jammed state, viz., it is not extensive.

Figures (8)

• Figure 1: The average lifetime $T$ versus $N$ for various values of $p$. Fits with Eq. \ref{['lifetime']} are present with dots. Theoretical predictions \ref{['lifetime']} qualitatively agree with simulation results when $p\leq 1/2$. When $p > 1/2$, a non-linear logarithmic growth $T \propto (\ln N)^{\alpha(p)}$ with exponent satisfying $\alpha(p)>1$ and varying with $p$ better fits the data. For each $N$, we used $10^{3}$ Monte Carlo runs to estimate the average lifetime.
• Figure 2: The temporal decay of the monomer density $c_1(\tau)$ for $p=7/20, ~p=1/2, ~p=3/4$ (top to bottom) exemplifying the evolution in the steady state, critical, and jamming regimes. In the critical regime, the monomer density is given by explicit formula \ref{['c1:RW-sol']}. Generally the exact Laplace transform $\widehat{c}_1(s)$ is known, Eq. \ref{['Lap:c1c']}, and numerically inverting it we obtained $c_1(\tau)$ for $p=7/20$ and $p=3/4$. In the jamming regime, the monomer density vanishes at finite modified time $\tau_\text{max}(p)$ corresponding to $t=\infty$; e.g., $\tau_\text{max}(3/4)\approx 1.75703$. The inset shows that the final cluster density $C(p)$ and the final monomer density $C_1(p)$ undergo the continuous phase transition at $p_c=1/2$. The final monomer density, $C_1(p)$, is given by \ref{['final:mon']}. The final cluster density is known in the steady state regime, Eq. \ref{['final:density']}. In the jamming regime, the exact Laplace transform $\widehat{c}(s)$ is also known, Eq. \ref{['Lap:c1c']}. Instead of numerically inverting the Laplace transform and also finding $\tau_\text{max}(p)$ it is easier to determine $C(p)$ using direct integration.
• Figure 3: For each $N$, the average lifetime is obtained by averaging $10^{3}$ Monte Carlo simulations. An exponential growth is observed, and the amplitude $A(p)$ appearing in \ref{['T:SS']} is extracted from the numerical data. The inset shows this amplitude together with an uncontrolled approximation \ref{['A-bar']} for the amplitude. An approximation gives a qualitatively correct dependence of the amplitude $A(p)$ on $p$.
• Figure 4: The average number of different island species obtained by averaging $10^3$ Monte Carlo runs for each value of $N$. When $p=1/2$, simulation results are in good agreement with the theoretical prediction \ref{['D:asymp']}. For $p < 1/2$, we could not verify the theory (see the inset): The system size seems insufficient for reaching the $\mathcal{D} \simeq D_{-}(p)\, \ln N$ asymptotic.
• Figure 5: The average number of different island species obtained by averaging $10^3$ Monte Carlo runs for each value of $N$ in the jammed state for several values of $p>1/2$. Numerical results agree with \ref{['D:asymp']} and demonstrate the dependence of $D_{+}(p)$ on $p$. The number of island species grows when $p$ decreases.