Coarsening, condensates, and extremes in aggregation-fragmentation models

We use extreme value statistics to study the dynamics of coarsening in aggregation-fragmentation models which form condensates in the steady state. The dynamics is dominated by the formation of local condensates on a coarsening length scale which grows in time in both the zero range process and conserved mass aggregation model. The local condensate mass distribution exhibits scaling, which implies anomalously large fluctuations, with mean and standard deviation both proportional to the coarsening length. Remarkably, the state of the system during coarsening is governed not by the steady state, but rather a preasymptotic state in which the condensate mass fluctuates strongly.

Figure 1

Illustrating ZRP and CMAM moves in the left and right panels, respectively. The mapping to exclusion models is depicted in the lower part of each panel.

Figure 2

Scaled distribution of the condensate mass in CMAM in steady state. $M_0\left(L\right)=126,247,488$ for $L=200,400,800$, respectively. Inset: Unscaled distributions.

Figure 3

During coarsening, $G(r,t)$ is a function of the scaled separation $r/\mathcal{L}\left(t\right)$ with $\mathcal{L}\left(t\right)=t^{1/2}$. Top panel: asymmetric ZRP ($b=3.5,{\rho }_c=2/3,\rho =2$), with $\mathcal{L}\left(t\right)=256,400,613,800$. Bottom panel: CMAM ($w=1,D=1,{\rho }_c=\sqrt{2}-1,\rho =1$) with $\mathcal{L}\left(t\right)=120,200,300,500$. Insets: Unscaled plots. The system size $L=10000$ for both models.

Figure 4

The probability distribution of the maxima in boxes of size $\mathcal{L}\left(t\right),P(m^{*},t)$ in the two models. Top panel: Asymmetric ZRP; $L=20000$, $\mathcal{L}\left(t\right)=400,800,1200,1500,2000$. Bottom panel: CMAM; $L=20000,\mathcal{L}\left(t\right)=200,600,1200,1600,2000$.

Figure 5

Scaled distribution of the largest mass in domains of size $\mathcal{L}\left(t\right)$. Top panel: Asymmetric ZRP; Bottom panel: CMAM. Model parameter values as in Fig. 3. $\mathcal{L}\left(t\right)=200,600,1200,1500,2000$ and system size $L=20000$. Gray Crossover region, $(m^{*}\approx m_{\times }^{*})$. The region $(m^{*}>m_{\times }^{*})$ describes the condensate. For $(m^{*}<m_{\times }^{*})$ the plots describe the coarsening critical fluid. Insets: Scaled plots for $(m^{*}<m_{\times }^{*})$; the dashed line is a Fréchet distribution.

Figure 6

The mass distribution $\mathcal{P}(m,t)$ for the $\mathrm{ZRP}$ (top) and the CMAM (bottom) for various $\mathcal{L}\left(t\right)$. For $\mathrm{ZRP}$, the system size $L=10000$, and for CMAM, $L=9600$.

Figure 7

Distribution of the local maximum mass $P(m^{*},t)$ for $\mathrm{ZRP}$ with $\mathcal{L}\left(t\right)=800,1200,2000$, and $L=20000$. This is consistent with the additive scaling form in Eq. (5) (shown with dashed line).

Figure 8

Variance of the global maximum mass with different system sizes $L=300,500,700,900$. Top panel: ZRP. Bottom panel: CMAM. Model parameter values as in Fig. 3. Insets show that the mean increases smoothly to its steady-state value $\sim O\left(L\right)$. The variance increases rapidly and overshoots to a value $\sim O\left(L^2\right)$, finally settling to the much smaller steady-state value $\sim O\left(L^{2\beta }\right)$, with $\beta =1/2$ for ZRP and $\simeq 0.75$ for CMAM. Note the discrepancy between fluctuations during coarsening and those in steady state.

Figure 9

The time evolution of the variance of the global maximum mass in ZRP and CMAM shows distinct regimes: the coarsening regime (I) and the relaxation regime (II).

Figure 10

The scaling analysis of the variance of the global maximum mass for different system sizes. The top panels are for ZRP $(L=440,512,700,900)$ and the bottom panels are for CMAM $(L=500,700,900,1200)$. The coarsening and relaxation regimes satisfy Eqs. (8) and (9).

Figure 11

Variance of the maximum mass in 2D asymmetric $\mathrm{ZRP}$ (top panel) and the CMAM (bottom panel) as a function of time. The data is for system sizes $A=L\times L=20\times 20,30\times 30,35\times 35,40\times 40$ for $\mathrm{ZRP}$ and $A=20\times 20,50\times 50,70\times 70$ for CMAM. The dashed line indicates growth of the variance as $\mathcal{A}{\left(t\right)}^2$.

Figure 12

In the exclusion models, $\mathrm{\Gamma }(r,t)$ is a function of the scaled variable $r/\mathcal{L}\left(t\right),\mathcal{L}\left(t\right)=t^{1/2}$. The model parameters are the same as in Fig. 2 in the main text. Top panel: EM-ZRP, with $\mathcal{L}\left(t\right)=245,300,346,424,500$. Bottom panel: EM-CMAM, with $\mathcal{L}\left(t\right)=200,300,400,500,600$. $\mathrm{\Gamma }(r,t)$ initially falls linearly, parallel to the dashed lines, indicating the validity of Porod's law. Insets: Unscaled plots.

Figure 13

Mean and variance of the total mass within domains of size $\mathcal{L}\left(t\right)$ in the coarsening regime, for $\mathrm{ZRP}$ and CMAM.

Figure 14

The scaling analysis of the average global maximum mass for different system sizes. The top panel for the ZRP and the bottom panel for the CMAM. The coarsening regime for CMAM shows the lack of scaling and the deviation from the $\sqrt{t}$ behavior is shown by the dashed lines.

Figure 15

Distribution and scaling of the maxima in a domain of size $\mathcal{L}\left(t\right)$ for the symmetric zero range process in the coarsening regime. The inset shows an unscaled plot.