Dynamics of condensate formation in stochastic transport with pair-factorized steady states: Nucleation and coarsening time scales

Driven diffusive systems such as the zero-range process (ZRP) and the pair-factorized steady states (PFSS) stochastic transport process are versatile tools that lend themselves to the study of transport phenomena on a generic level. While their mathematical structure is simple enough to allow significant analytical treatment, they offer a variety of interesting phenomena. With appropriate dynamics, the ZRP and PFSS models feature a condensation transition where, for a supercritical density, the translational symmetry breaks spontaneously and excess particles form a single-site or spatially extended condensate, respectively. In this paper we numerically study the typical time scales of the two stages of this condensation process: Nucleation and coarsening. Nucleation is the first stage of condensation where the bulk system relaxes to its stationary distribution and droplet nuclei form in the system. These droplets then gradually grow or evaporate in the coarsening regime to coalesce in a single condensate when the system finally relaxes to the stationary state. We use the ZRP condensation model to discuss the choice of the estimation method for the nucleation time scale and present scaling exponents for the ZRP and PFSS condensation models with respect to the choice of the typical droplet nuclei mass. We then proceed to present scaling exponents in the coarsening regime of the ZRP for partially asymmetric dynamics and the PFSS model for symmetric and asymmetric dynamics.

Figure 1

Schematics of the discussed models. From a randomly selected site $i$ a particle is taken with probability $u_{\text{ZRP}}=u\left(m_i\right)/u_{\text{max,ZRP}}$ or $u_{\mathrm{PFSS}}=u\left(m_i\right|m_{i-1},m_{i+1})/u_{\max ,\mathrm{PFSS}}$ and then moved to sites $i-1$ or $i+1$ with probability $p_{\text{left}}$ or $1-p_{\text{left}}$, respectively.

Figure 2

The nucleation (bottom right) and coarsening (top left) processes for various system sizes in the ZRP as observed in the average mass of the largest droplet at time $t$. The two groups of curves show the same data. The upper group, plotted versus the top and left axes, is merely rescaled in time by $1/M^2$ and in mass by $1/M$ in order to collapse the trajectories of different system sizes to a single master curve for late times, as shown in the upper inset. The lower group, plotted versus the lower and right axes, is shown without rescaling. The lower inset shows the weak dependence of droplet growth on system size in the early condensation dynamics. The curves were determined for totally asymmetric dynamics $p_{\text{left}}=1$ using 1000 trajectories per size $L=M$.

Figure 3

Sample time series of a PFSS condensation process in a system of $L=500$ sites and $M=500$ particles for totally asymmetric dynamics ($p_{\text{left}}=1$, “particles cyclically move down”) and interaction parameters $U=J=1$. Each vertical slice in the plot corresponds to the occupation number vector at time $t$. The time axis is logarithmic to show the different stages of the condensation process. Starting with a disordered state in the leftmost slice, small droplets emerge and gradually coarsen on every time scale until a single condensate remains at about $t\approx {10}^7$.

Figure 4

Droplet counts $n_{\text{c}}\left(t\right)$ for the ZRP with $b=5$ in the nucleation regime for thresholds in the droplet mass that are constant $[m_{\text{t,const}}=10$ in (a) and (d)], grow as the square root $[m_{\text{t,sqrt}}=\sqrt{(\rho -{\rho }_{\text{c}})L}$ in (b) and (e)] of, or linearly $[m_{\text{t,lin}}=(1/40)(\rho -{\rho }_{\text{c}})L$ in (c) and (f)] to the total number of particles. Results for totally asymmetric dynamics ($p_{\text{left}}=1$) are shown with empty symbols in the top row, and those for symmetric dynamics ($p_{\text{left}}=1/2$) with filled symbols in the bottom row. Symbols represent the different system sizes $L=320,640,1280$, and 2560; colors indicate different particle densities $\rho =1,2,4,8$, and 12. For readability, symbols are only sparsely plotted and mark the corresponding lines that fully reflect the available data. The resulting time scales on the $x$ axis roughly correspond to the heuristic approximation.

Figure 5

Droplet counts $n_{\text{c}}\left(t\right)$ for the PFSS with $U=J=1$ in the nucleation regime for thresholds in the droplet mass that are constant $[m_{\text{t,const}}=20$ in (a) and (d)], grow as the square root $[m_{\text{t,sqrt}}=(5/2)\sqrt{(\rho -{\rho }_{\text{c}})L}$ in (b) and (e)] of, or linearly $[m_{\text{t,lin}}=(1/20)(\rho -{\rho }_{\text{c}})L$ in (c) and (f)] to the total number of particles. Results for totally asymmetric dynamics ($p_{\text{left}}=1$) are shown with empty symbols in the top row, and those for symmetric dynamics ($p_{\text{left}}=1/2$) with filled symbols in the bottom row. Symbols represent the different system sizes $L=400,800,1600$, and 3200; colors indicate different particle densities $\rho =1,2$, and 4.

Figure 6

Estimation of the coarsening time-scale exponent ${\delta }_{\text{fps}}$ for the ZRP using the first-passage method with scaling ansatz (6).

Figure 7

ZRP coarsening: Overview of rescaled largest condensate mass trajectories fitted to collapse to master curves in the final coarsening dynamics. Groups of curves with identical strength of asymmetry are shifted on the vertical axis for readability, as indicated by the right-hand side labels for the strength of asymmetric dynamics $p_{\text{left}}$. Curves (blue to green) correspond to time series of smaller (from $L=100$) to larger (towards $L=1000$ for symmetric and partially asymmetric and 8000 for totally asymmetric dynamics) systems with overall density $\rho =1$. Each curve is computed from ${10}^4$ trajectories.

Figure 8

Estimation of the scaling exponent ${\delta }_{\text{fps}}$ in the coarsening regime of the PFSS by fitting the first-passage time ${\tau }_{\text{fps}}$ to the assumed power law (6). The obtained values are collected in Table 3.

Figure 9

PFSS coarsening: Overview of rescaled largest condensate mass trajectories fitted to collapse to master curves in the final coarsening dynamics. Groups of curves with identical values of $p_{\text{left}}$ are shifted vertically as indicated by the right-hand side labels. Curves (blue to green) correspond to time series of smaller (from $L=100$) to larger (towards $L=1000$ for symmetric and partially asymmetric and 4000 for totally asymmetric dynamics) systems. The particle density is $\rho =1$ for all systems. Each curve is computed from ${10}^4$ trajectories.

Figure 10

Scaling exponents $\delta$ of the coarsening times versus strength of asymmetric dynamics $p_{\text{left}}$ for both ZRP and PFSS transport processes.