Phase ordering dynamics of reconstituting particles

We consider the large-time dynamics of one-dimensional processes involving adsorption and desorption of extended hard-core particles (dimers, trimers, ..., $k$-mers), while interacting through their constituent monomers. Desorption can occur whether or not these latter adsorbed together, which leads to reconstitution of $k$-mers and the appearance of sectors of motion with nonlocal conservation laws for $k\ge 3$. Dynamic exponents of the sector including the empty chain are evaluated by finite-size scaling analyses of the relaxation times embodied in the spectral gaps of evolution operators. For attractive interactions it is found that in the low-temperature limit such time scales converge to those of the Glauber dynamics, thus suggesting a diffusive universality class for $k\ge 2$. This is also tested by simulated quenches down to $T=0$, where a common scaling function emerges. By contrast, under repulsive interactions the low-temperature dynamics is characterized by metastable states which decay subdiffusively to a highly degenerate and partially jammed phase.

Figure 1

Schematic description of adsorbing-desorbing $k$-mers (within brackets), and their transition rates as defined in Eq. (6). Filled and empty circles denote spins with opposite orientations in turn forming kinks (vertical lines) on the dual chain. These former involve (a) pairing processes with rates $r_{{}_{\pm }}=\frac{1}{2}(1\pm tanh2K)$, as well as (b) hopping events with no energy changes.

Figure 2

Schematic random walk of irreducible characters for $k=4$ and $L=12$ with PBC's. The deletion procedure described in the text is here depicted by the boxes around reducible groups of spins. At each step the identification of these latter is made, say, from left to right. Larger spins denote the irreducible characters whose ordering is left invariant by the dynamics. Dots signal the locations of updated spins with rates indicated at the rightmost column [see Eq. (6) and Fig. 1].

Figure 3

Finite-size scaling of spectral gaps of the evolution operator ($J>0$) on approaching low-temperature regimes in NS sectors for (a) $k=3$ and (b) $k=4$. From top to bottom solid and dashed lines stand in turn for sizes $L\propto k$ with (a) $L=30,...,15$, and (b) $L=36,...,20$. The slopes of the insets estimate the typical finite-size decay of these gaps in the limit of $T\to 0$, thus suggesting a common dynamic diffusive exponent. For displaying convenience, in (a) and (b) the vertical scales of main panels were normalized by factors ${30}^2{\lambda }_1^{*}\left(30\right)$ and ${36}^2{\lambda }_1^{*}\left(36\right)$, respectively.

Figure 4

(a) Low-temperature scaling of spectral gaps ($J>0$) for several $k$-mer sizes. As $T\to 0$, all gaps collapse onto the exact solution of the monomer case (Glauber dynamics), provided the chain length of this latter is rescaled by a factor $1/k$. Other reachable values of $L/k$ (not shown) also follow the Glauber solution in that limit. (b) Scaling plot of spin-spin correlation functions ($J>0$) for $L=1.2\times {10}^5$ after a quench down to $T=0$. The data collapse was attained upon renormalizing $r$ distances as $r/k$ while choosing dynamic exponents $z\in (1.96,2.07)$. Squares ($t=5\times {10}^3$ steps) and circles ($t=5\times {10}^4$) depict the case $k=3$, whereas rhomboids ($t={10}^3$) and triangles ($t={10}^4$) stand for $k=10$. Error bars are smaller than twice the symbol sizes. The dashed line represents the case $k=2$ after ${10}^4$ steps, while the solid one denotes the exact scaling function of Ref. [23].

Figure 5

Schematic decay from a trimer metastable state (top) under $J<0$. After surmounting the initial energy barrier, kinks separating opposite spin orientations can diffuse freely until an interval of nine parallel spins shows up. This allows one to decrease the initial energy by creating two kink pairs (bottom). Brackets show updated locations, whereas numbers in diffusion events indicate the ordering of those updates.

Figure 6

Scaling of normalized gaps [ Eq. (19), $L\propto k$ ] for NS spaces and $J<0$. Solid and dashed lines in upward order show alternately the cases of (a) $L=15,...,30$ for $k=3$, and (b) $L=20,...,36$ for $k=4$. The data collapse in the low-temperature regime was obtained using dynamic exponents read off from the slope of the insets (also see Table 1). Just as in Fig. 3, for displaying convenience the vertical scales of main panels were respectively normalized by factors ${30}^z{\mathrm{\Lambda }}_1^{*}\left(30\right)$ and ${36}^z{\mathrm{\Lambda }}_1^{*}\left(36\right)$. The ${\mathrm{\Lambda }}_1$ plateaus are consistent with the Arrhenius regimes referred to in the text.

Figure 7

Main panels: logarithmic growth of average jamming lengths ${\xi }_{{}_J}$ with the system size ($L\propto k$) in the ground states of NS sectors with $J<0$ for (a) $k=3$ ($\alpha \simeq 1.6$), and (b) $k=4$ ($\alpha \simeq 1.7$). The distribution of these length scales in our largest accessible chains are shown in the upper histograms. Vertical arrows signal the mean values of ${\xi }_{{}_J}$. Lower insets illustrate the exponential degeneracy of the corresponding ground states. In turn, the slopes indicate a residual entropy per site $S_0/L\sim 0.41$ for $k=3$, and $\sim 0.43$ for $k=4$.

Figure 8

Spin-pair correlations in the ground states of NS sectors $(J<0)$ obtained by the sampling procedure described in the text for (a) $k=3$ and (b) $k=4$. Squares, circles, and triangles stand respectively for $L=600,900,1200$ in (a), and $L=1800,2400,3000$ in (b). Error bars are smaller than twice the symbol sizes. The insets illustrate the cases of largest $L$'s, whereas for displaying clarity only local extrema are shown in the main panels. The inverse slopes of solid lines are fitted with correlation lengths $\xi$ which grow with the chain size.