Power laws, anisotropy and center-of-mass conservation in mass transport processes

Abstract

We present exact results for steady-state density correlation functions in conserved-mass transport processes with {\it anisotropic}, reflection-symmetric hopping on a $d-$dimensional hypercubic lattice. In addition to mass conservation, we consider center-of-mass (CoM) conservation, imposed either along a specific axis or along all axes. CoM-conserving dynamics is implemented through coordinated {\it multidirectional} hopping of two equal chunks of masses in {\it opposite} directions. While anisotropy and mass conservation are known to generate power-law density correlations $C({\bf x}) \sim 1/|{\bf x}|^d$ at large distance $|{\bf x}| \gg 1$ {\it [Phys. Rev. A {\bf 42}, 1954 (1990)]}, an additional CoM conservation can qualitatively alter the nature of the power law. Indeed, when CoM is conserved in {\it all} directions, the correlations decay faster $-$ typically as $C({\bf x}) \sim 1/|{\bf x}|^{(d+2)}$, regardless of the presence (or absence) of anisotropy. Consequently, the systems exhibit an extreme {\it hyperuniformity} (``class I''), where the long-wavelength density fluctuations, despite the slow power-law decay, are anomalously suppressed. When CoM is conserved along particular ({\it not} all) directions, the slower $1/|{\bf x}|^{d}$ power-law decay is recovered. The above behavior can be understood from an analogy between the correlation function and an electrostatic potential: While a (rank-$2$) quadrupolar charge distribution gives rise to the $1/|{\bf x}|^{d}$ power law, the $1/|{\bf x}|^{(d+2)}$ power law originates from a higher-order (rank-$4$) multipolar charge distribution. These findings reveal a rich interplay between anisotropy and CoM conservation in nonequilibrium steady states.

Figures (5)

• Figure 1: Schematic representation of the four variants of anisotropic mass chipping models (MCMs). With unit rate, a fraction of mass $m_{i,j}$ is chipped off from a site $(i, j)$, fragmented into four chunks, and transferred with each of the fragments to one of its four nearest neighbors. The anisotropy arises from different chipping (or retention) parameters $\zeta_1$ and $\zeta_2$ along the $x$ and $y$ directions, respectively. Depending on the amount of the fragmented masses, we can define three versions of MCMs. (a) MCM I: Random unequal fractions of mass, $\xi_1 \tilde{\zeta}_1m_{i, j}/2$ and $\tilde{\xi}_1 \tilde{\zeta}_1m_{i, j}/2$, are transferred to neighbors $(i+1, j)$ and $(i-1, j)$ respectively, along the $x$-axis; a similarly fragmented mass is distributed along the $y$-axis. (b) CoMC IA: We implement a center-of-mass-conserving move in which equal amount of mass, $\xi_1\tilde{\zeta}_1m_{i, j}/4$, is transferred to both neighbors along the $x$-axis, and $\xi_2\tilde{\zeta}_2m_{i,j}/4$ is transferred to both neighbors along the $y$-axis. (c) CoMC IB: We implement a center-of-mass conserving move along the $x$-axis ($\xi_3\tilde{\zeta}_1m_{i, j}/4$ to each), but transfer of unequal chunks of mass (distributed symmetrically) along the $y$-axis. Here, $\xi_{1},\xi_{2}\in [0,1]$ are independent random variables drawn uniformly from the unit interval. (d) MCM II: A single chunk of mass is transferred either in the horizontal direction (to the left) with rate $p_{x}$ or in the vertical direction (to the right) with rate $p_{y}$. In each allowed move, the mass chips to one of its two nearest neighbors with equal probability ($1/2$). In all panels, arrows of the same color indicate equal amounts of mass being transferred, illustrating conservation of the center of mass under the specific chipping rules.
• Figure 2: Structure factor and real-space density correlation functions for several variants of mass chipping models (MCMs). Panels (a)–(c) show the structure factor $S(\mathbf{q})$ in the first Brillouin zone for (a) MCM-I (without center-of-mass conservation; see Eq. \ref{['eq:sqI']}), (b) CoMC-IA (center-of-mass conservation along both $x-$ and $y-$ directions; see Eq. \ref{['sIA']}), and (c) CoMC-IB (center-of-mass conservation along $x-$direction only; see Eq. \ref{['sB']}). Panels (d)–(f) display the corresponding real-space heat maps of the density–density correlation function in steady state. We take global density $\rho=4$ and chipping parameters $\zeta_1=0.2$ and $\zeta_2=0.4$ in all panels.
• Figure 3: Static density correlations in model variant MCM I: (a) Spatial dependence of the correlation function $C^{mm}(\mathbf{r})$ along the principal axes. Note that the correlations exhibit sign anisotropy: they are negative along the $x$-direction (blue circles) and positive along the $y$-direction (red squares). (b) the density correlations along the $x$-axis, plotted as $-C^{mm}(x, 0)$. Similarly, density correlation $C^{mm}(0, y)$ is plotted along the $y$-axis, $C^{mm}(0, y)$ in panel (c). In all panels, symbols represent simulation results for a system size of $300 \times 300$, while colored dashed lines correspond to the exact theoretical prediction from Eq. \ref{['eq:cor_mcmI']}. The black dashed lines in (b) and (c) indicate the asymptotic power-law decay predicted by Eqs. \ref{['eq:mcm_x_assymp']} and \ref{['eq:mcm_y_assymp']}, respectively, showing excellent agreement with the data. Parameters used: $D_{xx} = 0.2$, $D_{yy} = 0.15$ and global density $\rho=4$.
• Figure 4: CoM-conserving models $-$ CoMC IA and CoMC IB: Density–density correlations along the $x$- and $y$-directions for two different models are shown. Panels (a) and (b) correspond to the model CoMC IA (center-of-mass conservation along both spatial directions), where the correlations along the $x$- and $y$-directions are plotted, respectively. The black dashed lines represent the guiding power laws $1/x^{4}$ and $1/y^{4}$, obtained from Eqs. \ref{['eq:MCM1A_cx']} and \ref{['eq:MCM1A_cy']}, respectively. Panels (c) and (d) show the density correlations along the $x$- and $y$-directions, respectively, for the model CoMC IB (center-of-mass conservation only along the $x$-direction). In this case, the black dashed lines correspond to the analytical asymptotic expressions, which scale as $\sim 1/x^{2}$ and $\sim -1/y^{2}$, obtained from Eqs. \ref{['eq:cx_cOmc1b']} and \ref{['eq:cy_cOmc1b']}, respectively. In all panels, the analytically obtained asymptotic forms agree well with the numerical simulation data (colored hollow circles and squares). The simulations were performed on a system of size $300 \times 300$ at global density $\rho = 4$, with parameters $\zeta_1 = 0.2$ and $\zeta_2 = 0.4$ for all panels.
• Figure 5: The macroscopic (hydrodynamic) mobility is plotted as a function of density for (a) MCM I and (b) CoMC IB. The symbols represent simulations data, while the dashed lines (in the corresponding colors) denote the analytical predictions for the respective models, as given in Eqs. \ref{['eq:FDR-mobility-alpha']}--\ref{['eq:FDR-mobility-beta']} and Table \ref{['tab:mobility_tensor']}. The simulations are performed on a system of size $L = 150 \times 150$ with total simulation time $T = 400$. In all panels, the chipping parameters were fixed at $\zeta_1 = 0.2$ and $\zeta_2 = 0.4$.