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

TL;DR

This paper derives exact results for density correlations in anisotropic, conserved-mass transport processes, showing how center-of-mass conservation qualitatively alters the power-law decay and hyperuniformity of correlations.

Contribution

It reveals how full or partial center-of-mass conservation changes the decay of density correlations and the resulting hyperuniformity in anisotropic mass transport models.

Findings

Full CoM conservation leads to faster decay $1/|x|^{d+2}$ of correlations.

Partial CoM conservation recovers the $1/|x|^{d}$ power-law decay.

Correlations relate to electrostatic multipolar potentials, revealing complex interplay.

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.