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

Published 2 Apr 2026 in cond-mat.stat-mech | (2604.02167v1)

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.

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Summary

The paper develops an exact theory for steady-state correlation functions in anisotropic mass transport models subject to center-of-mass conservation.
It reveals that enforcing full CoM conservation changes the decay scaling from 1/|x|^d to 1/|x|^(d+2), leading to hyperuniformity in driven systems.
Employing both microscopic derivations and hydrodynamic analysis, the study unifies previous heuristic observations under a rigorous theoretical framework.

Power Laws, Anisotropy, and Center-of-Mass Conservation in Mass Transport Processes

Introduction

The characterization of long-range spatial correlations in nonequilibrium steady states is fundamental to understanding driven lattice systems. The paper "Power laws, anisotropy and center-of-mass conservation in mass transport processes" (2604.02167) develops an exact theory for steady-state correlation functions in a broad class of mass transport models where microscopic dynamics are both anisotropic and subject to additional conservation laws beyond global mass. The study investigates the interplay between anisotropy—implemented via direction-dependent hopping rates—and center-of-mass (CoM) conservation enforced along specific axes, uncovering the universal features and critical differences between models with different constraint structures.

Problem Setting and Model Classes

The authors consider a general family of mass chipping models (MCMs) defined on $d$ -dimensional hypercubic lattices. At each site $(i, j)$ resides a continuous, non-negative mass variable. The primary processes are local fragmentation (chipping) and redistribution of mass via stochastic hopping, producing mass-conserving Markov jump processes.

Anisotropy is introduced via distinct hopping/chipping rates along different lattice axes ( $x$ , $y$ , etc.), while CoM conservation is imposed through paired, symmetric updates where equal masses are moved simultaneously in opposite directions, thereby preserving the CoM either along all or selected axes. Four main variants are examined:

MCM I: Multidirectional, anisotropic hopping, only mass conservation.
CoMC IA: Full CoM conservation along all directions.
CoMC IB: Partial CoM conservation (e.g., only along $x$ -axis).
MCM II: Anisotropic, unidirectional hopping.

The analysis is executed for $d=2$ , then generalized to higher dimensions.

Exact Correlation Structure: Main Findings

Structure Factor and Real-Space Decay

The paper delivers exact real-space and Fourier-space (structure factor) expressions for the steady-state two-point correlation function in each model class. These results confirm and refine earlier heuristic and numerical observations regarding algebraic decays in nonequilibrium conservative systems, with crucial refinements and theoretical unification.

Anisotropy with Mass Conservation Alone (MCM I):
- The steady-state correlation decays generically as $C(\mathbf{x}) \sim 1/|\mathbf{x}|^d$ ( $d=2$ : $1/r^2$ ), matching the quadrupolar spatial structure. Anisotropy manifests in non-universal amplitude and sign, sensitive to parameter choice.
Full CoM Conservation (CoMC IA):
- Key finding: Imposing CoM conservation along all axes qualitatively changes the scaling to $C(\mathbf{x}) \sim 1/|\mathbf{x}|^{d+2}$ (e.g., $(i, j)$ 0 in 2D), regardless of anisotropy in hopping. The structure factor vanishes as $(i, j)$ 1 for $(i, j)$ 2, producing "class I" hyperuniformity: long-wavelength density fluctuations are anomalously suppressed.
- The result is robust: CoM conservation dominates anisotropy, fundamentally altering power-law exponents.
Partial CoM Conservation (CoMC IB):
- When CoM is conserved only along a subset of directions, the system reverts to the slower $(i, j)$ 3 decorrelation associated with mass conservation and anisotropy, though with modified anisotropic amplitudes.
Unidirectional Chipping (MCM II):
- Emergence or absence of power-law correlations depends on more subtle details than hopping rate anisotropy alone, highlighting the necessary role of asymmetry in chipping fractions.

Electrostatic Analogy and Multipole Expansion

The correlation decay exponents are interpreted via the Green’s function of Poisson-like equations. Mass conservation removes monopole contributions, and the presence of reflection symmetry removes dipoles, so leading decay is quadrupolar ( $(i, j)$ 4). Full CoM conservation further suppresses the quadrupole, elevating the rank of the leading non-vanishing term—a multipole of rank-4—thus giving rise to $(i, j)$ 5 algebraic decay.

Hydrodynamic Structure and Fluctuation-Dissipation

Transport Coefficients

The authors use both fluctuating hydrodynamics and direct microscopic derivations to obtain model-dependent diffusion tensors and mobility (Onsager) tensors. Notably, for the considered class of "gradient" models, the transport coefficients do not depend on local density, enabling exact closure and computation.

Nonequilibrium Fluctuation-Dissipation Relation

Anisotropic systems with broken detailed balance generally lack equilibrium FDRs, but the authors derive a nonequilibrium FDR of the form:

$(i, j)$ 6

where $(i, j)$ 7 is the coarse-grained mobility (from space- and time-integrated current fluctuations), $(i, j)$ 8 is bulk diffusivity, and $(i, j)$ 9 is the small- $x$ 0 limit of the structure factor along direction $x$ 1. This relation governs the connection between spontaneous and induced responses, and is shown to have path dependence in higher dimensions—a hallmark of anisotropic nonequilibrium settings.

Theoretical and Practical Implications

Theoretical Insights

Classification of Power Laws: The work prescribes clear criteria under which different power law decays can emerge in stationary nonequilibrium systems, unifying the effect of both anisotropy and higher-moment conservation laws.
Generic Emergence of Hyperuniformity: The results establish that full CoM conservation—a minimal extension of simple mass conservation—enforces hyperuniformity even in the presence of significant anisotropy, extending the list of generic routes to hyperuniform states well beyond isotropic and equilibrium models.
Path Dependence and Universality: The explicit demonstration that certain fluctuation-response quantities (e.g., the structure factor) are path-dependent in reciprocal space for anisotropic systems sharpens the theoretical understanding of universality classes in nonequilibrium statistical mechanics.
Hierarchy and Solvability: For the class of models considered (with density-independent transport coefficients), the solution for two-point correlation functions closes exactly, avoiding the typical BBGKY hierarchy.

Broader Connections

Relevance to Absorbing-State Phase Transitions: The models and mechanisms are closely related to sandpile and conserved lattice-gas models, foundational in self-organized criticality and non-equilibrium critical phenomena.
Hydrodynamics and Fluctuations in Driven Systems: The electrostatic analogy and the link between microscopic conservation laws and emergent multipolar structures have analogs in the hydrodynamics of active matter, fracton fluids, and quantum many-body systems with dipole/momentum conservation.
Materials Design and Hyperuniformity: Identification of dynamical routes to hyperuniformity, particularly in driven but disordered systems, has implications for targeted materials—photonic crystals, jammed packings, and random organization protocols.

Future Directions

While the theoretical results presented are comprehensive for the class of models considered, natural extensions include:

Generalization to higher-order conserved moments (e.g., quadrupole conservation or fractonic constraints).
Investigation of the effect of including nonlinearities and density-dependent noise in the hydrodynamic description.
Nonequilibrium systems with more complex topologies or long-range interactions, where the presented mechanism could interact nontrivially with domain geometry or disorder.
Application in synthetic or living matter, where emergent hyperuniformity from local rules could impact function and robustness.

Conclusion

This work provides a complete and exact characterization of the correlation structure in anisotropic conservative mass transport models under varying conservation laws. It establishes that while anisotropy alone gives rise to long-range algebraic correlations, the enforcement of center-of-mass conservation can dramatically alter the scaling, universally producing hyperuniformity characterized by $x$ 2 decay. The results unify previously scattered observations, elucidate the role of symmetries and conservation laws, and offer a paradigmatic framework for understanding the statistical mechanics of driven, conservative systems well beyond equilibrium (2604.02167).