Diffusion with conserved marginal distributions and information theory in fracton hydrodynamics

TL;DR

This paper explores how subsystem symmetries in fracton hydrodynamics lead to nonlinear diffusion equations that preserve initial marginal distributions, with implications for information decay and maximum-entropy states.

Contribution

It demonstrates that subsystem symmetries produce nonlinear hydrodynamics with shear-only transport and derives the associated equations and equilibrium distributions.

Findings

Subsystem symmetries lead to nonlinear hydrodynamic equations.

Marginal distributions are preserved at long times.

Maximum-entropy distributions are derived under marginal constraints.

Abstract

Diffusion with multipole-moment conservation gives rise to transport laws that generalize Fick's law and has attracted growing attention following experimental advances in strongly tilted optical lattices. It was recently shown that conserving complete multipole-moment groups leads to subdiffusive dynamics governed by a nonlinear diffusion equation, raising the question of whether hydrodynamic equations would also be nonlinear when the conservation law is imposed only at the subsystem level. Here we show that subsystem symmetries generically produce nonlinear hydrodynamic equations with shear-only transport, in which any localization present in the initial marginal distributions is preserved at long times by the conservation of those marginals. A linear regime emerges only as a limiting case for small fluctuations around a uniform background. We derive the deterministic and fluctuating parts of the hydrodynamic equations in arbitrary dimensions and obtain the corresponding maximum-entropy equilibrium distributions under constrained marginals. We also show that marginal-conserving diffusion provides a concrete hydrodynamic realization of partial multipole-moment conservation, and we offer an information-theoretic interpretation in which total correlation decays monotonically even when pairwise mutual information does not.