Diffusion with conserved marginal distributions and information theory in fracton hydrodynamics

This paper demonstrates that subsystem symmetries in fracton hydrodynamics generically lead to nonlinear diffusion equations with shear-only transport, where conserved marginal distributions preserve initial localization and provide an information-theoretic framework in which total correlation decays monotonically despite non-monotonic pairwise mutual information.

The Gist

Imagine a crowded dance floor where people (particles) are trying to move around. In a normal crowd, people wander randomly, bumping into each other, and eventually spread out evenly across the room. This is standard diffusion, like a drop of ink spreading in water.

But this paper explores a very specific, unusual type of dance floor with strict rules. Here, the dancers can't just move anywhere; they are bound by a set of "subsystem symmetries."

The "Shear" Dance Move

The authors introduce a microscopic model (a set of tiny rules for how particles move) that acts like a shear motion.

Imagine a square table with four corners. In this dance, two people standing at opposite corners (say, top-left and bottom-right) can swap places with the two empty corners (top-right and bottom-left). They don't move individually; they move as a coordinated pair.

The Magic Rule: Because of this specific swap, something strange happens:

• If you look at just the rows of the table, the number of people in each row never changes.
• If you look at just the columns, the number of people in each column never changes.
• However, the total arrangement of people across the whole table does change.

This is like having a grid of lights where the total brightness of every horizontal line and every vertical line stays fixed, but the individual lights can flicker and swap as long as those line totals remain constant.

The "Frozen" Margins

The paper calls these unchanging row and column totals "marginal distributions."

Think of it like a shadow. If you shine a light from the side, the shadow of the crowd on the wall (the row totals) never changes shape, even though the people inside the room are dancing wildly. The paper shows that because these "shadows" are frozen, the particles get stuck in a way that prevents them from spreading out normally.

Instead of spreading out smoothly (like ink in water), the particles spread slowly and non-linearly. The authors found that the math describing this isn't a simple straight line; it's a complex, curved equation. The particles tend to get "localized" or stuck in clumps, preserving the initial shape of their shadows forever.

The "Information" Puzzle

The paper also looks at this through the lens of information theory (how much we know about the system).

• Total Correlation: Imagine you have a 3D cube of dancers. The paper shows that the "total messiness" or connection between all three dimensions (X, Y, and Z) steadily decreases as they dance. They are slowly becoming independent of each other.
• The Twist: However, if you only look at two dimensions at a time (say, just X and Y), their connection doesn't always get simpler. Sometimes, as the system tries to settle down, the connection between just X and Y might actually get stronger for a while before it finally fades away.

It's like two people in a crowded room who seem to be ignoring each other, then suddenly start dancing in sync for a moment, before finally going their separate ways. The paper proves that while the whole group is slowly losing its complex connections, pairs of people can have weird, temporary spikes in their connection.

The "Equilibrium" State

Eventually, the system settles down. The paper calculates what the final state looks like. Because the row and column totals are frozen, the final arrangement is simply the product of the initial rows and columns.

Imagine you have a photo of a crowd. If you take the "shadow" of the crowd from the side and the "shadow" from the front, and you multiply those two shadows together mathematically, you get the exact picture of where everyone ends up after they stop dancing. The complex 2D or 3D pattern collapses into a simple combination of 1D lines.

Summary

In short, this paper describes a new kind of "traffic jam" in physics where particles are forced to move in coordinated pairs. This creates a system where:

1. Spread is slow and weird: It doesn't follow the standard rules of diffusion.
2. Shadows stay fixed: The total count in every row and column is preserved forever.
3. Information behaves oddly: While the whole system slowly becomes "uncorrelated," small pairs of variables can temporarily become more connected before settling down.

The authors provide the exact mathematical formulas (hydrodynamic equations) to predict how this strange, slow-motion dance evolves over time, showing that it is a non-linear, complex process that only looks simple when the crowd is very uniform to begin with.

Technical Summary

1. Problem Statement

The paper addresses the hydrodynamic description of diffusion in systems governed by subsystem symmetries, specifically where marginal distributions (integrals of the density over subsets of coordinates) are conserved.

• Context: Standard diffusion follows Fick's law ($\partial_t \rho = \nabla \cdot (D \nabla \rho)$). However, recent experiments in tilted optical lattices and theoretical work on fractons have revealed subdiffusive dynamics where higher-order multipole moments (e.g., dipole, quadrupole) are conserved.
• The Gap: Previous work (e.g., Ref. [8]) established that conserving complete groups of multipole moments leads to nonlinear hydrodynamic equations. However, the hydrodynamic behavior of partial multipole conservation (e.g., conserving $x^2$ and $y^2$ but not $xy$) or conservation restricted to subsystems (lines/planes) remained an open question. Specifically, it was unclear if the resulting hydrodynamic equations would be linear or nonlinear and how they relate to information theory.

2. Methodology

The authors employ a multi-faceted approach combining microscopic modeling, continuum field theory, and information theory:

• Microscopic Modeling: They construct a lattice model in $d$ dimensions involving shear-only transport.
  • In 2D, particles on opposite corners of a $2\times2$ square hop to the other two corners. This move conserves the number of particles in every row and column (1D marginals) but allows exchange between them.
  • This is generalized to $d$ dimensions with $k$-dimensional subspaces, where $2k$ particles undergo a parity-flip move within a hypercube.
• Hydrodynamic Derivation: Using a master equation and a gradient expansion (leading order in lattice constant), they derive the continuum limit. They distinguish between the deterministic part and the fluctuating part (thermal noise) consistent with the Fluctuation-Dissipation Theorem.
• Maximum Entropy Principle: They derive the equilibrium distribution by maximizing Shannon-Jaynes entropy subject to the constraints of conserved marginal distributions.
• Information-Theoretic Analysis: They analyze the time evolution of Shannon entropy, Kullback-Leibler (KL) divergence, mutual information, and total correlation to characterize the approach to equilibrium.

3. Key Contributions and Results

A. Nonlinear Hydrodynamic Equations

The central finding is that subsystem symmetries (conservation of marginals) generically lead to nonlinear hydrodynamic equations, contrasting with the linear equations often assumed in previous literature.

• 2D Case (Conserved 1D Marginals): The density $\rho(x,y,t)$ evolves according to:
  $$\partial_t \rho = -D \partial_x \partial_y \left[ \rho^2 \partial_x \partial_y \log \rho \right]$$
  This equation features shear-only transport, meaning the diagonal components of the current tensor ($J_{xx}, J_{yy}$) vanish, and only off-diagonal currents ($J_{xy}$) exist.
• General $d$-Dimensional Case: For a system conserving $(k-1)$-dimensional marginals (where $k$ is the dimension of the subspace involved in the shear move), the equation is:
  $$\partial_t \rho = -\sum_{|S|=k} \left( \prod_{s \in S} \partial_s \right) J_S$$
  $$J_S = (-1)^k D \rho^{2k-1} \left( \prod_{s \in S} \partial_s \right) \log \rho$$
• Linear Limit: The previously known linear equation (e.g., $\partial_t \rho \propto \partial_x^2 \partial_y^2 \rho$) is recovered only as a limiting case for small fluctuations around a uniform background density.

B. Equilibrium Distribution

The system relaxes to a Maximum Entropy state constrained by the initial marginal distributions.

• 2D Result: The equilibrium distribution factorizes into the product of the initial marginals: $\rho_{eq}(x,y) = \rho(x)\rho(y)$.
• General Result: The equilibrium distribution is a product of exponentials of functions depending on $(k-1)$-dimensional subsets of coordinates:
  $$\rho_{eq}(r) = \frac{1}{Z} \exp\left[ \sum_{|S|=k-1} \phi_S(r_S) \right]$$
  This generalizes Jaynes' principle to subsystem constraints.

C. Resolution of Partial Multipole Conservation

The paper resolves the open question regarding partial multipole conservation (e.g., conserving $x^2, y^2$ but not $xy$).

• Finding: Diffusion with conserved $(k-1)$-dimensional marginals is the hydrodynamic realization of partial multipole conservation.
• Dominant Dynamics: The authors demonstrate that the order of the spatial derivatives in the hydrodynamic equation is determined solely by the highest order of the multipole moment group that is completely conserved. Partially conserved moments (higher-order mixed terms) do not affect the leading-order subdiffusive scaling.

D. Information-Theoretic Interpretation

The authors establish a rigorous link between fracton hydrodynamics and information theory:

• Monotonicity of Total Correlation: For systems with $(d-1)$-dimensional subsystem symmetries (conserving all 1D marginals), the Total Correlation (multivariate mutual information) between all spatial coordinates decays monotonically to zero. This serves as a Lyapunov functional for the dynamics.
• Non-Monotonic Pairwise Mutual Information: Crucially, while total correlation decreases monotonically, the pairwise mutual information between specific coordinates (e.g., $I[X(t), Y(t)]$ in 3D) does not necessarily decay monotonically.
  • The authors construct a counterexample using a 3D Gaussian distribution where pairwise mutual information transiently increases before decaying, even as the total correlation decreases.
• KL Divergence: The KL divergence between the time-dependent distribution and the equilibrium distribution is shown to be exactly equal to the entropy gap ($S[\rho_{eq}] - S[\rho]$) and decays monotonically.

4. Significance

• Theoretical Advancement: The work corrects the assumption that subsystem-symmetric diffusion is inherently linear, providing the exact nonlinear hydrodynamic equations for arbitrary dimensions.
• Experimental Relevance: The derived equations and scaling exponents ($z=2k$) provide concrete predictions for experiments in tilted optical lattices and quantum gas microscopes. The non-monotonic behavior of pairwise mutual information offers a specific, testable signature of fracton hydrodynamics.
• Conceptual Unification: By linking marginal conservation to partial multipole conservation and interpreting the dynamics through information-theoretic functionals (Total Correlation vs. Mutual Information), the paper bridges statistical mechanics, hydrodynamics, and information theory. It clarifies that "fracton" behavior is fundamentally driven by the conservation of specific marginal constraints rather than just global multipole moments.

5. Conclusion

Mohanty and Ro demonstrate that diffusion constrained by subsystem symmetries leads to nonlinear, shear-only transport equations. These equations drive the system toward a product-state equilibrium determined by the initial marginals. The work highlights a subtle information-theoretic phenomenon where global decorrelation (total correlation) proceeds monotonically, while local correlations (pairwise mutual information) can exhibit transient growth, offering a new perspective on the relaxation dynamics of fractonic matter.