The chiral random walk: A quantum-inspired framework for odd diffusion

Chirality in active and passive fluids gives rise to odd transport properties, most notably the emergence of robust edge currents that defy standard dissipative dynamics. While these phenomena are well-described by continuum hydrodynamics, a microscopic framework connecting them to their topological origins has remained elusive. Here, we present a lattice model for an isotropic chiral random walk that bridges the gap between classical stochastic diffusion and unitary quantum evolution. By equipping the walker with an internal degree of freedom and a tunable chirality parameter, $p$, we interpolate between a standard diffusive random walk and a deterministic, topologically non-trivial quantum walk. We show that the topological protection characteristic of the unitary limit ($p=1$) remarkably persists into the dissipative regime ($p<1$). This correspondence allows us to theoretically ground the robustness of edge flows in classical chiral systems using the bulk-boundary correspondence of Floquet topological insulators. Our results provide a discrete microscopic description for odd diffusion, offering a powerful toolkit to predict transport in confined geometries and disordered chiral media.

💡 Research Summary

This paper introduces a novel discrete lattice model termed the “chiral random walk” (CRW), which provides a unified microscopic framework for understanding “odd diffusion” in chiral systems. Odd diffusion, characterized by an antisymmetric off-diagonal component in the diffusion tensor, is a hallmark of systems with broken time-reversal symmetry, such as active microswimmers, colloidal particles under a Lorentz force, or chiral fluids. While continuum hydrodynamic descriptions exist, a foundational discrete stochastic model connecting this phenomenon to its potential topological origins has been lacking.

The core innovation of the CRW model is the incorporation of an internal degree of freedom (IDF)—akin to the “coin” in quantum walks—that represents a directional state (right, left, up, down). The walker’s state is a combination of its lattice position and this IDF. Time evolution proceeds in two sequential operations: a “coin flip” that stochastically updates the internal state, followed by a “step” that moves the particle one lattice site in the direction indicated by its current IDF.

The model’s versatility stems from a single tunable chirality parameter, p (0 ≤ p ≤ 1). The coin operator is defined as a convex combination of a fully randomizing matrix (representing standard diffusion) and a deterministic cyclic permutation matrix (representing perfect chiral motion). Thus, p=0 recovers the classical isotropic random walk, p=1 yields a deterministic chiral walk (formally equivalent to a quantum walk), and 0<p<1 describes a stochastic, chiral diffusive process.

Analysis of the model shows that for p<1, the long-time mean-squared displacement exhibits normal Brownian scaling, but with a diffusion coefficient that decreases with increasing p. Crucially, the derived diffusion tensor features an antisymmetric component proportional to p, precisely matching the form of the odd diffusion tensor observed in continuous theories. This validates the CRW as a genuine microscopic model for odd diffusion.

The most significant finding concerns behavior in confined geometries. Numerical simulations demonstrate that for p > 0, a walker initialized near a reflective boundary exhibits a pronounced probability accumulation along the edge, a signature of chiral edge following. The authors show that in the deterministic limit (p=1), this system can be mapped onto a Floquet topological insulator. In this mapping, the edge-following modes correspond to topologically protected edge states, as evidenced by a gap in the bulk spectrum and localized states at the boundary. Remarkably, the robustness of these edge modes persists into the dissipative regime (p < 1), providing a theoretical foundation—via the bulk-boundary correspondence—for the observed robustness of edge flows in classical chiral systems.

In conclusion, the chiral random walk model bridges the gap between classical stochastic dynamics and quantum-inspired topological phenomena. It offers a powerful discrete toolkit to predict and analyze transport in complex chiral environments, including confined geometries and disordered media, by leveraging insights from topological matter.