Odd Diffusion in Three-Dimensional Isotropic Media

Technical Summary: Odd Diffusion in Three-Dimensional Isotropic Media

Problem Statement
Odd transport coefficients, which generate currents transverse to thermodynamic forces, are a defining feature of nonequilibrium chiral matter. In two dimensions, odd diffusion is well-established, described by a linear constitutive law involving the two-dimensional Levi-Civita tensor ($\epsilon_{ij}$), which rotates density gradients by ninety degrees. However, a fundamental question remains regarding the existence of such transport in three-dimensional isotropic systems. Conventional symmetry arguments suggest that isotropic odd diffusion is impossible in three dimensions because rotational invariance forbids the existence of an isotropic antisymmetric rank-two tensor required for a linear odd diffusion law. This obstruction parallels restrictions on Hall-like responses in isotropic media, leading to the prevailing view that odd diffusion is intrinsically a lower-dimensional phenomenon.

Methodology
The authors employ a dual approach combining macroscopic symmetry analysis with microscopic derivation:

1. Symmetry Construction: The paper first analyzes the constraints imposed by rotational invariance, parity, and conservation laws on transport coefficients in three dimensions. It demonstrates that while linear odd diffusion is forbidden, nonlinear constitutive laws involving multiple conserved fields may circumvent this obstruction. The authors identify the unique lowest-order isotropic parity-odd term permitted by symmetry, utilizing the three-dimensional Levi-Civita tensor ($\epsilon_{ijk}$) and gradients of distinct species.
2. Microscopic Derivation: To validate the symmetry-based constitutive law, the authors derive it from a microscopic model of overdamped particles. They consider a system of three conserved species interacting via nonreciprocal three-body forces. Using the Dean–Kawasaki coarse-graining procedure, they map the stochastic particle dynamics to a continuum hydrodynamic description, showing how the nonlinear odd current emerges naturally from the microscopic interactions.

Key Contributions and Results

• Existence of Nonlinear Odd Diffusion: The primary result is the demonstration that odd diffusion can exist in isotropic three-dimensional systems, provided the system is multicomponent and the transport law is nonlinear. The authors derive the constitutive law for the flux $J^A_i$ of species $A$:
  $$J^A_i = -D\partial_i\rho^A - \omega\epsilon_{ijk}\epsilon_{ABC}(\partial_j\rho^B)(\partial_k\rho^C)$$
  where $\omega$ is a nonlinear odd diffusivity, and $\epsilon_{ABC}$ is the antisymmetric tensor in species space. This term represents the lowest-order isotropic odd transport allowed in 3D.

• Boundary-Driven Rotational Transport: Unlike conventional odd diffusion, which modifies bulk density relaxation, the 3D nonlinear odd current is divergence-free in the bulk ($\nabla \cdot J^A_{odd} = 0$). Consequently, the steady-state density profiles satisfy the standard Laplace equation ($\nabla^2 \rho^A = 0$) and remain identical to those of ordinary diffusion. The odd transport manifests exclusively as rotational currents (circulation) whose structure is entirely determined by boundary conditions.

• Generation of Vorticity and Enstrophy: Despite the absence of external torques, preferred directions, or momentum conservation, the system generates finite vorticity ($\Omega^A = \nabla \times J^A_{odd}$) and enstrophy. The vorticity arises from the spatial variation in the orientation of concentration gradients (specifically, the curvature of harmonic profiles). The enstrophy serves as a global measure of this rotational transport, scaling quadratically with the odd coefficient $\omega$ and vanishing only when concentration gradients are spatially uniform.

• Microscopic Origin: The paper establishes that this transport arises from nonreciprocal three-body interactions. Specifically, a force of the form $F_\alpha \propto \lambda \epsilon_{ABC} \sum_{\beta,\gamma} S^A_\alpha S^B_\beta S^C_\gamma (\mathbf{r}_{\alpha,\beta} \times \mathbf{r}_{\alpha,\gamma})$ breaks detailed balance and time-reversal symmetry at the microscopic level. Upon coarse-graining, this interaction yields the macroscopic cross-product structure of the current, linking microscopic nonreciprocity to macroscopic rotational transport without requiring external fields or intrinsic particle rotation.

Significance
The paper claims to establish a minimal framework for odd transport in isotropic three dimensions, challenging the notion that odd diffusion is strictly a two-dimensional phenomenon. By showing that symmetry obstructions to linear responses can be bypassed through nonlinear multicomponent couplings, the work extends the scope of odd hydrodynamics. It reveals that in three dimensions, odd transport does not alter density relaxation but reorganizes transport into boundary-controlled rotational structures. This provides a theoretical basis for realizing rotational nonequilibrium transport in isotropic active matter, such as multicomponent active colloids or synthetic reaction-diffusion media, where parity-breaking arises dynamically through species coupling rather than geometric constraints.