The Role of Odd Diffusivity in Multipoint Statistics of State-Dependent Observables

Abstract

Odd diffusivity is a transverse transport coefficient that appears when time-reversal and parity symmetries are broken. Its most distinctive signature is a probability flux perpendicular to a density gradient, generated by the antisymmetric part of the diffusion tensor. Here we show that for an arbitrary number of state-dependent observables measured at arbitrary times, their joint statistics are independent of the antisymmetric part of the diffusion tensor. Thus, the Lorentz flux does not contribute to any multipoint state-observable measurements. The result clarifies the separate roles of two effects that are often linked together by fluctuation--dissipation relations, i.e., odd mobility and odd diffusivity. This observation demonstrates that the anomalous correlations in odd-diffusive systems originate solely from odd mobility. It also allows the statistics of state-dependent observables in odd-diffusive systems to be computed using conventional Langevin dynamics without the antisymmetric diffusion part while preserving the antisymmetric mobility tensor. It implies that universal relations associated with state-dependent observables alone, such as nonlinear fluctuation--dissipation relations and generalized Green--Kubo relations, remain valid in odd-diffusive systems without modification. We verify our findings through mean back relaxation, the diffusion coefficient, the nonlinear fluctuation--dissipation relation, and a generalized Green--Kubo relation in diverse systems.