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The authors formulate "odd stochastic density functional theory" (odd SDFT) to capture the dynamics of many-body systems exhibiting chiral overdamped limits, such as magnetized electrolytes and chiral active matter. By deriving the framework from fluctuating hydrodynamics of underdamped particles under Lorentz forces, they compute the non-reciprocal collective mobility tensor using both direct linear response and Green-Kubo relations. The resulting theory extends the relaxation component of Debye-Hückel-Onsager theory to finite magnetic fields, successfully capturing Hall conductivity beyond mean-field approximations.
Broken time-reversal symmetry in fluctuating many-body systems can now be tractably modeled at the continuum level, solving a long-standing limitation in stochastic density functional theory for chiral and magnetized matter.
Stochastic density functional theory (SDFT) provides a powerful framework for studying the dynamics of many-body systems both in and out of equilibrium. In its standard form, it can be viewed as a density-based formulation of coupled Langevin equations with white noise. Here, we extend SDFT to a class of systems, ranging from magnetized electrolytes to chiral active matter, that possess a chiral overdamped limit in which a relaxation timescale and an oscillation period are simultaneously taken to zero. We derive this formulation, which we dub ``odd SDFT", by taking the overdamped limit of a fluctuating hydrodynamic description of underdamped particles subject to a Lorentz force. We establish the validity and utility of the resulting framework by calculating the collective mobility tensor of interacting particles using both direct linear response and a Green-Kubo relation. We then apply the theory to the Hall conductivity of electrolytes in magnetic fields. Going beyond mean-field theory and using two distinct Green-Kubo relations, we derive the full conductivity tensor, extending the relaxation part of the Debye-H\"uckel-Onsager theory to the case of finite magnetic fields.