Inverse Clausius Thermodynamics in Run-and-Tumble Dynamics

We establish a mapping between one-dimensional run-and-tumble particle dynamics in the presence of thermal noise, and overdamped Brownian motion in a spatially inhomogeneous temperature field. The approach is formulated as an inverse-Clausius thermodynamic framework, where the effective temperature is inferred from the steady-state density. Within this mapping, the local entropy flux and entropy production rate can be extracted directly from steady-state observables, without requiring explicit knowledge of the full position-velocity distribution. The framework introduces a closure that, to leading order, relates entropy flow to spatial variations of effective temperature, yielding a picture of entropy transfer from hotter to colder regions. We apply the approach to two-state and multistate run-and-tumble models in harmonic and nonlinear potentials. For harmonic confinement, the closure is exact in the two-state case and very accurate in multistate models. For nonlinear potentials that are locally confining (positive curvature at the origin), comparable accuracy is observed. In contrast, potentials with vanishing or negative curvature require higher-order corrections and reveal a correspondence between the spatial structure of the entropy production rate and that of the confining potential.