Thermodynamic Uncertainty Relation For a Multi-Phase Alternating Renewal Random Walk

Abstract: Thermodynamic Uncertainty Relations (TURs) impose universal bounds linking current precision to entropy production in nonequilibrium systems. While general bounds like the Proesmans--Van den Broeck (PV) relation provide a broad framework, they often remain loose for processes characterized by renewal events. In this work, we derive a generalized entropic bound for current fluctuations in renewal-reward processes. By utilizing a rigorous variance decomposition within the framework of renewal-reward theory, we obtain a model-independent bound that is not only rigorous but tighter than the standard PV relation. Notably, in the linear-response regime, our bound correctly scales with the renewal rate and identifies a precision penalty arising from cycle-length fluctuations. These results provide a physically informative constraint on the precision of run-and-tumble-type dynamics and highlight the universal limits of transport in complex stochastic walkers.