Thermodynamic Uncertainty Relations in Chaotic Andreev Billiards

We investigate how particle-hole symmetry, quantum interference, and tunnel barriers shape thermodynamic uncertainty relations in chaotic Andreev billiards. Using random-matrix theory and the Mahaux-Weidenmüller scattering approach, we study charge conductance and shot noise across the four Altland-Zirnbauer symmetry classes, from the single-channel extreme quantum limit to the multichannel semiclassical regime and from ideal to opaque contacts. We characterize thermodynamic precision through two complementary ensemble observables: either by averaging the sample-resolved noise-to-conductance ratio, or from the separately averaging noise and conductance. Their pronounced discrepancy in the extreme quantum regime reveals the non-self-averaging character of mesoscopic transport and persists over a broad range of barrier transparencies. For ideal contacts, the first provides a sensitive fingerprint of the Altland-Zirnbauer symmetry class. In the opaque regime, this hierarchy changes. Despite these strong symmetry- and barrier-dependent effects, the standard thermodynamic uncertainty relation remains satisfied throughout all regimes investigated. Our results establish thermodynamic uncertainty as a symmetry-sensitive probe of universal transport in chaotic normal-superconducting systems.