Non-monotonic Irreversibility in Polytropic Steering

The efficient manipulation of thermodynamic states within the finite time is fundamentally constrained by the intrinsic dissipative cost. While the slow-driving regime is well-characterized by a universal $1/\tau$-scaling of irreversibility, the physics governing fast, non-adiabatic transitions remains elusive. Here, we propose the polytropic steering protocols that provide an exact analytical bridge between the isothermal and adiabatic limits for Brownian particles far-from-equilibrium. We demonstrate that for any protocol duration $\tau$, the system can be precisely steered along a prescribed polytropic trajectory, revealing a striking non-monotonic dependence of irreversibility on the driving rate. Contrary to the near-equilibrium paradigm where faster driving necessitates higher energetic costs, we identify a most-irreversible timescale, beyond which dissipation is anomalously suppressed by rapid driving. By mapping these protocols onto a broad class of controllable thermodynamic cycle, we establish power-efficiency tradeoffs and position the polytropic index as a genuine thermodynamic control knob for the rational design of high-speed, high-performance microscopic thermal machines.