Slow is fast: raising barriers to accelerate thermal relaxation

Abstract

For a reversible system relaxing to equilibrium, the obvious fastest strategy is to lower all kinetic barriers (open all gates). We find that such intuition holds at three levels: the all-open-gate strategy achieves the highest local conductance, it maximizes the instantaneous speed of approach in every $f$-divergence, and it simultaneously maximizes all relaxation eigenvalues. Nevertheless, we show that a counter-intuitive finite-time optimum lies beyond this intuition and operates at a fourth level, invisible to all three: eigenvector rotation. Noncommutativity enables timed schedules to reproject residual amplitudes across relaxation modes, thereby achieving faster relaxation. Optimal schedules are bang--bang. In our illustrative example, the best-found schedule also employs counter-gating, transiently raising selected barriers, and reduces the terminal residual by a factor of $130$ relative to all-open, and by $7$ relative to the best static landscape. A no-go theorem shows that noncommutativity is necessary: commuting generators collapse every schedule to a static time-averaged landscape, worse than the intuitive static control. In the reverse problem, the dual schedule preserves nonequilibrium free energy far more effectively than intuitively keeping all barriers at maximum heights. Whether accelerating or delaying relaxation, barrier control performs no work on the reduced Markov system; it only re-times a fixed total dissipation budget.