Does geometry secretly govern how much energy dissipates?

Entropy production isn't just a number — it turns out to define a precise geometry on the space of physical states. By casting relaxation in classical diffusions, discrete Markov chains, and quantum Lindblad systems all as 'gradient flows,' the authors derive tight bounds on how much energy any process must waste. Crucially, they show that quantum coherence and classical inertia both shrink the effective distance between states, quantifying exactly how much dissipation can be saved — a foundation for designing truly optimal thermodynamic protocols.