Maximal power output of a stochastic thermodynamic engine

Classical thermodynamics aimed to quantify the efficiency of thermodynamic engines, by bounding maximal amount mechanical energy produced, compared heat required. While this was accomplished early on, Carnot and Clausius, more practical problem limits power that can be delivered, remained elusive due fact quasistatic processes require infinitely slow cycling, resulting in a vanishing output. Recent insights, drawn from stochastic models, appear bridge gap between theory practice they lead physically meaningful expressions for dissipation cost operating engine over finite time window. Indeed, optimize expressed as control problem. Building on framework we derive bounds cycling an overdamped ensemble particles via time-varying potential while alternating contact with baths different temperature (Tc cold, Th hot). Specifically, assuming suitable bound M spatial gradient controlling potential, show achievable is bounded M8(ThTc?1). Moreover, reached within factor (ThTc?1)?(ThTc+1) cyclic process quadratic potential.