Curzon-Ahlborn-type efficiency in a Brownian heat engine with exponential temperature profile

We investigate a Brownian heat engine wherein a particle moves through a periodic ratchet potential under an exponentially decreasing temperature profile, a spatial configuration that closely resembles experimentally realizable conditions such as laser-induced thermal gradients and thermoplasmonic heating. This model yields exact analytical expressions for the particle current, thermodynamic efficiency, entropy production, and coefficient of performance (COP), and uniquely recovers the Curzon-Ahlborn efficiency and the corresponding endoreversible COP exactly in the quasistatic limit. These findings provide a rare and rigorous realization of endoreversible thermodynamics at the mesoscopic scale because they are derived directly from microscopic stochastic dynamics without recourse to phenomenological assumptions, asymptotic approximations, or coarse-graining techniques. Although the derived efficiency and COP are exact, they remain strictly below the Carnot limit, reflecting the inherent irreversibility embedded within the endoreversible framework. Furthermore, we show that in comparison to linear and piecewise-constant temperature profile cases, the exponential temperature profile leads to significantly higher particle velocities and higher entropy production, but lower thermodynamic efficiency, which underscores the fundamental trade-off between transport speed and energy cost. We further extend our analysis to networks of interacting Brownian motors operating in spatially nonuniform thermal environments. Numerical simulations confirm our analytical predictions and reveal the critical roles of temporal dynamics and external load in shaping motor performance, as well as transport directionality. Importantly, the exponential temperature profile is not only analytically tractable but also experimentally viable, providing a powerful platform for probing the emergence of macroscopic thermodynamic behavior from the underlying microscopic nonequilibrium dynamics.