(untitled) (Contemporary Mathematics, 2025)
In this study, we advance the understanding of nonequilibrium systems by deriving thermodynamic relations for a heat engine operating under an exponentially decreasing temperature profile. Such thermal configurations closely mimic spatially localized heating, such as laser-induced thermal gradients. Using exact analytical solutions, we show that this arrangement results in significantly higher velocity, entropy production, and extraction rates than piecewise thermal profiles, while exhibiting reduced irreversibility and complexity relative to linear or quadratic gradients. We further examine the thermodynamic behavior of the Brownian particles in the networks. Our study reveals that the velocity and entropy production rates remain independent of the network size; in contrast, extensive quantities, such as total entropy, depend on the number of microstates. Additionally, we show that a Brownian particle in a ratchet potential with spatially varying temperature achieves directed motion, even without external forces driven solely by thermal asymmetry. These findings highlight the critical role of temperature asymmetry in controlling transport processes and optimizing particle dynamics. This will have promising applications in microfluidic devices and nanoscale sensors. Finally, we explore the influence of the system parameters on the efficiency and performance of the heat engine. The exponential temperature profiles enable faster velocities while simultaneously exhibiting higher efficiency compared to other thermal arrangements. Moreover, by addressing key questions on entropy production, we provide insights into the transition between nonequilibrium and equilibrium systems and contribute tools for optimizing energy-efficient systems in both natural and engineered settings.