Time-dependent solutions for efficiency and velocity of a Brownian heat engine that operates in a two-dimensional lattice coupled with a nonuniform thermal background

In this work, by obtaining exact time-dependent solutions, we study how the velocity, the efficiency, and the coefficient of performance of the refrigerator behave not only at steady-state but also at any time t for a particle moves on M Brownian ratchets arranged on a network. Far from a steady state, all of the thermodynamic quantities explicitly depend on network size. However, at steady state, the velocity, the efficiency, and the coefficient of performance of the refrigerator become independent of the network size M. We then study the thermodynamic features of the model system by including heat loss due to particle recrossing (heat loss due to kinetic energy) at the boundary between the hot and cold reservoirs. When the heat exchange via kinetic energy is included, the system becomes irreversible or less efficient even at the quasistatic limit. Furthermore, the velocity, efficiency, and coefficient of performance of the refrigerator of the motor that operates between the hot and cold baths are also compared and contrasted with a system that operates in a heat bath where its temperature linearly decreases along with the reaction coordinate. Regardless of the network size, a system that operates between the hot and cold baths has significantly lower velocity but a higher efficiency in comparison with a linearly decreasing temperature case. For a linearly decreasing temperature case, we show that the efficiency of such a Brownian heat engine is far less than Carnot’s efficiency even at the quasistatic limit. Our analysis indicates that depending on the thermal ratchets arrangement, not only a fast unidirectional speed can be achieved but also the particle can be transported to the desired location in a network with many dangling bonds and loops. Thus the present paradigmatic model serves as a basic tool to study the thermodynamic features of the realistic system such as protein-based molecular motors. Graphical abstract