Thermodynamic irreversibility in underdamped Brownian motion with spatial temperature gradients

In this work, we extensively explore the thermodynamic properties of Brownian particles moving in an underdamped medium. Extending our previous analysis [Phys. Rev. E 101, 012131 (2020)10.1103/PhysRevE.101.012131], we examine systems subjected to different thermal arrangements, such as quadratically and linearly decreasing temperature profiles, as well as piecewise constant-temperature distributions. By conducting rigorous derivations, we derived several thermodynamic relations that provide insights into nonequilibrium thermodynamics and the underlying mechanisms governing entropy production and extraction. Furthermore, we address a fundamental question that includes whether the vanishing entropy production or extraction rate necessarily implies a thermodynamic equilibrium. Our analytical findings reveal that for a Brownian particle in an underdamped medium (free from external forces or ratchet potentials), both the entropy production and extraction rates decrease to zero, even in the presence of a spatially varying temperature gradient. However, the total entropy production (E_{P}>0) and total entropy extraction (H_{d}>0) remain finite, indicating that the system retains intrinsic irreversibility driven by heat exchange via kinetic energy transfer. This also implies that the zero-entropy production rate does not signify an equilibrium. Our results further indicate that in the absence of external loads and potentials, most thermodynamic rates asymptotically decay to zero. However, the system's irreversibility persists because of the continuous heat flow from regions of higher to lower temperatures. We also consider the motion of a Brownian particle that moves in a ratchet potential coupled with a spatially varying temperature. Such particles move unidirectionally, even in the absence of force. By imposing asymmetry in the thermal arrangements, one can manipulate the direction of the particles, indicating that this finding has significant applications in microscale and nanoscale transport systems. We show that, because the velocity of a particle depends on its mass, barrier height, load, and noise intensity, the particles can be sorted along the reaction coordinate depending on their physical properties.