Anomalous Thermodynamics at the Microscale

Particle motion at the microscale is an incessant tug-of-war between thermal fluctuations and applied forces on one side and the strong resistance exerted by fluid viscosity on the other. Friction is so strong that completely neglecting inertia—the overdamped approximation—gives an excellent effective description of the actual particle mechanics. In sharp contrast to this result, here we show that the overdamped approximation dramatically fails when thermodynamic quantities such as the entropy production in the environment are considered, in the presence of temperature gradients. In the limit of vanishingly small, yet finite, inertia, we find that the entropy production is dominated by a contribution that is anomalous, i.e., has no counterpart in the overdamped approximation. This phenomenon, which we call an entropic anomaly, is due to a symmetry breaking that occurs when moving to the small, finite inertia limit. Anomalous entropy production is traced back to futile phase-space cyclic trajectories displaying a fast downgradient sweep followed by a slow upgradient return to the original position.

Figure 1

Anomalous entropy is produced during futile cycles of fast downgradient sweeping and slow upgradient climbing. Along a microscopic trajectory that starts and ends at the same temperature (solid red line), the overdamped contribution to entropy vanishes according to Eq. (7). However, anomalous entropy is produced because of heat absorption at higher temperatures, followed by heat release at smaller ones. The reversed cycle (dashed blue line) with negative anomalous entropy production is much less likely to occur in agreement with Eq. (14).