Geometry-driven thermodynamic anomaly in overdamped Langevin dynamics.

Changing geometry creates its own overdamped thermodynamic anomaly. This is the geometric analog of the underdamped-overdamped anomaly known for Langevin systems in nonuniform or time-dependent temperature fields. We show this for Langevin dynamics in R^{3} under a time-dependent potential and a moving holonomic constraint. The moving constraint imposes a normal control velocity on the overdamped motion. Starting from a one-parameter family of underdamped parent models indexed by χ, with exact hard constraints, we show that the small-mass overdamped limit leaves a finite geometric contribution proportional to the surface divergence of the control velocity. This term is absent from the naive overdamped heat-work split. It is therefore a genuine geometric anomaly. Once restored, it reorganizes the first law into a modified heat-work pair. On the heat side, the detailed fluctuation theorem recovers its Clausius form. On the work side, the Jarzynski equality is closed on effective work. The same identification also clarifies the relative-entropy balance. Our results identify changing geometry as a source of thermodynamic structure in the overdamped Langevin dynamics.