Thermodynamic uncertainty relation for Langevin dynamics via time contraction

The thermodynamic uncertainty relation (TUR) quantifies the relationship between fluctuation and response in out-of-equilibrium systems, and is the natural counterpart of the fluctuation-dissipation theorem in equilibrium statistical mechanics. For both overdamped and underdamped Langevin dynamics, we derive a TUR by considering a virtual time contration on the discretized equations of motion. The TUR emerges naturally from the contraction principle of large deviation theory and makes no recourse to information-theoretic inequalities. We compare our TUR with known results in the overdamped and underdamped regimes and explore their similarities and differences.