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Abstract

Thermodynamic uncertainty relations (TURs) bound the dissipation in non-equilibrium systems
from below by fluctuations of an observed current. Contrasting the elaborate techniques employed
in existing proofs, we here prove TURs directly from the Langevin equation. This establishes
the TUR as an inherent property of overdamped stochastic equations of motion. By including
current-density correlations we, moreover, derive a new sharpened TUR for transient dynamics.
Our arguably simplest and most direct proof allows us to systematically determine conditions under
which the different TURs saturate and thus allows for a more accurate thermodynamic inference.