  Stochastic thermodynamics provides an important framework to explore small
physical systems where thermal fluctuations are inevitable. In the studies of
stochastic thermodynamics, some thermodynamic quantities, such as the
trajectory work, associated with the complete Langevin equation (the Kramers
equation) are often assumed to converge to those associated with the overdamped
Langevin equation (the Smoluchowski equation) in the overdamped limit under the
overdamped approximation. Nevertheless, a rigorous mathematical proof of the
convergence of the work distributions to our knowledge has not been reported so
far. Here we study the convergence of the work distributions explicitly. In the
overdamped limit, we rigorously prove the convergence of the extended
Fokker-Planck equations including work using a multiple timescale expansion
approach. By taking the linearly dragged harmonic oscillator as an exactly
solvable example, we analytically calculate the work distribution associated
with the Kramers equation, and verify its convergence to that associated with
the Smoluchowski equation in the overdamped limit. We quantify the accuracy of
the overdamped approximation as a function of the damping coefficient. In
addition, we experimentally demonstrate that the data of the work distribution
of a levitated silica nanosphere agrees with the overdamped approximation in
the overdamped limit, but deviates from the overdamped approximation in the
low-damping case. Our work fills a gap between the stochastic thermodynamics
based on the complete Langevin equation (the Kramers equation) and the
overdamped Langevin equation (the Smoluchowski equation), and deepens our
understanding of the overdamped approximation in stochastic thermodynamics.
