Effects of the kinetic energy in heat for overdamped systems

In the derivation of the thermodynamics of overdamped systems, one ignores the kinetic energy contribution, since the velocity is a slow variable. In this paper, we show that the kinetic energy needs to be present in the calculation of the heat distribution to have a correct correspondence between the underdamped and overdamped cases, meaning that the velocity can not be fully ignored in the thermodynamics of these systems. We do this by investigating in detail the effect of the kinetic energy for three different systems, the harmonic potential, the logarithm potential, and an arbitrary non-isothermal process.