  In this work, we study the nonequilibrium statistical properties of the
relaxation dynamics of a nanoparticle trapped in a harmonic potential. We
report an exact time-dependent analytical solution to the Langevin dynamics
that arises from the stochastic differential equation of our system's energy in
the underdamped regime. By utilizing this stochastic thermodynamics approach,
we are able to completely describe the heat exchange process between the
nanoparticle and the surrounding environment. As an important consequence of
our results, we observe the validity of the heat exchange fluctuation theorem
(XFT) in our setup, which holds for systems arbitrarily far from equilibrium
conditions. By extending our results for the case of $N$ noninterating
nanoparticles, we perform analytical asymptotic limits and direct numerical
simulations that corroborate our analytical predictions.
