  We present an extensive numerical study of the time irreversibility of the
dynamics of heavy inertial particles in three-dimensional, statistically
homogeneous and isotropic turbulent flows. We show that the probability density
function (PDF) of the increment, $W(\tau)$, of a particle's energy over a
time-scale $\tau$ is non-Gaussian, and skewed towards negative values. This
implies that, on average, particles gain energy over a period of time that is
longer than the duration over which they lose energy. We call this
$\textit{slow gain}$ and $\textit{fast loss}$. We find that the third moment of
$W(\tau)$ scales as $\tau^3$, for small values of $\tau$. We show that the PDF
of power-input $p$ is negatively skewed too; we use this skewness ${\rm Ir}$ as
a measure of the time-irreversibility and we demonstrate that it increases
sharply with the Stokes number ${\rm St}$, for small ${\rm St}$; this increase
slows down at ${\rm St} \simeq 1$. Furthermore, we obtain the PDFs of $t^+$ and
$t^-$, the times over which $p$ has, respectively, positive or negative signs,
i.e., the particle gains or loses energy. We obtain from these PDFs a direct
and natural quantification of the the slow-gain and fast-loss of the particles,
because these PDFs possess exponential tails, whence we infer the
characteristic loss and gain times $t_{\rm loss}$ and $t_{\rm gain}$,
respectively; and we obtain $t_{\rm loss} < t_{\rm gain}$, for all the cases we
have considered. Finally, we show that the slow-gain in energy of the particles
is equally likely in vortical or strain-dominated regions of the flow; in
contrast, the fast-loss of energy occurs with greater probability in the latter
than in the former.
