  An interesting result recently reported for Mrk 421 is the detection of a
significant excess at hard X-ray energies, which could provide useful
information for investigating particle acceleration and emission mechanisms in
the relativistic jet. Considering a two-injection scenario, we develop a
self-consistent one-zone leptonic model to understand the origin of the hard
X-ray excess in Mrk 421 during the period of extremely low X-ray and very high
energy (VHE) flux in 2013 January. In the model, two populations of
mono-energetic ultrarelativistic electrons are injected into the emission
region that is a magnetized plasmoid propagating along the blazar jet. We
numerically calculate the emitting electron energy distribution by solving a
kinetic equation that incorporates both shock acceleration and stochastic
acceleration processes. Moreover, we infer analytic expressions relating
electrons acceleration, cooling, escape and injection to the observed spectra
and variability. For the injection luminosity in particular, we derive a new
approximate analytical expression for the case of continual injection with a
mono-energetic distribution. Based on a comparison between the theoretical
predictions and the observed SED, we conclude that the hard X-ray excess
observed in Mrk 421 may be due to the synchrotron radiation emitted by an
additional electrons population, which is co-spatial with an electron
population producing simultaneous Optical/UV, soft X-ray, and $\gamma$-ray
emissions. The stochastic acceleration may play a major role in producing the
observed X-ray spectrum.
