  Considering the relativistic scenario, we dedicate our study to the
relativistic quantum description of one-dimensional Majorana fermions. Thus, we
focus on aspects related to exciton-like particles. Seeking to reach our
purpose, one analyzes the relativistic quantum mechanical system characterized
by an effective mass distribution. In this context, we adopt an exciton-like
position-dependent mass without impurity, i.e., without electromagnetic
interactions. From this perspective, one notes results of noteworthy interest
as consequences of the theory adopted. For instance, we highlight that, even
without interaction, exciton-like Majorana fermions manifest theoretically
bound states. Also, we construct a Majorana fermion ensemble with effective
mass immersed in a thermal reservoir. That allows for a thorough investigation
of the thermodynamic properties of the system. Among the thermodynamic
characteristics studied in the canonical ensemble, we focus on the Helmholtz
free energy, mean energy, entropy, and heat capacity. The numerical results
obtained for these thermodynamic properties corroborate the validity of the
Dulong-Petit law for our system.
