  We compute the Casimir thermodynamic quantities for a massive fermion field
between two parallel plates with the MIT boundary conditions, using three
different general approaches and present explicit solutions for each. The
Casimir thermodynamic quantities include the Casimir Helmholtz free energy,
pressure, energy and entropy. The three general approaches that we use are
based on the fundamental definition of Casimir thermodynamic quantities, the
analytic continuation method represented by the zeta function method, and the
zero temperature subtraction method. We include the renormalized versions of
the latter two approaches as well, whereas the first approach does not require
one. Within each general approach, we obtain the same results in a few
different ways to ascertain the selected cancellations of infinities have been
done correctly. We then do a comparative study of the three different general
approaches and their results, and show that they are in principle not
equivalent to each other and they yield, in general, different results. In
particular, we show that the Casimir thermodynamic quantities calculated only
by the first approach have all three properties of going to zero as the
temperature, the mass of the field, or the distance between the plates
increases.
