  The possibility of the existence of mesons with two or more quark-antiquark
pairs is investigated with a new application of the Thomas-Fermi (TF)
statistical quark model. Quark color couplings are treated in a mean field
manner similar to a previous application to baryons, and short and concise
expressions for energies are derived. We find that, on average, quarks only
interact with antiquarks in such systems. The TF differential equation is
constructed and systems with heavy-light quark content are examined. Three
types of mesonic systems are defined. In the case of charm quarks,
multi-charmonium, multi-$Z$ meson and multi-$D$ meson family types are
examined. System analogs for bottom quarks are also constructed. Quantitative
trends for system energies of mesonic quark matter are extracted as a function
of the number of quark pairs. We find indications from energy plots that
multi-$Z$ type mesons (and their bottom quark analogs) are actually stable for
a range of quark number pairs. At this initial stage we have not yet included
explicit spin interaction couplings between quarks, but we can take one level
of degeneracy into account in our two-inequivalent TF function construction.
