  We develop a formalism to incorporate exchange symmetry considerations into
the calculation of collisional frequency shifts and blackbody radiation effects
for atomic clock transitions using a density matrix formalism. The formalism is
developed for both fermionic and bosonic atomic clocks. Results for a finite
temperature ${}^{87}$Sr ${}^1S_0$ ($F = 9/2$) atomic clock in a magic
wavelength optical lattice are presented.
