Thermodynamical properties of bosons and fermions for lattice motivated dispersion relationships

In this thesis, we have studied the coupled simple harmonic oscillator physics of one-dimensional strings, for acoustical phonons, and n-dimensional physics for the optical case (for arbitrary n). For the acoustical modes, we have calculated only the Bosonic Simple Harmonic Oscillator (BSHO), for the optical mode, we have included both BSHO and Fermionic Simple Harmonic Oscillator (FSHO). For the optical modes, for both BSHO's and FSHO's, we have found that the problem is reduced to the physics of a single oscillator with a partial vibration frequency. For the FSHO's in the optical cases, we have encountered the presence of negative temperatures for all n. In both cases, we have discovered the presence of correct asymptotical behavior as T → 0 and T → ∞. In the acoustical case, we have used Mellin Transform and a summation formula due to Euler to calculate the partition function and the thermodynamical properties thereof. Again we have obtained the correct asymptotical behavior as T → 0 and T → ∞. We note that whenever possible, we have included more than one method of calculation and obtained similar results