# Dispersion relation statistical mechanics

Luca2018
Homework Statement:
Consider general wave-excitations with a dispersion relations \omega(\vec(k)) =c |\vec(k) |^(\gamma) canpropagateind-dimensions(d=1,2,3)with wave-vector \vec(k).
a) Determine the density of states g(\omega) for each of the dimensions d =1,2,3and arbitrary γ>0.
b) What is the corresponding power-law at low temperatures for the energy
E \propto T^(\alpha) and the specific heat? (Hint: The independent modes of the wave- excitations can be described by the single-particle partition function for a harmonic oscillator).

Relevant Equations:
I think you have to use the delta distribution to solve part a).
I know that this is the general case of the Einstein and Debye Model.

Ishika_96_sparkles
Homework Statement:: Consider general wave-excitations with a dispersion relations \omega(\vec(k)) =c |\vec(k) |^(\gamma) canpropagateind-dimensions(d=1,2,3)with wave-vector \vec(k).
a) Determine the density of states g(\omega) for each of the dimensions d =1,2,3and arbitrary γ>0.
b) What is the corresponding power-law at low temperatures for the energy
E \propto T^(\alpha) and the specific heat? (Hint: The independent modes of the wave- excitations can be described by the single-particle partition function for a harmonic oscillator).

• 