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).

Can you please help me? :)

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).

Can you please help me? :)
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.

Please see this link and rewrite your question for people to be able to understand.

 

[dRic2]

It seems to me like a generalization of the Debye Model. If you study the Debye Model (for example check Ziman's "theory of solid" chapter 2, but you can find it everywhere) you will see that it is assumed a dispersion relation of the form $\omega = ck$; just substitute that relation with the one you are given and do the exactly same calculations.