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 P: 10 As for motivation: I'm a biochemistry student about to start work on a more physics intensive project and I got some old course notes form my friends to learn from. There is some discussion of the Debye theory of specific heats in arbitrary dimensions and I just want to make sure that I understand all of the details. The notes contain some mysterious passages, like the question I originally posted. I can get the phonon and electron specific heats in the low and high T limits. Then there is a question asking if these are ever comparable in magnitude. I think that I can answer this as well: in different temperature ranges you can obtain constraints on the dimension $$d$$ and exponent of the electron dispersion $$g$$ for which the two contributions have the same temperature dependence. The next part of the notes asks to think about how all of this (DOS, specific heats, etc) changes when particle number is held constant instead of chemical potential. There is also a thermodynamic relation that relates chemical potential to specific heat $$C_p= -NT(\frac{\partial^2 \mu}{\partial T^2})_p$$ and $$C_v= VT(\frac{\partial^2 P}{\partial T^2})_v-NT(\frac{\partial^2 \mu}{\partial T^2})_p$$ I'm not quite sure what that second derivative of chemical potential means and how to reproduce the results of the Debye theory for this case. Thanks.