Thermal properties in ID 

(a) Derive the formula (77) for the low temperature heat capacity of a single 1D phonon mode within the Debye approximation. 

(b) Derive the relation for the thermal conductance (78) of a 1D phonon mode between two reservoirs by calculating the energy flow out of one reservoir at a temperature T, and subtracting the energy flow from the other reservoir at a temperature T₂. Use an approach analogous to that used to obtain (20) and (24) for the electrical conductance.