Answer Happy

This question is super important for my final revision,
thank you a million for detailed solution in advance.
C.1 [3] This question concerns a gas of hypothetical particles that we will call doublons. Their nature is such that no more than two doublons can simultaneously occupy the same quantum state. In the following, consider a gas of weakly interacting doublons in contact with a bath of heat and particles at temperature T and chemical potential p. a) Calculate the grand canonical partition function of a single quantum state i with the energy ei per doublon. b) Demonstrate that the average number of doublons in a single quantum state is given by 61-6] + 2 exp(-2 exp kBT ni = 1+exp (- 1] + exp[-2] € - kBT where kb is the Boltzmann constant. c) Sketch the average occupation number ñi at zero temperature as a function of €. d) Assume that the density of states g(e) for doublons is given by [5] [3] 2m 3/2 g(€)de = 4V (mm)"Vede, where e is the energy, m is the doublon mass, h is the Planck constant, and V is the volume of the system. Calculate the chemical potential u(T = 0) at zero temperature assuming that the average number of doublons in the system is N. [4] e) Show that the average energy of the system at zero temperature is given by 3 ET = 0) EN(T = 0). [4] f) Estimate the heat capacity Cy of a doublon gas at a very low but finite temperature. You can assume that u(T +0) = u(T = 0). - [6]