For a gas of particles confined inside a two-dimensional box, the density of states is constant, independent of ϵ (see Problem). Investigate the behaviour of a gas of non interacting bosons in a two-dimensional box. You should find that the chemical potential remains significantly less than zero as long as T is significantly greater than zero, and hence that there is no abrupt condensation of particles into the ground state. Explain how you know that this is the case, and describe what does happen to this system as the temperature decreases. What property must g(ϵ) have in order for there to be an abrupt Bose-Einstein condensation?
Consider a free Fermi gas in two dimensions, confined to a square area A = L2.
(a) Find the Fermi energy (in terms of N and A), and show that the average energy of the particles is ϵF/2
(b) Derive a formula for the density of states. You should find that it is a constant, independent of ϵ.
(c) Explain how the chemical potential of this system should behave as a function of temperature, both when. kT ≪ ϵF and when T is much higher.
(d) Because g(ϵ) is a constant for this system, it is possible to carry out the integral for the number of particles analytically. Do so, and solve for µ as a function of N. Show that the resulting formula has the expected qualitative behavior.
(e) Show that in the high-temperature limit, kT ≫ϵF, the chemical potential of this system is the same as that of an ordinary ideal gas.
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