[UPSC IFoS 2021] Statistical Physics (Q260, 8M)

A gas of N spinless Bose particles of mass m is enclosed in a volume V at a temperature T.

(i) Find an expression for the density of single-particle state D(\varepsilon) as a function of the single-particle energy \varepsilon. Sketch the result.

(ii) Write down an integral expression which implicitly determines \mu(T). Referring to your sketch in (i), determine in which direction \mu(T) moves as T is lowered.