Find P for fast rotation \u03a9 \u03c9 and its asymptotic form in the limit of ultrafast

Find p for fast rotation ω ω and its asymptotic

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Find P for fast rotation (Ω = ω ) and its asymptotic form in the limit of ultrafast rotation (Ω ω ), and interpret your results. 2
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Useful reminders: The canonical partition function of a classical gas of N identical particles is Z = 1 h 2 N N ! (R d 2 r d 2 p e - βh ( r , p ) ) N , where h ( r , p ) is the single-particle Hamiltonian. R R 0 re - λr 2 d r = 1 - e - λR 2 2 λ (recall the Jacobian in polar coordinates is d x d y = r d r d θ ) R -∞ e - λu 2 d u = p π λ A · ( B × C ) = B · ( C × A ) = C · ( A × B ) A × ( B × C ) = ( A · C ) B - ( A · B ) C 3
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Problem 2: Statistical Mechanics II Consider non-interacting spin-1 / 2 fermions in two dimensions (2D) with a linear dis- persion relation ± ( ~ k ) = ± ~ v | ~ k | . Positive energy states (with energy + ) define the conduction band and negative energy states (with energy - ) define the valence band. Assume that the allowed wavevectors ~ k = { k x , k y } correspond to periodic boundary conditions over a square region of area A . At temperature T = 0 the valence band is completely filled and the conduction band is completely empty. At finite T , excitations above this ground state correspond to adding particles (occupied states) in the conduction band or holes (unoccupied states) in the valence band. (a) (25 points) Find the single-particle density of states D ( ) as a function of the energy in terms of ~ , v, A . Sketch D ( ) over both the negative and positive energy region. In the next two parts we will argue that the chemical potential μ ( T ) = 0 at any temperature T (which can be assumed so for the rest of the problem). (b) (20 points) Using the Fermi-Dirac distribution, show that if μ ( T ) = 0 then the prob- ability of finding a particle at energy is equal to the probability of finding a hole at energy - . (c) (15 points) Particle number conservation requires that the number of particles in the conduction band must equal the number of holes in the valence band, N p = N h . Show that if μ ( T ) = 0 then this condition holds at all T . (Do not worry if the integrals are formally divergent – they will be cut off in any physical system.) (d) (25 points) Find the total internal energy of the excitations above the T = 0 state, U ( T ) - U (0), expressed in terms of A, v, ~ , k B . Note that since we are subtracting U (0), in the valence band you only need to count the energy associated with holes. Useful integral: 1 n ! Z 0 x n e x + 1 = (1 - 2 - n ) ζ ( n + 1) , where ζ ( n + 1) = k =1 k - n - 1 is the Riemann zeta function. (e) (15 points) Use your answer in (d) to find the heat capacity at constant area C A ( T ) T α . What is the exponent α ? 4
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