10.1 Master equation(25) Consider Pauli’s equation (also known as the kinetic balance or master equation) awi = – at

(1)

where R,j > 0 is the rate of transition from state j to state i, and to; are probabilities for the system states = P = E tv,

The rates are symmetric Pis = Pj,. (a) Prove that the normalization of probabilities S. governed by equation (1) is conserved (b) Prove that the entropy of the system described by equation (1) can only increase monotonically. (c) Find the time-dependent probabilities w,(t) for a two-level system with P12 = P21 = p and initial wi(t = 0) = 1.

10.2 Frequency-dependent conductivity(25) Use the relaxation time approximation to generalize the formula for the conductivity of a degenerate Fermi gas a(w = 0) = neer fin (Drude’s formula) for the case of alternating current and voltage with frequency

= 0) o(w) 1 – kat-Discuss the Joule heat generation in the limit of large frequency to > r -1.

(2)

10.3 Transport phenomena in ideal gas(25) Calculate the electric conductivity o, the thermal conductivity K, and the thermoelectric coefficient S Olt OT thermo e.m.f. 79-x = S S. (3) for a classical ideal gas of charged particles in relaxation time approximation with the relaxation time r. Assume that the Coulomb interaction is completely screened, and the relaxation time is caused by short-range interactions.

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