Blundell Thermal Physics · Problem 33.3

Brownian motion and fluctuations

Physics — Statistical Mechanics · kramers-kronig-relations

Problem

Repeat the derivation that led to eqn 33.72 (one of the Kramers–Kronig relations), but this time set

y(-t) = y(t)

, so that

\tilde{y}(\omega)

is purely real. In this case, prove the other Kramers–Kronig relation, which states that

\tilde{\chi}''(\omega) = -\mathcal{P}\int_{-\infty}^{\infty} \frac{\mathrm{d}\omega'}{\pi} \frac{\tilde{\chi}'(\omega')}{\omega' - \omega}. \tag{33.121}