Blundell Thermal Physics · Problem 33.2

Brownian motion and fluctuations

Physics — Statistical Mechanics · linear-response-power-dissipation

Problem

A system has displacement x~(ω)=χ~(ω)f~(ω)\tilde{x}(\omega) = \tilde{\chi}(\omega)\tilde{f}(\omega) in response to a force f~(ω)\tilde{f}(\omega). Show that if the force is given by f(t)=f0cos⁡ωtf(t) = f_0 \cos \omega t, the average power dissipated is 12f02ωχ~′′(ω)\frac{1}{2} f_0^2 \omega \tilde{\chi}''(\omega).

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