= Solution
<Equipartition> gives mean <energy> $k_BT/2$ for each <independent> quadratic term in a classical canonical <Hamiltonian>. For example, $p^2/(2m)$ and $kx^2/2$ each contribute this amount in a <harmonic oscillator>. It is a thermal average, not an instantaneous equality; constraints, nonquadratic terms or quantum level spacings large compared with $k_BT$ require a different calculation.
The <fluctuation-dissipation theorem> connects equilibrium fluctuations with the dissipative part of <linear response>. If an applied <force> couples as $-f(t)x$, use response to $e^{-i\omega t}$ and define the two-sided angular-frequency spectrum $S_{xx}(\omega)=\int e^{i\omega t}\langle\delta x(t)\delta x(0)\rangle dt$. The classical relation is
$$
\boxed{S_{xx}(\omega)=\frac{2k_BT}{\omega}\operatorname{Im}\chi(\omega).}
$$
For a viscously damped particle, $m\dot v=-\zeta v+\xi(t)$, the corresponding white-noise <force> satisfies $\langle\xi(t)\xi(t')\rangle=2\zeta k_BT\delta(t-t')$. Noise replenishes <energy> lost by <friction>; their strengths cannot be varied independently at fixed equilibrium <temperature>.
Electrically, the one-sided voltage-noise spectrum per hertz of an equilibrium <resistor> is $4k_BTR$. A <capacitor> in parallel filters it to
$$
S_V^{(1)}(f)=\frac{4k_BTR}{1+(2\pi fRC)^2},\qquad
\int_0^\infty S_V^{(1)}(f)\,df=\frac{k_BT}{C}.
$$
The same equilibrium <voltage> <variance> follows from <equipartition>. Increasing R makes the spectrum taller and narrower without changing its integrated area. This thermodynamic connection between dissipative conductance and thermal electrical fluctuations is the content of the classical noise calculation.<Thermal Agitation of Electric Charge in Conductors by Harry Nyquist>[1928, abstract: equilibrium thermodynamic and statistical calculation of thermal electromotive force]{ref}
\Image[/past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2013/paper-1-tsp2013-equilibrium-noise.png]
{title=Original normalized resistor-capacitor noise spectra: increasing resistance changes the spectral width and height while preserving the full equilibrium variance}
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