The classical fluctuation-dissipation theorem states that the power spectral density $S_{\eta\eta}(\omega)$ of a classical random variable $\eta(t)$ is given by $$S_{\eta\eta}(\omega)=\frac{2k_BT}{\omega}{\rm Im}\left[\chi(\omega)\right]\tag{1}$$ where $\chi(\omega)$ is the Fourier ransform of the response function $\chi(t)$ called the generalized susceptibility.
The Johnson-Nyquist noise in a resistor arises as a special case of this theorem. Similarly, the relation between the strength of the random force acting on a Brownian particle and the damping can also arise as an application of this theorem.
However, there is another fluctuation-dissipation relation (See Eq.$1.27$ of David Tong's lecture notes on Statistical Physics, and the paragraph following it) in statistical mechanics: the thermal fluctuation in energy is given by$$\sigma_E^2=k_BT^2C_V\tag{2}$$ where $c_v$ is the heat capacity at constant volume. Can Eq.$(2)$ be derived from Eq.$(1)$ as a special case of it? I asked a similar question here but it does not answer how relation $(2)$ could arise from a more general relation.
