# Noise spectrum of two systems and interacting Hamiltonian

I've been discovering recently the concept of noise spectrum, defined as: $$S_{xx}[\omega] = \int dt \langle x(t)x(0)\rangle \text{e}^{-i\omega t}$$ Roughly the Fourrier transform of the two-point function. Apparently it represents, the probability that the system has to absorb (or emit) energy for positive (negative) $\omega$. I am not familiar with this new object to me, but let's say that I look at composite system, for example an oscillator in a bath. The Hamiltonian pictures says that the exchange of energy between both are caused by the existence of an interacting Hamiltonian: $$H_{int} = g\;\hat{F}.\hat{x}$$ quantum in this example. From what I said about the noise spectrum, I would say that the exchanges of energy would be motivated by an overlap of the two noise spectrum functions $S_{xx}$ and $S_{FF}$. However, I don't see yet how to conciliate these two points?

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The volumes in the Landau-Lifshits series are devoting a lot of space to this type of arguments, which you could read with much profit. It is called the linear response theory, and it is extensively discussed in vols. "Statistical Mechanics", "Electrodynamics of Continua" but is popping up now and again in the entire series. They call your noise spectrum a "generalized susceptivity" and prove strong results like the Kramers-Krönig and the Fluctuation-Dissipation theorems. – Lupercus Dec 18 '12 at 0:09

The quantity you are describing ($S(\omega)$) is called the power spectral density.I can't say if your interpretation of the power spectral density in this case is mistaken or not, because I haven't encountered it myself.
But in context of a stationary physical process, the power spectral density describes how the total power in the system is distributed over various frequencies. Here power is taken to mean the square of the signal, i.e, if the signal is $f(t)$, then the total power is given as $$P = \frac{1}{T} \int_{0}^{T} |f(t)|^2 dt$$
The function that you have described is actually the fourier transform of the autocorrelation function, a result given by the Weiner-Khinchin theorem. If your $x(t)$ is a stochastic variable, then $S(\omega)$ is the spectral power distribution for that variable/process.