Fluctuation-dissipation (aka Nyquist) theorem

 

This theorem states that if a passive linear system in thermal equilibrium dissipates (i.e. its impulse response is dumped, or its poles in the transfer function are displaced from the imaginary axis), than its output should be fluctuating even without an input signal. All happens as if the friction that transform the energy of a signal stored in the system into thermal energy could also bound the system to the energy fluctuations of the thermal reservoir, so that they can drive the system. Mathematically, there is a precise relation between the power spectrum density  of this intrinsic noise source, the temperature and the dissipative part of a system, that is the real part of the complex impedance :

where  is the Boltzmann constant.

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