Fluctuation-dissipation Theorem - General Formulation

General Formulation

The fluctuation-dissipation theorem can be formulated in many ways; one particularly useful form is the following:

Let be an observable of a dynamical system with Hamiltonian subject to thermal fluctuations. The observable will fluctuate around its mean value with fluctuations characterized by a power spectrum . Suppose that we can switch on a scalar field which alters the Hamiltonian to . The response of the observable to a time-dependent field is characterized to first order by the susceptibility or linear response function of the system

where the perturbation is adiabatically switched on at .

Now the fluctuation-dissipation theorem relates the power spectrum of to the imaginary part of the Fourier transform of the susceptibility ,

.

The left-hand side describes fluctuations in, the right-hand side is closely related to the energy dissipated by the system when pumped by an oscillatory field .

This is the classical form of the theorem; quantum fluctuations are taken into account by replacing with (whose limit for is ). A proof can be found by means of the LSZ reduction, an identity from quantum field theory.

The fluctuation-dissipation theorem can be generalized in a straightforward way to the case of space-dependent fields, to the case of several variables or to a quantum-mechanics setting.