Editing Fluctuation–dissipation theorem (section)

==General formulation==
The fluctuation–dissipation theorem can be formulated in many ways; one particularly useful form is the following:{{Citation needed|date=October 2010}}.

Let <math>x(t)</math> be an [[observable]] of a [[dynamical system]] with [[Hamiltonian mechanics|Hamiltonian]] <math>H_0(x)</math> subject to thermal fluctuations.
The observable <math>x(t)</math> will fluctuate around its mean value <math>\langle x \rangle_0</math> with fluctuations characterized by a [[power spectrum]] <math>S_x(\omega) = \langle \hat{x}(\omega)\hat{x}^*(\omega) \rangle</math>.
Suppose that we can switch on a time-varying, spatially constant field <math>f(t)</math> which alters the Hamiltonian to <math>H(x) = H_0(x) - f(t)x</math>.
The response of the observable <math>x(t)</math> to a time-dependent field <math>f(t)</math> is 
characterized to first order by the [[Linear response function|susceptibility]] or [[linear response function]]
<math>\chi(t)</math> of the system
<math display="block">
 \langle x(t) \rangle = \langle x \rangle_0 + \int_{-\infty}^t f(\tau) \chi(t - \tau)\,d\tau,
</math>
where the perturbation is adiabatically (very slowly) switched on at <math>\tau = -\infty.</math>

The fluctuation–dissipation theorem relates the two-sided power spectrum (i.e. both positive and negative frequencies) of <math>x</math> to the imaginary part of the [[Fourier transform]] <math>\hat{\chi}(\omega)</math> of the susceptibility <math>\chi(t)</math>:
<math display="block">
 S_x(\omega) = -\frac{2 k_\text{B} T}{\omega} \operatorname{Im}\hat{\chi}(\omega),
</math>
which holds under the Fourier transform convention <math>f(\omega)  =\int_{-\infty}^\infty f(t) e^{-i\omega t}\, dt</math>. The left-hand side describes ''fluctuations'' in <math>x</math>, the right-hand side is closely related to the energy ''dissipated'' by the system when pumped by an oscillatory field <math>f(t) = F \sin(\omega t + \phi)</math>. The spectrum of fluctuations reveal the linear response, because past fluctuations cause future fluctuations via a linear response upon itself.

This is the classical form of the theorem; quantum fluctuations are taken into account by replacing <math>2 k_\text{B} T / \omega</math> with <math>\hbar \, \coth(\hbar\omega / 2k_\text{B}T)</math> (whose limit for <math>\hbar \to 0</math> is <math>2 k_\text{B} T/\omega</math>). A proof can be found by means of the [[LSZ reduction]], an identity from quantum field theory.{{Citation needed|date=August 2013}}

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.<ref name=Callen-Welton/>