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The fluctuation-dissipation theorem (FDT) or fluctuation-dissipation relation (FDR) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the theorem is a general proof that thermodynamic fluctuations in a physical variable predict the response quantified by the admittance or impedance of the same physical variable (like voltage, temperature difference, etc.), and vice versa. The fluctuation-dissipation theorem applies both to classical and quantum mechanical systems.
The fluctuation-dissipation theorem says that when there is a process that dissipates energy, turning it into heat (e.g., friction), there is a reverse process related to thermal fluctuations. This is best understood by considering some examples:
If an object is moving through a fluid, it experiences drag (air resistance or fluid resistance). Drag dissipates kinetic energy, turning it into heat. The corresponding fluctuation is Brownian motion. An object in a fluid does not sit still, but rather moves around with a small and rapidly-changing velocity, as molecules in the fluid bump into it. Brownian motion converts heat energy into kinetic energy--the reverse of drag.
If electric current is running through a wire loop with a resistor in it, the current will rapidly go to zero because of the resistance. Resistance dissipates electrical energy, turning it into heat (Joule heating). The corresponding fluctuation is Johnson noise. A wire loop with a resistor in it does not actually have zero current, it has a small and rapidly-fluctuating current caused by the thermal fluctuations of the electrons and atoms in the resistor. Johnson noise converts heat energy into electrical energy--the reverse of resistance.
When light impinges on an object, some fraction of the light is absorbed, making the object hotter. In this way, light absorption turns light energy into heat. The corresponding fluctuation is thermal radiation (e.g., the glow of a "red hot" object). Thermal radiation turns heat energy into light energy--the reverse of light absorption. Indeed, Kirchhoff's law of thermal radiation confirms that the more effectively an object absorbs light, the more thermal radiation it emits.
Examples in detail
The fluctuation-dissipation theorem is a general result of statistical thermodynamics that quantifies the relation between the fluctuations in a system that obeys detailed balance and the response of the system to applied perturbations.
For example, Albert Einstein noted in his 1905 paper on Brownian motion that the same random forces that cause the erratic motion of a particle in Brownian motion would also cause drag if the particle were pulled through the fluid. In other words, the fluctuation of the particle at rest has the same origin as the dissipative frictional force one must do work against, if one tries to perturb the system in a particular direction.
A simple circuit for illustrating Johnson-Nyquist thermal noise in a resistor.
This observation can be understood through the lens of the fluctuation-dissipation theorem. Take, for example, a simple circuit consisting of a resistor with a resistance and a capacitor with a small capacitance . Kirchhoff's law yields
In the low-frequency limit , its imaginary part is simply
which then can be linked to the auto-correlation function of the voltage via the fluctuation-dissipation theorem
The Johnson-Nyquist voltage noise was observed within a small frequency bandwidth centered around . Hence
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 time-varying, spatially constant field which alters the Hamiltonian
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 (very slowly) switched on at .
The fluctuation-dissipation theorem relates the two-sided power spectrum (i.e. both positive and negative frequencies) 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.
We derive the fluctuation-dissipation theorem in the form given above, using the same notation.
Consider the following test case: the field f has been on for infinite time and is switched off at t=0
where is the Heaviside function.
We can express the expectation value of by the probability distribution W(x,0) and the transition probability
The probability distribution function W(x,0) is an equilibrium distribution and hence
given by the Boltzmann distribution for the Hamiltonian
For a weak field , we can expand the right-hand side
here is the equilibrium distribution in the absence of a field.
Plugging this approximation in the formula for yields
where A(t) is the auto-correlation function of x in the absence of a field:
Note that in the absence of a field the system is invariant under time-shifts.
We can rewrite using the susceptibility
of the system and hence find with the above equation (*)
To make a statement about frequency dependence, it is necessary to take the Fourier transform of equation (**). By integrating by parts, it is possible to show that
The fluctuation-dissipation theorem relates the correlation function of the observable of interest (a measure of fluctuation) to the imaginary part of the response function (a measure of dissipation), in the frequency domain. A link between these quantities can be found through the so-called Kubo formula
which follows, under the assumptions of the linear response theory, from the time evolution of the ensemble average of the observable in the presence of a perturbing source. The Kubo formula allows us to write the imaginary part of the response function as
where in the second equality we re-positioned using the cyclic property of trace (in this step we have also assumed that the operator is bosonic, i.e. does not introduce a sign change under permutation). Next, in the third equality, we inserted next to the trace and interpreted as a time evolution operator with imaginary time interval . We can then Fourier transform the imaginary part of the response function above to arrive at the quantum fluctuation-dissipation relation 
where is the Fourier transform of and is the Bose-Einstein distribution function. The "" term can be thought of as due to quantum fluctuations. At high enough temperatures, , i.e. the quantum contribution is negligible, and we recover the classical version.
Violations in glassy systems
While the fluctuation-dissipation theorem provides a general relation between the response of systems obeying detailed balance, when detailed balance is violated comparison of fluctuations to dissipation is more complex. Below the so called glass temperature, glassy systems are not equilibrated, and slowly approach their equilibrium state. This slow approach to equilibrium is synonymous with the violation of detailed balance. Thus these systems require large time-scales to be studied while they slowly move toward equilibrium.
To study the violation of the fluctuation-dissipation relation in glassy systems, particularly spin glasses, Ref.  performed numerical simulations of macroscopic systems (i.e. large compared to their correlation lengths) described by the three-dimensional Edwards-Anderson model using supercomputers. In their simulations, the system is initially prepared at a high temperature, rapidly cooled to a temperature below the glass temperature, and left to equilibrate for a very long time under a magnetic field . Then, at a later time , two dynamical observables are probed, namely the response function
where is the spin living on the node of the cubic lattice of volume , and is the magnetization density. The fluctuation-dissipation relation in this system can be written in terms of these observables as
Their results confirm the expectation that as the system is left to equilibrate for longer times, the fluctuation-dissipation relation is closer to be satisfied.
In the mid-1990s, in the study of dynamics of spin glass models, a generalization of the fluctuation-dissipation theorem was discovered  that holds for asymptotic non-stationary states, where the temperature appearing in the equilibrium relation is substituted by an effective temperature with a non-trivial dependence on the time scales.
This relation is proposed to hold in glassy systems beyond the models for which it was initially found.
The Rényi entropy as well as von Neumann entropy in quantum physics are not observables since they depend nonlinearly on the density matrix. Recently, Ansari and Nazarov proved an exact correspondence that reveals the physical meaning of the Rényi entropy flow in time. This correspondence is similar to the fluctuation-dissipation theorem in spirit and allows the measurement of quantum entropy using the full counting statistics (FCS) of energy transfers.