Fluctuation-dissipation theorem

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 (to be intended in their general sense, not only in electromagnetic terms) 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 was proven by Herbert Callen and Theodore Welton in 1951 
and expanded by Ryogo Kubo. There are antecedents to the general theorem, including Einstein's explanation of Brownian motion
during his "annus mirabilis" and Harry Nyquist's explanation in 1928 of Johnson noise in electrical resistors.
Qualitative overview and examples.
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:
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.
Brownian motion.
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.
From this observation Einstein was able to use statistical mechanics to derive the Einstein–Smoluchowski relation
which connects the diffusion constant "D" and the particle mobility "μ", the ratio of the particle's terminal drift velocity to an applied force. "k""B" is the Boltzmann constant, and "T" is the absolute temperature.
Thermal noise in a resistor.
In 1928, John B. Johnson discovered and Harry Nyquist explained Johnson–Nyquist noise. With no applied current, the mean-square voltage depends on the resistance formula_2, formula_3, and the bandwidth formula_4 over which the voltage is measured:
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 formula_2 and a capacitor with a small capacitance formula_7. Kirchhoff's law yields
and so the response function for this circuit is
In the low-frequency limit formula_10, its imaginary part is simply
which then can be linked to the auto-correlation function formula_12 of the voltage via the fluctuation-dissipation theorem
The Johnson–Nyquist voltage noise formula_14 was observed within a small frequency bandwidth formula_15 centered around formula_16. Hence
General formulation.
The fluctuation–dissipation theorem can be formulated in many ways; one particularly useful form is the following:
Let formula_18 be an observable of a dynamical system with Hamiltonian formula_19 subject to thermal fluctuations.
The observable formula_18 will fluctuate around its mean value formula_21
with fluctuations characterized by a power spectrum formula_22.
Suppose that we can switch on a time-varying, spatially constant field formula_23 which alters the Hamiltonian
to formula_24.
The response of the observable formula_18 to a time-dependent field formula_23 is 
characterized to first order by the susceptibility or linear response function
formula_27 of the system
where the perturbation is adiabatically (very slowly) switched on at formula_29.
The fluctuation–dissipation theorem relates the two-sided power spectrum (i.e. both positive and negative frequencies) of formula_30 to the imaginary part of the Fourier transform formula_31 of the susceptibility formula_27:
The left-hand side describes fluctuations in formula_30, the right-hand side is closely related to the energy dissipated by the system when pumped by an oscillatory field formula_35.
This is the classical form of the theorem; quantum fluctuations are taken into account by
replacing formula_36 with formula_37 (whose limit for formula_38 is formula_36). 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.
Derivation.
Classical version.
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 formula_41 is the Heaviside function.
We can express the expectation value of formula_30 by the probability distribution "W"("x",0) and the transition probability formula_43
The probability distribution function "W"("x",0) is an equilibrium distribution and hence
given by the Boltzmann distribution for the Hamiltonian formula_45
where formula_47.
For a weak field formula_48, we can expand the right-hand side
here formula_50 is the equilibrium distribution in the absence of a field.
Plugging this approximation in the formula for formula_51 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 formula_53 using the susceptibility
of the system and hence find with the above equation (*)
Consequently,
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
Since formula_56 is real and symmetric, it follows that
Finally, for stationary processes, the Wiener–Khinchin theorem states that the two-sided spectral density is equal to the Fourier transform of the auto-correlation function:
Therefore, it follows that
Quantum version.
The fluctuation-dissipation theorem relates the correlation function of the observable of interest formula_60 (a measure of fluctuation) to the imaginary part of the response function formula_61 in the frequency domain (a measure of dissipation). 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 formula_63 in the presence of a perturbing source. Once Fourier transformed, the Kubo formula allows writing the imaginary part of the response function as
In the canonical ensemble, the second term can be re-expressed as
where in the second equality we re-positioned formula_66 using the cyclic property of trace (in this step we have also assumed that the operator formula_67 is bosonic, i.e. does not introduce a sign change under permutation). Next, in the third equality, we inserted formula_68 next to the trace and interpreted formula_69 as a time evolution operator formula_70 with imaginary time interval formula_71. The imaginary time shift turns into a formula_72 factor after Fourier transform
and thus the expression for formula_74 can be easily rewritten as the quantum fluctuation-dissipation relation 
where the power spectral density formula_76 is the Fourier transform of the auto-correlation formula_77 and formula_78 is the Bose-Einstein distribution function. The same calculation also yields
thus, differently from what obtained in the classical case, the power spectral density is not exactly frequency-symmetric in the quantum limit. Consistently, formula_60 has an imaginary part originating from the commutation rules of operators. The additional "formula_81" term in the expression of formula_82 at positive frequencies can also be thought of as linked to spontaneous emission. An often cited result is also the symmetrized power spectral density
The "formula_84" can be thought of as linked to quantum fluctuations, or to zero-point motion of the observable formula_67. At high enough temperatures, formula_86, 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 formula_87, 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 formula_88 below the glass temperature formula_89, and left to equilibrate for a very long time formula_90 under a magnetic field formula_91. Then, at a later time formula_92, two dynamical observables are probed, namely the response function
and the spin-temporal correlation function 
where formula_95 is the spin living on the node formula_30 of the cubic lattice of volume formula_97, and formula_98 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.
Quantum version.
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.