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 proof that thermodynamic fluctuations in a physical variable predict the response quantified by the admittance or impedance (in their general sense, not only in electromagnetic terms) of the same physical variable (like voltage, temperature difference, etc.
The fluctuation–dissipation theorem was proven by Herbert Callen and Theodore Welton in 1951[1] and expanded by Ryogo Kubo.
There are antecedents to the general theorem, including Einstein's explanation of Brownian motion[2] during his annus mirabilis and Harry Nyquist's explanation in 1928 of Johnson noise in electrical resistors.
This is best understood by considering some examples: 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.
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.
In 1928, John B. Johnson discovered and Harry Nyquist explained Johnson–Nyquist noise.
over which the voltage is measured:[4] This observation can be understood through the lens of the fluctuation-dissipation theorem.
Kirchhoff's voltage law yields and so the response function for this circuit is In the low-frequency limit
, its imaginary part is simply which then can be linked to the power spectral density function
The fluctuation–dissipation theorem relates the two-sided power spectrum (i.e. both positive and negative frequencies) of
, 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
A proof can be found by means of the LSZ reduction, an identity from quantum field theory.
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 The fluctuation-dissipation theorem relates the correlation function of the observable of interest
A link between these quantities can be found through the so-called Kubo formula[5] which follows, under the assumptions of the linear response theory, from the time evolution of the ensemble average of the observable
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
can be easily rewritten as the quantum fluctuation-dissipation relation [6] where the power spectral density
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.
" can be thought of as linked to quantum fluctuations, or to zero-point motion of the observable
, 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, researchers have performed numerical simulations of macroscopic systems (i.e. large compared to their correlation lengths) described by the three-dimensional Edwards-Anderson model using supercomputers.
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.
[9] This relation is proposed to hold in glassy systems beyond the models for which it was initially found.
In systems subjected to an external driving force, which could be an electromagnetic field or a mechanical shear flow, the standard fluctuation-dissipation theorem gets modified because the statistics of the bath is influenced by the driving field.
As a result, the thermal noise becomes biased and the fluctuation-dissipation relation becomes intrinsically non-Markovian, typically with a memory related to the time-autocorrelation of the external field (for the case of a time-dependent external drive).
