Fluctuation–dissipation theorem

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 $${\displaystyle D=\mu k_{\text{B}}T,}$$ 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 ${\displaystyle R}$, ${\displaystyle k_{\text{B}}T}$, and the bandwidth ${\displaystyle \mathrm{\Delta }\nu }$ over which the voltage is measured: $${\displaystyle ⟨V^2⟩\approx 4R{k}_{\text{B}}T\mathrm{\Delta }\nu .}$$

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 ${\displaystyle R}$ and a capacitor with a small capacitance ${\displaystyle C}$. Kirchhoff's voltage law yields $${\displaystyle V=-R\frac{dQ}{dt}+\frac{Q}{C},}$$ and so the response function for this circuit is $${\displaystyle \chi (\omega )\equiv \frac{Q(\omega )}{V(\omega )}=\frac{1}{\frac{1}{C}-i\omega R}.}$$

In the low-frequency limit ${\displaystyle \omega \ll (RC{)}^{-1}}$, its imaginary part is simply $${\displaystyle \mathrm{Im}\left[\chi (\omega )\right]\approx \omega R{C}^2,}$$ which then can be linked to the power spectral density function ${\displaystyle S_V(\omega )}$ of the voltage via the fluctuation-dissipation theorem: $${\displaystyle S_V(\omega )=\frac{S_Q(\omega )}{C^2}\approx \frac{2k_{\text{B}}T}{C^2\omega }\mathrm{Im}\left[\chi (\omega )\right]=2R{k}_{\text{B}}T.}$$

The Johnson–Nyquist voltage noise ${\displaystyle ⟨V^2⟩}$ was observed within a small frequency bandwidth ${\displaystyle \mathrm{\Delta }\nu =\mathrm{\Delta }\omega /(2\pi )}$ centered around ${\displaystyle \omega =\pm {\omega }_0}$. Hence $${\displaystyle ⟨V^2⟩\approx S_V(\omega )\times 2\mathrm{\Delta }\nu \approx 4R{k}_{\text{B}}T\mathrm{\Delta }\nu .}$$

General formulation

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

Let ${\displaystyle x(t)}$ be an observable of a dynamical system with Hamiltonian ${\displaystyle H_0(x)}$ subject to thermal fluctuations. The observable ${\displaystyle x(t)}$ will fluctuate around its mean value ${\displaystyle ⟨x{⟩}_0}$ with fluctuations characterized by a power spectrum ${\displaystyle S_x(\omega )=⟨\hat{x}(\omega ){\hat{x}}^{\ast }(\omega )⟩}$. Suppose that we can switch on a time-varying, spatially constant field ${\displaystyle f(t)}$ which alters the Hamiltonian to ${\displaystyle H(x)=H_0(x)-f(t)x}$. The response of the observable ${\displaystyle x(t)}$ to a time-dependent field ${\displaystyle f(t)}$ is characterized to first order by the susceptibility or linear response function ${\displaystyle \chi (t)}$ of the system $${\displaystyle ⟨x(t)⟩=⟨x{⟩}_0+{\int }_{-\mathrm{\infty }}^{t}f(\tau )\chi (t-\tau )d\tau ,}$$ where the perturbation is adiabatically (very slowly) switched on at ${\displaystyle \tau =-\mathrm{\infty }.}$

The fluctuation–dissipation theorem relates the two-sided power spectrum (i.e. both positive and negative frequencies) of ${\displaystyle x}$ to the imaginary part of the Fourier transform ${\displaystyle \hat{\chi }(\omega )}$ of the susceptibility ${\displaystyle \chi (t)}$: $${\displaystyle S_x(\omega )=-\frac{2k_{\text{B}}T}{\omega }\mathrm{Im}\hat{\chi }(\omega ),}$$ which holds under the Fourier transform convention ${\displaystyle \hat{f}(\omega )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)e^{-i\omega t}dt}$. The left-hand side describes fluctuations in ${\displaystyle x}$, the right-hand side is closely related to the energy dissipated by the system when pumped by an oscillatory field ${\displaystyle f(t)=F\sin (\omega t+\varphi )}$. 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 ${\displaystyle 2k_{\text{B}}T/\omega }$ with ${\displaystyle \hslash \coth (\hslash \omega /2k_{\text{B}}T)}$ (whose limit for ${\displaystyle \hslash \to 0}$ is ${\displaystyle 2k_{\text{B}}T/\omega }$). 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

$${\displaystyle f(t)=f_0\theta (-t),}$$ where ${\displaystyle \theta (t)}$ is the Heaviside function. We can express the expectation value of x by the probability distribution W(x,0) and the transition probability ${\displaystyle P(x^{\prime },t|x,0)}$

$${\displaystyle ⟨x(t)⟩=\int d{x}^{\prime }\int dx{x}^{\prime }P(x^{\prime },t|x,0)W(x,0).}$$

The probability distribution function W(x,0) is an equilibrium distribution and hence given by the Boltzmann distribution for the Hamiltonian ${\displaystyle H(x)=H_0(x)-x{f}_0}$

$${\displaystyle W(x,0)=\frac{\exp (-\beta H(x))}{\int d{x}^{\prime }\exp (-\beta H(x^{\prime }))},}$$ where ${\displaystyle {\beta }^{-1}=k_{\text{B}}T}$. For a weak field ${\displaystyle \beta x{f}_0\ll 1}$, we can expand the right-hand side

$${\displaystyle W(x,0)\approx W_0(x)[1+\beta f_0(x-⟨x{⟩}_0)],}$$

here ${\displaystyle W_0(x)}$ is the equilibrium distribution in the absence of a field. Plugging this approximation in the formula for ${\displaystyle ⟨x(t)⟩}$ yields

where A(t) is the auto-correlation function of x in the absence of a field:

$${\displaystyle A(t)=⟨[x(t)-⟨x{⟩}_0][x(0)-⟨x{⟩}_0]{⟩}_0.}$$

Note that in the absence of a field the system is invariant under time-shifts. We can rewrite ${\displaystyle ⟨x(t)⟩-⟨x{⟩}_0}$ using the susceptibility of the system and hence find with the above (1)

$${\displaystyle f_0{\int }_0^{\mathrm{\infty }}d\tau \chi (\tau )\theta (\tau -t)=\beta f_{0}A(t)}$$

Consequently,

To make a statement about frequency dependence, it is necessary to take the Fourier transform of (1). By integrating by parts, it is possible to show that $${\displaystyle -\hat{\chi }(\omega )=i\omega \beta {\int }_0^{\mathrm{\infty }}{e}^{-i\omega t}A(t)dt-\beta A(0).}$$

Since ${\displaystyle A(t)}$ is real and symmetric, it follows that $${\displaystyle 2\mathrm{Im}[\hat{\chi }(\omega )]=-\omega \beta \hat{A}(\omega ).}$$

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: $${\displaystyle S_x(\omega )=\hat{A}(\omega ).}$$

Therefore, it follows that $${\displaystyle S_x(\omega )=-\frac{2k_{\text{B}}T}{\omega }\mathrm{Im}[\hat{\chi }(\omega )].}$$

Quantum version

The fluctuation-dissipation theorem relates the correlation function of the observable of interest ${\displaystyle ⟨\hat{x}(t)\hat{x}(0)⟩}$ (a measure of fluctuation) to the imaginary part of the response function ${\displaystyle \text{Im}\left[\chi (\omega )\right]=\left[\chi (\omega )-{\chi }^{\ast }(\omega )\right]/2i}$ in the frequency domain (a measure of dissipation). A link between these quantities can be found through the so-called Kubo formula

$${\displaystyle \chi (t-t^{\prime })=\frac{i}{\hslash }\theta (t-t^{\prime })⟨[\hat{x}(t),\hat{x}(t^{\prime })]⟩}$$

which follows, under the assumptions of the linear response theory, from the time evolution of the ensemble average of the observable ${\displaystyle ⟨\hat{x}(t)⟩}$ in the presence of a perturbing source. Once Fourier transformed, the Kubo formula allows writing the imaginary part of the response function as

$${\displaystyle \text{Im}\left[\chi (\omega )\right]=\frac{1}{2\hslash }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}⟨\hat{x}(t)\hat{x}(0)-\hat{x}(0)\hat{x}(t)⟩e^{i\omega t}dt.}$$

In the canonical ensemble, the second term can be re-expressed as

$${\displaystyle ⟨\hat{x}(0)\hat{x}(t)⟩=\mathrm{Tr}{e}^{-\beta \hat{H}}\hat{x}(0)\hat{x}(t)=\mathrm{Tr}\hat{x}(t)e^{-\beta \hat{H}}\hat{x}(0)=\mathrm{Tr}{e}^{-\beta \hat{H}}\underset{\hat{x}(t-i\hslash \beta )}{\underbrace{e^{\beta \hat{H}}\hat{x}(t)e^{-\beta \hat{H}}}}\hat{x}(0)=⟨\hat{x}(t-i\hslash \beta )\hat{x}(0)⟩}$$

where in the second equality we re-positioned ${\displaystyle \hat{x}(t)}$ using the cyclic property of trace. Next, in the third equality, we inserted ${\displaystyle e^{-\beta \hat{H}}{e}^{\beta \hat{H}}}$ next to the trace and interpreted ${\displaystyle e^{-\beta \hat{H}}}$ as a time evolution operator ${\displaystyle e^{-\frac{i}{\hslash }\hat{H}\mathrm{\Delta }t}}$ with imaginary time interval ${\displaystyle \mathrm{\Delta }t=-i\hslash \beta }$. The imaginary time shift turns into a ${\displaystyle e^{-\beta \hslash \omega }}$ factor after Fourier transform

$${\displaystyle {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}⟨\hat{x}(t-i\hslash \beta )\hat{x}(0)⟩e^{i\omega t}dt=e^{-\beta \hslash \omega }{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}⟨\hat{x}(t)\hat{x}(0)⟩e^{i\omega t}dt}$$

and thus the expression for ${\displaystyle \text{Im}\left[\chi (\omega )\right]}$ can be easily rewritten as the quantum fluctuation-dissipation relation

$${\displaystyle S_x(\omega )=2\hslash \left[n_{\text{BE}}(\omega )+1\right]\text{Im}\left[\chi (\omega )\right]}$$

where the power spectral density ${\displaystyle S_x(\omega )}$ is the Fourier transform of the auto-correlation ${\displaystyle ⟨\hat{x}(t)\hat{x}(0)⟩}$ and ${\displaystyle n_{\text{BE}}(\omega )={\left(e^{\beta \hslash \omega }-1\right)}^{-1}}$ is the Bose-Einstein distribution function. The same calculation also yields

$${\displaystyle S_x(-\omega )=e^{-\beta \hslash \omega }{S}_x(\omega )=2\hslash \left[n_{\text{BE}}(\omega )\right]\text{Im}\left[\chi (\omega )\right]\ne S_x(+\omega )}$$

thus, differently from what obtained in the classical case, the power spectral density is not exactly frequency-symmetric in the quantum limit. Consistently, ${\displaystyle ⟨\hat{x}(t)\hat{x}(0)⟩}$ has an imaginary part originating from the commutation rules of operators. The additional "${\displaystyle +1}$" term in the expression of ${\displaystyle S_x(\omega )}$ at positive frequencies can also be thought of as linked to spontaneous emission. An often cited result is also the symmetrized power spectral density

$${\displaystyle \frac{S_x(\omega )+S_x(-\omega )}{2}=2\hslash \left[n_{\text{BE}}(\omega )+\frac{1}{2}\right]\text{Im}\left[\chi (\omega )\right]=\hslash \coth \left(\frac{\hslash \omega }{2k_{B}T}\right)\text{Im}\left[\chi (\omega )\right].}$$

The "${\displaystyle +1/2}$" can be thought of as linked to quantum fluctuations, or to zero-point motion of the observable ${\displaystyle \hat{x}}$. At high enough temperatures, ${\displaystyle n_{\text{BE}}\approx (\beta \hslash \omega {)}^{-1}\gg 1}$, 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 ${\displaystyle T_{\text{g}}}$, 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. In their simulations, the system is initially prepared at a high temperature, rapidly cooled to a temperature ${\displaystyle T=0.64T_{\text{g}}}$ below the glass temperature ${\displaystyle T_{\text{g}}}$, and left to equilibrate for a very long time ${\displaystyle t_{\text{w}}}$ under a magnetic field ${\displaystyle H}$. Then, at a later time ${\displaystyle t+t_{\text{w}}}$, two dynamical observables are probed, namely the response function $${\displaystyle \chi (t+t_{\text{w}},t_{\text{w}})\equiv {\frac{\mathrm{\partial }m(t+t_{\text{w}})}{\mathrm{\partial }H}|}_{H=0}}$$ and the spin-temporal correlation function $${\displaystyle C(t+t_{\text{w}},t_{\text{w}})\equiv \frac{1}{V}{\sum _x⟨S_x(t_{\text{w}})S_x(t+t_{\text{w}})⟩|}_{H=0}}$$ where ${\displaystyle S_x=\pm 1}$ is the spin living on the node ${\displaystyle x}$ of the cubic lattice of volume ${\displaystyle V}$, and $m(t)\equiv \frac{1}{V}\sum _x⟨S_x(t)⟩$ is the magnetization density. The fluctuation-dissipation relation in this system can be written in terms of these observables as $${\displaystyle T\chi (t+t_{\text{w}},t_{\text{w}})=1-C(t+t_{\text{w}},t_{\text{w}})}$$

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. In particular, the violation of the fluctuation–dissipation theorem is observed for systems in the critical state – at the points of the second-order phase transition and in the low-temperature phase of systems with the BKT phase transition.

Nonequilibrium baths

For a slow probe weakly coupled to a driven or active bath, the reduced dynamics contains a friction and a noise term which are not related via the standard fluctuation-dissipation relation of the second kind. The friction can be decomposed into two contributions, one entropic and one frenetic, following the splitting of the path-space action of the bath. The violation of the fluctuation-dissipation theorem (also known as the Einstein relation) is due to the inequality between the entropic and frenetic parts in the friction, which is typical for nonequilibrium baths.

For the modified fluctuation-dissipation relation of the first kind (yielding nonequilibrium response relations), see Linear response function.

Nonequilibrium driven systems

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). These modified fluctuation-dissipation relations can be derived from a Caldeira-Leggett Hamiltonian for a particle interacting with a thermal bath, where both the particle and the bath respond to the external field.