Fluctuation-dissipation theorem explained

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 (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^[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.^[3]

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:

• Drag and Brownian motion

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.

• Resistance and Johnson noise

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.

• Light absorption and thermal radiation

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.

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

D = {\mu k _{\rm} 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

R , k _{\rm} T , and the bandwidth \Delta\nu over which the voltage is measured:^[4] \langle V ^{2} \rangle ≈ 4 Rk _{\rm} T\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

R and a capacitor with a small capacitance C . Kirchhoff's voltage law yieldsV =- R (dQ)/(+ dt) (Q)/(C)

and so the response function for this circuit is

\chi(\omega)\equiv (Q(\omega))/(= V(\omega)) (1)/((1)/(- i\omega R C))

In the low-frequency limit

\omega\ll (RC) ^{-1} , its imaginary part is simplyIm \left[\chi(\omega)\right] ≈ \omega RC ^{2}

which then can be linked to the power spectral density function

S _{V(\omega)} of the voltage via the fluctuation-dissipation theoremS _{V(\omega) = (S _{Q(\omega)})/(C ^{2})} ≈ (2 k _{\rm} T)/(C ^{2\omega}) Im \left[\chi(\omega)\right] =2 Rk _{\rm} T

The Johnson–Nyquist voltage noise

\langle V ^{2} \rangle was observed within a small frequency bandwidth \Delta \nu = \Delta\omega/( 2 \pi) centered around \omega = \pm \omega _{0} . Hence\langle V ^{2} \rangle ≈ S _{V(\omega) x} 2 \Delta \nu ≈ 4 Rk _{\rm} T\Delta \nu

General formulation

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

Let

x(t) be an observable of a dynamical system with Hamiltonian H _{0(x)} subject to thermal fluctuations.The observable x(t) will fluctuate around its mean value \langle x\rangle _{0} with fluctuations characterized by a power spectrum S _{x(\omega)} = \langle \hat{x}(\omega)\hat{x} ^{*(\omega)} \rangle .Suppose that we can switch on a time-varying, spatially constant field f(t) which alters the Hamiltonianto H(x) = H _{0(x)-f(t)x} .The response of the observable x(t) to a time-dependent field f(t) is characterized to first order by the susceptibility or linear response function\chi(t) of the system\langle x(t) \rangle = \langle x \rangle _{0} + \int ^{t} _{-inf ty} f(\tau) \chi(t - \tau)d\tau,

where the perturbation is adiabatically (very slowly) switched on at

\tau =-inf ty .

The fluctuation–dissipation theorem relates the two-sided power spectrum (i.e. both positive and negative frequencies) of

x to the imaginary part of the Fourier transform \hat{\chi}(\omega) of the susceptibility \chi(t) :$$S\_x(\backslash omega)\; =\; -\backslash frac\; \backslash operatorname\backslash hat(\backslash omega).$$

which holds under the Fourier transform convention

f(\omega) = \int ^{inf ty} _{-inf ty} f(t) e ^{-i\omega} dt . The left-hand side describes fluctuations in x , the right-hand side is closely related to the energy dissipated by the system when pumped by an oscillatory field f(t) = F \sin(\omega t + \phi) . 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

2 k _{B} T / \omega with \hbar \coth(\hbar\omega / 2 k _{BT)} (whose limit for \hbar\to 0 is 2 k _{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.^[1]

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

f(t) = f _{0} \theta( - t) , where \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 P(x',t | x, 0 ) \langle x(t) \rangle = \int dx' \int dx x' P(x',t|x, 0 ) W(x, 0 ) .

The probability distribution function W(x,0) is an equilibrium distribution and hencegiven by the Boltzmann distribution for the Hamiltonian

H(x) = H _{0(x)} - x f _{0} W(x, 0 ) = (\exp( - \beta H(x)))/(\int dx' \exp( - \beta H(x'))) , where \beta ^{-1} = k _{\rm} T .For a weak field \beta x f _{0} \ll 1 , we can expand the right-hand sideW(x, 0 ) ≈ W _{0(x)} [ 1+ \beta f _{0} (x - \langle x \rangle _{0)],}

here

W _{0(x)} is the equilibrium distribution in the absence of a field.Plugging this approximation in the formula for \langle x(t) \rangle yields

Notes and References

1. H.B. Callen . T.A. Welton . Herbert Callen . Theodore A. Welton . 1951 . Irreversibility and Generalized Noise . . 83 . 1 . 34 - 40 . 1951PhRv...83...34C . 10.1103/PhysRev.83.34.
2. Einstein . Albert . Albert Einstein . Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen . . 322 . 8 . 549–560 . May 1905 . 10.1002/andp.19053220806 . 1905AnP...322..549E. free .
3. Nyquist H . Harry Nyquist . 1928 . Thermal Agitation of Electric Charge in Conductors . . 32 . 1 . 110 - 113 . 1928PhRv...32..110N . 10.1103/PhysRev.32.110 .
4. Book: Blundell . Stephen J. . Blundell . Katherine M. . Concepts in thermal physics . 2009 . OUP Oxford.