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Wikipedia / 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.), 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:

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 which connects the diffusion constant D and the particle mobility , the ratio of the particle's terminal drift velocity to an applied force; is the Boltzmann constant, and is the absolute temperature.

Thermal noise in a resistor

In 1928, John B. Johnson discovered and Harry Nyquist explained Johnson–Nyquist noise 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 and a capacitor. Kirchhoff's and so the response function

was observed within a small frequency bandwidth

General formulation

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

be an observable of a dynamical system with Hamiltonian with fluctuations characterized by a power spectrum is characterized to first order by the susceptibility or linear response function

to the imaginary part of the Fourier transform. The left-hand side describes fluctuations, the right-hand side is closely related to the energy dissipated. The spectrum of fluctuations reveal the linear response, because past fluctuations cause future fluctuations via a linear response upon itself.

). 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 has been on for infinite time and is switched off at

is the Heaviside function. We can express the expectation value of by the probability distribution

The probability distribution function W(x,0) is an equilibrium distribution and hence given by the Boltzmann distribution

, we can expand the right-hand side

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

using the susceptibility of the system and hence find with the above

Consequently,

To make a statement about frequency dependence, it is necessary to take the Fourier transform of

Finally, for stationary processes, the Wiener–Khinchin theorem states that the two-sided spectral density is equal to the Fourier transform

Quantum version

The fluctuation-dissipation theorem relates the correlation function (a measure of fluctuation) to the imaginary part of the response function 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 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

with imaginary time factor after Fourier transform

can be easily rewritten as the quantum fluctuation-dissipation relation

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 has an imaginary part originating from the commutation rules of operators. at positive frequencies can also be thought of as linked to spontaneous emission. An often cited result is also the symmetrized power spectral density

" can be thought of as linked to quantum fluctuations, or to zero-point motion, 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, 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., two dynamical observables are probed, namely the response function and the spin-temporal correlation function

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.