涨落耗散定理

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.

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=μ k_(B)T,

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 H₀(x) subject to thermal fluctuations.

The observable x(t) will fluctuate around its mean value ⟨ x⟩ ₀ with fluctuations characterized by a power spectrum Sₓ(ω)=⟨ x(ω)x^(*)(ω)⟩.

Suppose that we can switch on a time-varying, spatially constant field f(t) which alters the Hamiltonian to H(x)=H₀(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 χ (t) of the system ⟨ x(t)⟩ =⟨ x⟩ ₀+∫ _(-∞)^(t)f(τ)χ (t-τ) dτ,

where the perturbation is adiabatically (very slowly) switched on at τ =-∞.

The fluctuation–dissipation theorem relates the two-sided power spectrum (i.e.

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)=f₀θ (-t),

where θ (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)

⟨ ⟨ x(t)⟩ =∫ dx'∫ dx x'P(x',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 H(x)=H₀(x)-xf₀

W(x,0)=(exp(-β H(x)))/(∫ dx' exp(-β H(x'))),

where β ⁻¹=k_(B)T.

For a weak field （公式略）, we can expand the right-hand side W(x,0)≈ W₀(x)[1+β f₀(x-⟨ x⟩ ₀)],

here W₀(x) is the equilibrium distribution in the absence of a field. Plugging this approximation in the formula for ⟨ x(t)⟩ yields

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 T_(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.

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.