Fluctuation-dissipation theorem

In statistical physics, the fluctuation-dissipation theorem is derived within the framework of the so-called "linear response" theory quantitatively - rigorous from the statistical operator of the system from, and indeed the best with the help of the so-called LSZ reduction or the related Källén - Lehmann representation. It is one of the most fundamental and most difficult results of quantum statistics, which can be reproduced here in full generality not.

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Content implies the theorem that the response of a system in thermal equilibrium to a small external perturbation is the same as its response to spontaneous fluctuations, and that especially the so-called " dissipative part" of this reaction ( that is, the " friction component " ) directly to is proportional to the fluctuations. This can be used to produce an explicit relationship between molecular dynamics in thermal equilibrium and the response to small macroscopic time-dependent disorders, which can be observed in dynamic measurements. This allows the fluctuation-dissipation theorem to use microscopic models of equilibrium statistics in order to make quantitative predictions of material properties, even if these deviations from equilibrium describe.

Dissipative and reactive portion of the reaction function ( odd and even share in the frequency spectrum) are so-called Kramers -Kronig relations are linked.

In its original form, the said fluctuation-dissipation theorem, to the friction of a particle suspended in a solvent is in a quantitative relation to the molecules of the liquid caused by the particles fluctuations.

The theorem is called but, inter alia, in the following respects a substantial intensification: It affects not only thermal but also quantum fluctuations, even in completely precise, but very complex way. It is only noted that relationships covered by the theorem, the fluctuation spectrum of two quantum mechanically measurable quantities A and B formed in a certain way and the associated Dissipationsspektrum as a function of angular frequency and Kelvin temperature T as follows

That is, the two variables and are proportional to each other in a precise manner. It is assumed ergodicity (ie, the theorem does not apply to glass systems). Further, substantially the reciprocal temperature and indeed applies, with the Boltzmann constant. The function coth (x ) is the hyperbolic cotangent, is Planck's constant divided by. For high temperatures, low frequencies, or generally under classical conditions, the prefactor simplifies to before

After predecessors were known for some time, Herbert B. Callen and Theodore Welton proved in 1951 a general fluctuation-dissipation theorem.

Applications of the theorem

Einstein relation

Einstein noted 1905 to its release to Brownian motion, that the same random forces which cause the erratic motion of a particle due to the Brownian motion, causing a resistance when the particles are pulled through the liquid. In other words, the fluctuations of the particle is at rest, actually have the same origin as the dissipative friction force against which you have to work when you pull the particles in a certain direction. ( A similar result reached Marian Smoluchowski 1906).

Based on this observation, it was possible for them to use deriving statistical mechanics an unexpected relationship, the Einstein - Smoluchowski relation:

It links the diffusion constant D (corresponding to the fluctuating force) with the mobility μ of the particles (corresponding to the dissipation). Here, μ = VD / F, the ratio of final velocity Vd can reach the particle under the action of an external force F to the force F. Next kB is the Boltzmann constant and T is the absolute temperature.

Langevin equation

The following applies to the fluctuating force n (t ) in a Langevin equation as a "white noise" designated Act:

Thermal noise in an electrical resistance

Flows at a resistance no electricity, so true

Here, V is the voltage, R is the resistance and the bandwidth over which the voltage is measured. This Johnson - Nyquist noise was discovered in 1928 by John B. Johnson, and declared by Harry Nyquist.