Fluctuation-dissipation theorem

In statistical physics , within the framework of the so-called "linear response" theory, the fluctuation-dissipation theorem is derived quantitatively and rigorously from the statistical operator of the system, preferably with the help of the so-called LSZ reduction or the associated Källén - Lehmann representation. The fluctuation-dissipation theorem establishes the connection between spontaneous fluctuations of a system in equilibrium and the reaction of the system to external disturbances. It is one of the most fundamental and difficult results of quantum statistics, which cannot be reproduced here in full generality.

Overview

In terms of content, the theorem states that the reaction of a system in thermal equilibrium to a small external disturbance is the same as its reaction to spontaneous fluctuations and that especially the so-called “dissipative part” of this reaction (ie the “frictional part”) is directly proportional to the fluctuations is. This can be used to establish an explicit relationship between molecular dynamics in thermal equilibrium and the macroscopic response to small time-dependent perturbations that can be observed in dynamic measurements. Thus, the fluctuation-dissipation theorem allows microscopic models of equilibrium statistics to be used to make quantitative predictions about material properties, even if these describe deviations from equilibrium.

Dissipative and reactive components of the reaction function (even and odd components in the frequency spectrum) are linked to one another via so-called Kramers-Kronig relationships .

In its original form, the fluctuation-dissipation theorem states that the friction of a particle suspended in a solvent is quantitatively related to the particle fluctuations caused by the liquid molecules.

But the theorem mentioned is u. a. A substantial tightening in the following respect: It affects not only thermal, but also quantum fluctuations, and indeed in a very precise, but very complex way. It should only be noted that according to the theorem, the fluctuation spectrum formed from two quantum mechanically measurable quantities and in a certain way and the associated dissipation spectrum as a function of angular frequency and temperature (in Kelvin ) are related as follows ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \phi _{A,B}}$ ${\displaystyle \chi ''_{A,B}}$ ${\displaystyle \omega }$ ${\displaystyle T}$

${\displaystyle \phi _{A,B}(\omega )={\hbar }\cdot \coth \left({\frac {\hbar \omega }{2k_{\mathrm {B} }T}}\right)\cdot \chi ''_{A,B}(\omega )\,,}$

d. H. the two quantities and are precisely proportional to one another. In this case, ergodicity provided (i. E., The theorem is valid for. Example, not for glass systems). The function is the hyperbolic cotangent , the Boltzmann constant and is Planck's quantum of action , divided by . For high temperatures, low frequencies or generally under classical conditions, the pre-factor is simplified to ${\displaystyle \phi }$ ${\displaystyle \chi ''}$ ${\displaystyle \coth(x)}$ ${\displaystyle k_{\mathrm {B} }}$ ${\displaystyle \hbar }$ ${\displaystyle 2\pi }$ ${\displaystyle \hbar \to 0}$ ${\displaystyle \chi ''}$ ${\displaystyle {\frac {2k_{\mathrm {B} }T}{\omega }}\,.}$

After the forerunners had long been known, Herbert B. Callen and Theodore Welton proved a general fluctuation-dissipation theorem in 1951. Ryogo Kubo's article provides an overview of the complexity of the mathematical requirements .

Applications of the theorem

Einstein relation

Einstein noted in his 1905 publication on Brownian Molecular Motion that the same random forces that cause a particle to move aimlessly due to Brownian motion create resistance as the particle is pulled through the liquid. In other words: the fluctuations of the particle that is actually at rest have the same origin as the dissipative frictional force, against which one has to work when one pulls the particle in a certain direction. ( Marian Smoluchowski achieved a similar result in 1906).

Based on this observation, they were able to use statistical mechanics to derive an unexpected relationship, the Einstein-Smoluchowski relationship :

${\displaystyle D=\mu \,k_{\mathrm {B} }T}$

It links the diffusion constant (according to the fluctuating force) with the mobility of the particles (according to the dissipation ). Here is the ratio of the final speed ( drift speed ) that the particle can reach under the action of an external force . Next is the Boltzmann constant and the absolute temperature . ${\displaystyle D}$ ${\displaystyle \mu }$ ${\displaystyle \mu =v_{\mathrm {d} }/F}$ ${\displaystyle v_{\mathrm {d} }}$ ${\displaystyle F}$ ${\displaystyle k_{\mathrm {B} }}$ ${\displaystyle T}$

Langevin's equation

For the fluctuating force in a Langevin equation , the law known as "white noise" applies: ${\displaystyle n(t)}$

${\displaystyle \langle n(t)n(t')\rangle ={\frac {2k_{\mathrm {B} }T}{\mu }}\delta (t-t')}$.

Thermal noise in an electrical resistor

If there is no current flowing through a resistor , the following applies

${\displaystyle \langle V^{2}\rangle =4Rk_{\mathrm {B} }T\Delta \nu }$

Here is the voltage , resistance and bandwidth over which the voltage is measured. This Johnson-Nyquist noise was discovered by John B. Johnson in 1928 and explained by Harry Nyquist . ${\displaystyle V}$ ${\displaystyle R}$ ${\displaystyle \Delta \nu }$

Web links

• Gallavotti Fluctuations , Scholarpedia

Individual evidence

1. ↑ Yehuda B. Band, Yshai Avishai: Quantum Mechanics with Applications to Nanotechnology and Information Science . Academic Press, 2012, ISBN 978-0-444-53786-7 .
2. ↑ HB Callen, TA Welton: Irreversibility and Generalized Noise . In: Phys. Rev. Band 83 , 1951, pp. 34 , doi : 10.1103 / PhysRev.83.34 , bibcode : 1951PhRv ... 83 ... 34C .
3. ^ R. Kubo: Statistical Mechanical Theory of Irreversible Processes. In: JPS Journals. 1957, Vol. 12, Issue 6, Pages 570-586, doi : 10.1143 / JPSJ.12.570