## Fluctuation-Dissipation theorem ### Classical FDT The classical picture can be considered from the particle diffusion. For a randomly moving particle, its displacement does not have specific preference, which might give the average displacement $\bar{x}$ being zero, but the collection of these particles could have some statistical behavior giving non-zero values, like [[Statistical quantities#^010768|MSD]]. And this big 'jump' from micro picture of individual particles to collection of particles, or [[Ensemble|ensembles]], is connected by FDT. General form writes as $\braket{\delta A(t)\delta A(0)}=2k_{B}T\chi''(\omega) $ The derivation is given in [wiki page](https://en.wikipedia.org/wiki/Fluctuation-dissipation_theorem#Classical_version). If we write $\delta A(t)$ as velocity, it is possible to obtain the relation as $\langle v(0)v(\tau) \rangle = 2d\lambda k_B T \delta(\tau)$ The same form can be obtained by considering the autocorrelation shown in [Langevin equation](https://en.wikipedia.org/wiki/Langevin_equation). This $\lambda$ is a damping coefficient, $d$ is dimensions, and the left part may be written as the form of MSD by integration. But here we just consider the straight forward picture, from MSD and diffusion coefficient. If we interpret the damping coefficient as mobility $\mu$, a quantity showing how particles randomly move, and using [[Einstein relation]], we'll have $\mathrm{MSD} = 2dDt$ $D$ acts as the $\lambda k_{B}T$ in Langevin equation, showing the result of 'damping' together withe thermal motion, or the **dissipation**, to compensate/reflect the **fluctuation** of displacement, the MSD here. >[!Notice] >The Langevin equation was suggested in 1908, which is later than Einstein relation, in 1905. Above text just aim to provide some insights on how to understand FDT, and this is not what happened in history and not how people are thinking to get such relation. ### DFT under linear response theory TBA. >[!Info] >See more on the follow links: >- [Langevin equation](https://en.wikipedia.org/wiki/Langevin_equation) >- [Fluctuation-dissipation theorem - Wikipedia](https://en.wikipedia.org/wiki/Fluctuation-dissipation_theorem) >- https://web.stanford.edu/~peastman/statmech/friction.html