# Fluctuation dissipation theorem

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In statistical physics, the **fluctuation dissipation theorem** is derived from the assumption that the response of a system in thermodynamic equilibrium to a small external perturbation is the same as its response to a spontaneous fluctuation. There is therefore a direct relation between the fluctuation properties of the thermodynamic system and its linear response properties.

## Example: Brownian motion

For example, Einstein in his 1905 paper on Brownian motion noted that the same random forces which 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 he was able to use statistical mechanics to derive a previously unexpected connection, the Einstein-Smoluchowski relation:

linking *D*, the Diffusion constant, and *μ*, the mobility of the particles. (*μ* is the ratio of the particle's terminal drift velocity to an applied force, *μ = v _{d} / F*).

*k*is Boltzmann's constant, and

_{B}≈ 1.38065 x 10^{-23}m^{ 2}kg s^{-2}K^{-1}*T*is the absolute temperature.

## Example: Thermal noise in a resistor

In 1928, John B. Johnson discovered and Harry Nyquist explained Johnson–Nyquist noise. With no applied current, the mean-square voltage depends on the resistance *R*, *k*_{B}*T*, and the bandwidth Δν over which the voltage is measured:

## General applicability

The example above turns out to be just one example of a very general result in statistical thermodynamics, the *fluctuation dissipation theorem*, which can be used to give an explicit relationship between molecular dynamics at thermal equilibrium, and the macroscopic response that is observed in a dynamic measurement. It thus allows molecular scale models (microscopic models) to be used quantitatively to predict material properties in the context of linear response theory.

The essence of fluctuation-dissipation theorem is that it relates equilibrium fluctuations to out-of-equilibrium quantities, like noise power is related to resistance. The theorem is based on fields that are weak relative to the potential of molecular interaction so that rates of relaxation are not affected by the applied field. "Out-of-equilibrium" in the above sentence should be understood as close to equilibrium or stationary states.

Categories: Statistical mechanics | Non-equilibrium thermodynamics