# Why is the correlation function of fluctuation force in Brownian motion related to a delta function?

For the first step to derive fluctuation-dissipation theorem, I find

$$\langle F(t)F(t')\rangle=2B\delta(t-t')$$

where $B$ is a constant, and $F(t)$ is a random fluctuating force with Gaussian distribution, which is being called white noise.

Why is the average value related to a delta function? How to derive or verify the equation?

"Why is the average value related to a delta function?"

The delta is indicating that if $F(t) = \mathcal X$, then $F(t+\delta t)$ = a random variable with a Gaussian distribution; no matter how small $\delta t$ is.

In more technical terms, the delta function is the temporal autocorrelation function corresponding to a physical process that has no memory, ie. one "time frame" is completely independent to the next.

And the average is across the ensemble of all possible F(t)s.

"How to derive or verify the equation?"

It is a characteristic of systems without memory. It is an assumption/approximation.

• Thank you, but what if there is memory effect, how to rewrite and derive $\langle F(t)F(t')\rangle$? Commented Jun 15, 2018 at 0:41
• In many systems, the memory is well-described by a decaying exponential with a characteristic correlation time, ie. $\langle F(t) F(t´) \rangle \propto \exp(-t/ \tau)$. An example of this is the magnetic field sensed by a spin on a molecule in a liquid (Redfield relaxation theory). In this case, the correlation time is related to the size and shape of the molecule, which determine how fast it rotates and moves around. Another example is the velocity correlation times in a gas, which are very short (10s of picoseconds, sci-hub.tw/10.1103/PhysRevA.40.2860). Commented Jun 22, 2018 at 21:11
• Deriving these correlation functions from first principles can be done, but it is not a straightforward thing. As an example you can check this out (deriving the velocity autocorrelation function in a dilute hard sphere gas mixture): aip.scitation.org/doi/10.1063/1.439801 Commented Jun 22, 2018 at 21:15
• Thx, I suppose Redfield relaxation theory in NMR you mentioned above has the same physics with the Redfield theory in quantum dynamics, which is up to 2nd order perturbation. In general Redfield has Markovian approx., but it also use an exponential decay memory. A lil bit confused. But I think different model can have different expression for memory kernel. Commented Jun 22, 2018 at 21:34

I know, the prove may be like this (may be in velocity case):

$$v(t)=\dot{x}(t)=\frac{dx}{dt}=\lim_{t \to 0}\frac{\Delta x}{\Delta t}$$

and

$$\Delta x \sim N(0,\Delta t)$$

so

$$=\lim_{t\to0}<\frac{(\Delta x)^2}{(\Delta t)^2}>=\lim_{t\to0}\frac{\Delta t}{(\Delta t)^2}=\lim_{t\to0}\frac 1{\Delta t}=\infty$$

• If you have a new question, please ask it by clicking the Ask Question button. Include a link to this question if it helps provide context. - From Review Commented Apr 19, 2023 at 6:57

[...] and F(t) is a random fluctuating force with Gaussian distribution, which is being called white noise.

(emphasis mine)
White noise is defined as noise that has the same spectral density at all frequencies, i.e., $$S(\omega)=\int dt e^{i\omega \tau}\langle F(t+\tau)F(t)\rangle=const$$ Evaluating the reverse Foruier transform we obtain \langle F(t+\tau)F(t)\rangle=\int \frac{d\omega}{2\pi}e^{-i\omega \tau}S(\omega)=const \times \delta(\tau)$.$