This started as my answer (below) to a question I now cannot find (8^((.  The issue is that the gaussian Diffusion Approximation is being used on a alrge scale in areas in which it is invalid...see attached paper.  There is a trivial diagnostic...if you are doing scattering, and your relaxation is a simple exponential in time, you are not close to the Gaussian Approximation limit.

There is overwhelming experimental evidence that the Langevin equation/central limit theorem image seen in, e.g., Berne and Pecora Chapter 5 is not significant for complex fluids. *Berne and Pecora knew this*, and assumed that their readers (I am roughly of their generation) knew this, which is why they gave readers the later chapters in their book on projection operators and the Mori-Zwanzig theorem.

The core issue is very simple. If we have a random walk composed of a large number of random steps, then the familiar consequence is that the displacements are covered by the Central Limit Theorem. What is much less well understood is that a walk composed of a long series of random steps is described by *two* fundamental results, namely the Central Limit Theorem and Doob's theorem.

The Central Limit Theorem gives you

P(x, t) \sim exp(- x^2/ ) (1)

Here x is the displacement during t, and is the mean-square diffusion during t. I suspect almost no readers are surprised by the above.

However, and equally valid under these conditions, Doob's theorem *guarantees* under the same conditions that

= 2 D t (in one dimension) (2)

where D IS A CONSTANT. The last statement is in Berne and Pecora, Chapter 5, but they do not make a big point of it. After all, everyone knew it was true. You can read Doob in the Dover book on noise and stochastic processes. There are large numbers of literature papers that do not agree with (2).

As a result, if your does not increase linearly in t ("subdiffusion"), then your system is not described by the simple Langevin/Central limit model, and equation 1 is inapplicable. Correspondingly, if you study single-particle motion by scattering (key word 'optical probe diffusion'), and equation 1 is correct, your spectrum (well, dynamic structure factor) must be a single exponential. If your spectrum is not a single exponential, e.g., the spectrum is a sum of exponentials or stretched exponentials, then equation 1 is invalid in your system.

The simple fluids at very short times people apparently knew this for a long time.This result is in Boone and Yip.

Getting the math to come out is a bit tricky. If you just say you have memory (nontrivial memory kernel, random noise still random but with correlations, you get memory kernels and colored noise BUT you generally still get equations 1 and 2. There are nonphysical beautiful elegant math models 'percolation on lattices" that are different than this.

To model a complex fluid, e.g., a colloid suspension or polymer solution, you need to handle the random forces correctly, namely you have one random force with a delta correlation that gives -fv and another random force with a long memory that gives long time displacement correlations. The two random forces have different distributions so their sum IS NOT covered by the central limit theorem. If you go to my researchgate page you can download my 1 1/2 papers on this. (1/2? massive rewrite).

A large number of experimental techniques are being interpreted based on using the central limit theorem result (1) under conditions when it is certainly wrong, e.g., for complex fluids pulsed field gradient NMR, microrheology, biophysics subdiffusion, diffusing wave spectroscopy, QELSS/DLS using the formula g^(1s)(q,t) = exp(- q^2 ) with x(t0^2 not increasing linearly in t. I could go on. I would liek a more complete list.