Any microscopic particle placed in a fluid will move about randomly. This effect, known as Brownian motion, is caused by collisions between the particle and the surrounding molecules. Two hallmarks of Brownian motion are that particles spread out linearly over time, and the probability of finding a particle at a certain position at a given time is mathematically described by a Gaussian function (a bell curve). However, in certain situations, such as individual nematodes (a type of roundworm) or microscopic beads on lipid tubes, this probability behaves quite differently - sometimes as an exponential function. At first, this phenomenon, now observed in a large range of systems, seems to violate a universal mathematical law known as the central limit theorem, which predicts that this probability should converge to a Gaussian function. Here, I will discuss a physical minimal model for such "Brownian yet non-Gaussian" diffusion that we recently proposed [1].
Using analytical calculations and simulations, we show that both the linear spread of particles and an exponential probability distribution can be reconciled when the intensity of the random jiggling of the particles itself becomes a random function of time. We augment the standard Langevin equation - a differential equation that describes the Brownian motion of a particle - with a random noise strength. This "diffusing diffusivity" has an inherent correlation time that defines a crossover from the non-Gaussian probability seen on short time scales to a long-time Gaussian.

[1] A.V. Chechkin, F. Seno, R, Metzler and I.V. Sokolov Brownian yet Non-Gaussian Diffusion: From Superstatistics to Subordination of Diffusing Diffusivities Phys. Rev. X 7, 021002 (2017)