Brownian yet Non-Gaussian Diffusion: From Superstatistics to Subordination of Diffusing Diffusivities

A growing number of biological, soft, and active matter systems are observed to exhibit normal diffusive dynamics with a linear growth of the mean-squared displacement, yet with a non-Gaussian distribution of increments. Based on the Chubinsky-Slater idea of a diffusing diffusivity, we here establish and analyze a minimal model framework of diffusion processes with fluctuating diffusivity. In particular, we demonstrate the equivalence of the diffusing diffusivity process with a superstatistical approach with a distribution of diffusivities, at times shorter than the diffusivity correlation time. At longer times, a crossover to a Gaussian distribution with an effective diffusivity emerges. Specifically, we establish a subordination picture of Brownian but non-Gaussian diffusion processes, which can be used for a wide class of diffusivity fluctuation statistics. Our results are shown to be in excellent agreement with simulations and numerical evaluations.

Popular Summary

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, we establish a physical minimal model for such “Brownian yet non-Gaussian” diffusion.

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.

Our minimal model for the diffusing-diffusivity approach to Brownian yet non-Gaussian diffusion is very versatile, and we believe that it will help establish this approach as a new paradigm in the physics of stochastic processes.

Figure 1

Probability density function $P(x,t)$ in $d=n=1$ at longer times in dimensionless form ($\sigma =\tau =D_{\star }=1$). We compare results from simulations (Sim) of the set of Langevin equations (20a)–(20c), represented by the symbols, with the direct inverse Fourier transform (IFT) of result (44). Excellent agreement is observed.

Figure 2

Mean-squared displacement $⟨x^2(t)⟩$ obtained from simulation of the set of Langevin equations (20a)–(20c) at $d=n=1$, corresponding to the red symbols (because of the density of points, these appear as the thick red line), showing excellent agreement with the Brownian law (1) shown by the (thin) full black line. In the bottom panel, we show $⟨x^2(t)⟩/t$, demonstrating that the deviations from the expected behavior are fairly small. The grey lines (lower panel) show an interval [0.975, 1.024] around unity, based on $10^6$ trajectories.

Figure 3

The kurtosis $K$ for $d=n=1$ defined in Eq. (57), based on relations (1) and (55), is shown by the solid line; simulation results are represented by the symbols. The crossover occurs at the correlation time $\tau =1$. Over the entire displayed time range, the agreement between theory and simulations is excellent.

Figure 4

Probability density function $P(x,t)$ for $d=n=1$ at short times in dimensionless form ($\sigma =\tau =D_{\star }=1$). We compare results from simulations of the set of Langevin equations (20a)–(20c), represented by the symbols, with the explicit short-time solution (63). Excellent agreement is observed; only for the longest time $t=0.5$ do the wings start to show some deviations.

Figure 5

Probability density function $P(x,t)$ for $d=n=1$ from simulations of the Langevin equations (20) for three different times, in dimensionless form ($\sigma =\tau =D_{\star }=1$). Comparison with the Gaussian distribution (79) demonstrates the strongly non-Gaussian behavior at short times and the almost fully Gaussian shape at longer times.

Figure 6

Top panel: Superstatistical probability density function $P_s(x,t)$ according to Eq. (a3) from numerical integration, for exponents $\kappa =0.5$, 1, and 2 (see the figure key). Bottom panel: Convergence of the full numerical solution to the analytical asymptotic form (a8) for $\kappa =0.5$ and 2. All distributions are drawn for $t=1$ (a.u.).