Anomalous yet Brownian

Abstract

We describe experiments using single-particle tracking in which mean-square displacement is simply proportional to time (Fickian), yet the distribution of displacement probability is not Gaussian as should be expected of a classical random walk but, instead, is decidedly exponential for large displacements, the decay length of the exponential being proportional to the square root of time. The first example is when colloidal beads diffuse along linear phospholipid bilayer tubes whose radius is the same as that of the beads. The second is when beads diffuse through entangled F-actin networks, bead radius being less than one-fifth of the actin network mesh size. We explore the relevance to dynamic heterogeneity in trajectory space, which has been extensively discussed regarding glassy systems. Data for the second system might suggest activated diffusion between pores in the entangled F-actin networks, in the same spirit as activated diffusion and exponential tails observed in glassy systems. But the first system shows exceptionally rapid diffusion, nearly as rapid as for identical colloids in free suspension, yet still displaying an exponential probability distribution as in the second system. Thus, although the exponential tail is reminiscent of glassy systems, in fact, these dynamics are exceptionally rapid. We also compare with particle trajectories that are at first subdiffusive but Fickian at the longest measurement times, finding that displacement probability distributions fall onto the same master curve in both regimes. The need is emphasized for experiments, theory, and computer simulation to allow definitive interpretation of this simple and clean exponential probability distribution.diffusion ͉ fluorescence ͉ imaging ͉ probability distribution ͉ actin T he supposition that Brownian motion follows Gaussian statistics is seldom tested experimentally. In fact, there are few physical situations in which the statistics of displacement distribution can be measured directly. More often, one simply verifies that mean-square displacement is proportional to time, which defines a diffusion coefficient, D. However, a finite diffusion coefficient does not necessarily stem from a Gaussian distribution, because D only holds information regarding second-order cumulants but not the full probability distribution of displacements (1). Relaxation functions in Fourier space are also heavily relied upon in experimental tests of theories that assume Gaussian statistics, but the limited signal-to-noise ratio in measuring these quantities does not typically allow one to deduce distribution functions in real space with the large dynamic range presented by the experiments described below. It is easy to show formally that other zero-centered distributions than Gaussian, in particular exponentially decaying probability distributions of displacement amplitude, would also lead formally to meansquared displacement proportional to time.Here, using single-particle tracking (2), we report the full displacement probability during Brownian m...

Rectification of asymmetric surface vibrations with dry friction: An exactly solvable model

“…In fact, vibrations with PSN statistics should in principle be easier to implement than Gaussian noise, since the fluctuations appear on a separate timescale. Dry friction has recently been shown to be relevant in particular to biological soft matter systems, which opens up the possibility of realizing the transport properties of our model in a nanoscale system [19,[30][31][32]. The thermal environment would provide an additional noise source, but overall the qualitative transport features should remain unchanged since thermal noise does not exert any net force.…”

Effect of Coulombic friction on spatial displacement statistics

“…The other example is the one-dimensional diffusive motion of colloidal polystyrene particles on top of straight bilayer membrane tubes [25]. These tubes were adsorbed on a solid substrate and not moving in the lateral direction.…”

“…(In other words, on the experimental time scales the exponential tails due to an effective Coulombic friction may have left the observation window already, and the exponential tails observed in ref. [25] may have a different source). At this stage, a direct connection between our results and the above experimental results is therefore not possible.…”

Characterizing anomalous diffusion in crowded polymer solutions and gels over five decades in time with variable-lengthscale fluorescence correlation spectroscopy

“…4B), indicating that the MSD is also linear. The fact that mildly entangled fluids can give rise to "anomalous, yet Brownian" diffusion, with a non-Gaussian propagator and yet a linear MSD, ties in with recent observations that this type of behaviour is frequent in biological systems 83,84 . It remains nevertheless a surprising observation, because a linear MSD is often considered to be associated with a Gaussian propagator, and because most classical anomalous diffusion models predict an MSD deviating from linearity.…”

“…We also note that whether diffusion in complex media is characterized by Gaussian propagators is a current topic of interest [80][81][82] . However most of the data available so far has been obtained from single-particle experiments, and therefore is restricted to slow diffusion processes [83][84][85] .…”