Simple Brownian Diffusion: An Introduction to the Standard Theoretical Models

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In very dense two-dimensional lattice gas systems, ultraslow diffusion emerges, as well. Successive waiting times are thus correlated: a short waiting time is followed by a similarly short one, and vice versa for a long waiting time. In the correlated case the gradual increase of the waiting times is distinct from the occasional very long waiting times of the uncorrelated model.

Chapter 3: Introduction to Brownian Motion

Also the trajectories of the two models are very different: without correlations, the long waiting times effect distinct immobilisation events, while for the correlated waiting times, the motion appears almost Brownian, albeit with a gradual increase of the waiting times.

In this process the waiting time on average is an increasing function and diverges in the limit of many steps. Moreover the process ages, as shown via the decaying response of the process to a sinusoidal driving force. The first way is to modify the distribution of jump lengths. This is particularly relevant in the two-dimensional world, in which effectively most human and animal motion occurs.

Stochastic processes

The alternative approach is to introduce a coupling between jump lengths and waiting times. This phenomenon may be referred to as ultraweak ergodicity breaking. To leading order, the time averaged MSDs 51 and 52 do not exhibit ageing in the sense that the measurement time t does not appear explicitly, in contrast to the corresponding forms for the subdiffusive CTRW processes discussed above. As shown in ref. Concurrently, the friction term becomes a convolution integral such that the noise kernel balances the non-local noise to fulfil the generalised fluctuation dissipation relation.

The friction kernel is then equally of power-law form. Note that due to the coupling of the friction kernel and the fractional Gaussian noise via the fluctuation dissipation relation a large instantaneous value of the noise couples to a high effective friction. The fact that the friction increases with how strongly we push a substance is indeed an everyday experience when we deal with viscoelastic substances such as toothpaste, honey, or liquid concrete. If we hit it hard or jump onto the toothpaste tube, the response is that of a very highly elastic substance, causing the tube to explode.

In this regime, that is, the motion is subdiffusive for persistent noise. The FLE can be shown to govern the effective dynamics of a tagged particle in a single file or the motion of a monomer in a long polymer chain. A behaviour similar to that of eqn 71 is observed for FLE motion. Here, however, the noise is internal, i. The power-law relaxation for the time averaged MSD in a viscoelastic environment was indeed observed experimentally by optical tweezers single particle tracking in wormlike micellar solution, as shown in Fig.

For regular Brownian motion physical observables are independent of the ageing time t a. For sufficiently long trajectories, that is, the randomness in single trajectories of SBM is deterministically decreased with time and the process becomes practically reproducible. Equipping the Langevin eqn 77 for SBM with an additional external potential force F x , one can derive the associated Fokker—Planck equation.

The force-free propagator encoded by eqn 82 is exactly that of free FBM, eqn 56 , despite the fundamental difference between the two processes. Clearly this corresponds to a far from thermal equilibrium state. For its interesting behaviour, we mention the associated time averaged MSD in the harmonic confinement.

In a way, the behaviour is opposite to that of subdiffusive CTRWs, for which we observe the thermal plateau for the MSD and a continuing power-law growth for the time averaged MSD, 45 confirmed by experiment. In view of these results SBM represents a very simple model for sub- and superdiffusive anomalous diffusion. However, its physicality is somewhat questionable for most experimental settings, in which the system is connected to a heat bath, or when the system is stationary.

SBM-type dynamics in fact occurs in free granular gases. HDPs represent a deterministic in contrast to random quenched environment. The particle, that is, has the same diffusivity K x each time it returns to the point x. Interestingly, despite the different nature of HDPs they share some common features such as weak ergodicity breaking with renewal CTRW processes.

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Even when the space dependence of the diffusivity becomes annealed, many of these effects are preserved. In that case the MSD assumes an exponential time dependence. This is opposite to the behaviour observed for subdiffusive and superdiffusive fractional diffusion equations. The linear scaling of is nicely fulfilled, and individual realisations show pronounced amplitude scatter around this mean. Concurrently, the ageing behaviour of HDPs with power-law form of K x was numerically shown to be in good agreement with the form 30 of the ageing depression of the CTRW approach.

We note that different forms for the position dependent diffusivity K x such as an exponential and logarithmic dependence were studied in ref. For more details see ref. The fluctuations in the HDP model were analysed in more detail in ref.


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In Fig. Finally, Fig.

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As can be seen from the graph, on finer scales the motion appears to be more like normal diffusion, due to the broad distribution of K x magnitudes, on a coarser resolution the trajectory appears to be similar to that of a subdiffusive CTRW with a diverging characteristic time scale shown in Fig. It will be interesting to compare these results to the behaviour of stochastic HDPs in annealed and quenched environments. We distinguish mathematical fractals with their strict building rules and exact self-similarity from statistical fractals, for which self-similarity is present only in a certain average sense.

Let us briefly address these two concepts. First, mathematical fractals are constructed by iteration. An equilateral triangle is divided into four equilateral triangles and the central one removed. This subdivision rule is repeated, ideally an infinite amount of times. In particular, we see that the generated object contains empty triangles on all scales. To come from one given sector to another, a random walker on such a geometry needs to first locate and then traverse narrow causeways, as sketched in Fig. This considerably slows down the particle propagation in the embedding space.

As said, natural objects are not exact mathematical fractals.

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However, within a certain upper length scale—of the order of the extension of the country—and a lower length scale— e. An important approach to the description of porous or crowded media is the percolation model. Note that when the averaged motion of random walkers placed on all clusters, a different scaling exponent characterises the MSD, for more details see ref. Both overlap perfectly, corroborating the ergodicity of this anomalous diffusion process.

The straight line in Fig. In ref.

Fractal percolation clusters are often used for simulations of free diffusive processes as well as facilitated diffusion processes in the crowded cytoplasm of living biological cells. A fractal support was also diagnosed to be superimposed onto the subdiffusive CTRW motion for the diffusion of potassium channels in the plasma membrane of living human cells in ref. It is important to note that when we consider the motion on all clusters the motion is a forteriori no longer ergodic: a walker moving on a finite, disconnected cluster cannot explore the entire phase space.

While the increments of random walk processes on fractal structures are stationary 31 and the infinite percolation cluster simulations of ref. A second open question is what happens in the presence of a topological bias, for instance, a bias away from the backbones of a diffusion cluster. In that case at least transient non-ergodicity would be expected. Recent experimental studies on the active transport of polystyrene beads in living cells exhibit a particular type of strong anomalous diffusion. Such a behaviour is sometimes called bi-fractal scaling, as the simplest case of multi-fractal behaviour.

For example, FBM, FLE, sub-diffusive decoupled CTRW, and fractional diffusion equations, 42,56 do not predict the piecewise bi-linear scaling 98 of the moments, in other words these popular stochastic models considered above cannot describe the active transport found in ref. In particular, we should not universally accept the special role of the second moment, beyond the fact that it indicates certain deviations from normal behaviour. For more details on the mathematical treatment of the following, see ref.

The position of the particle is then simply. The main ingredients of the stochastic model are the power-law distributed waiting times These can be justified from first principles models or from observations, at least in some systems, compare the discussions in ref.

Hence a hand-waving argument yields the PDF of the position of the particle using the generalised central limit theorem.


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Our argument is flawed since we have neglected the correlation between the jump sized and time in the problem. At least on the stochastic level this may imply that non-linear relations between jump size and waiting times are important. The experimental finding of non-ballistic scaling of large- q moments in experiment the largest q was 8 is an indication for the insights one can achieve by analysis of time dependence of moments. The identity of the observable of interest is thus crucial, e. However, is the infinite density merely a mathematical construction with which we obtain statistical information on the moments of the process, or does it actually contain information on the particle PDF?

Since at all times, one may wonder why a non-normalised solution emerges? In the first way of plotting this curve will be the infinite density, and thus we can estimate this density from numerical or experimental data. Since the latter is very common we believe that infinite densities also have some general validity. In mathematics, infinite densities, briefly discussed here, are a subject of research for many years, in the context of infinite ergodic theory.

The small behaviour of D is not sensitive to the shape of F v provided it is symmetric and in that sense it is universal. For example, for a Gaussian F v , the infinite density is plotted in Fig. We believe that further work on infinite densities is required since only recently these have attracted some attention in statistical physics 83,,— In general, the scatter distribution and the ergodicity breaking parameter of both processes are relatively similar to those of regular Brownian motion.