Anomalous diffusion from Brownian motion with random confinement
We present a model of anomalous diffusion consisting of an ensemble of particles undergoing homogeneous Brownian motion except for confinement by randomly placed reflecting boundaries. For power-law distributed compartment sizes, we calculate exact and asymptotic values of the ensemble averaged mean squared displacement and find that it increases subdiffusively, as either a power or the logarithm of time. Numerical simulations show that the probability density function of the displacement is non-Gaussian. We discuss the relevance of the model for the analysis of single-particle tracking experiments and its relation to other sources of subdiffusion. In particular we discuss an intimate connection with diffusion on percolation processes.
In the study of diffusion in physical systems it is well known that a variety of processes lead to diffusion that deviates from pure Brownian motion Havlin and Ben-Avraham (1987); Bouchaud and Georges (1990); Metzler and Klafter (2004); Barkai et al. (2012); Höfling and Franosch (2013); Metzler et al. (2014). The most commonly studied signature of anomalous diffusion is the mean squared displacement (MSD), which often takes the form of a power law,
where is the displacement of a particle at time . For Brownian motion , and for subdiffusive motion . Subdiffusive systems may be successfully modeled by various random processes such as continuous time random walk (CTRW) Scher and Montroll (1975); Klafter et al. (1987); Klafter and Sokolov (2011), fractional Brownian motion Kolmogorov (1940); Mandelbrot and Ness (1968) (FBM), diffusion on percolation Stauffer and Aharony (1991) (DOP), Lorentz models Höfling and Franosch (2013), or by a combination of these Meroz et al. (2010); Weigel et al. (2011). The problem of distinguishing which processes contribute to or modify anomalous diffusion is an active area of research Magdziarz et al. (2009); Magdziarz and Weron (2011); Jeon et al. (2013); Meroz et al. (2013). In particular, one may need to model complicating factors such as confinement. Confinement plays an important role in modifying or attenuating the subdiffusion manifested by the power-law behavior in (1) Meilhac et al. (2006); Condamin et al. (2007); Destainville et al. (2008); Condamin et al. (2008); Denisov et al. (2008); Burada et al. (2009); Neusius et al. (2009); Magdziarz and Klafter (2010); Jeon and Metzler (2012); Bruna and Chapman (2014). But, in this paper we take a different view of the relation of confinement to subdiffusion: We show that random confinement may in fact be the sole cause of observed subdiffusion. While the effect is quite general, the model and analysis are motivated specifically by single particle tracking (SPT) experiments, in particular studies of the biophysics of cells Tolić-Nørrelykke et al. (2004); Golding and Cox (2006); Burov et al. (2011); Giannone et al. (2013). We present a scenario in which disordered confinement produces quantitative signatures of subdiffusion in a typical analysis of SPT data. Given that heterogeneous confining boundaries are often observed in biophysics, it follows that in experiments searching for contributions to the subdiffusion exponent , confinement may not be discarded a priori as a candidate.
Experimental scenario— Consider a number of Brownian particles uniformly distributed in a space that is partitioned into compartments by a random arrangement of reflecting barriers. Apart from the presence of the boundaries, the motion is diffusive, with parameters homogeneous is space and time. We collect an SPT trajectory (time series) for each of several particles sampled uniformly from the space. The trajectory consists of the displacement of each particle from its starting point recorded at a series of times. Typically, the trajectories are first analyzed via the ensemble averaged mean squared displacement (EMSD), or the time-ensemble averaged mean squared displacement (TEMSD), where the TEMSD consists, operationally, of first computing a time (sliding) averaged MSD (TMSD) for each trajectory and then averaging the result over the trajectories. For the minimal model introduced below, we find
The EMSD is unbounded and subdiffusive in the sense of (1) if the distribution of the linear size of the compartments has a sufficiently heavy-tail.
There is no weak ergodicity breaking (if the system is in equilibrium): the EMSD is equal to the TEMSD.
The TMSD (ie for a single trajectory) tends to a constant. That is, the observables of a trajectory are not self-averaging over disorder.
The model— We choose a minimal model that captures the essential features and displays asymptotically subdiffusive motion. We refer to the particular model presented below as the random scale-free confinement model (RSFC). We state the results for the MSD before giving the detailed calculations. In this paper, we treat in detail only one spatial dimension, stating some results for higher dimension at the end, but leaving details to a subsequent paper. The probability density for the displacement of a particle diffusing on a line segment of length with reflecting boundary conditions obeys
with . We assume that the probability for a particle to be found in a segment of length is Pareto-distributed
We find that the asymptotic EMSD is, for
so that the EMSD either decays to a constant for or grows without bound for . For we find
In (4) and (5) angle brackets denote averaging over particle trajectories, the tilde denotes averaging over the disorder (ie over ), and are constants that depend on dimension, and the initial distribution of particles. Examples of curves for three values of showing both bounded and unbounded EMSD are shown in Fig. 1.
Comparison with other sources of subdiffusion— Our random confinement model does not manifest weak ergodicity breaking J. P. Bouchaud (1992), a characteristic whose presence is known to imply dramatic differences between EMSD and TEMSD observed in SPT experiments He et al. (2008); Lubelski et al. (2008); Sokolov (2008); Jeon et al. (2011); Khoury et al. (2011); Kusumi et al. (2012). On the contrary, the model is in equilibrium at , and the increments of are stationary so that the TEMSD and EMSD are equal at all times. Thus, the subdiffusion is due to correlated increments, as is the case for both DOP and FBM. Our simulations show that the PDF of the displacement for large times is non-Gaussian as shown in Fig. 2, a feature shared by DOP, and CTRW, but not FBM. RSFC and DOP (but not FBM and CTRW) are examples of diffusion on disordered, or more broadly, heterogeneous media. A model of this type that is closely related to RSFC considers domains of random scale-free size and diffusivity Massignan et al. (2014). Despite this similarity, the latter model has deeper similarities to other models of heterogeneous transport coefficients Dentz and Berkowitz (2005); Dentz and Bolster (2010); Kang et al. (2011); Kühn et al. (2011); Manzo et al. (2015); Cherstvy and Metzler (2013); Cherstvy et al. (2014); Metzler et al. (2014). When these models do show subdiffusion, it arises from non-stationary increments, and they are thus more closely related to CTRW.
RSFC and percolation— It is interesting to compare RSFC more closely to DOP. We distinguish two cases: The first is diffusion of a particle starting at a randomly chosen site on a percolation process at the critical threshold Stauffer and Aharony (1991) (DOP I). In the second case the initial site is a random site on the critical infinite cluster (DOP II). Both DOP I and DOP II show subdiffusive EMSD, but with differing values of the anomalous diffusion exponent in (1) Ben-Avraham and Havlin (1982); Gefen et al. (1983); Stauffer and Aharony (1991). For DOP II, subdiffusion is due to the fractal properties of the infinite cluster. However, because the volume fraction of the critical infinite cluster is zero, it gives no weight in DOP I where the initial position may be any point on the lattice. As is the case with the pure random confinement of RSFC, in DOP I every particle is confined to a region of finite size. Thus, for both RSFC and DOP I, the time averaged mean squared displacement (TMSD) —the time average of a single trajectory— tends to a constant at long times. In DOP I, the size of the confinement regions (the finite clusters) has a heavy-tailed, power-law distribution. Thus, as for RSFC, averaging over an ensemble of uniformly distributed particles gives an unbounded, subdiffusive MSD, with an additional contribution from the fractal structure of large clusters. In other words, The fact that differs between (I) the walk on all clusters and (II) the walk on the infinite cluster, is due to random, scale-free confinement and RSFC represents an abstraction of this phenomenon.
We address a potential issue in simulations and experiment. In CTRW, a finite number of trajectories show a subdiffusive EMSD no matter how long their duration, though this implies an ever increasing spatial domain. In DOP I and RSFC, the average over a finite number of trajectories will tend to a constant EMSD at long times, even for . But, simulations and SPT trajectories are typically limited to times shorter than the time required to explore the entire experimental domain. Since a heavy-tail () implies that there are compartments whose area is of the order of the experimental domain, the time required to cross these domains will be of the same order as , and thus power-law subdiffusion will be observed as shown in Fig. 1. Note that: In CTRW, no particle is trapped for an infinite time, yet the step rate decreases toward zero. In RSFC, the motion of every particle is bounded for an infinite time, yet the EMSD grows without bound.
Calculation of EMSD— We denote by the solution to (2) with and initial condition . We translate the origin to , and average over to get the density for displacement from initial position for a trajectory sampled from the equilibrium distribution, which we denote by . For this average is
Because is symmetric in , we have
We denote by the EMSD averaged over for a segment of length . For the segment of unit length, this is given by
The EMSD averaged over the disorder is given by
Separating time and space variables, the solution to (2) may be written as an eigenfunction expansion
with coefficients determined by the initial condition and orthogonality of the eigenfunctions. Using the initial condition , we find
Setting shows that
We now consider a segment of length , rather than , and use (10) to write
where we have absorbed the initial term into the sum in order to cancel two diverging quantities below. Averaging over the disorder we have
where and we have defined a generalized entire exponential integral by
The series obtained by expanding the exponential111 is easily seen to be analytic for except for simple poles at . In particular, (13) is analytic at , where , the usual entire exponential integral. This corresponds to the critical value of the power-law exponent . It can be shown that for , , where is the lower incomplete gamma function. Large corresponds to large , so that, using , we find the asymptotic form
where is the Hurwitz Zeta function. Eq. (14) shows that for , converges to a constant at long times, while for , shows subdiffusion. For , we use and to find the asymptotic solution
Non-equilibrium initial conditions— Although we assumed equilibrium above, subdiffusion is also observed for other initial distributions. We performed calculations analogous to those above for the initial distribution , that is, each particle begins at the center of the confinement domain, and obtained the asymptotic form for
where is the Dirichlet eta function. For we found
where is the Riemann zeta function. In two dimensions, ie diffusion on disks of random radius, we obtained the asymptotic forms, for
where is the th positive zero of the Bessel function of the first kind , and is
and is the radial density, ie averaged over azimuthal angle. For , we found
Conclusion— We have demonstrated that heavy-tailed random scale-free confinement gives rise to a subdiffusive EMSD. We have discussed its relation to other sources of anomalous diffusion, in particular its presence in certain diffusive processes on percolation. A number of questions remain, including: Are there random potential fields that lead to RSFC ? What is nature of the propagator (PDF) of RSFC ? …or of autocorrelation of the displacement? What is the anomalous exponent when RSFC is combined with other sources of anomalous diffusion ?
Acknowledgements.Acknowledgments— This work was supported by the European Research Council (ERC) through the project MHetScale (Contract number 617511), Fundació Cellex, ERC AdG Osyris, and Spanish Ministry Project FOQUS (FIS2013-46768-P)
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