Reply to Comment on 'Anomalous diffusion originated by two Markovian hopping-trap mechanisms'

The contents of this comment are correct [1]. We acknowledge V P Shkilev for pointing out them. Thus, here we focus on a re-modulation of our conclusions that we estimate to be still valuable for feeding the investigation of anomalous diffusion: we are grateful to the journal for this opportunity. In particular, since the emerging of anomalous diffusion from a pair of Brownian motions—when properly mixed—is proved also by studies after our [2, 3], the general conclusions of our study are not to be rejected but have to be exploited within the specific adopted framework of the continuous-time random walk (CTRW) that is indeed profitable with respect to other approaches, see, e.g. [2–4].

Into its merit, the comment [1] concerns the mean square displacement (MSD) of random walkers and it was already analytically provided for an equivalent system by V P Shkilev [4]. Unfortunately, we were not aware of that paper [4], which indeed would have been relevant for designing our study and interpreting the outputs displayed by our simulations. The MSD from our simulations was reported in figure 4 [5] and, by revising the code, we found that, because of a sampling issue, simulations were actually ran by setting the initial distribution of the walkers according to a Gaussian distribution, that is also the jump-length distribution, rather than according to a Dirac-delta function, as it would have been. Hence, the power-law scaling of the MSD observed in figure 4 [5] is due to such initial condition. When the initial distribution of the walkers is set equal to a Dirac-delta function, then formula (3) from the comment [1] by V P Shkilev fits the outputs. Thus, the resulting process is mainly a Brownian yet non-Gaussian diffusion. In terms of modelling, a post-interpretation of that code command is compatible with the generalisation provided, for example, by Weiss [6, page 50], where the first-step survival probability is different from the survival probability during the rest of the process. In our case, the first-step survival probability follows from a Dirac-delta waiting-time distribution peaked on the time-step of the sampling, and thus different from the double exponential used during the rest of the process.

Clarified this, the rich transport dynamics showed by our model still holds: specifically the statistics of single-trajectory, which are not available from other models [2–4], and are in fact a profitable feature of the CTRW approach. In particular, we would like to explicitly remind the trend of the p-variation test: which may show a monotonic-continuous growing consistently with a typical signature of the motion inside living cells, see figure 8 and related caption [5]. Moreover, we would like to explicitly remind also the behaviour of the time-averaged MSD, see figure 9 and related caption [5]. We observe that the ensemble of walkers' trajectories are characterised by a population of diffusion coefficients and these last may span within a range with a wideness that depends on the probability of occurrence of each hopping-trap mechanism. This fact is responsible for the weak ergodicity breaking.

Finally, our study was theoretically based on the idea that a power law may be approximated by a finite sum of exponentials: this assumption was done in operative analogy with [7, 8] and mathematically supported by [9]. However, while two exponentials and the following weak approximation of a power-lawed waiting-times distribution are sufficient for the emerging of many anomalous aspects of diffusion, the emerging of a power-lawed MSD requires a much more solid approximation. Hence, by reminding from the comment by Shkilev [1] that the studied process is mainly a Brownian yet non-Gaussian diffusion, the corresponding single-trajectory statistics highlight the fact that the origin of a power-lawed MSD can indeed be different in nature from the origin of other features of anomalous diffusion. A re-modulated conclusion of our study is that the Brownian yet non-Gaussian diffusion may indeed be distinct from a genuine anomalous diffusion and this calls for further investigation.

No new data were created or analysed in this study.