Area under subdiffusive random walks

TL;DR

The paper investigates the area $A(t)=\int_0^t x(\tau)\,d\tau$ and absolute area $\mathcal{A}(t)=\int_0^t |x(\tau)|\,d\tau$ for subdiffusive processes described by SBM, fBM, CTRW, and HBM. It derives the first two moments, PDFs scaling, and ergodicity-breaking parameters for these functionals, with exact results available in Gaussian cases and robust approximations for non-Gaussian models, all validated by Monte Carlo simulations. A key finding is that while the temporal scaling of $\langle Z(t)^2\rangle$ (with $Z=A$ or $\mathcal{A}$) follows a universal $t^{2+\gamma}$ pattern governed by the MSD exponent $\gamma$, the prefactors and tail behaviors depend sensitively on the underlying subdiffusion mechanism, enabling discrimination between models. The results have practical implications for experimental characterization of subdiffusive motion (e.g., via NMR), providing a pathway to infer microscopic transport mechanisms from stochastic functionals of trajectories.

Abstract

We study the statistical properties of the area and the absolute area under the trajectories of subdiffusive random walks. Using different frameworks to describe subdiffusion (as the scaled Brownian motion, fractional Brownian motion, the continuous-time random walk or the Brownian motion in heterogeneous media), we compute the first two moments, the ergodicity breaking parameter for the absolute area and infer a general scaling for the probability density functions of these functionals. We discuss the differences between the statistical properties of the area and the absolute area for the different subdiffusion models and illustrate the experimental interest of our results. Our theoretical findings are supported by Monte Carlo simulations showing an excellent agreement.

Figures (10)

• Figure 1: Schematic picture of the area and absolute area under a random path up to time $t$.
• Figure 2: Second moment of the area $\langle A^{2}(t) \rangle$ as a function of time $t$. (a) Scaled Brownian motion (SBM). The lines correspond to the theoretical prediction given by Eq. \ref{['A2sbm']}. (b) Fractional Brownian motion (fBM). The lines correspond to the theoretical prediction given by Eq. \ref{['A2fbm']}. (c) Continuous Time Random Walk (CTRW). The lines correspond to the theoretical prediction given by Eq. \ref{['a2sct']}. (d) Heterogeneous Brownian motion (HBM). The lines correspond to the theoretical prediction given by Eq. \ref{['A2dx']}. The parameters used in the simulations are listed in \ref{['app:simulations']}.
• Figure 3: Scaled Brownian Motion (SBM). (a) First moment of the absolute area $\langle \mathcal{A}(t) \rangle$ as a function of time $t$. The lines correspond to the theoretical prediction given by Eq. \ref{['aiaaaa']}. (b) Second moment of the area $\langle \mathcal{A}^{2}(t) \rangle$ as a function of time $t$. The lines correspond to the theoretical prediction given by Eq. \ref{['aiaaaa']}. (c) Ergodicity breaking parameter EB as a function of the parameter $\alpha$. The line corresponds to the theoretical prediction given by Eq. \ref{['EBaasbm']}. The simulations details are provided in \ref{['app:simulations']}.
• Figure 4: Fractional Brownian Motion (fBM). (a) First moment of the absolute area $\langle \mathcal{A}(t) \rangle$ as a function of time $t$. The lines correspond to the theoretical prediction given by Eq. \ref{['a1a2aa']}. (b) Second moment of the area $\langle \mathcal{A}^{2}(t) \rangle$ as a function of time $t$. The lines correspond to the theoretical prediction given by Eq. \ref{['a1a2aa']}. (c) Ergodicity breaking parameter EB as a function of the parameter $H$. The line corresponds to the theoretical prediction given by Eq. \ref{['EB_fbm']}. The simulations details are provided in \ref{['app:simulations']}.
• Figure 5: Continuous Time Random Walk (CTRW). (a) First moment of the absolute area $\langle \mathcal{A}(t) \rangle$ as a function of time $t$. The lines correspond to the theoretical prediction given by Eq. \ref{['aa12']}. (b) Second moment of the area $\langle \mathcal{A}^{2}(t) \rangle$ as a function of time $t$. The lines correspond to the theoretical prediction given by Eq. \ref{['aa12']}. (c) Ergodicity breaking parameter $EB$ as a function of the parameter $\alpha$. The line corresponds to the theoretical prediction given by Eq. \ref{['EB_ctrw']}. The simulations details are provided in \ref{['app:simulations']}.