Establishing the kinetics of ballistic-to-diffusive transition using directional statistics

We establish the kinetics of ballistic-to-diffusive (BD) transition observed in two-dimensional random walk using directional statistics. Directional correlation is parameterized using the walker's turning angle distribution, which follows the commonly adopted wrapped Cauchy distribution (WCD) function. During the BD transition, the concentration factor (ρ) governing the WCD shape is observed to decrease from its initial value. We next analytically derive the relationship between effective ρ and time, which essentially quantifies the BD transition rate. The prediction of our kinetic expression agrees well with the empirical datasets obtained from correlated random walk simulation. We further connect our formulation with the conventionally used scaling relationship between the walker's mean-square displacement and time.

Figure 1

Examples of the random walk simulated using a WCD function with ${\rho }_1=0.95$. The walker's trajectories observed under normalized timescale $\tau /{\tau }_1=1$ and ${10}^2$ are colored in gray and black, respectively.

Figure 2

(a) Normalized time-averaged mean-square displacement $\langle {\delta }^2\rangle /{\delta }_1^2$ as a function of normalized timescale $\tau /{\tau }_1$ for the random walk simulated with ${\rho }_1=0.95$. Straight lines in the log-log plot have slopes of 2 and 1, corresponding to the values of scaling exponent γ for ballistic motion and diffusion, respectively. (b) Evolution of the walker's turning angle distribution P(θ) with $\tau /{\tau }_1$ increasing from 4 to 128. Circles represent the P(θ) datasets empirically obtained from the CRW simulation. Red lines follow the WCD function [Eq. (1)] parameterized by the ρ values shown in each subpanel.

Figure 3

(a) The random walker is seen at five locations (black dots) when the motion is observed with timescale ${\tau }_i$, which gives rise to three successive turning angles ${\theta }_{i,1}$, ${\theta }_{i,2}$, and ${\theta }_{i,3}$. (b) When the timescale increases by two, that is ${\tau }_{2i}$, the walker could only be seen at three locations (blue dots). Correspondingly one turning angle ${\theta }_{2i}$ is conceived. Vectors shown in black ${\delta }_i$ and blue ${\delta }_{2i}$ represent the net displacements of the walker during ${\tau }_i$ and ${\tau }_{2i}$, respectively. The magnitude of ${\delta }_i$ is assumed to be constant.

Figure 4

Relationship between ${\rho }_{2i}$ and ${\rho }_i$. Solid line follows the solution to the analytical Eq. (6). Circles represent empirical datasets obtained from CRW simulation by performing least square fitting to the measured P(θ) at changing τ. Dotted line follow Eq. (7). The dash-dot line represents a hypothetical relationship ${\rho }_{2i}={\rho }_i$.

Figure 5

(a–c) The decrease in the effective value of ρ for random walker starting with ${\rho }_1=0.99$, 0.95, and 0.50, respectively. (d–f) The corresponding normalized $\langle {\delta }^2\rangle /{\delta }_1^2\sim {\left(\tau /{\tau }_1\right)}^{\gamma }$ scaling relationship. (g–i) The evolution of the scaling exponent γ. In (a–c), solid lines represent solutions to equation Eq. (6). Circles represent empirical datasets obtained from CRW simulation. Squares follow Eq. (7). In (d–i), solid line represents solution to Eqs. (6) and (10). Circles represent empirical datasets calculated from CRW simulation using Eq. (2). Squares represent solution to Eqs. (7) and (10).

Figure 6

Relationship between γ and ρ. Solid line represents the solution to Eqs. (11) and (6). Circles represent empirical datasets obtained from CRW simulation. Dotted line represents the solution to Eqs. (11) and (7).

Figure 7

Contour plots of γ as a function of ${\rho }_1$ and normalized timescale $\tau /{\tau }_1$.

Figure 8

When the walker is observed with timescale ${\tau }_i$, it is seen at five successive locations O, A, B, C, and D.

Figure 9

Enumeration of ${\theta }_{i,1}$.

Figure 10

Enumeration of ${\theta }_{i,2}$.