Nonergodicity Mimics Inhomogeneity in Single Particle Tracking

Most statistical theories of anomalous diffusion rely on ensemble-averaged quantities such as the mean squared displacement. Single molecule tracking measurements require, however, temporal averaging. We contrast the two approaches in the case of continuous-time random walks with a power-law distribution of waiting times $\psi (t)\propto t^{-1-\alpha }$, with $0<\alpha <1$, lacking the mean. We show that, contrary to what is expected, the temporal averaged mean squared displacement leads to a simple diffusive behavior with diffusion coefficients that strongly differ from one trajectory to another. This distribution of diffusion coefficients renders a system inhomogeneous: an ensemble of simple diffusers with different diffusion coefficients. Taking an ensemble average over these diffusion coefficients results in an effective diffusion coefficient $K_{\mathrm{eff}}\sim T^{\alpha -1}$ which depends on the length of the trajectory $T$.

Figure 1

Three single-particle trajectories calculated from a CTRW with an asymptotic power-law distribution $\psi (y)\sim t^{-3/2}$. The inhomogeneous nature of the system is obvious and is further emphasized in Fig. 4. Inset: The CTRW model on a two-dimensional lattice. The waiting times are symbolized by circles whose area is proportional to the waiting time a walker spends at a lattice point before the next jump event occurs. The jumps lengths are equidistant.

Figure 2

Time-averaged and ensemble-averaged MSDs for $\alpha =0.8$, $T=2\times {10}^6$, and $t=500$. The red crosses, green stars, and cyan triangles show the time-averaged MSDs of three different single trajectories. The time-averaged MSDs increase linearly with time, $⟨X^2(t){⟩}_T=2Kt$ with different $K$ values. The trajectories have to be long enough to allow for many steps. In the time averaging simulations, a distribution of exponents $\alpha$ is obtained around $\alpha =1$ manifesting another aspect of inhomogeneity. The magenta solid curve is a time-averaged curve with the average diffusion coefficient, and the black dashed curve is the ensemble-averaged MSD which follows $⟨X^2(t){⟩}_{\mathrm{ens}}\sim t^{0.8}$.

Figure 3

The distribution $P(K)$ of diffusion coefficients obtained from time-averaged single trajectories for the case of $\alpha =0.8$, $T=2\times {10}^6$, and $t=500$. Note the inhomogeneous nature of observation, leading to relatively broad distribution of $K$ values.

Figure 4

For each time span $T$, we obtained the average diffusion coefficient ($\overline{K}$) from the distribution $P(K)$. We checked the behavior of the averaged diffusion coefficients for different time spans and obtained good agreements with Eq. (7). We present calculations for $\alpha =0.5$, 0.6, and 0.8. The slopes $b=-0.52$, $-0.40$, and $-0.19$ of the straight lines $y=bx+c$ as obtained by the least squares fits to the points are compatible with the theoretical prediction $b=\alpha -1$.