Time averages in continuous time random walks

Abstract

We investigate the time averaged squared displacement (TASD) of continuous time random walks with respect to the number of steps $N$, which the random walker performed during the data acquisition time $T$. We prove that the TASD, and as well the apparent diffusion constant, grow linearly with $N$, provided the steps possess a fourth moment and can not accumulate in small intervals. Consequently, the fluctuations of the latter are dominated by the fluctuations of $N$, and fluctuations of the walker's thermal history are irrelevant. Furthermore, we show that the relative scatter decays as $1/\sqrt{N}$, which suppresses all non-linear features in a plot of the TASD against the lag time. Parts of our arguments also hold for continuous time random walks with correlated steps.

6 pages, 2 figures

References in corpus (6)

• Random Time-Scale Invariant Diffusion and Transport Coefficients
• Diffusion and subdiffusion of interacting particles on comb-like structures
• Cages and anomalous diffusion in vibrated dense granular media
• Ergodic properties of continuous-time random walks: finite-size effects and ensemble dependences
• Anomalous diffusion in a quenched-trap model on fractal lattices
• Phase Diagram in Stored-Energy-Driven Lévy Flight

Cited by in corpus (1)

• Power spectral density of a single Brownian trajectory: What one can and cannot learn from it