Subdiffusive behavior in a trapping potential: Mean square displacement and velocity autocorrelation function

A theoretical framework for analyzing stochastic data from single-particle tracking in viscoelastic materials and under the influence of a trapping potential is presented. Starting from a generalized Langevin equation, we found analytical expressions for the two-time dynamics of a particle subjected to a harmonic potential. The mean-square displacement and the velocity autocorrelation function of the diffusing particle are given in terms of the time lag. In particular, we investigate the subdiffusive case. Using a power-law memory kernel, exact expressions for the mean-square displacement and the velocity autocorrelation function are obtained in terms of Mittag-Leffler functions and their derivatives. The behaviors for short-, intermediate-, and long-time lags are investigated in terms of the involved parameters. Finally, the validity of usual approximations is examined.

Figure 1

(Color online) MSD vs time lag for $\lambda =1/2$, ${\gamma }_{\lambda }=1$, and ${\omega }_0=1.4$. Solid line corresponds to the exact solution (37), dotted line to the approximate solution (44), and dashed line to the asymptotic behavior (48).

Figure 2

(Color online) Idem Fig. 1 for ${\omega }_0=0.8$.

Figure 3

(Color online) $C_V$ vs time lag for $\lambda =1/2$, ${\gamma }_{\lambda }=1$, and ${\omega }_0=1.4$. Solid line corresponds to the exact solution (38), dotted line to the approximate solution (45), and dashed line to the asymptotic behavior (49).

Figure 4

(Color online) Idem Fig. 3 for ${\omega }_0=0.8$.