Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials

We consider an overdamped Brownian particle moving in a confining asymptotically logarithmic potential, which supports a normalized Boltzmann equilibrium density. We derive analytical expressions for the two-time correlation function and the fluctuations of the time-averaged position of the particle for large but finite times. We characterize the occurrence of aging and nonergodic behavior as a function of the depth of the potential, and we support our predictions with extensive Langevin simulations. While the Boltzmann measure is used to obtain stationary correlation functions, we show how the non-normalizable infinite covariant density is related to the superaging behavior.

Figure 1

Typical trajectories (thin gray) and their time average (thick red) for the Brownian particle moving in an asymptotically logarithmic potential $U\left(x\right)=(U_0/2)\ln (1+x^2)$ (top panel, inset). The parameter $\alpha$ measures the ratio of the depth of the potential $U_0$ and the temperature [see Eq. (10)]; here we used $k_{B}T=0.5$, $\gamma =1$. The top panel shows the nonergodic phase ($U_0=2k_{B}T$), where the time average $\overline{x}\left(t\right)$ does not converge to the ensemble average $\langle x\rangle$ (which is zero, dashed line). The bottom panel shows the ergodic phase ($U_0=4k_{B}T$); here the time average does tend to the ensemble average for long times. Also note the different scales for the $x$ axes.

Figure 2

The correlation function $C(t,t_0)$ for $U\left(x\right)=(U_0/2)\ln (1+x^2)$ [thus $Z=\sqrt{\pi }\Gamma (\alpha -1)/\Gamma (\alpha -1/2)$] and $\alpha =2.5$ ($U_0=1$, $k_{B}T=0.25$, and $\gamma =1$). The solid black line is the analytical expression (44). The red (dark gray) and green (light gray) dots are the results of Langevin simulations for two different times, $t_0=1000$ and $t_0=\infty$ (i.e., starting from the equilibrium distribution). For $t_0=1000$, we observe a transition from the stationary behavior Eq. (47) to the aging form Eq. (46). For $t_0=\infty$, we observe the stationary correlation function (47) at all times $t$.

Figure 3

The normalized correlation function $C(t,t_0)/\langle x^2\left(t_0\right)\rangle$ for $U\left(x\right)=(U_0/2)\ln (1+x^2)$ and $\alpha =1.5$ ($U_0=1$, $k_{B}T=0.5$, and $\gamma =1$). The solid black line is the analytical result (48). The numerical simulations for $t_0=300$ (red/dark gray) and $t_0=3000$ (green/light gray) perfectly confirm the superaging behavior.

Figure 4

Power-law exponent of the long-time variance ${\overline{\sigma }}^2\left(t\right)\simeq t^X$, Eq. (55) (red solid line), and numerical data obtained from Langevin simulations by fitting the long-time behavior up to $t=4000$ (black dots), as a function of the parameter $\alpha =U_0/\left(2k_{B}T\right)+1/2$, for the potential $U\left(x\right)=(U_0/2)\ln (1+x^2)$.

Figure 5

Asymptotic long-time behavior of the variance ${\overline{\sigma }}^2\left(t\right)$ simulated for various values of the parameter $\alpha =U_0/\left(2k_{B}T\right)+1/2$ for the potential $U\left(x\right)=(U_0/2)\ln (1+x^2)$. The dashed lines are the analytical predictions given by Eq. (55). The simulation data are for $k_{B}T=1$, $\gamma =1$.

Figure 6

Prefactor $c_{\alpha }$ of ${\overline{\sigma }}^2\left(t\right)\simeq c_{\alpha }{\left(4Dt\right)}^X$, Eq. (55), as a function of $\alpha$ for $U\left(x\right)=(U_0/2)\ln (1+x^2)$. Note that it is zero at the transition from normal to subdiffusion ($\alpha =1$) and diverges at $\alpha =3$. Inset: Detail for $\alpha <2$. The result for $\alpha >3$ was taken from [6].