Correlations between rare events due to long-term memory

Abstract

Rare events refer to qualitatively unlikely events whose realization can nevertheless have important consequences. Typically, the prediction of the kinetics of these events relies on Arrhenius laws, with exponentially distributed waiting times, and no correlations between successive occurrences. However, this description breaks down in the presence of long-term memory, as has been observed in the contexts of geophysical time series or protein dynamics. So far, existing analytical approaches do not quantify the correlations between rare events due to long-term memory. Here, for non-Markovian Gaussian processes, we determine analytically the impact of long-term memory on the distribution of first and second passage times to a rarely reached threshold. This distribution is non-exponential, thus going beyond the Arrhenius paradigm. We obtain an explicit expression for the covariance between the first and second passage times, and we predict how the mean time to the next extreme event depends on the previous passage time, illustrating the phenomenon of clustering of extreme events. These analytical results, validated through extensive stochastic simulations, shed lights on the strong correlation between successive occurrences of extreme events due to long-term memory.

Figures (4)

• Figure 1: Sketch of the problem. For a non-smoooth Gaussian stochastic process $x(t)$ with long-term memory, what is the distribution of the first passage time $T$ to a rarely reached threshold $L$ ? More generally, what are the correlations between $T$ and the second passage time, $T_2$ (defined as the time to reach again the threshold, measured after a time $\tau_w$ after the first passage) ? The shown trajectory corresponds to a process $x(t)$ satisfying Eq. (\ref{['GLE']}), with $\alpha=1/2$.
• Figure 2: (a) The distribution $F(t)$ of FPTs, renormalized by $F(0)$ for better readability, is plotted for the processes described by Eq. (\ref{['GLE']}), for different values of $L$ (in legend) and $\alpha=1/2$. Symbols: numerical simulations. Dashed lines: theoretical prediction $F(t)/F(0)=e^{-v}\left\{1+ \varepsilon \left[v^{1-\alpha}\left(-2+\alpha+v\right)-\Gamma(3-\alpha)v \right]/[(2-\alpha)(1-\alpha)] \right\}$ with $v=t/\langle T\rangle$ and $\varepsilon=AL^2/(l^2\langle T\rangle^{\alpha})$, which is compatible at order $\varepsilon$ with Eq. (\ref{['Result_h']}). (b) The quantity $d_el^2/L^2$, measuring the deviation of $F(t)$ with respect to the exponential distribution, is plotted against $\langle T \rangle$, for the process described by the generalized Langevin Equation, with $\alpha=1/2, 3/4$ and $9/10$. Symbols are simulation results, for each value of $\alpha$ several values of the time step $dt$ were used, which are indicated in the legend. Lines represent the analytical prediction (\ref{['Predict_de']}). Units were chosen so that $K_\alpha=k=k_B\mathcal{T}=1$.
• Figure 3: (a) Normalized covariance between the first and second passage times $T$ and $T_2$ versus $\langle T\rangle$ for the passive process described by the GLE (\ref{['GLE']}). Symbols: simulations. Lines: analytical prediction (\ref{['CovT1T2']}). Here, $\tau_w=1$. (b) Mean second passage time $T_2$, conditional on $T$, against $T/\langle T\rangle$ for the same process. Symbols: simulation results. Line: prediction (\ref{['T1CondT2']}), with $\varepsilon=L^2A/(l^2\langle T\rangle^{\alpha})$. Parameters: $\tau_w=1.0$, $L/l=3.4$, $dt=2.38\times 10^{-5}$. Units were chosen so that $K_\alpha=k=k_B\mathcal{T}=1$.
• Figure 4: Same quantities as in Figs \ref{['FigFPTDistribution']} and \ref{['Fig_Cov_T1_T2']}, but for the active process described by Eq. (\ref{['ActiveProcess']}). (a) Distribution of FPTs, $F(t)$ (b) Parameter $d_e$ quantifying the deviation to the exponential distribution. (c) Rescaled covariance between the first and second passage times. (d) Mean second passage conditioned on the first passage time. Symbols are simulation results and lines are corresponding theoretical predictions with $\varepsilon=L^2A/(l^2\langle T\rangle^{\alpha})$. Parameters are indicated in the graphs, except for $\tau_w=1.0$ [(c),(d)] and $L/l=2.8$, $dt=1.2\times 10^{-2}$ [(d)].