You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We study the statistical properties of return intervals $r$ between successive energy dissipation rates above a certain threshold $Q$ in three-dimensional fully developed turbulence. We find that the distribution function $P_Q(r)$ scales with the mean return interval $R_Q$ as $P_Q(r)=R_Q^{-1}f(r/R_Q)$ except for $r=1$, where the scaling function $f(x)$ has two power-law regimes. The return intervals are short-term and long-term correlated and possess multifractal nature. The Hurst index of the return intervals decays exponentially against $R_Q$, predicting that rare extreme events with $R_Q\to\infty$ are also long-term correlated with the Hurst index $H_\infty=0.639$.'
author:
- Chuang Liu
- 'Zhi-Qiang Jiang'
- Fei Ren
- 'Wei-Xing Zhou'
bibliography:
- 'E:/Papers/Bibliography.bib'
title: 'Scaling and memory in the return intervals of energy dissipation rate in three-dimensional fully developed turbulence'
---

Extreme events are ubiquitous in nature and society and understanding their dynamics is of crucial importance [@Bunde-Kropp-Schellnhuber-2002; @Kondratyev-Varotsos-Krapivin-2006]. However, extreme events are usually rare, which makes it difficult to investigate their occurrence properties. Recently, there is an increasing interest in the study of return intervals or reoccurrence times $r$ between successive events above (or below) some threshold $Q$, aiming at unveiling the laws governing the occurrence of extreme events by studying the statistics of the return intervals for increasing threshold $Q$.

Recent studies show that the long-term correlation structure has essential influence on the statistics of return intervals [@Bogachev-Eichner-Bunde-PAG-2008]. For long-term power-law correlated monofractal records with exponent $\gamma$, numerical analysis illustrates that the distribution density function of the return intervals follows a stretched exponential $P_Q(r)\sim
\exp\left[-b (r/R_Q)^{\gamma}\right]$ with the same exponent $\gamma$ and the return intervals are also long-term correlated, again with the same exponent $\gamma$, where $R_Q$ is the mean return interval associated with threshold $Q$ [@Bunde-Eichner-Havlin-Kantelhardt-2003-PA; @Bunde-Eichner-Kantelhardt-Havlin-2005-PRL; @Altmann-Kantz-2005-PRE]. Uncorrelated records with $\gamma=1$ is a special case, whose return intervals are exponentially distributed. When $0<\gamma\leqslant1$, theoretical analysis with certain approximation shows that the distribution of return interval is Weibull: $P_Q(r) \sim
r^{\gamma-1} \exp(-c r^{\gamma})$ [@Olla-2007-PRE; @Santhanam-Kantz-2008-PRE]. This seems consistent with recent numerical results [@Eichner-Kantelhardt-Bunde-Havlin-2007-PRE].

For multifractal records in the presence or absence of linear correlations, extensive simulations based on the multiplicative random cascade (MRC) model [@Meneveau-Sreenivasan-1987-PRL] and the multifractal random walk (MRW) model [@Bacry-Delour-Muzy-2001-PRE] unveil that the return intervals have a power-law decay in the distribution and are long-term correlated governed by power laws whose exponents depend explicitly on the threshold $Q$, and the conditional return intervals increase as a power-law function of the previous return interval [@Bogachev-Eichner-Bunde-2007-PRL; @Bogachev-Eichner-Bunde-EPJST-2008; @Bogachev-Bunde-2008-PRE]. These results are of particular interest since a variety of time series exhibit multifractal nature. For instance, the returns of common stocks can be well modeled by the multifractal random walk [@Muzy-Delour-Bacry-2000-EPJB; @Bacry-Delour-Muzy-2001-PA], and the statistics of the associated return intervals are found to comply with the numerical prediction [@Bogachev-Eichner-Bunde-2007-PRL; @Bogachev-Bunde-2008-PRE], which can be used to significantly improve risk estimation [@Bogachev-Eichner-Bunde-2007-PRL].

Another important issue of multifractal records concerns the possible scaling behavior of the return interval distributions $P_Q(r)$ over different thresholds $Q$. Numerical simulations of MRC and MRW time series find no evidence of such scaling [@Bogachev-Eichner-Bunde-2007-PRL; @Bogachev-Eichner-Bunde-EPJST-2008; @Bogachev-Bunde-2008-PRE]. However, empirical return interval analysis of financial volatility gives miscellaneous results. Several studies reported that there is a scaling law in the return interval distributions [@Yamasaki-Muchnik-Havlin-Bunde-Stanley-2005-PNAS; @Wang-Yamasaki-Havlin-Stanley-2006-PRE; @Wang-Weber-Yamasaki-Havlin-Stanley-2007-EPJB; @Jung-Wang-Havlin-Kaizoji-Moon-Stanley-2008-EPJB; @Qiu-Guo-Chen-2008-PA], while others argued that the cumulative distributions of return intervals had systematic deviations from scaling and showed multiscaling behaviors [@Wang-Yamasaki-Havlin-Stanley-2008-PRE; @Lee-Lee-Rikvold-2006-JKPS; @Ren-Zhou-2008-EPL; @Ren-Guo-Zhou-2009-PA].

In this Letter, we perform return interval analysis of the energy dissipation rate in three-dimensional fully developed turbulence based on a high-Reynolds turbulence data set collected at the S1 ONERA wind tunnel by the Grenoble group from LEGI [@Anselmet-Gagne-Hopfinger-Antonia-1984-JFM]. The size of the velocity time series $\{v_i: i=1,2,\cdots,N\}$ is about $1.73 \times
10^7$. Using Taylor’s frozen flow hypothesis which replaces a spatial variation of the fluid velocity by a temporal variation measured at a fixed location, the rate of kinetic energy dissipation at position $i$ is $ \epsilon_i \sim \left[ \left(v_{i+1} - v_i
\right) / \delta_\ell \right] ^2$, where $\delta_\ell$ is the resolution (translated in spatial scale) of the measurements. The energy dissipation rate time series exhibit multifractal nature [@Zhou-Sornette-2002-PD] and its Hurst index is $H =
1-\gamma/2=0.81$. Contrary to previous studies, we find that the return interval distributions show two power-law regimes and collapse onto a single curve for different thresholds $Q$. The scaling phenomenon is also observed for conditional interval distributions.

We have calculated the return interval time series $r_i$ for different thresholds $Q$, which can be mapped nonlinearly to the mean return intervals $R_Q$. Logarithmic binning is adopted to construct the distribution density functions $P_Q(r)$. In order to ensure that the bins cover the whole $r$-axis, we use the following procedure. First, the interval $[1,\max(r_i)]$ is partitioned logarithmically into $n-1$ subintervals whose edges are $x_1<x_2<\cdots<x_n$. Then we obtain the sequence $y_i=[x_i]$, where $[x_i]$ is a round function of $x_i$. We discard duplicate integers in the $y_i$ sequence and obtain a new sequence $w_j$. The edge sequence of the bins are determined by $\{e_i\}=\{0.5,\{w_i+0.5\}\}$. For each bin $(e_i,e_{i+1}]$, the empirical density function can be calculated by $$P_Q(r_i) = \frac{\#(e_i<r<e_{i+1})}{\#(r>0)}\frac{1}{e_{i+1}-e_i},
 \label{Eq:Pq:r}$$ where $r_i=(e_i+e_{i+1})/2$ and $\#()$ is the number of return intervals that satisfies the condition in the parenthesis. The empirical distribution of the return intervals is depicted in Fig. \[Fig:RI:PDF\] for $R_Q=50$, $150$ and $500$. We find that the three distributions collapse onto a single curve except for $r=1$, in remarkable contrast to the simulation results for MRC and MRW time series. Two power-law regimes are observed $$R_QP_Q(r)\sim \left\{
 \begin{array}{lllll}
    A_1(r/R_Q)^{-\delta_1} && {\rm{if}}& 1<r<r_c\\
    A_2(r/R_Q)^{-\delta_2} && {\rm{if}}& r>r_c\\
 \end{array}
 \right.,
 \label{Eq:RI:PDF}$$ where $A_1=0.107$ and $A_2=33.4$ are prefactors, the crossover return interval $r_c\approx7R_Q$, and $\delta_1=0.987\pm0.013$ and $\delta_2=3.88 \pm 0.09$. For the shuffled data, the $R_QP_Q(r)$ curves collapse to a single exponential curve.

![\[Fig:RI:PDF\] (color online). Log-log plot of the scaled distribution $R_QP_Q(r)$ as a function of $r/R_Q$ for three different values of $R_Q$. The straight lines are the best fits to power laws. The power-law exponents are $\delta_1=0.987\pm0.013$ and $\delta_2=3.88 \pm 0.09$. For the shuffled data, the $R_QP_Q(r)$ curves collapse to a same exponential curve, as expected.](Fig_RI_PDF.eps){width="7cm"}

In risk estimation, a quantity of great interest is the probability $W_Q(\Delta{t},t)$ that an extreme event occurs after a short time $\Delta{t}\ll t$ from now on, conditioned that the time elapsed $t$ after the occurrence of the previous extreme event [@Bogachev-Eichner-Bunde-2007-PRL]: $$W_Q(\Delta{t}|t)=\frac{\int_t^{t+\Delta{t}}P_Q(t)dt}{\int_t^{\infty}P_Q(t)dt}.
 \label{Eq:Wq}$$ When $t>r_c$, simple algebraic manipulation leads to $$W_Q(\Delta{t}|t)\simeq(\delta_2-1){\Delta{t}}/{t}.
 \label{Eq:Wq2}$$ The probability $W_Q(\Delta{t}|t)$ is found to be proportional to $\Delta{t}$ and inversely proportional to $t$. An intriguing feature is that $W_Q(\Delta{t}|t)$ is independent of the threshold $Q$, which is a direct consequence of the scaling behavior of $P_Q(r)$ shown in Fig. \[Fig:RI:PDF\]. When $t<r_c$, we obtain that $$\begin{aligned}
 W_Q(\Delta{t}|t)&\approx&\frac{(\delta_1-1)\left(\frac{t}{R_Q}\right)^{-\delta_1}\frac{\Delta{t}}{R_Q}}
                       {\left(\frac{t}{R_Q}\right)^{1-\delta_1}
                       +\frac{A_2}{A_1}\frac{\delta_1-1}{\delta_2-1}\left(\frac{r_c}{R_Q}\right)^{1-\delta_2}
                       -\left(\frac{r_c}{R_Q}\right)^{1-\delta_1}
                       }\nonumber\\
                 &\approx& \frac{(\delta_1-1)\left(\frac{t}{R_Q}\right)^{-\delta_1}\frac{\Delta{t}}{R_Q}}
                       {\left(\frac{t}{R_Q}\right)^{1-\delta_1}-1.0308}.
 \label{Eq:Wq1}\end{aligned}$$ We find that $W_Q(\Delta{t}|t)$ is proportional to $\Delta{t}$. However, $W_Q(\Delta{t}|t)$ also depends on $R_Q$.

In order to test the memory effects of the return intervals, we first investigate the conditional PDF $P_Q(r|r_0)$, which is the distribution of return intervals immediately after $r_0$. To gain better statistics, we study $P_Q(r|r_0)$ for a range of $r_0$ rather than individual $r_0$ values. For each threshold $Q$ or $R_Q$, the return intervals sequence are sorted in an increasing order and then divided into eight groups $G_1,\cdots,G_8$ with approximately equal size. An empirical conditional distribution is determined for each $r_0$ group. Figure \[Fig:RI:CondPDF\] shows $P_Q(r|r_0)$ for $r_0\in G_1$ and $r_0\in G_8$. For each group, the three distributions collapse onto a single curve, indicating evident scaling behavior. The figure shows that the probability of finding small (large) $r$ in $G_1$ is enhanced (decreased) compared with $G_8$. This discrepancy in the two groups of distributions unveils the memory effect that large (small) return intervals tend to follow large (small) return intervals. This is true since the distributions associated with different $G_i$ should not exhibit significant discrepancy if there is no memory in the return intervals [@Yamasaki-Muchnik-Havlin-Bunde-Stanley-2005-PNAS].

![\[Fig:RI:CondPDF\] (color online). Scaling and memory in the conditional distributions. The scaled conditional distribution $R_QP_Q(r|r_0)$ is plotted as a function of $r/R_Q$ with $r_{0}$ in $G_1$ and $G_8$ for three different values of $R_Q$.](Fig_RI_CondPDF.eps){width="7cm"}

The memory effect in the conditional distribution $P_Q(r|r_0)$ can also be illustrated by the mean conditional return interval $\langle
r|r_0\rangle$. If there is no memory in the return intervals, $\langle r|r_0\rangle$ does not depend on $r_0$ and is equal to $R_Q$. Figure \[Fig:RI:Mean\] plots the mean conditional return interval $\langle r|r_0\rangle$ as a function of $r_0$ for $R_Q=50$, 150 and 500. It is shown that $\langle r|r_0\rangle$ is a power-law function of $r_0$ when $r_0$ is larger than certain value and the power-law exponent decreases with $R_Q$, which is consistent with many simulational and empirical studies. Also shown in Fig. \[Fig:RI:Mean\] is the mean conditional return interval of the shuffled energy dissipation rate, which does not depend on $r_0$.

![\[Fig:RI:Mean\] (color online). Scaled mean conditional return interval $\langle r|r_0\rangle/R_Q$ versus $r_0/R_Q$ of return interval sequence in turbulence for various $R_Q$. The filled symbols represent the results for the shuffled records without memory. ](Fig_RI_Mean.eps){width="7cm"}

We now study the long-term correlation in the return intervals using the multifractal detrended fluctuation analysis (MFDFA), which is able to extract long-term power-law correlation in non-stationary time series [@Peng-Buldyrev-Havlin-Simons-Stanley-Goldberger-1994-PRE; @Kantelhardt-Bunde-Rego-Havlin-Bunde-2001-PA; @Kantelhardt-Zschiegner-Bunde-Havlin-Bunde-Stanley-2002-PA]. The MFDFA considers the cumulative time series $R_i=\sum_{i=1}^m
(r_i-\langle{r}\rangle)$, which is partitioned into $N_s$ disjoint boxes with the same size $s$. In each box $k$, the local trend is removed from the subseries by a polynomial function and the local rms fluctuation $f_k(s)$ is determined. The overall detrended fluctuation is calculated by $$F_q(s) = \left\{\frac{1}{N_s}\sum_{k=1}^{N_s}[f_k(s)]^q
 \right\}^{1/q}.
  \label{Eq:Fq}$$ By varying the value of $s$, one can expect the detrended fluctuation function $F_q(s)$ scales with the size $s$: $$F_q(s) \sim s^{h(q)},
  \label{Eq:scaling}$$ where $h(q)$ is the generalized Hurst index. The return interval series possesses multifractal nature if and only if $h(q)$ is a nonlinear function of $q$. When $q = 2$, $h(2)$ is nothing but the Hurst index $H$ and the MFDFA reduces to the DFA. The Hurst index $H$ is related to the autocorrelation exponent $\gamma$ by $\gamma =
2-2H$.

We first investigate the linear long-term correlation property of the return intervals using DFA. The dependence of the fluctuation function $F_2$ is drawn in Fig. \[Fig:RI:DFA\] against $s$ for $R_Q=50$, 150, and 500. In all cases, we find nice power-law relation and the scaling range decreases with the increase of $R_Q$. The inset shows the dependence of the Hurst index with respect to $R_Q$, which has an exponential decay: $$H = H_{\infty} + b e^{-R_Q/R_c} = 0.639 + 0.158 e^{-R_Q/69.9},
  \label{Eq:H:Rq}$$ where $R_c=69.9$ is the characteristic scale. For extreme events with very large $Q$, $R_Q$ tends to infinity, and the Hurst index can be predicted as $H=H_{\infty}=0.639$. This implies that the return intervals of those extreme events also exhibit long-term memory.

![\[Fig:RI:DFA\] (color online). Detrended fluctuation analysis of the return interval time series for different $R_Q$. The inset shows the exponential decay of the Hurst index $H$ against $R_Q$.](Fig_RI_DFA.eps){width="7cm"}

We also apply the MFDFA to investigate the multifractal nature of the return intervals. Figure \[Fig:RI:MFDFA\](a) shows the power-law dependence of the overall fluctuation $F_q(s)$ on the scale $s$ for $R_Q=50$. When $R_Q$ increases, the $F_q(s)$ function becomes more noisy, especially for negative $q$. The slopes of the straight lines are the linear least-squares estimates of the generalized Hurst indexes $h(q)$, which are drawn in Fig. \[Fig:RI:MFDFA\](b). The mass scaling exponent function $\tau(q)=qh(q)-1$ and the multifractal spectrum $f(\alpha)$ calculated according to the Legendre transform of $\tau(q)$ are illustrated respectively in Fig. \[Fig:RI:MFDFA\](c) and Fig. \[Fig:RI:MFDFA\](d). The sound nonlinearity in $h(q)$ and $\tau(q)$ is a hallmark of multifractality in the return intervals, whose singularity strength increases with $R_Q$.

![\[Fig:RI:MFDFA\] (color online). Multifractal detrended fluctuation analysis of the return interval time series for different $R_Q$. (a) MFDFA fluctuation function for different $q$ when $R_Q=50$. (b) Generalized Hurst indexes $h(q)$. (c) Mass scaling exponents $\tau(q)$. (d) Multifractal singularity spectra $f(\alpha)$.](Fig_RI_MFDFA_Fq.eps "fig:"){width="4cm"} ![\[Fig:RI:MFDFA\] (color online). Multifractal detrended fluctuation analysis of the return interval time series for different $R_Q$. (a) MFDFA fluctuation function for different $q$ when $R_Q=50$. (b) Generalized Hurst indexes $h(q)$. (c) Mass scaling exponents $\tau(q)$. (d) Multifractal singularity spectra $f(\alpha)$.](Fig_RI_MFDFA_hq.eps "fig:"){width="4cm"} ![\[Fig:RI:MFDFA\] (color online). Multifractal detrended fluctuation analysis of the return interval time series for different $R_Q$. (a) MFDFA fluctuation function for different $q$ when $R_Q=50$. (b) Generalized Hurst indexes $h(q)$. (c) Mass scaling exponents $\tau(q)$. (d) Multifractal singularity spectra $f(\alpha)$.](Fig_RI_MFDFA_tau.eps "fig:"){width="4cm"} ![\[Fig:RI:MFDFA\] (color online). Multifractal detrended fluctuation analysis of the return interval time series for different $R_Q$. (a) MFDFA fluctuation function for different $q$ when $R_Q=50$. (b) Generalized Hurst indexes $h(q)$. (c) Mass scaling exponents $\tau(q)$. (d) Multifractal singularity spectra $f(\alpha)$.](Fig_RI_MFDFA_falpha.eps "fig:"){width="4cm"}

In summary, we have studied the statistical properties of return intervals of the energy dissipation rate in three-dimensional fully developed turbulence. The interval distribution is found to exhibit scaling behavior across different $R_Q$ and two power-law regimes, except for intervals $r=1$. We found that the conditional interval distributions also collapse onto a single curve for same $r_0$, but deviate for different $r_0$, and the mean conditional interval increases as a power law of $r_0$, which indicates the presence of short memory in the return intervals. The long-term memory and the multifractal nature in the return intervals are confirmed by the DFA and MFDFA. The Hurst index of the return intervals decays exponentially against $R_Q$, which allows us to predict the asymptotic Hurst index of return intervals between rare extreme events as $H_\infty=0.639$. Our results signify discrepancy with the numerical results using MRC and MRW models [@Bogachev-Eichner-Bunde-2007-PRL], implying that the energy dissipation process can not be modeled using these models.

This work was partially supported by the National Basic Research Program of China (2004CB217703), the Program for Changjiang Scholars and Innovative Research Team in University (IRT0620), and the Program for New Century Excellent Talents in University (NCET-07-0288).