Exact Results for First-Passage-Time Statistics in Biased Quenched Trap Models

TL;DR

This paper derives exact formulas for the mean and variance of first-passage times in biased quenched trap models, providing insights into diffusion in heterogeneous environments with disorder.

Contribution

It presents the first exact analytical results for FPT statistics in biased quenched trap models, applicable to large systems and nearly all disorder realizations.

Findings

Exact formulas for FPT mean and variance in large systems

Results applicable to nonperiodic heterogeneous environments

Provides a basis for estimating diffusivity in disordered media

Abstract

We provide exact results for the mean and variance of first-passage times (FPTs) of making a directed revolution in the presence of a bias in heterogeneous quenched environments where the disorder is expressed by random traps on a ring with period $L$. FPT statistics are crucially affected by the disorder realization. In the large-$L$ limit, we obtain exact formulae for the FPT statistics, which are described by the sample mean and variance for waiting times of periodically arranged traps. Furthermore, we find that these formulae are still useful for nonperiodic heterogeneous environments; i.e, the results are valid for almost all disorder realizations. Our findings are fundamentally important for the application of FPT to estimate diffusivity of a heterogeneous environment under a bias.

Equations

$${\mu }_L=\frac{1}{L}\sum _{i=0}^{L-1}{\tau }_i$$

$$⟨T{⟩}_L$$

$$⟨\delta T^2{⟩}_L$$

$${\sigma }_L^2=\frac{1}{L}\sum _{i=0}^{L-1}{\tau }_i^2-{\mu }_L^2.$$

$$⟨T_{\mathrm{ctrw}}{⟩}_L$$

$$⟨\delta T_{\mathrm{ctrw}}^2{⟩}_L$$

$$\mathrm{SA}(L;⟨\mathcal{O}{⟩}_L)$$

$$\mathrm{SA}(L;⟨T{⟩}_L)=\frac{⟨{\mu }_L^2{⟩}_{\mathrm{dis}}-⟨{\mu }_L{⟩}_{\mathrm{dis}}^2}{⟨{\mu }_L{⟩}_{\mathrm{dis}}^2}=\frac{⟨{\tau }^2{⟩}_{\mathrm{dis}}-⟨\tau {⟩}_{\mathrm{dis}}^2}{L⟨\tau {⟩}_{\mathrm{dis}}^2}.$$

$$T_{\mathrm{M}}\equiv \frac{⟨T{⟩}_L}{T_{\mathrm{MFPT}}^{\prime }}\mathrm{and}{T}_{\mathrm{V}}\equiv \frac{⟨\delta T^2{⟩}_L}{T_{\mathrm{VFPT}}^{\prime }},$$

$${\eta }_{\nu ,\epsilon }\left(\frac{L}{N_{\mathrm{dis}}}\right)\equiv \frac{1}{N_{\mathrm{dis}}}\sum _{j=1}^{N_{\mathrm{dis}}}{I}_{[1-\epsilon ,1+\epsilon ]}(T_{\nu }^{(j)}),\nu =\mathrm{M},\mathrm{V}$$

$$⟨T_{\mathrm{rw}}{⟩}_L=\frac{L}{p-q}\mathrm{and}⟨\delta T_{\mathrm{rw}}^2{⟩}_L=\frac{4pqL}{(p-q{)}^3}.$$

$$T_{\mathrm{rw}}=\sum _{i=0}^{L-1}{k}_i,$$

$$Z(\lambda )\to \frac{p-q}{e^{-\lambda }-2q},$$

$$⟨k_i⟩\sim 1/(p-q)\mathrm{and}⟨k_i^2⟩-⟨k_i{⟩}^2\sim 2q/(p-q{)}^2$$

$$C_l\equiv ⟨k_i{k}_{i+l}⟩-⟨k{⟩}^2\to \frac{{\epsilon }^l}{(p-q{)}^2}(L\to \infty ),$$

$$⟨T_{\mathrm{rw}}{⟩}_L\sim L⟨k⟩\mathrm{and}⟨T_{\mathrm{rw}}^2{⟩}_L\sim L(⟨\delta k^2⟩+⟨k{⟩}^2)+\sum _{i\ne j}⟨k_i{k}_j⟩,$$

$$⟨\delta T_{\mathrm{rw}}^2{⟩}_L$$

$$C=\frac{q}{(p-q{)}^3}.$$

$$T=\sum _{i=0}^{L-1}{T}_i\mathrm{with}{T}_i=\sum _{l=1}^{k_i}{\tau }_m^{(i)},$$

$$\frac{⟨\delta T^2{⟩}_L}{L}\sim \frac{2q({\sigma }_L^2+{\mu }_L^2)}{(p-q{)}^2}+\frac{⟨\delta {\tau }^2{⟩}_L}{p-q}+\frac{2q{\mu }_L^2}{(p-q{)}^3},$$

$$⟨\delta {\tau }^2{⟩}_L=\frac{1}{L}\sum _{i=0}^{L-1}(⟨({\tau }_l^{(i)}{)}^2⟩-⟨{\tau }_l^{(i)}{⟩}^2).$$

$$⟨\delta x_t⟩\sim L⟨n_t⟩=\frac{(p-q)t}{{\mu }_L}(t\to \infty ).$$

$$⟨\delta x_t^2⟩-⟨\delta x_t{⟩}^2\sim \left(\frac{(p-q)({\sigma }_L^2+{\mu }_L^2)}{{\mu }_L^3}+\frac{2q}{{\mu }_L}\right)t.$$

$$q_z^L=\frac{{\epsilon }^L-{\epsilon }^z}{{\epsilon }^L-1},$$

$$T_{\mathrm{rw}}=\sum _{i=0}^{L-1}{k}_i,$$

$$S(n)\sim p_{L-i}^L\sum _{k=0}^n{}_n{C}_k(p{q}_1^{L-i}{)}^k(q{p}_{L-1}^L{)}^{n-k}p{p}_1^{L-i}=p_{L-i}^L(p{q}_1^{L-i}+q{p}_{L-1}^L{)}^{n}p{p}_1^{L-i}$$

$$S(n)\sim q_{L-i}^L\sum _{k=0}^{n-1}(p{q}_1^L+q{p}_{L-1}^L{)}^{k}p{p}_1^L(p{q}_1^{L-i}+q{p}_{L-1}^L{)}^{n-k-1}p{p}_1^{L-i}+p_{L-i}^L(p{q}_1^{L-i}+q{p}_{L-1}^L{)}^{n}p{p}_1^{L-i}$$

$$S(n)\to (p-q)(2q{)}^n(L\to \infty ),$$

$$Z(\lambda )=\sum _{n=0}^{\infty }S(n)e^{\lambda (n+1)}\to \frac{p-q}{e^{-\lambda }-2q}$$

$$⟨k_i⟩={\frac{\partial Z(\lambda )}{\partial \lambda }|}_{\lambda =0}=\frac{1}{p-q}$$

$$⟨k_i^2⟩-⟨k_i{⟩}^2={\frac{{\partial }^2\ln Z(\lambda )}{{\partial }^2\lambda }|}_{\lambda =0}=\frac{2q}{(p-q{)}^2},$$

Full text

I Introduction

Encountering a reactive molecule or finding a reactive site by a molecule is the first step in chemical reactions. Therefore, finding a specific target in stochastic processes is a fundamental problem in the context of chemical as well as biological reactions Redner (2001). In particular, this target-search problem attracts significant interests in biomolecular reactions in cells such as transcription factors searching for a specific DNA sequence Berg et al. (1981); von Hippel (2007); Mirny (2008). Many stochastic models have been utilized to unravel how biomolecules can efficiently reach the targets in cells Loverdo et al. (2008); Mirny et al. (2009); Lomholt et al. (2009); Bénichou et al. (2011), where a combination of 3D free diffusion and 1D sliding motion on DNA plays a vital role in reducing the first-passage time (FPT) to the target.

The 1D sliding motion is crucially affected by interactions between a searching molecule and DNA sequences Shimamoto (1999). DNA sequences exhibit anomalous fluctuations, such as long correlations and $1/f$ fluctuations Li and Kaneko (1992). Thus, a 1D sliding motion on DNA is described according to the diffusion in a quenched heterogeneous environment. In experiments, the diffusion coefficients of a repressor protein diffusing on DNA are obtained by single-particle-tracking measurements and show large trajectory-to-trajectory fluctuations Wang et al. (2006); Granéli et al. (2006). These fluctuations are evidence of the heterogeneity of the environment. In fact, intrinsic fluctuations of the diffusion coefficients are observed in diffusion on heterogeneous environments such as the quenched trap model (QTM), which represents a random walk (RW) on a random energy landscape, and an annealed model of the QTM, which presents continuous-time RW (CTRW) He et al. (2008); Miyaguchi and Akimoto (2011); *Miyaguchi2015.

The FPT statistics in heterogeneous environments are key quantities in target-search problems Slutsky et al. (2004); Slutsky and Mirny (2004) and play an important role in estimating the diffusion coefficient Zwanzig (1988). In higher dimensions, the CTRW provides a good description of diffusion in quenched heterogeneous environments, and hence the FPT statistics with the CTRW have intensively been studied Rangarajan and Ding (2000); Barkai (2001); Condamin et al. (2007); Krüsemann et al. (2014); *krusemann2015. However, in 1D systems, the diffusion in quenched potentials cannot be approximated with the CTRW, and the FPT statistics is not well understood. It is known that 1D quenched systems exhibit distinct behaviors of diffusion Bouchaud and Georges (1990); Miyaguchi and Akimoto (2011); *Miyaguchi2015. Hence, filling the lacuna on the FPT statistics in 1D systems is critical to update fundamental understanding of diffusion in the quenched potential.

In this paper, we clarify several properties of the FPT statistics inherent to the quenched potential by looking at the 1D biased QTM. We first consider a periodic random potential and derive exact FPT statistics. We next show that the formulae are available to understand the nonperiodic potential also. Although the periodic potential landscape is employed to simplify the setup in this paper, stochastic dynamics in the periodic potential have been intensively studied analytically Reimann et al. (2001); *Reimann2002; Dean et al. (2014) and also experimentally Reimann and Eichhorn (2008); Hayashi et al. (2015); Ma et al. (2015); *Ma2017; Kim et al. (2017). Moreover, a bias in diffusion processes induces surprising phenomena such as giant acceleration of diffusivity in periodic potentials Reimann et al. (2001); *Reimann2002; Reimann and Eichhorn (2008), field-induced superdiffusion Shlesinger (1974); Margolin and Berkowitz (2002); Berkowitz et al. (2006); Burioni et al. (2013); *Burioni2014, and distinct initial ensemble dependence of diffusivity in disordered media such as the CTRW Akimoto et al. (2018a); Hou et al. (2018). Our analysis also unravels several indications of the effects caused by quenched disorder on these phenomena.

II Model and Main Results

We consider effects of bias on the FPT statistics in a quenched heterogeneous environment. In particular, we use a biased RW on a 1D quenched random energy landscape, which is periodically arranged, i.e., a biased QTM with a periodic landscape (see Fig. 1(a)) Bouchaud and Georges (1990). The target is located only at site $L$ only while the energy landscape is periodic. Probabilities of the right and left jumps are given by $p$ and $q=1-p$, respectively. A biased RW implies $p\neq 1/2$, and here we consider $p>1/2$. We assume that the tops of the potentials are flat; i.e., the tops are the same height, implying that probability $p$ does not affect the shape of the random energy landscape. This physical situation is relevant to a biased diffusion in heterogeneous comb-like structures Berezhkovskii et al. (2015), e.g., porous media Berkowitz et al. (2006) and neuronal dendrites Méndez and Iomin (2013); Jose et al. (2018), in which the bias is considered to be a flow in the backbone.

We assume that the lattice constant is set to unity and number $L$ of lattice sites with different energies is finite ($L<\infty$). At each site, depth $E~{}(>0)$ of the energy trap is randomly assigned. In particular, the depths are independent and identically distributed (IID) random variables with an exponential distribution, $\rho(E)=T^{-1}_{g}\exp(-E/T_{g})$, where $T_{g}$ is called the glass temperature. A particle can escape from a trap and jump to one of the nearest neighbors. A waiting time when a particle escapes from the $i$th trap is a random variable, and the distribution follows the exponential distribution with mean $\tau_{i}$: $\psi^{(i)}(\tau)=\tau_{i}^{-1}e^{-\tau/\tau_{i}}$ Mel’nikov (1991). Mean waiting time $\tau_{i}$ follows the Arrhenius law, i.e., $\tau_{i}\propto\exp(E_{i}/T)$, where $E_{i}$ is the depth of the energy at the $i$th trap and $T$ denotes the temperature. Through the Arrhenius law and the energy distribution $\rho(E)$, the probability density function of $\tau_{i}$ follows a power law, i.e., $\psi_{\alpha}(\tau)\propto\tau^{-1-\alpha}$ with $\alpha\equiv T/T_{g}$ Bardou et al. (2002); Akimoto et al. (2018b). When the temperature is below $T_{g}$, i.e., $\alpha<1$, the mean of $\tau_{i}$ diverges, inducing anomalous features such as anomalous diffusion and aging Bouchaud and Georges (1990); Bertin and Bouchaud (2003); Miyaguchi and Akimoto (2011); *Miyaguchi2015. Note that sample mean waiting time

[TABLE]

for a fixed disorder in the QTM with a periodic landscape never diverges when $L<\infty$.

We consider the FPT, i.e., a time when a particle starting from the origin reaches the target (site $L$) for the first time. As the main results of this study, we show the mean FPT (MFPT) and the variance of the FPT (VFPT) for a given quenched periodic landscape for large $L$:

[TABLE]

where $\delta T\equiv T-\langle T\rangle_{L}$ and $\sigma_{L}^{2}$ is the sample variance, i.e.,

[TABLE]

Sample variance $\sigma_{L}^{2}$ quantifies the degree of heterogeneity. The VFPT for $\alpha<1$ becomes $\langle\delta T^{2}\rangle_{L}\sim L(\sigma^{2}_{L}+\mu_{L}^{2})/(p-q)^{2}$ because $\mu_{L}^{2}=o(\sigma^{2}_{L})$ for large $L$. In this paper, we discuss physics behind the exact results and provide a brief sketch of the derivation. We note that the MFPT diverges when $p=q$ because the mean return time to the origin diverges. Therefore, the results include case $p=q$. When the bias is small, $p-q$ can be expressed as $p-q\cong F/T$, where $F$ is an external field. In this situation, the leading orders for small $F$ dependencies of the FPT statistics are represented as $\langle T\rangle_{L}\propto 1/F$ and $\langle\delta T^{2}\rangle_{L}\propto 1/F^{3}$.

A crucial aspect of the FPT statistics is that they are expressed by the statistics of the waiting times in the quenched heterogeneous environment. Note that the results do not explicitly include parameter $\alpha$. Instead, they depend on $\mu_{L}$ and $\sigma_{L}$, which are finite and depend on each realization of the disorder. In addition, the VFPT are connected to the diffusivity in the biased QTM on a ring Reimann et al. (2001); *Reimann2002. Therefore, the results play a significant role in estimating diffusivity, as discussed later.

The comparison of properties of FPTs in the QTM with those in the CTRW is intriguing. In the CTRW the waiting-time distribution is identical for all sites. Although the FPT distribution in CTRW has already been studied Condamin et al. (2007), the explicit forms of the FPT statistics have never been obtained so far. Importantly, our results also lead to the exact expressions of the biased CTRW sup , which are given by

[TABLE]

where $\mu$ and $\sigma^{2}$ are the mean and the variance of the waiting-time distribution. The MFPT and VFPT diverge for $\alpha<1$ and $\alpha<2$, respectively, while the QTM results are finite for all regimes of $\alpha$. The VFPT of the biased-CTRW is not given by a straightforward extension obtained from that of the biased QTM. This is unexpected because the CTRW is believed to be a good approximation of the QTM when a bias is added.

III Sample-to-sample fluctuations

We present numerical verifications of exact results (2) and (3) to see how these formula work for each disorder realization. Figure 2 shows the MFPT and VFPT for $100$ disorder realizations, where the numerical values are plotted as a function of the theoretical values. All the results are collapsed on the $y=x$ line, which shows a perfect agreement between the theory and numerical results. Note that the numerical results are provided for parameter $\alpha=0.5$, where the disorder realizations show large sample-to-sample fluctuations. The MFPT and VFPT also exhibit strong sample-to-sample fluctuations because the disorder strongly affects $\mu_{L}$ and $\sigma_{L}^{2}$. Nevertheless, the theoretical values for different realizations are remarkably correct.

Next, we discuss sample-to-sample fluctuations by considering a disorder average. In general, when the heterogeneity of a disorder realization is sufficiently weak, physical observables comprise self-averaging (SA) properties Bouchaud and Georges (1990). Here, we quantify sample-to-sample fluctuations by the SA parameters defined as follows:

[TABLE]

where $\langle\cdot\rangle_{\rm dis}$ indicates the disorder average and observable $\mathcal{O}$ is $T$ or $\delta T^{2}$. The vanishing of these quantities implies a perfect realization of SA, and hence these parameters systematically quantify a degree of the SA property. This definition is analogous to that of the diffusivity in the QTM discussed in Akimoto et al. (2016); *Akimoto2018.

As $\mu_{L}$ is a random variable, the SA parameter for the MFPT, i.e., $\mathcal{O}=T$, can be rewritten as

[TABLE]

For $\alpha<2$, the SA parameter is infinite because $\langle\tau^{2}\rangle_{\rm dis}$ diverges, while for $\alpha>2$, it vanishes in the large-$L$ limit, implying that the SA is satisfied for $\alpha>2$, while it is violated for $\alpha<2$. Hence, the transition between SA and non-SA occurs at $\alpha_{c}=2$ for quantity $\langle T\rangle_{L}$. Similarly, the transition from SA to non-SA for quantity $\langle\delta T^{2}\rangle_{L}$ can be discussed through Eq. (7). From a similar calculation, the critical value can be easily obtained as $\alpha_{c}=4$. That is, the VFPT has an SA property for $\alpha>4$, while it is broken for $\alpha<4$.

IV Numerical argument of the biased QTM with nonperiodic landscape

We discuss the FPT statistics in the biased QTM in which the potentials are arranged randomly in the infinite line (Fig. 1(b)). For a finite bias and large $L$, a particle will experience deep potentials mainly in the positive regime $i>0$. Therefore, the FPT might be dominated by waiting times for regime $i>0$. If this is true, the exact results for the periodic QTM may still be useful to understand the FPT statistics in the infinite 1D systems. To discuss the validity of this theory, we define the following two quantities

[TABLE]

where $T_{\rm MFPT}^{\prime}$ and $T_{\rm VFPT}^{\prime}$ are the same expressions as in Eqs. (2) and (3), respectively. We should note that $\mu_{L}$ and $\sigma_{L}^{2}$ in these expressions are calculated from $\tau_{i}$ for $i=0,\cdots,L-1$ (in other words, we do not use the information of potentials for $i<0$). In addition, to quantify how our prediction works well, we introduce the following ratio:

[TABLE]

where $T_{\nu}^{(j)}$ is a numerical value of $T_{\nu}$ for the $j$th realization of disorder and $I_{A}(x)$ is an indicator function, i.e., $I_{A}(x)=1$ if $x\in A$ and $I_{A}(x)=0$ otherwise. This quantifies a ratio that $T_{\nu}^{(j)}$ is within the corresponding theory with a $\varepsilon$-dependent accuracy. As shown in Fig. 3(a), the ratio approaches to 1 with increasing $L$.

To understand more details at the level of each disorder realization, we next consider each $T_{\nu}^{(j)}$. In Fig. 3(b), we present numerical data of $T_{\nu}^{(j)}$as a function of $T_{\rm MFPT}^{\prime}$ or $T_{\rm VFPT}^{\prime}$ depending on $\nu={\rm M}$ or ${\rm V}$, respectively. Figure 3(b) shows that $T_{\rm MFPT}^{\prime}$ and $T_{\rm VFPT}^{\prime}$ are very good approximations of $\langle T\rangle_{L}$ and $\langle\delta T^{2}\rangle_{L}$ for almost all realizations, except for small number of realizations with extremely large deviations. In such rare samples, significantly large waiting times are assigned for $i<0$ and small $|i|$ (see sup ). Except for such rare samples, the biased-QTM results with a periodic landscape are surprisingly useful in nonperiodic landscapes.

V Derivation of Main results

We now briefly describe the derivation of our results. We divide our explanation into two steps: step 1 explains about the FPT in the standard biased RW (without random traps), and step 2 explains about the FPT in the biased QTMs with periodic landscapes.

V.1 Step 1: Statistics of the numbers of visits

In the first step, we outline our main strategy to derive the main results for the QTM, which gives us another derivation for known results of the FPT statistics in the classical RW. The main strategy is to use statistics of the number of visits at each site. A similar quantity has also been employed to study diffusion of nonbiased motions Bouchaud and Georges (1990); Burov and Barkai (2011). A biased RW was studied in the context of the classical ruin problems Feller (1968). In the ruin problems, a gambler with a capital wins or loses a dollar with probabilities $p$ or $q$, respectively. The FPT from the origin to $L$ site in the RW, i.e., $T_{\rm rw}$, correspond to a duration of the game in the ruin problems, in which the game is over when one of the two players is ruined. Here, we consider that one of the players has infinite capital and his win probability is $p>1/2$. The generating function for the FPT in the classical ruin problems was derived in Feller (1968), and the MFPT and VFPT are respectively obtained as

[TABLE]

While these results are exact for any $L$, we will consider the large-$L$ limit to derive the biased-QTM results. Note that $\langle T_{\rm rw}\rangle$ is a special case of Eq. (2), while $\langle\delta T_{\rm rw}^{2}\rangle_{L}$ is not a special case of Eq. (3) (see Eq. (18) for the general result, which reproduces $\langle\delta T_{\rm rw}^{2}\rangle_{L}$).

The FPT from the origin to $L$ site can be represented by the sum of the numbers of visits at each site, i.e.,

[TABLE]

where $k_{i}$ is the number of visits to the $i$th site until the particle reaches site $L$. Note that $k_{i}$ includes the number of the visits at sites $+i-nL$ ($n=1,2,\cdots$). To obtain the moments and correlation function of $k_{i}$, we consider the large-$L$ limit. In this limit, the probability that a particle reaches site $-L$ becomes zero; i.e., a particle never visits the site $-L$. In the large-$L$ limit, one can obtain the generating function of $k_{i}$ until the particle reaches site $L$ (see Appendix A):

[TABLE]

which yields the followings:

[TABLE]

for $L\to\infty$. Moreover, correlation $\langle k_{i}k_{j}\rangle$ can be obtained exactly as follows:

[TABLE]

where $\varepsilon=q/p$, $\langle k\rangle=1/(p-q)$, and $l\equiv|i-j|$. This correlation is derived in Appendix B. Note that $C_{l}$ does not depend on $i$, and $l$ is an arbitrary integer satisfying $l<L-i$. The MFPT and second moment of the FPT can be respectively represented as

[TABLE]

where $\langle\delta k^{2}\rangle=2q/(p-q)^{2}$. Thus, the VFPT becomes

[TABLE]

where $C=\sum_{l=1}^{L-1}C_{l}$. Note that $\sum_{l=1}^{L-1}lC_{l}$ converges to a constant for $L\to\infty$ because $C_{l}$ decays exponentially to zero. In the large-$L$ limit, we have

[TABLE]

This leads to the desired results of the MFPT and VFPT. Note that the RW results of the FPT statistics are exact for any $L$ Feller (1968).

V.2 Step 2: Derivation of the QTM results

By using the same technique used in step 1, the biased-QTM results can be derived. Note that the FPT in the QTM can be obtained by

[TABLE]

where $\tau_{m}^{(i)}$ is the waiting time for the $m$th visit to site $i$ and $T_{i}$ is the occupation time at site $i$. The mean of $T_{i}$ can be calculated as $\langle T_{i}\rangle=\langle\tau_{1}^{(i)}+\cdots+\tau_{k_{i}}^{(i)}\rangle=\langle k\rangle\tau_{i}.$ Thus, the MFPT in the large-$L$ limit is given by $\langle T\rangle_{L}\sim\langle k\rangle\sum_{i=0}^{L-1}\tau_{i}\sim L\mu_{L}/(p-q)$, i.e, Eq. (2). This is a simple extension of the MFPT for a biased RW and is easily obtained by multiplying $\langle T_{\rm rw}\rangle_{L}$ by $\mu_{L}$.

Using $\langle T_{i}^{2}\rangle$, we also have the VFPT in the biased QTM (the details are given in Appendix C). In the large-$L$ limit, the VFPT for $\alpha>1$ becomes

[TABLE]

where

[TABLE]

For $\alpha<1$, $\mu_{L}^{2}$ can be ignored because $\mu_{L}^{2}=o(\sigma_{L}^{2})$. In the QTM, $\langle\delta\tau^{2}\rangle_{L}=\sigma_{L}^{2}+\mu_{L}^{2}$, which gives our main claim, i.e., Eq. (3). Note that Eq. (18) is a more general expression of the VFPT than Eq. (3), which includes the VFPT in the classical RW, $\langle\delta T_{\rm rw}^{2}\rangle_{L}$. Moreover, it is straightforward to derive exact results for the biased CTRW. In the CTRW, the waiting-time distribution is identical for all sites. Thus, sample variance $\sigma_{L}^{2}$ in CTRWs is zero. Replacing $\mu_{L}$ and $\langle\delta\tau^{2}\rangle_{L}$ with $\mu$ and $\sigma^{2}$ gives the exact expressions, i.e., Eqs. (5) and (6).

VI discussion

We derived the MFPT and VFPT in the QTM with a random periodic potential in the presence of bias. In the large-$L$ limit, our formulae provide the exact expressions of the FPT statistics in the biased CTRW. Unexpectedly, the VFPT values of the biased CTRW and QTM are distinctive. Furthermore, the results for the biased QTMs with periodic landscapes are still surprisingly useful even when the energy landscape is not periodically arranged in the 1D line.

Finally, we briefly discuss the diffusion coefficient in the biased QTM on a ring. Here, we apply our formulae to diffusion in the system with period $L$. Let $n_{t}$ be the number of events in which a particle makes a directed revolution (biased direction). As the time intervals between the events are IID random variables, the process of $n_{t}$ is described as a renewal process. By renewal theory Cox (1962), the mean of $n_{t}$ is given by $\langle n_{t}\rangle=t/\langle T\rangle_{L}$. Displacement $\delta x_{t}$ is represented by $\delta x_{t}=Ln_{t}+C_{L}$, where $C_{L}$ is a random variable, and the mean has the order of $L$, i.e., $\langle C_{L}\rangle=O(L)$. Thus, $\langle\delta x_{t}\rangle$ becomes

[TABLE]

Moreover, using the variance of $n_{t}$ Cox (1962), we have

[TABLE]

Acknowledgement

We thank E. Barkai and T. Miyaguchi for giving us fruitful comments. This work was supported by JSPS KAKENHI Grant Number 16KT0021, 18K03468 (TA), and JP17K05587 (KS).

Appendix A Details for a biased random walk

A biased RW was studied in the context of the classical ruin problems Feller (1968) in which a gambler with a finite/infinite capital wins or loses a dollar with probabilities $p$ and $q$, respectively. Let us consider the probability of his ruin when the initial capital is $z$ and the other player’s capital is $L-z$. In the language of the RW, this probability corresponds to the probability that a particle starting from site $z$ ($0\leq z\leq L$) reaches site [math] without visiting site $L$. This probability denoted by $q_{z}^{L}$ is known as Feller (1968)

[TABLE]

where $\varepsilon=q/p$. Moreover, the probability of his win is given by $p_{z}^{L}=1-q_{z}^{L}$ because the game will end in the future with probability 1.

Here, we assume $p>1/2$ and the random walker starts at the origin. The FPT from the origin to $L$ site can be represented by the sum of the numbers of the visits at each site, i.e.,

[TABLE]

where $k_{i}$ is the number of visits to the $i$th site until the particle reaches the site $L$. We note that $k_{i}$ includes the number of the visits at sites $+i-nL$ ($n=1,2,\cdots$).

To obtain the moments and the correlation function of $k_{i}$, we consider the large-$L$ limit. In this limit, the probability that a particle starting at site $-1$ reaches site $-L$ is zero, i.e., $q_{L-1}^{L}\to 0$ for $L\to\infty$. Thus, a particle never visit site $-L$. This follows that the probability that a random walker visits site $i$ $(n+1)$th times is given by

[TABLE]

for $i\ll L$ because $q^{L}_{L-i}\to 0$ for $i\ll L$ , and

[TABLE]

for $i\sim L$. In the large-$L$ limit,

[TABLE]

for both cases $i\ll L$ and $i\sim L$. Therefore, the generating function of the number of visits at site $i$ until the particle reaches site $L$ is given by

[TABLE]

in the large-$L$ limit. The generating function does not comprise $i$-dependence, that is, the distribution of $k_{i}$ is the same for all $i$. Thus, the moments of $k_{i}$ do not depend on the site. In particular, the mean and variance of $k_{i}$ are given by

[TABLE]

and

[TABLE]

respectively. Because the moments do not depend on site $i$, we use the following notations: $\langle k\rangle=1/(p-q)$ and $\langle\delta k^{2}\rangle=2q/(p-q)^{2}$.

Appendix B Correlation function $\langle k_{0}k_{l}\rangle$

To obtain correlation function $\langle k_{0}k_{l}\rangle$, we consider the generating function defined by

[TABLE]

where $p(n_{0},n_{l})$ is the joint probability of $k_{0}=n_{0}$ and $k_{l}=n_{l}$. Counting $k_{0}$ and $k_{l}$ for $l>1$, we have

[TABLE]

where $P^{(l)}$ is the probability that a random walker starting from the site $l$ will return to site $l$ without visiting sites [math] and $L$, i.e., $P^{(l)}=pq_{1}^{L-l}+qp_{l-1}^{l}$. In the large-$L$ limit, the generating function becomes

[TABLE]

which satisfies the normalization, i.e., $Z(0,0)=1$.

The correlation function is given by

[TABLE]

where $\varepsilon\equiv q/p<1$. Thus, the correlation function decays exponentially. We note that this expression is valid for $l=1$ because $q_{0}^{1}=p_{1}^{1}=1$ and $p_{0}^{1}=0$.

Next, we consider $\langle k_{i}k_{i+l}\rangle$. Because the probability that a random walker starting from the origin visits site $-L+i+l$ for $i+l\ll L$ becomes zero in the large-$L$ limit, for $i+l\ll L$ the generating function $Z(\lambda_{i},\lambda_{i+l})$ is given by

[TABLE]

where $P_{i}^{(l)}=pq_{1}^{L-i-l}+qp_{l-1}^{l}$, which is equivalent to $P^{(l)}$ in the large-$L$ limit. Moreover, $p_{1}^{L-i-l}=p_{1}^{L-l}$ in the large-$L$ limit. Therefore, $Z(\lambda_{i},\lambda_{i+l})$ is the same as $Z(\lambda_{0},\lambda_{l})$ in the large-$L$ limit. It follows that correlation function $\langle k_{i}k_{i+l}\rangle$ does not depend on $i$ and is the same as $\langle k_{0}k_{l}\rangle$.

For $i\sim L$, a random walker will visit site $-L+i+l$ with probability $q_{L-i-l}^{L-l}$, which is nonzero even in the large-$L$ limit. Thus, the generating function becomes

[TABLE]

where $P_{-L+i}=q+pq_{1}^{l}$ and $P_{i}^{(l)}=pq_{1}^{L-i-l}+qp_{l-1}^{l}$. By a straightforward calculation, we have

[TABLE]

Therefore, the correlation function does not depend on $i$.

Appendix C Derivation of the VFPT in the biased QTM

The second moment of $T_{i}$ can be calculated as $\langle T_{i}^{2}\rangle=\langle(\tau_{1}^{(i)}+\cdots+\tau_{k_{i}}^{(i)})^{2}\rangle=\langle k^{2}\rangle\langle(\tau_{l}^{(i)})\rangle^{2}+\langle k\rangle(\langle(\tau_{l}^{(i)})^{2}\rangle-\langle(\tau_{l}^{(i)})\rangle^{2})$, where $\langle k^{2}\rangle=\langle\delta k^{2}\rangle+\langle k\rangle^{2}$, and $\langle(\tau_{l}^{(i)})^{2}\rangle=2\tau_{i}^{2}$ when the waiting-time distribution is the exponential distribution. Therefore, the second moment of the FPT in the QTM can be represented as

[TABLE]

where

[TABLE]

The third term is given by $\sum_{i\neq j}\langle k_{i}k_{j}\rangle\tau_{i}\tau_{j}=2\sum_{l=1}^{L-1}C_{l}\sum_{i=0}^{L-1-l}\tau_{i}\tau_{i+l}+\langle k\rangle^{2}\sum_{i\neq j}\tau_{i}\tau_{j}=2\sum_{l=1}^{L-1}C_{l}\sum_{i=0}^{L-1-l}\tau_{i}\tau_{i+l}+\langle k\rangle^{2}[(\sum_{i=0}^{L-1}\tau_{i})^{2}-\sum_{i=0}^{L-1}\tau_{i}^{2}]$, where we set $C_{l}\equiv\langle k_{i}k_{i+l}\rangle-\langle k\rangle^{2}$ because $\langle k_{i}k_{i+l}\rangle$ does not depend on $i$. Therefore, the second moment of the FPT becomes

[TABLE]

The combination of Eqs. (28), (29), and (49) gives Eq. (18).

Appendix D Some distribution functions

Here, we define the probability density functions (PDFs) of $\tau_{i}\tau_{i+1}$ and $\tau_{i}^{2}$, where $\tau_{i}$ and $\tau_{i+1}$ are independent and identically distributed random variables with a power-law distribution of $\psi_{\alpha}(\tau)=\alpha\tau^{-1-\alpha}$ ($\tau\geq 1$). First, the probability that $\tau_{i}^{2}$ is smaller than $x$ is given by

[TABLE]

Therefore, the PDF of $\tau_{i}^{2}$ is given by $\psi_{\alpha/2}(x)$. Next, the probability that $\tau_{i}\tau_{i+1}$ is smaller than $x$ is given by

[TABLE]

Thus, the PDF $\psi_{\tau_{i}\tau_{i+1}}(x)$ of $\tau_{i}\tau_{i+1}$ becomes

[TABLE]

Appendix E Asymptotes

Here, we consider the asymptotic behavior of $\sum_{l=1}^{L-1}C_{l}\sum_{i=0}^{L-1-l}\tau_{i}\tau_{i+l}$. First, we show that $a_{l}(L)=\sum_{i=0}^{L-1-l}\tau_{i}\tau_{i+l}$ satisfies $a_{1}\sim a_{m}$ for $m\ll L$ in the large-$L$ limit. In the large-$L$ limit,

[TABLE]

where $\Delta\tau_{i,m}=\tau_{i+1}-\tau_{i+m}$. Because $\tau_{i}$ and $\Delta\tau_{i,m}$ are independent and $\sum_{i=0}^{L-m-1}\Delta\tau_{i,m}/L\to 0$ for $L\to\infty$, the order of $a_{1}(L)-a_{m}(L)$ is at most that of $\sum_{i=0}^{L-1}\tau_{i}$, i.e, $O(L^{1/\alpha+1})$. It follows that $a_{1}(L)-a_{m}(L)=o(a_{1}(L))$ because the PDF of $\tau_{i}\tau_{j}$ follows

[TABLE]

where $\tau_{i}$ and $\tau_{j}$ are independent. Therefore, we have $a_{1}\sim a_{m}$ for $m\ll L$ in the large-$L$ limit.

Because $C_{l}$ decays exponentially to zero and $a_{1}\sim a_{l}$ for $l\ll L$, $\sum_{l=1}^{L-1}C_{l}a_{l}(L)$ can be approximated by

[TABLE]

where $m$ is small and does not depend on $L$ and $C\equiv\sum_{l=1}^{L-1}C_{l}$.

Next, we show $\mu_{L}^{2}\sim a_{1}/L$ for $\alpha>1$ in the large-$L$ limit. Because the sum of $\mu_{L}-\tau_{i+1}$ is a small order of $L$, i.e.,

[TABLE]

we have

[TABLE]

For $\alpha>1$, $\mu_{L}^{2}-a_{1}(L)/L$ becomes small (and is a small order of $\mu_{L}^{2}$) because $\sum_{i=1}^{L}\tau_{i}\tau_{i+1}/L\cong\mu_{L}^{2}$ for large $L$. Because $\mu_{L}$ is a small order of $a_{1}/L$, we have $\mu_{L}^{2}-a_{1}(L)/L=o(a_{1}(L)/L)$, i.e.,

[TABLE]

For $\alpha>1$,

[TABLE]

For $\alpha<1$, the generalized central limit theorem states that

[TABLE]

where $\Rightarrow$ implies the convergence in distribution and $Y_{\alpha}$ is a random variable with a stable distribution with index $\alpha$ and

[TABLE]

Thus, $a_{1}(L)/\sigma_{L}^{2}\to 0$ for $L\to\infty$. Therefore, both $\mu_{L}^{2}$ and $a_{l}(L)$ are small orders of $\sigma_{L}^{2}$. Thus, these terms in Eq. (37) can be ignored.

Appendix F Disorder average and correlation plot for the MFPT and VFPT in the infinite 1D systems with nonperiodic landscapes

To quantify the degree of the disorder average, we introduce a sample-number-dependent variance:

[TABLE]

where $T_{\nu}^{(j)}$ is a numerical value of $T_{\nu}$ for the $j$th realization of disorder. This quantifies sample-to-sample fluctuations of $T_{\nu}^{(j)}$ as a function of $N_{\rm dis}$. With increasing $N_{\rm dis}$, the average approaches the exact disorder average. From the indication of the CTRW results (5) and (6), the exact disorder averages of FPT statistics will diverge for small $\alpha$. In Fig. 4, we show the $N_{\rm dis}$-dependence of $\sigma_{\nu}^{2}(N_{\rm dis})$. The figure clearly shows that $\sigma_{\rm M}^{2}(N_{\rm dis})$ and $\sigma_{\rm V}^{2}(N_{\rm dis})$ become divergent behaviors for $\alpha<1$ and $\alpha<2$, respectively, as increasing $N_{\rm dis}$.

Figure 5 presents numerical data of $\langle T\rangle_{L}$ and $\langle\delta T^{2}\rangle_{L}$. Numerical results on the $y=x$ line imply that $T_{\rm MFPT}^{\prime}$ and $T_{\rm VFPT}^{\prime}$ are very good approximations of the MFPT and VFPT, respectively. However, there are a few realizations that deviate from the $y=x$ line. This figure explicitly explains a mechanism of the divergence according to the disorder average; i.e., the divergence is caused by a small proportion of samples with extremely large deviations. In such rare samples, significantly large waiting times are assigned for $i<0$ and small $|i|$. Except for such rare samples, the biased-QTM results with a periodic landscape are surprisingly useful in nonperiodic landscapes.

Appendix G Derivation of the CTRW results

We derive the MFPT and VFPT in the CTRW following Ref. Condamin et al. (2007), and use the same notations as in Condamin et al. (2007) to present the MFPT as follows:

[TABLE]

The Laplace transform of $\psi_{n}(t)$ can be given by $\hat{\psi}_{n}(s)=\hat{\psi}(s)^{n}$. It follows that

[TABLE]

Based on the classical RW result, we obtain

[TABLE]

Similarly, we have

[TABLE]

Thus, the VFPT of the biased CTRW becomes Eq. (5).