Ergodicity breaking and ageing of underdamped ...

Journal of Statistical Mechanics: Theory and Experiment

PAPER: DISORDERED SYSTEMS, CLASSICAL AND QUANTUM

Ergodicity breaking and ageing of underdamped Brownian dynamics with quenched disorder To cite this article: Wei Guo et al J. Stat. Mech. (2018) 033303

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J

ournal of Statistical Mechanics: Theory and Experiment

PAPER: Disordered systems, classical and quantum

Wei Guo1, Yong Li2, Wen-Hua Song2 and Lu-Chun Du2 1

Department of Physics, Kunming University, Kunming 650214, People’s Republic of China 2 Department of Physics, Yunnan University, Kunming 650091, People’s Republic of China E-mail: [email protected] (Wei Guo) and [email protected] (Lu-Chun Du) Received 7 July 2017 Accepted for publication 10 February 2018 Published 20 March 2018 Online at stacks.iop.org/JSTAT/2018/033303 https://doi.org/10.1088/1742-5468/aab04d

Abstract. The dynamics of an underdamped Brownian particle moving in

one-dimensional quenched disorder under the action of an external force is investigated. Within the tailored parameter regime, the transiently anomalous diusion and ergodicity breaking, spanning several orders of magnitude in time, have been obtained. The ageing nature of the system weakens as the dissipation of the system increases for other given parameters. Its origin is ascribed to the highly local heterogeneity of the disorder. Two kinds of approximations (in the stationary state), respectively, for large bias and large damping are derived. These results may be helpful in further understanding the nontrivial response of nonlinear dynamics, and also have potential applications to experiments and activities of biological processes.

Keywords: ergodicity breaking, transport properties, driven diusive systems, stochastic processes

© 2018 IOP Publishing Ltd and SISSA Medialab srl

1742-5468/18/033303+19$33.00

J. Stat. Mech. (2018) 033303

Ergodicity breaking and ageing of underdamped Brownian dynamics with quenched disorder


Contents 1. Introduction

2

2. System

4

3. Observables

5

5. Conclusions 16 Acknowledgments............................................................................................. 17 References 18

1. Introduction Over the past few decades, the diusive transport of a single driven Brownian particle through a potential landscape has flourished as one of the most active fields due to the possible applications in many dierent contexts of physics, chemistry and biology [1–5]. This framework works well, especially in more complex dynamic situations, e.g. Abrikosov and Josephson vortices in superconductors, cell migration, particle mixing or separation tasks [1–5]. The potential landscapes may be spatially periodic [1, 3–5], random [2, 6–12], and even spatiotemporally random [13]. Customarily, the spatially static (without time-dependent) random potential is also called the quenched disorder [2, 6, 7, 9]. Genetic material in nature has been already foreseen as an aperiodic disordered crystal [14], and practically in many experimental realizations on solid state surfaces, a small amount of randomness is unavoidable [7–10]. Theoretically, this randomness can be generally modeled by quenched disorder, a crude but serviceable approximation for many experimental situations [15]. In a random potential (referring to the quenched disorder, the same as follows), the anomalous diusion of a Brownian particle can be observed when the dispersion of potential fluctuations exceeds the thermal energy (temperature) [6, 7, 11, 12]. In this case, the corresponding mean squared displacement (MSD) shows the scaling forms with power laws for long time periods x(t)2 ∝ tα, and the phenomenological exponent α deviates from the value α = 1 for normal Brownian diusion: α < 1 named subdiusion and α > 1 superdiusion motion [2, 16, 17]. When a periodic potential is corrugated by a small amount of random potential, the dramatic diusion enhancement in the stationary state occurs near the critical tilt associated with the transition from locked to running solutions [8]. This behavior is in excellent agreement with experimental observations by tracking the motion of colloidal spheres [8]. Meanwhile, transiently anomalous behavior in the form of subdiusion and superdiusion, over very long time periods, has also been observed by tuning the correlation length of the random potential [9] or by the bias [10]. Experimentally, https://doi.org/10.1088/1742-5468/aab04d

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4. Results and discussion 7 4.1. Algorithm procedure.........................................................................................7 4.2. Results..............................................................................................................8


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a Gaussian random potential can be created by laser light fields, based on a special diuser or a holographic optical set-up [18–20]. As it should be, Gaussian disorder emerges naturally according to the central limit theorem, such as in molecularly-doped polymers with dipolar disorder [21], the behavior of RNA, and the proteins on DNA and transmembrane helices [22–24]. Weak ergodicity breaking and ageing originate from non-stationarity or ultra-slow relaxation of the associated process [2, 17, 25, 26]. Weak ergodicity breaking was originally introduced in the context of spin-glasses and asserted that the state or phase space of the system is not divided into mutually inaccessible regions (in contrast to strong ergodicity breaking [25]) but a relevant trajectory cannot sample it fully even at infinite time [2, 17, 25, 26]. Since the state space of the system is not known, one practically uses the equality between the ensemble average and the time average for an observable, e.g. MSD characteristic, to identify the ergodicity of a stochastic system. The ergodicity is vital for interpreting information provided by the experiments of single particle tracking [17]. In systems with weak ergodicity breaking, time-averaged quantities remain notably random variable [17]. In contrast, in systems with ergodicity, we evaluate some observables with only minor deviations from a few suciently long time trajectories. Consequently, for the latter, the information obtained from the time average of a single trajectory can be representative for that from the ensemble average. Previous works have shown that experimentally, weak ergodicity breaking has been observed, such as for the blinking dynamics of quantum dots [27], protein motion in human cell walls [28], lipid granule diusion in yeast cells [29] and the pathogen-recognition receptors on living-cell membranes [30]. Theoretically, weak ergodicity breaking has also been achieved in dierent popular models. The continuous time random walk processes [31, 32], with diverging characteristic waiting time, exhibit weak ergodicity breaking and the time-averaged MSDs remain random even for a long time limit. The fractional Brownian and fractional Langevin equation motion driven by longrange correlated Gaussian noise are transiently non-ergodic [26]. The heterogeneous diusion process [34] (only position-dependent diusion), the scaled Brownian motion [33] (only time-dependent diusion) or even the generalized diusion process with both the position- and time-dependence [35], exhibit weak ergodicity breaking. In particular, the generalized diusion process provides an attractive concept to model anomalous diusion, for instance, in microscopic biological systems. In generic Hamiltonian systems [36], weak ergodicity breaking for superdiusive transport arises. In Lévy walks [37] with constant velocity, nonergodic behavior only occurs in the ballistic diusion case. In quenched disorder systems, a transient breaking of ergodicity only survives within the finite characteristic breaking length [22] (as mentioned below). Significantly, these models may provide theoretical insight into the physical origin of weak ergodicity breaking in biological systems [17, 26], such as modeling nonergodic subdiusion in living cells [28–30]. However, weak ergodicity breaking is commonly connected to the ageing of a system, which refers to the potential dependence of the physical observables on the time span between initialization of the system and the start of the measurement. Both ergodicity breaking and ageing originate from non-stationarity or ultraslow relaxation of the associated process—they are two sides of the same coin [17, 26]. Undamped Brownian dynamics is recognized as a fundamentally conceptual paradigm in nonequilibrium statistical physics. As is often the case with biological and


2. System As an archetypal working model, we consider an underdamped particle driven by a bias through a Gaussian random potential. The time evolution of the system is governed by the dynamical equation expressed in dimensionless form [12] dx dV (x) d2 x − + f + ξ(t), 2 = −γ dt dt dx

(1)

where x is the displacement of the particle at time t; γ denotes Stokes’ friction coecient; f is an applied external static force; and ξ(t) is Gaussian white noise with the fluctuation-dissipation relation ξ(t)ξ(t ) = 2γT δ(t − t ), where T is the noise intensity (the temperature of the heat bath). The quenched random potential V (x) obeys unbiased Gaussian distribution and Gaussian correlation https://doi.org/10.1088/1742-5468/aab04d

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most artificial suspensions, the Brownian particle dynamics in the bulk can be regarded as overdamping (low Reynolds number) [4]. This corresponds to the inertial term to zero (the zero mass), or equivalently, the friction strength tends to infinity [38, 39]. However, some properties that are allowed to occur in systems with inertia can completely disappear when the inertial term is settled to zero precisely. For instance, it is known that the inertial term is one of the key ingredients for the occurrence of anomalous transport—the absolute negative mobility is absent for the overdamped dynamics [40, 41]. Moreover, Brownian motion on a periodic potential is shown to be extremely sensitive to the friction strength for a larger damping situation. The particle mobility and the diusion coecient develop sharp correlated peaks as the the friction strength varies [42]. We have also reported similar results for ballistic diusion as well as nondispersive transport in a vibrational motor [43]. In a purely random surface, anomalous diusion ranging from subdiusion to superdiusion as the friction coecient changes has been found in [7]. In corrugated channels [38, 39] it has been demonstrated that the overdamped dynamics assumption for Brownian diusion through pores is reasonable only in the case when the width of the channel bottleneck is larger than an appropriate diusion length of the particle. Certainly, a general approach would include the inertial term, and vary the friction coecient. For this archetypal model, the undamped Brownian motion with quenched disorder, a natural question arises, i.e. does the dissipation term, or other factors (such as thermal temperature and static force), of a system enhance or restrain the weak ergodicity breaking and ageing, in the context of anomalous diusion? To the best of our knowledge, this has not been reported yet. In this work, we investigate the ergodicity breaking and ageing in anomalous diusion for underdamped Brownian dynamics with a Gaussian random potential. The outline is as follows. In section 2, the dynamical equation and some detailed characterization of the potential are presented. In section 3, a number of definitions of observables are introduced. In section 4, the numerical simulation results for characterizing ergodicity breaking and ageing are displayed. Related discussions are also conducted. In section 5, a conclusion of the results ends the paper.


|x − x |2 V (x)V (x )L = σ g(|x − x |), g(|x − x |) = exp − 2λ2

2

.

(2)

3. Observables Over many dierent time scales in a wide range of systems, the scaling form of the ensemble-averaged mean squared displacement (MSD) is characterized by 2 2 α [x (t) − x(t) ] ∝ t .

(4)

α = 1 is just for the normal Brownian motion and, accordingly, the diusion coecient D(t) = M SD/(2t) is constant. Deviations from the normal Brownian motion (α = 1) are then considered as a sign of anomalous diusion [2, 7, 9, 17, 26, 45]: α < 1 is subdiusive and α > 1 superdiusive motion [2, 17]. Representative examples for subdiusion include the charge carrier motion in amorphous semiconductors, the spreading of tracer chemicals in subsurface aquifers or in convection rolls, as well as biological cells in the crowded world [17, 26, 45, 46]. The superdiusion is known from


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Here, σ is the dispersion of the potential fluctuations, and λ is the characteristic correlation length of the spatial variations. The · · ·L indicates a spatial average over the length L of the system [12, 13]. The V (x), in a broad sense, is stationary and there exists a typical ergodicity length Lerg only above which the self-averaging occurs [22]. The typical ergodicity length can be determined by the the spatially ergodicity breaking parameter (SEB), and is governed by equation [22] 1 (1 − y) exp [σ 2 g(yLerg )/T 2 ]dy = 1. 0 (3) √ For example, as T = 0.5, σ = 2/2, the value of the ergodicity length obtained by numerically solving equation (3) is Lerg 11.0793λ. Equation (3) is obtained by the following procedure. The normal transport coecients renormalized by disorder can be found by a standard trick with the spatial period L artificially imposed [2], D = T /(γCL+ · CL− ), with the spatial average L CL± = L1 0 exp [±V (x)/T ]dx, here, C ± = exp [±V (x)/T ]. Considering the limit L −→ ∞ according to the spirit for handling the disordered system (at the end of calculation), the CL± identifies the statistical ensemble average C ± over random realizations of V (x). This results in D = Tγ exp[−σ 2 /(T 2 )], which is precisely the same result obtained earlier for noncorrelated potentials [44]. Actually, there exists a typical ergodicity length Lerg above which this equivalence between CL± and C ± would occur. At this typical ergodicity length Lerg, the relative ensemble variance of the spatial average CL± , i.e. 2 the SEB ≡ [C ± − C ± 2 ]/[C ± 2 ], equals 1. By performing a tedious simplification, equation (3) can be obtained [22]. With the ergodicity length given by equation (3), it is predicted that diusion of regulatory proteins on DNAs becomes essentially anomalous on a typical length of genes, with a large scatter in single-trajectory averages [22].


where t is the lag time and tw is the length of the time series. In certain circumstances, the TMSDs are still random variables from one trajectory to another. A combination of time and ensemble averages is inevitable, yielding the time- and ensemble-averaged MSD (TEMSD) tw −t 1 2 2 δ (t) = [x(t + t) − x(t )] dt . (6) tw − t 0 Moreover, weak ergodicity breaking is quantified in terms of an ergodicity breaking (EB) parameter, a kind of ‘Fano factor’, 2 2 2 2 δ (t) − δ (t) , 2 EB = (7) δ 2 (t)

which provides a good measure for describing the magnitude of fluctuations of TMSDs. For a stationary process, a vanishing EB is a sucient condition of ergodicity. Nevertheless, it should be noted that for non-stationary processes, the vanishing of EB may not imply ergodicity [48]. Therefore, one proposes calling EB the heterogeneity parameter, or ensemble heterogeneity [48], to avoid slightly misleading the reader. Also, if, for any fixed time length tw, this relative dispersion stays finite or diverges, this may be considered to be the nonexistence of the limit in equation (6). Thus, a test for the stationarity of processes or a comparison between the ensemble and the time averages must precede the usage of EB [17, 48] (our system is stationary for a long time). The ratio of the time and ensemble averaged MSDs can be employed 2 δ (t) EB = (8) . x2 (t) 1 are, respectively, for and against the ergodicity of the The values EB = 1 and EB = system following the Boltzmann–Khinchin ergodic hypothesis [17, 49]. In addition, there exists a potential dependence of physical observables, e.g. TEMSD, on the time span ta between the initialization of the system and the start of the measurement ta +tw −t 1 [x(t + t) − x(t )]2 dt , δ 2 (t; ta ) = (9) tw − t ta https://doi.org/10.1088/1742-5468/aab04d

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the tracer motion in turbulent flows and weakly chaotic systems, or for randomly searching, actively moving creatures such as micro-organisms and bacteria, albatrosses, or humans [17, 26, 45, 46]. Generally speaking, anomalous diusion processes arise because of the loss of independence of the random variables, divergence of the variance of the step length, or the mean of the step time distribution, as well as the tortuosity of the embedding space [47]. We characterize the time average MSD (TMSD) from a single trajectory tw −t 1 2 2 [x(t + t) − x(t )] dt , δ (t) = (5) tw − t 0


where ta is the ageing time. An ageing factor can be defined as δ 2 (t; ta ) . Λ = δ 2 (t; 0)

(10)

Λ = 1 implies that there is no ageing in the system.

4. Results and discussion

Numerical simulations of equation (1) are carried out by the Runge–Kutta method of the second order with time step ∆t 0.01λ × min(γ, 1/γ)/f following [12], where min(γ, 1/γ) denotes taking the smaller value. In most cases, 100 particles over 50 dierent realizations of the potential are simulated, with initially uniform distribution of positions along a spatial interval 2500λ to avoid statistical correlations, and with the initial Maxwellian distribution of velocities at the temperature T. Thus, the statistical ensemble averages, equivalently, involve N = 5000 independent realizations. Meanwhile, the potential landscape covers n = 223 lattice points with a lattice constant δx = λ/10, i.e. the spatial length scale of the system L = nδx. The random forces in space with respect to the potential are evaluated by standard linear interpolation of the forces at the lattice points, and explicit details are given in [50, 51]. To check the algorithm for the generation of the random potential, the numerical results and the theoretically analytical expression (equation (2)) of the spatial correlation function are plotted in figure 1(a). It shows an excellent agreement between them and thus the reliability of the numerical algorithm. Figure 1(b) shows that for a larger value of the correlation length λ, the distance (or width) between the two adjacent barriers of the random potential increases (the average distance 2λ). The potential becomes flatter with λ and the degree of local heterogeneity (roughness) becomes weaker. Note that the potential average heights (amplitude) are ±σ and shown as solid lines on the top and the bottom for each realization. Consequently, superficially, the arguments below involving λ might be expected to be opposite to those invoked for σ; however, things could be far more complicated than this, e.g. diusion exponent k determined by σ and T, not by λ [2, 6]. Moreover, to explore the ergodicity breaking and ageing of the dierent anomalous regimes, we resort to the static force f, the temperature T and the friction coecient γ as control parameters in the following. The characteristic length √ of the spatial variations λ = 1 and the dispersion of the potential fluctuations σ = 2/2 are fixed throughout (except for special instructions). Note that strictly speaking, x(t) should be written as x(t) − x(0) for MSD [10, 17], but we omit this in signing. The dimensionless times in the underdamped regime tun, and the strictly overdamped limit tov, obey tun /tov = γ [12]. Hence, the observables above and the time t are rescaled by γ so as to facilitate the comparison of results. Our numerical results related to the observable quantifiers are presented as follows.


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4.1. Algorithm procedure

Ergodicity breaking and ageing of underdamped Brownian dynamics with quenched disorder (a)

(b) λ=2

2

0.5

1

0.3

0 −1 −2

λ=1

2 1

V (x)

V (x)V (0)

L

0.4

V (x)

λ = 0.5 λ = 0.5 λ=1 λ=1 λ=2 λ=2

0 −1

0.2

−2

V (x)

1 0 −1

0

−2 0

1

2

3

4

5

x

6

7

8

9

10

−60

−40

−20

0

x

20

40

60

Figure 1. (a) Comparison of the autocorrelation function of the random potential, from the numerical simulation results (symbols) and the analytical function equation (2) (line). (b) Three realizations of the random potential, for dierent values of the correlation length, λ = 0.5, 1, 2. The dotted lines on the top and the bottom for each realization in (b) are, respectively, corresponding to V (x) = ±σ ,√which are the potential average heights, and a guide for the eye. Parameter: σ = 2/2. 4.2. Results

In the absence of the bias ( f = 0) for equation (1), the potential and stochastic force of the system satisfy spatial and temporal symmetries. The particle may be trapped in a few potential wells of the potential, and located in a locked state. In the presence of the bias, the symmetries of the system are broken and the diusion of that is rather complicated for small and moderate biases. For the large biases, if the potential of equation (1) has strict spatial periodicity, the periodic portion of the potential becomes unimportant and the particle follows the force direction, with a diusion coecient determined by the Einstein relation D = T /γ [1]. It can be deduced that analogous properties exist when the potential is strictly spatially stochastic in our case (see figure 2 for f > 0.6 and the discussions). Moreover, the anomalous diusion of a Brownian particle in a pure random potential can be observed when thermal temperature T is less than the dispersion of the potential fluctuations σ [6, 7, 11, 12]. From the above, to explore the nontrivial response of the dynamics, extensive numerical simulations are performed just for small and moderate bias f or the thermal temperature T. The anomalous diusion and ergodicity breaking are shown in figures 2 and 3. The ageing eects of the system are displayed in figure 4. To gain insight into their underlying mechanism, the TEMSDs as a function of the length of the time series tw are presented in figure 5, and the evo lution of the trajectories, the distributions, as well as the ageing factor, are provided in figures 6 and 7. The representative eects of the external bias f and temperature T on the anomalous diusion and ergodicity breaking are shown in figures 2 and 3. For each f presented, figure 2 (a) shows that the γT EM SD/(2t) (symbols) cannot reach stationary https://doi.org/10.1088/1742-5468/aab04d

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λ = 0.5

2 0.1

Ergodicity breaking and ageing of underdamped Brownian dynamics with quenched disorder (a) ∼ (γt)0.9994

5

10

f = 0.05 f = 0.15 f = 0.3

4

10

f = 0.6 f = 0.05

3

10

∼ (γt)1.0050

f = 0.15 f = 0.3 f = 0.5

2

10

∼ (γt)0.06

f = 0.6

1

10

0

10

∼ (γt)1.02

−1

10

−1

10

0

10

1

10

2

γt

10

3

10

4

10

(b)

6

10

EB, f = 0.05

∼ (γt)1.1500

EB, f = 0.15 EB, f = 0.3 4

10

EB, f = 0.5 EB, f = 0.6 EB, f = 0.05 EB, f = 0.15

2

10

EB, f = 0.3

EB, EB

EB, f = 0.5 EB, f = 0.6 0

10

−2

10

−4

10

−6

10

−1

10

0

10

1

10

2

γt

10

3

10

4

10

Figure 2. The anomalous diusion and ergodicity breaking of an underdamped Brownian particle governed by equation (1), for dierent values of the external bias f. (a) Time- and ensemble-averaged mean squared displacement (TEMSD, marked by symbols) with equation (6), and the ensemble mean squared displacement (MSD, marked by lines) with equation (4). (b) The corresponding ergodicity breaking parameters EB (symbols) with equation (7) and EB (lines) with equation (8). The √ remaining parameters: γ = 0.1, σ = 2/2, λ = 1, n = 223, δx = 0.1, T = 0.5 and the length of the time series γtw = 104. The representative dynamic properties are indicated by the (dotted) fitted lines.

values. Similarly, in the time scale γt ∼ [103 , 104 ], γM SD/(2t) within time decreases for 0.6 > f > 0.52 and increases for f 0.7, γM SDs/(2t) reach stationary values, implying normal diusion. Yet again, weak ergodicity breaking can be confirmed by EB and EB in figure 3 (b). Roughly speaking, from figures 2 (b) and 3 (b), one can perceive that both the bias and thermal temperature tend to make the EB (relative dispersion of TMSDs) drop, thus the degree of weak ergodicity breaking drops. Physically, this is because for the large bias, the potential becomes unimportant and the particle follows the force direction (homogenization), resulting in the ergodic process. In a similar vein, a higher temperature is more likely to help the particle overcome potential barriers and thus normal behaviors appear (diusion and ergodicity). The ageing properties of the system are presented in figure 4. The palpable dependence of γT EM SD/2t on the time span ta is shown in figure 4. The values of γT EM SD/2t decline, except for the initial stage of large ta, namely, the ageing eect. This is because for a longer ageing time, ta, the particle explores the random potential longer before the measurement. The particle can succeed in escaping the minima of the potential (see the inset of figure 1(a)), and, correspondingly, the deeper minima they encounter. For γta exceeding a certain value, e.g. 103 in figure 4(a), this eect became feeble due to this time scale being long enough for the system to be in, or close to, the stationary state [20]. After that, the values of γT EM SD/2t remain constant. From figures 4(a) and (b), one can perceive that even a very small bias can pull the particle, walking farther, and thus pass through a sequence of shallow and deeper minima of the potential. However, the ageing time in figure 4(d) hardly influences the values of γT EM SD/2t, namely, no ageing eects. After a careful inspection of figures 4(b)–(d), one can perceive that large damping weakens the ageing properties of the system. Both γT EM SD/2t and γM SD/2t in figure 4 (d) at long time periods approximately reach the same steady value γD ≈ 0.0677 (solid line) which is in accordance with the result of the overdamped case [44] https://doi.org/10.1088/1742-5468/aab04d

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−1

10


γD = T exp(−

σ2 ) T2

(11)

√ for σ = 2/2, T = 0.5. In addition, according to the relation L2erg = 2Dτerg where Lerg is governed by equation (3), the ergodicity recovery time (or anomalous diusion lifetime) τerg is expected to be in the time scale γτerg ≈ 103 in the overdamped case (solid line in figure 4(d)). It should be noted that weak ergodicity breaking with slow relaxation can be quantified by the Deborah number [17] τ D e=t , (12) w


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which is a ratio of the relaxation time τ of a given observable and the time of observation tw. If De diverges, De → ∞, then the system gives the impression of being ’frozen’. The motion of the particle eventually does not sample the full state space, i.e. weak ergodicity breaking [17]. This can happen not only because tw is short, but also because τ is extremely long. In our system weakly nonergodic behavior occurs when the ergodicity recovery time (or the lifetime of the weak ergodicity breaking or, equivalently, of the ageing eects) is suciently long, and by carefully choosing the values of parameters σ and T which allow appropriate values of Lerg (see equation (3)), so that the condition De 1 is satisfied. In order to explore the above properties in more detail, the TEMSD and the the corresponding EB as a function of the length of the time series γtw are displayed in figure 5. We can see that for each of the selected parameters, EB shows a clear decreasing tendency with increasing γtw . It means that the observed properties with regard to ergodicity breaking may be transient and the ergodicity can ultimately recover. Moreover, in a similar vein, most TEMSDs descend with γtw . Notice that from equation (6), the TEMSD at a fixed lag time t is highly nonlocal in time due to the contrib utions from dierent times, during the whole temporal evolution length tw of the system [20]. At an intermediate lag time t, larger tw leads to a larger weight of later times after t. Due to the evolution of the system towards its steady state at longer periods of time, this imposes slower dynamics with an increasing occupation of deep minima, resulting in a monotonic decrease of the TEMSD. Consistent with this, the decrease is more pronounced for the lower temperature T, the smaller static force f, or the smaller friction coecient γ, towards the steady state later. Combining figures 4 and 5, the length of the time series tw and the ageing time ta play the same roles in bringing about slower dynamics, resulting in a monotonic decrease of the TEMSD. To gain insight into the underlying mechanism of the above observations, let us look at typically the TMSDs, the trajectories, and the distributions of the particle in figure 6. The originating γT M SDs/(2t) form the single trajectory, maintain randomly, and disperse around γT EM SD/(2t). The typical orbits are divided into two parts: the first part displays almost weak oscillated motion due to the particle hopping between the adjoined minima of the random potential; the second part exhibits intermittent bursts into the positive direction due to the action of the static force f. The former is ascribing to the trap or obstacle eects created by the random potential (see figure 1) and the latter is the drafting eects of static force. The eect by force in figure 6(c) is more remarkable than that in figure 6(a). This has revealed that diusion in such environments is often anomalous [2, 7, 9, 17, 26, 45]. Furthermore, their position distributions exhibit


5. Conclusions We have reported a study on weak ergodicity breaking and ageing of an undamped Brownian particle moving in quenched disorder under the action of an external force. Within the tailored parameter regime, the transiently anomalous diusion in the form https://doi.org/10.1088/1742-5468/aab04d

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pronounced asymmetric peaks. Significantly, for subdiusion in figure 6(a), the distribution is non-Gaussian (exponential), but for superdiusion in figure 6(c), noticeable deviations from the Gaussianity are exhibited. Generally speaking, it is assumed that if the the distribution of the positions is non-Gaussian, then the diusion is anomalous [45, 53] (for the special case beyond this supposition, see [54]). Figures 6(b) and (d) display the velocity distributions, corresponding respectively, to the subdiusion of figure 6(a) and the superdiusion of figure 6(b), at typical times. The velocity distributions in the transient regime might have strong fluctuations. For the subdiusion, initially, the distributions are nearly Maxwellian around the locked state v = 0, with a width determined by the temperature T (=0.2), attributed to most of the particles trapped in the deep wells of the random potential. We recall the initial velocity with the Maxwellian distribution at T in the simulation. Subsequently, near γt = 5e − 3 (curve not shown), we observe a small portion of the distributions (the second peak) centered at the running state v = f /γ (v = 1 for f = 0.1, γ = 0.1), since a few particles are still pulled by the external force. The second small peak decreases rapidly with time, because friction dissipates the energy of the particles in the running state. These particles then become trapped in a few of the potential wells, that is, they join the locked population. For the superdiusion, with time the unimodal velocity distribution (Maxwellian), centered at a locked state v = 0 with a width determined by T (=0.5), becomes a bimodal structure that has the second maximum centered at the running state v = f /γ = 3 (f = 0.3, γ = 0.1). After that, the first small peak drops and even disappears ultimately (γt = 104). Meanwhile, the second peak may be dominant, indicating that there are more running particles than stuck particles, with the assistance of the force. These observations are consistent with the notion that the origin of the anomalies is the randomness of the barrier crossing processes: the dierent anomalies correspond to the relative dynamical weights of the locked state (dominant subdiusive regime) and the running state (dominant for superdiusion) [11, 12]. Note that except for the initial stage, for anomalous diusion, the local velocity distributions centered at the locked state and running state cannot arrive at Maxwellian type. This is slightly dierent from the observation reported in [11] and [12]. It should be noted that after a long time, for the normal diusion driven by large biases, the distribution of the velocity is Maxwellian near the running state [11, 12]. Eventually, we would like to confirm the eects of the local heterogeneity of the quenched disorder on ageing. Figure 7 shows that the ageing factor Λ exhibits approximatively a power law dependence on the time of measurement preparation (ageing time). Significantly, for a smaller correlation length of the potential, Λ deviates more obviously from the value 1. Namely, as the degree of local heterogeneity of the quenched disorder increased (see figure 1(b)), the ageing eects under the weak ergodicity breaking for the superdiusion (similarly for the subdiusion, not shown) are evidently enhanced. This confirms that the origin of ageing is ascribed to the highly local heterogeneity of the disorder.


Acknowledgments We wish to sincerely thank two anonymous referees for their insightful comments and suggestions to improve this work. We are grateful to Professor Guo-Jun Jin for https://doi.org/10.1088/1742-5468/aab04d

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of subdiusion and superdiusion, the weak ergodicity breaking and ageing of the system have been obtained. First, we find that the bias, the temperature, and the friction, can substantially modulate the weak ergodicity and ageing in the transiently anomalous diusion. The bias drives the particle into the running state (from the locked state), and the temperature kicks the particle to reach local equilibrium in the velocity space during potential barrier crossing figures 2, 3 and 6. The dierent anomalies correspond to the relative dynamical weights of the locked state (dominant in the subdiusive regime) and the running state (dominant for superdiusion). This is consistent with the prediction in works [11] and [12]. The ageing nature of the system weakens as the dissipation of the system increased (the fast relaxation of the system), for other given parameters (figures 4(b)–(d) and 5(c)). Second, from the view of the steady state, for the large bias the particle follows the direction of the external static force with a diusion coecient determined by the Einstein relation (see figure 2). For large damping, the diusion coecient of the particle is approximatively in accordance with the result of the overdamped limit (see equation (11) and figure 4 (d)). One can also forecast the ergodicity recovery time terg by the diusion coecient and the ergodicity length. For the other situation beyond the approximations, it may be deduced that transiently anomalous diusion, ergodicity breaking and ageing disappear in the long-time limit because the typical ergodicity length Lerg determined by equation (3) is finite and the space scale of the evolution of the system may ultimately exceed Lerg [22]. Lastly, it would of course also be desirable to extend these studies to higher dimensions. Work on anomalous diuse properties has recently been presented and, roughly speaking, similar properties exist, since the degrees of freedom for them are independent (e.g. see the anomalous diuse properties in figure 4 of [55]). Further investigation in detail will be presented in a separate publication. Finally, from the view of experiments, particular current interests are whether the transiently anomalous behaviors can appear in realistic observation time intervals, and whether in single particle tracking the information obtained from the time averages of a single trajectory is representative of that from the ensemble averages [17, 26]. Nonergodic diusion has been realized by tracking a single ultracold caesium atom in a periodic potential [56], and the Gaussian random potentials can be created by laser light fields and a special diuser (or a holographic optical set-up)[18–20]. The ergodicity breaking and ageing in the anomalous diusion above may be identified exper imentally. However, many relevant processes in nature occur on finite timescales during which ergodic behavior cannot be taken for granted [29, 30, 56]. Our models have been associated with relevant biophysical mechanisms such as trapping [30, 57] (associated with chemical binding to stationary cellular components e.g. an actin cytoskeleton or microtubules [30]), the viscoelastic properties of the environment [58], or the presence of barriers and obstacles to diusion [59]. Accordingly, our results may be helpful for further understanding biological processes such as a protein sliding by one base along DNA [22] without the need to invoke extraordinary fluctuations beyond the usual thermal description [28–30], and may inspire related experimental works.


stimulating and illuminating discussions. This work was partly supported by the National Natural Science Foundation of China (Grant No. 11547027 and No. 11505149), the Yunnan Province Applied Basic Research Project (Grant No. 2015FB105), and the Science Foundation of Kunming University (Grant No. YJL15005 and No. XJL15016). References

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