Aging dynamics in interacting many-body systems

Low-dimensional, complex systems are often characterized by logarithmically slow dynamics. We study the generic motion of a labeled particle in an ensemble of identical diffusing particles with hardcore interactions in a strongly disordered, one-dimensional environment. Each particle in this single file is trapped for a random waiting time $\tau$ with power law distribution $\psi(\tau)\simeq\tau^{-1- \alpha}$, such that the $\tau$ values are independent, local quantities for all particles. From scaling arguments and simulations, we find that for the scale-free waiting time case $0<\alpha<1$, the tracer particle dynamics is ultra-slow with a logarithmic mean square displacement (MSD) $\langle x^2(t)\rangle\simeq(\log t)^{1/2}$. This extreme slowing down compared to regular single file motion $\langle x^2(t)\rangle\simeq t^{1/2}$ is due to the high likelihood that the labeled particle keeps encountering strongly immobilized neighbors. For the case $1<\alpha<2$ we observe the MSD scaling $\langle x^2(t)\rangle\simeq t^{\gamma}$, where $\gamma<1/2$, while for $\alpha>2$ we recover Harris law $\simeq t^{1/2}$.

Ultraslow, logarithmic time evolution of physical observables is remarkably often observed, for instance, for paper crumpling in a piston [1], DNA local structure relaxation [2], frictional strength [3], grain compactification [4], glassy systems [5], record statistics [6], as well as magnetization, conductance, and current relaxations in superconductors, spin glasses, and field-effect transistors [7].Here we demonstrate how ultraslow dynamics of a labeled particle in a many-body system of excluded volume particles arises while without the excluded volume effects the dynamics is characterized by a power-law spreading.
Imagine a single colloidal particle diffusing in a narrow fluidic channel.Its mean squared displacement (MSD) x 2 (t) t will display the linear time dependence characteristic of Brownian motion.In contrast, if the same particle is made viscid by functionalization with "sticky" ends and the channel surface complementary coated, its motion will exhibit intermittent pausing events caused by transient binding to the channel surface.The distribution of pausing durations τ is of power-law form ψ(τ ) τ −1−α with 0 < α < 1 in a certain temperature window, effecting subdiffusive behavior x 2 (t) t α [8].This random motion belongs to the family of the Scher-Montroll-Weiss continuous time random walk (CTRW) [9,10], a renewal process with independent successive waiting times.Stochastic dynamics governed by powerlaw forms of ψ(τ ), with 0 < α < 1 were also shown to apply to tracer particle motion in the cytoplasm [11] in membranes [12] of living cells, in reconstituted actin networks [13], and determine the blinking dynamics of single quantum dots [14] as well as the dynamics involved in laser cooling [15].Physically, the form ψ(τ ) may arise from comb models [16] or random energy land- scapes [17].The divergence of the mean waiting time τ = ∞ 0 τ ψ(τ )dτ [18,19] leads to ageing phenomena [20] and weak ergodicity breaking [21], with profound consequences for, e.g., molecular cellular processes [22].
What will happen if we surround the colloidal particle with identical particles (Fig. 1)?As the channel is narrow, individual particles cannot pass each other, thus forming a single file [24][25][26][27].When the colloidal particles and the channel walls are not coated, it is well known that the mutual exclusion of the particles in the channel leads to the characteristic Harris scaling x 2 (t) t 1/2 of the MSD of the labeled particle [24][25][26][27].However, once we introduce sticky surfaces, we will show from scaling arguments and extensive simulations that the motion of the labeled, sticky particle becomes slowed down dramatically, its MSD following the logarithmic law FIG.2: MSD x 2 (t) for a single file system with waiting time distribution (1), and α = 2.2 and 3.2.In both cases the MSD follows the Harris 1/2-scaling (dashed line) for long times.Parameters: scaling factor τ = 1, lattice size L = 600, number of particles N = 201, so that the particle density is = N/L ≈ 0.3.The MSD was averaged from 3.5 × 10 3 simulations.
x 2 (t) log 1/2 (t).We also find that even when a characteristic waiting time τ exists, as long as 1 < α < 2 the motion of the labeled particle is still anomalous, with a dynamic exponent γ < 1/2.Only when α > 2, we return to the regular 1/2 Harris scaling exponent.
Motion rules for the walkers.A single walker is updated following the simple CTRW rules: on our onedimensional lattice, jumps occur to left and right with equal probability, and the waiting times between successive jumps are drawn from the probability density where τ is a scaling factor with unit of time.Practically, the waiting times become τ = τ [r −1/α − 1], where r is a uniform random number from the unit interval.After each jump the walker's clock is updated, algorithmically, T → T + τ , where initially T = 0.For the scale-free case, 0 < α < 1, i.e., infinite average waiting time τ , we obtain subdiffusive transport, is the anomalous diffusion coefficient with a the lattice spacing.For α > 1, τ is finite and we recover normal diffusion.
For the case of a single file of many excluded-volume walkers the motion of individual particles is updated in a similar fashion, with the convention that any jump leading to a double occupancy of lattice sites is canceled.More specifically: (i) Assign initial positions to all the N particles (indexed by i = 1, ..., N ).We position the labeled particle at the middle lattice point and randomly distribute equally many particles to the left and right.Each particle carries its own clock with timer T i , and all clocks are initiated simultaneously, T i = 0. (ii) Draw an independent random waiting τ i from Eq. ( 1) for each particle and add this to the timer, T i → T i + τ i .(iii) Deter- mine the particle j with the minimal value T j = min{T i } and move particle j with probability 1/2 to the left or right, unless the chosen site is already occupied by another particle.In this case cancel the move.(iv) Add a new waiting time τ j chosen from ψ(τ ) to the timer of particle j, i.e., T j → T j + τ j , and return to (iii).This is repeated until a designated stop time.
This motion scenario used in our stochastic simulations directly reflects the local nature of the physical problem (Fig. 1).Namely, when we follow individual, sticky particles in the channel, each binding and subsequent unbinding event will provide a different, random, waiting time.Even when the same position is revisited by the same particle, the waiting time will in general be different.To move the particle with the shortest remaining waiting time, i.e., whose timer first coincides with the laboratory (master) clock appears as a natural choice.
For the case of the power-law waiting time distribution (1) with exponent α > 2, we show results from extensive simulations based on above motion rules in Fig. 2. Our results reproduce the classical Brownian single-file scaling x 2 (t) t 1/2 [24]; the fitted scaling exponents are 0.49± 0.01 for α = 2.2 and 0.48 ± 0.01 for α = 3.2 [23].
Simulation results for 0 < α < 1 (diverging mean waiting time τ ) are depicted in Fig. 3.Note that we plot the square of the MSD versus time, such that according to the logarithmic time evolution, Eq. ( 4), of the MSD derived from scaling argument below, we would expect a linear dependence on the abscissa using linear-log scales.Indeed, the numerical results strongly support the predicted universal square root-logarithmic time evolution at long times for all exponents α.
To understand and quantify the system's dynamics we now obtain the MSD for the labelled particle from a scal-ing argument.Let us start with the case of a Poissonian (exponential) waiting time distribution with a welldefined characteristic waiting time and finite moments of all order.This is the scenario of regular single file diffusion with the famed Harris' law [24].If we consider the lattice dynamics of the single file [28], the MSD of the labeled particle reads where we follow the evolution of the MSD as function of the number n of steps performed by the tracer particle.Given the finite mean waiting time τ , the average number of steps is linearly related to the process time t through n = t/ τ .In Eq. ( 2) the prefactor is , where is the particle concentration (average number of particles per site).Consider now the case of functionalized particles and channel walls (non-Poissonian single file dynamics).The dynamics is then characterized by motility periods, i.e. unhindered CTRW motion described by Eq. ( 2), separated by blockage events when immobile neighbors are encountered.For scale-free waiting time distributions these blockage events are long compared with the duration of the motility periods, and thus the blockage events dominate the dynamics.We take this into account by converting the number of steps, n, in motility periods into the process time, t, measured by the laboratory master clock (subordination [29]).Formally, if we denote by H n (t) the probability that the tracer particle has taken n steps up to time t, we invoke the transformation Here we assumed that the subordinator H n (t) is slowly varying in n to replace the sum by an integral.
To proceed we employ a scaling argument to relate the number of steps n with laboratory time t in the limit of many jumps (long times).For a scale-free distribution ψ(τ ) of waiting times (0 < α < 1), longer and longer τ occur in the course of the process.In particular, individual τ may become of the order of the laboratory time.Thus, when the labeled particle meets a trapped neighbor, statistically the neighbor will experience one of these extremely long waiting time periods.Compared to these extreme blockage events the local motion of the tracer particle shuttling back and forth between immobilized neighbors will be negligible in the long time limit.The duration of the limiting steps for the motion of the labeled particle, i.e., to see its blocking neighbor resume its motion, are dominated by the probability for the next jump to occur.This exactly corresponds to the so-called forward waiting time of CTRWs [20].In the long time limit we thus face a process, in which every step n is governed by the forward waiting time.Such a process was considered recently, and there it was shown that the average number of steps taken at time t scales as log t [30].Furthermore, the spread of the corresponding probability distribution was shown to grow slower than log t, implying that in the long time limit we may consider n log t as a deterministic (scaling) relation between n and t.
The argument above leads us to the scaling ansatz for the subordinator H n (t) for 0 < α < 1, namely, it should be expressed in terms of a scaling function f (n/ log t).Imposing the normalization ∞ 0 H n (t)dn = 1, i.e., a jump necessarily occurs at some given time, we thus have the result H n (t) (log t) −1 f (n/ log t) valid in the limit of many jumps, n 1, which automatically implies t τ .Combining this scaling form with Eqs. ( 2) and ( 3) we obtain, after a change of variables n → n/ log t, that Interestingly, compared to the standard square root scaling of Brownian single file motion, the scale-free waiting time process introduces a logarithmic time.As we show in Fig. 3 this simple scaling argument combined with the results from Ref. [30] indeed accurately captures the dynamics of the many-body CTRW system.We discuss this result further below.
What happens when we turn to larger values of the anomalous exponent α, such that the characteristic waiting time τ becomes finite?Similar to the observations in Ref. [30] it turns out that we need to distinguish two cases.Let us start with the case α > 2. The results of [30] suggest a deterministic, linear scaling between n and t.Thus H n (t) t −1 f (n/t), i.e., Eq. ( 2) becomes and we recover the Brownian single file dynamics.This characteristic 1/2-scaling is indeed confirmed in Fig. 2.
For the intermediate case, 1 < α < 2, our type of single-file dynamics is rather subtle.As already shown in Ref. [30], despite the existence of the scale τ this regime behaves differently to the case α > 2 [40].Our previous results in [30] would imply that n t α−1 , suggesting the scaling ansatz H n (t) , where the prefactor is again due to normalization [31].This approach would yield the MSD x 2 (t) t γ(α) with γ(α) = (α − 1)/2.As shown in Fig. 4 (inset), this prediction for the scaling exponents does not agree well with the simulations.An improved argument goes as follows: since the random walk is unbiased, a given particle can equally well escape in either direction from an interval confined by two blocking particles.Thus this particle only needs to wait for the blocked neighbor that moves first, corresponding to the minimum of two waiting times drawn from the forward waiting time density ψ 1 (τ ) τ −α .The distribution of this minimum time will have a tail ψ1 (τ ) = 2ψ 1 (τ ) [32] The resulting MSD for the labeled particle thus scales as where γ(α) = α − 1 for 1 < α < 3/2 and γ(α) = 1/2 for α > 3/2.As seen from Fig. 4, this leads to an improved agreement with the fitted exponents.We note that this argument would become much more involved if we considered multiple escapes from blockage events to further improve the agreement with the simulations data.This argument using the minimum of two waiting times will not alter the MSD scaling for α < 1, as the log t scaling was a result of the aging of the waiting time distribution (i.e., its dependence on the time at which the waiting began), a property that will be carried over in the distribution of the minimum ψ1 .The MSD scaling for α > 2 is also unchanged by the modified argument above.
Discussion.We studied a physical model for the motion of interacting (excluded volume) particles in an aging system.Building on recent experiments of sticky particles moving along a complementary, functionalized surface, we assume that each particle performs a CTRW with a power-law waiting time distribution ψ(τ ).In particular, this scenario implies that each particle carries an individual clock whose timer triggers motion attempts according to this ψ(τ ).As function of the laboratory time t (master clock) we attempt to move the particle whose timer expires first.Thus, while the update of the timers for each particle is a renewal process, the excluded volume interactions lead to strong correlations between the motion of the particles: when one particle attempts to move and finds the neighboring lattice site occupied, typically the blocking particle is caught in a long waiting time period, and repeated attempts of motion by the mobile particle will be required.In the long time limit, we demonstrated from scaling arguments and extensive simulations that this many-body blockage scenario leads to an ultraslow logarithmic time evolution of the MSD of a labeled particle.
When the environment is less strongly disordered and the waiting time exponent α > 1, the associated characteristic waiting time τ is finite.However, similar to biased CTRW processes [33], there exists an intermediate regime for α > 1, which still exhibits anomalous scaling: the MSD has a power-law scaling with time, but the associated exponent is smaller than the value 1/2 for Brownian (Harris) single file motion.Only when the waiting time exponent α exceeds the value 2, the process returns to Harris-type single file motion with x 2 (t) t 1/2 .
In Refs.[34,35] another CTRW-based generalization of single file motion was considered.However, their update rules for particles colliding with a neighbor are very different.One way to view their process is that of a castling, i.e., particles are allowed to move through each other (phantom particles), while the labels of the particles switch in this castling.Thus the labels will stay in the same order in the file and the tracer following a specific label.Alternatively, the rule can be stated as particles switching their clocks when they collide.Refs.[34,35] found that with this rule the generalized single file dynamics acquires the MSD x 2 (t) t α/2 for 0 < α < 1.This result is fundamentally different from our ultraslow result (4), as we explicitly consider excluded volume effects.We also mention that in Ref. [36] a single file system of CTRW particles was considered, for which clustering of particles and the asymptotic log 2 (t) behavior of the MSD were found.This approach is different from ours, in particular, we do not observe any clustering.
Finally, we put the ultraslow time evolution discovered here in perspective to other stochastic models with logarithmic growth of the MSD.The most famous process is that of Sinai diffusion of a single particle in a quenched, random force field in one dimension, leading to a log 4 (t) scaling of the MSD [37].In Sinai diffusion, deep traps exist at certain position of one realization of the force field that cause the massive slow-down.In our aging single file system the strong interparticle correlations effect the logarithmic time evolution.Logarithmically slow time evolution is also found for a Markovian diffusion equation with exponential position-dependence of the diffusion constant leading to a rapid depletion of the fast-diffusivity region [38].The third class of stochastic systems with logarithmic time evolution are renewal CTRWs with logarithmic waiting time distributions [39].
We expect our work to stimulate new research in the field of interacting many-body systems in strongly disordered environments.
TA and LL are grateful for funding from the Swedish Research Council.RM acknowledges funding from the Academy of Finland (FiDiPro scheme).

FIG. 1 :
FIG. 1: Schematic of a narrow channel containing a single file of N colloidal particles particles.Regular particles in a bare channel perform single file motion characterized by Harris' law x 2 (t) t 1/2 (top), in the case of functionalized, sticky particles (bottom) the motion becomes ultraslow, x 2 log 1/2 (t).