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Counterintuitive Short Uphill Transitions in Single-File Diffusion Artem Ryabov, Dominik Lips, and Philipp Maass J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 08 Feb 2019 Downloaded from http://pubs.acs.org on February 8, 2019

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The Journal of Physical Chemistry

Counterintuitive Short Uphill Transitions in Single-File Diffusion Artem Ryabov,∗,† Dominik Lips,∗,‡ and Philipp Maass∗,‡ †Charles University, Faculty of Mathematics and Physics, Department of Macromolecular Physics, V Holeˇsoviˇckách 2, CZ-18000 Praha 8, Czech Republic ‡ Universität Osnabr¨ uck, Fachbereich Physik, Barbarastraße 7, D-49076 Osnabr¨ uck, Germany E-mail: [email protected]; [email protected]; [email protected]

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ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract A universal feature of single-file transport in micropores is the anomalous diffusion of a tracer particle. Here, we report on a new inherent property of the tracer dynamics in periodic free energy landscapes, which manifests itself in the kinetics of local transitions between neighboring energy minima: In the presence of a drift, transitions uphill against the drift are faster than along the more favorable downhill direction. The inequality between the uphill and downhill transition times holds for all densities and particle sizes. Dependent on the particle size, the transition times can be shorter or longer than the corresponding ones for a single particle. We provide a clear physical interpretation of these behaviors and show how they appear in various regimes of collective dynamics. Our findings provide a new robust method to probe collective effects by measurements of local tracer dynamics. They moreover demonstrate the complexity of extending Kramer’s diffusion model to an interacting many-body system.

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Introduction

One-dimensional diffusion in narrow channels with diameters comparable to the particle size is ubiquitous in biology and chemistry. It occurs naturally, for example, in zeolites, 1–3 nanotubes, 4–8 and in membrane channels and pores. 9–13 Because diffusing agents in these confined structures move without overtaking each other, this type of process is referred to as single-file diffusion. 14 The perhaps most investigated universal feature in single-file transport is the anomalous diffusion of a tracer particle, which manifests itself in a square-root longtime behavior of its mean-square displacement. 15–19 As the confining environment in single-file diffusion often has a periodic structure, theoretical modeling is frequently based on the overdamped Brownian motion in periodic free energy landscapes. 20 The landscape can be of entropic origin, e.g., reflecting variations in the pore crossection, 21–23 or/and of energetic nature arising from binding sites inside a membrane channel. 24–26 In fact, applicability of such models is not restricted to diffusion of passive 2


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Figure 1: Illustration of transition kinetics in driven single-file diffusion of hard spheres in a periodic potential. A tracer particle, marked in yellow, diffuses to a neighboring minimum in either uphill or downhill direction of the tilted periodic potential before passing the neighboring minimum in the opposite direction. The mean times for the corresponding uphill and downhill transitions are τ+ and τ− , respectively. agents. They are also used to describe the directed active motion of molecular motors. 27 In this work, we report on a further intriguing property of single-diffusion in periodic structures, which occurs if a flow is generated in the system by driving it out of equilibrium: The transition time for a tracer particle to move between two neighboring wells of the energy landscape is shorter against than in flow direction. This counterintuitive inequality of the transition times is valid for all densities and particle sizes. It is not only interesting as a mere fact, but allows one to draw important conclusions about the collective properties by an analysis of the local tracer dynamics.

2

Model

We consider a one-dimensional diffusion of N Brownian hard spheres of diameter σ illustrated in Fig. 1. Their centers evolve in time according to the Langevin equations √ dxi = µF (xi ) + 2Dξi (t), dt

i = 1, ..., N,

(1)

where ξi (t) are uncorrelated Gaussian white noises with hξi (t)i = 0 and hξi (t)ξj (t0 )i = δij δ(t− t0 ). The diffusion coefficient D is connected to the mobility µ via the fluctuation-dissipation theorem D = kB T µ, where kB T is the thermal energy. The total external force F (x) acting 3



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on each particle consists of [−U 0 (x)] and a constant drag force f > 0, dU F (x) = − + f, dx

U0 U (x) = cos 2

2π x λ

.

(2)

The system size L is a multiple of λ, L = M λ, and we apply periodic boundary conditions. As the N particles cover a fraction N σ/L ≤ 1 of the system, the number density ρ = N/L satisfies 0 ≤ ρ ≤ 1/σ. The model parameters correspond to Kramer’s limit, 28 i.e. the individual particles perform a hopping motion between well-defined minima of the tilted periodic potential, [U (x) − f x] as sketched in Fig. 1. This requires U0 to be larger than the thermal energy and the drag force f to be small enough that the minima do not disappear. All results discussed below remain qualitatively valid within these limits. For demonstrating our findings, we set U0 = U0 /(kB T ) = 6 and f = f λ/(kB T ) = 1. Quantities will be given in units of λ (length), λ2 /D (time), and kB T (energy). A system size L = 100 was sufficient for the results to be almost unaffected by finite size effects (corrections are below 2%).

3 3.1

Simulation Methods Algorithms

The singular nature of hard-core interactions prevents the direct use of standard textbook integration schemes for stochastic differential equations. Several advanced numerical algorithms were developed for the simulation of hard sphere systems with Brownian dynamics. 29–32 All algorithms use displacements generated by the Euler scheme for propagating particle trajectories, √ ∆xi = µF (xi )∆t +

2D∆t ζi ,

(3)

where ∆t is the time step, and ζi are Gaussian random numbers with zero mean and variance equal to one. This equation must be supplemented by the hard-sphere constraint |xi+1 −xi | ≥ 4


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σ. One method to handle this constraint is that the particle coordinates are propagated according to Eq. (3), but with displacements rejected that produce a configuration with an overlap of two or more particles. This method is the simplest to implement but gives accurate results only for very small time steps. 32 It thus requires a high computational effort for simulating many-body dynamics. A more practical and controlled simulation method is the event-driven scheme, 29,31 which is based on the solution of the one-dimensional Smoluchowski equation for free diffusion on the positive half line with a reflective boundary at the origin. Here, binary collisions between particles occurring in a time interval [t, t + ∆t] are treated as elastic collisions between freely propagating hard-spheres with fictive velocities v˜i = ∆xi /∆t. The correctness of this eventdriven scheme is, however, guaranteed only for spatially constant external forces. For incorporating a spatially varying external force, an extended algorithm was proposed. 31,32 It uses the exact solution for diffusion with a constant drift and a reflecting boundary at the origin to approximate corrections of the displacements generated by Eq. (3) for the case that an overlap occurs. The corresponding algorithm should be applicable up to moderate coverages ρσ only. In our simulations, we used both the event-driven and the extended scheme and did not find any significant deviations in the results for the investigated quantities (splitting probabilities, transitions times, mean velocity). All these quantities were determined in the non-equilibrium steady state emerging after some transient time (for any initial particle configuration).

3.2

Transition Times and Splitting Probabilities

For characterizing transitions of a tracer from one well to a neighboring one, quantities of primary interest are the splitting probabilities W± for moving downhill or uphill, and the respective mean times τ± needed for these movements. To determine W± and τ± in our 5



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simulations, we start to track a particle position when it passes through a minimum of the tilted periodic potential [U (x) − f x], and record the time of its first passage through a neighboring minimum. For n such transitions (n ' 105 in our simulations), we obtain two n

+ statistical ensembles of transition times: the ensemble of n+ transition times {τ+,k }k=1 for

n

− for uphill transitions. downhill transitions, and the ensemble of n− transition times {τ−,k }k=1 Pn± τ±,k /n± , and the splitting The averages give us the two mean transition times τ± = k=1

probabilities are given by W± (ρ, σ) = n± /n, n = n+ + n− . The mean exit time regardless of the direction of the jump, is 1 τ (ρ, σ) = n+ + n−

n+ X k=1

τ+,k +

n− X

! τ−,k0

k0 =1

= W+ (ρ, σ)τ+ (ρ, σ) + W− (ρ, σ)τ− (ρ, σ) .

(4)

We note that τ+ is the mean transition time under the condition that a downhill transition takes place. This condition means that, during the period τ+ , the particle does not pass through the neighboring minimum in uphill direction before reaching the minimum in downhill direction. A corresponding remark holds for τ− . Our results for τ± provide fundamental new insight into a difficult (non-Markovian) first-passage problem for a tracer particle in single-file diffusion. 33–35

3.3

Mean Velocity

In the following section we will see that the mean transition times and the splitting probabilities exhibit a remarkable behavior in dependence of σ. This dependence allows us to get physical insight into the collective particle motion with the average drift velocity v in the steady state. To establish the relation between these quantities, we replace ensemble averages by long-time averages. The mean velocity in the steady state is given by

v(ρ, σ) = lim λ t→∞

N+ (t) − N− (t) , t

6


(5)

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where N± (t) stands for the number of downhill/uphill transitions of the particle during the time interval (0, t). This expression is equivalent to the standard one, v = limt→∞ xi (t)/t (with arbitrary i), but for our purposes Eq. (5) is more convenient. The splitting probabilities and the average transition time can be expressed as N± (t) , t→∞ N+ (t) + N− (t)

W± (ρ, σ) = lim

(6)

and t , t→∞ N+ (t) + N− (t)

τ (ρ, σ) = lim

(7)

respectively. Combining the equations we get

v(ρ, σ) = λ

W+ (ρ, σ) − W− (ρ, σ) . τ (ρ, σ)

(8)

In addition to v computed from Eq. (8) (with W± and τ determined as described above), we calculated v also from Eq. (5) and thus verified Eq. (8).

4

Results and Discussion

Figure 2 gives an overview of the behavior of the mean transitions times τ± . In Fig. 2(a), τ± is shown as a function of σ for a low and high density, and in Fig. 2(b) the ratio τ− /τ+ for several densities ρ. Figure 3 shows, again as a function of σ for several ρ, (a) the splitting probability W+ = 1 − W− , (b) the mean exit rate 1/τ [Eq. (4)], and (c) the mean velocity [Eq. (8)]. In the following we discuss these results and their physical signficance.

4.1

Low-Density Limit

In the low-density limit (ρ → 0) the particles rarely meet, hence their interaction can be neglected and the tagged particle behaves as a single particle diffusing in a tilted-periodic

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potential. Denoting characteristics of a single-diffusing particle by a superscript “(0)”, this (0)

(0)

means: τ± (ρ, σ) → τ± , W± (ρ, σ) → W± , and v(ρ, σ) → v (0) as ρ → 0. Interestingly, for a single particle, the downhill and uphill transition times were proven to be identically distributed random variables. 36,37 Hence, their mean values are identical, 36,37 (0)

(0)

τ+ = τ− = τ (0) .

(9)

This equality of times is related to the time-reversibility of the single-particle trajectories. 38 It occurs in a broad class of results for so-called “last touch first touch times” 39–43 relevant in chemical kinetics of enzymes and molecular motors. 39,44–47 Recently, the equality was generalized to higher dimensions. 48 The single-particle splitting probabilities are independent of the concrete form of the periodic potential U (x), 37 (0)

W+ =

1 , 1 + e−βλf

(0)

W− =

e−βλf , 1 + e−βλf

(10)

where β = 1/kB T . Their difference determines the direction of the mean velocity [Eq. (8)], (0)

v

(0)

(0)

W −W = λ + (0) − . τ

(11)

The curves for ρ = 0.02 in Figs. 2, and 3 (blue symbols and lines) show a good agreement with the single-particle results. In particular, in this single-particle limit there is no dependence on the particle diameter σ. For higher densities effects of the inter-particle interactions become visible. The two times become different and splitting probabilities are no longer given by Eqs. (10).

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Figure 2: (a) Mean transition times τ± in dependence of σ for a small density close to the single-particle limit (the two curves for τ+ and τ− overlap) and a high density, demonstrating the large difference between τ+ and τ− around σ ' 0.75. (b) Ratio τ− /τ+ as a function of σ for various ρ, showing that the difference between τ+ and τ− becomes more pronounced with increasing σ and ρ.

4.2

Inequality of Transition Times

The identity of the uphill and downhill transition times of a single non-interacting particle is closely related to the microscopic reversibility of its trajectories, 38 i.e., to the detailed balance condition. In the N -particle case, the detailed balance condition holds for the trajectories in the configuration space of all particles. When considering a single tagged particle, i.e. integrating the probabilities of configurational changes over positions of the other N − 1 particles, the marginal tagged-particle probability density function does not satisfy the detailed balance because of the correlations arising due to interactions. Hence 9



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interactions break the microscopic reversibility of the tagged particle trajectories and, as a result, the equality of the transition times (9) is violated in the interacting system. Because the direction of average particle motion is determined by the direction of the external bias f , f > 0, intuitively one may expect that the downhill transition time τ+ in bias direction is shorter than the uphill transition time τ− against the bias. However, we find the opposite to be true, τ+ (ρ, σ) ≥ τ− (ρ, σ).

(12)

The mean transition time τ+ is never shorter than τ− . We have verified the inequality (12) for several values of ρ and σ in the range ρ ∈ [0, 1], σ ∈ [0, 1], and Fig. 2 demonstrates our findings. In Fig. 2(a), one can see that for the high density ρ = 0.9, τ+ (red circles) is larger than τ− (red squares) for all σ. The difference (τ+ − τ− ) becomes larger than one order of magnitude around σ = 0.75. Dependent on the particle size, both τ+ and τ− can be longer and shorter than τ (0) . This will be discussed further below. Collisions between particles become more frequent with increasing σ at fixed ρ and vice versa, and then their impact on the tagged particle dynamics gets more pronounced. This is seen in Fig. 2(b), where τ− /τ+ decreases with increasing σ and increasing ρ. This disparity of the transition times can be utilized as a tagged-particle measure to probe interactions in many-body systems. A significant difference in transition times signalizes that the non-interacting picture cannot be applied. Only if the disparity is sufficiently small, particle interactions should be negligible.

4.3

Mean Velocity: Three Dynamical Regimes

Figure 3(a) shows the mean velocity v(ρ, σ) of a particle as a function of σ for various ρ. According to Eq. (8), it is given by two factors: the rate 1/τ [Fig. 3(b)], which quantifies an effective attempt frequency, and the difference (W+ −W− ) in splitting probabilities [Fig. 3(c)],

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which gives the preferred direction of transitions. Comparing these two factors, we see that the rate 1/τ is governing the behavior of the mean velocity. Let us now discuss how the two factors behave in dependence of σ and ρ. We base this discussion on three competing effects, which govern the collective motion: (i) a barrier reduction, (ii) an exclusion, and (iii) an exchange symmetry effect. 49 These give rise to three corresponding regimes: Regime (i): Defect propagation (barrier reduction prevails). For small σ . 0.5, a single well can be occupied by more than one particle. In a multiple-occupied well, the left-most and the right-most particles are pushed by exclusion interactions to higher average energies compared to a particle in a single-occupied well. Consequently, these particles need to overcome a lower potential barrier for moving to a neighboring well. It implies that both times τ± are shorter than τ (0) , see Fig. 2(a) for σ . 0.5, and that the rate 1/τ according to Eq. (4) is increased compared to the non-interacting case, see Fig. 3(b). The rate 1/τ has a maximum around σ = 0.25, which results in the maximum of v in Fig. 3(a). Also the directional factor (W+ − W− ) is increased compared to the non-interacting case due to the multiple occupation of the wells. This can be explained as follows: Let us imagine a particle passing the minimum in a multiple-occupied well. For this particle to move to a neighboring well, other particles have first to escape the well, and this escape of the other particles in the well is more likely in the downhill direction. Hence, the enhancement of (W+ − W− ) is a result of the collective dynamics, which here reflects the difference in the escape probabiities of the other particles in the same well in downhill and uphill directions. Regime (ii): Cluster motion (exclusion prevails). For larger σ & 0.5, formation of double occupancies is still possible (up to σ = 1), yet it is very unlikely because of high energies required to assemble more than one particle in a well. In this regime, most clearly observed around σ = 0.75, the particle dynamics resembles hopping on a lattice with at most one particle per lattice site. The change of τ− and τ+ with ρ then is strongly different: while τ+ becomes very large for ρ → 1, τ− is only slightly increasing and remains of the order of τ (0) , see Fig. 2.

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Figure 3: (a) Mean velocity, (b) mean transition rate, and (c) difference of splitting probabilities as a function of particle size σ for various densities ρ. The velocity and mean transition rate are normalized to their values in the single-particle limit. In the inset of (b), the mean transition rate 1/τ (ρ, 1) is shown as a function of ρ for commensurate particle size σ = λ and compared to the rate 1/τ (ρ0 , 0) with ρ0 = ρ/(1 − ρ) in the point particle case [see Eq. (14)]. The inset in (c) shows (W+ − W− ) for these two cases.

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The reason for the different behaviors of τ− and τ+ lies in the different transition dynamics of a tagged particle, when larger clusters of occupied neighboring wells form upon increasing ρ. For a tagged particle inside a cluster to move downhill, it has to wait until the neighboring well in bias direction becomes empty. For this to happen, the particles of the cluster, which are in front of the tagged one, all have to move first. The corresponding waiting time increases with the mean cluster size, and accordingly τ+ with ρ. On the contrary, there is only a small probability that the particles of a cluster, which are behind of a tagged one, all move uphill and thereby vacate a well next to the tagged particle. An uphill jump rather occurs preferentially if the tagged particle first leaves an initial well by a downhill transition and thereafter makes an uphill transition before another particle occupies the initial well. The corresponding time for the uphill transition is of the order of τ (0) . The small probability of collective uphill motion leads only to a slight enhancement of τ− compared to τ (0) , see Fig. 2(a). As for the splitting probabilities, they become approximately equal to that in the single particle case for σ & 0.7, see Fig. 3(c). This is because multiple occupations of wells are negligible. Because (W+ − W− ) and τ− are close to their single-particle values, the strong increase of τ+ for large ρ leads to a significant suppression of the mean velocity v, see Fig. 3(a). Regime (iii): Exchange symmetry. For point particles (σ = 0) and for diameters commensurate with the period of the external potential (σ = λ = 1), we see from Fig. 3(a) that the mean velocity v(ρ, σ) is (i) independent of ρ and (ii) the same as for non-interacting particles. This at first view surprising result is a consequence of an exchange symmetry effect: 49 the case σ = 1 is reduced to σ = 0 by the coordinate transformation x0i = xi − iσ = xi − iλ in the Langevin equation (1) and a length reduction by L0 = L − N σ = L − N λ, which changes the density to ρ0 = ρ/(1 − ρ). Accordingly,

v(ρ, 1) = v(ρ0 , 0)

13

ρ0 =

ρ . 1−ρ


(13)


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Moreover, v(ρ, 0) is in fact independent of ρ. This is because for hard-core interacting point particles, all collective quantities independent of the particle labelling, i.e. that are invariant under particle exchange, are known to be identical to the corresponding ones in a system of non-interacting particles. 50 We hence have v(ρ, 1) = v(ρ0 , 0) = v (0) . Considering (non-collective) quantities of a tagged particle, their behavior in a system of hard-core interacting point particles is in general no longer the same as for non-interacting particles. Nevertheless, the relation (13) still must hold for the splitting probabilities and the mean transition rate. Indeed, this is confirmed by the simulated data shown in the inset of Fig. 3(c), where it is demonstrated that [W+ (ρ, 1) − W− (ρ, 1)] = [W+ (ρ0 , 0) − W− (ρ0 , 0)]. The mapping to the point-particle case allows us to interpret the strong increase of (W+ − W− ) with ρ for σ = 1, see the inset of Fig. 3(c). As ρ → 1 the transformed density ρ0 goes to infinity, i.e. the wells are occupied with many point particles in the transformed system for large ρ. Hence, the enhancement of (W+ − W− ) is caused by the same mechanism as described in our discussion of regime (i) above. Because the mean velocity v(ρ, 1) = v (0) = (W+ − W− )/τ is constant, this implies a strong decrease of the mean transition rate 1/τ with density at σ = 1, see the inset of Fig. 3(b). To conclude, there are three competing physical mechanisms (barrier reduction, blocking and exchange symmetry), which determine how the mean velocity is resulting from the splitting probability and mean transition rate of a tagged particle. Either one of these mechanisms prevails in three regimes of collective particle dynamics for particle sizes 0 ≤ σ ≤ 1. For larger particle sizes σ ≥ λ, the combined coordinate and density transformation explained above can be used for relating any of the investigated quantities A(ρ, σ) (A = W± , τ± , v) to the behavior found for σ < 1. The corresponding relation can be written in the general form A(ρ, σ) = A(ρ0 , σ − mλ) ,

ρ0 =

ρ , 1 − mρ

where m = int(σ/λ) denotes the integer part of σ/λ. 14


(14)

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5

Conclusions and Perspectives

In summary, we have evidenced a surprising transition kinetics of tracers, which occurs when single-file diffusion through periodic potentials is driven out of equilibrium. It is a consequence of three competing collective transport mechanisms. The most intriguing feature is that transitions between neighboring potential wells are on average faster uphill than downhill, as quantified by the inequality (12) of the corresponding mean transition times τ+ and τ− . This behavior is strikingly different from the universal equality of the two times in the absence of interactions and is a consequence of a breakage of time-reversibility of single-particle trajectories in the interacting system. As a result of a blocking effect, the difference (τ+ − τ− ) becomes most pronounced at intermediate sphere diameters σ around three quarters of the period length of the potential. Dependent on σ and the number density ρ, the uphill and downhill mean transition times can be both shorter and longer than for a single particle. Splitting probabilities, giving how likely a transition is uphill or downhill, are governed by the extent of multiple-occupation of potential wells. In conjunction with the mean transition times they provide the mean velocity v of the particles [Eq. (8)] in the single-file transport, i.e. the collective flow in the system. The magnitude of v turned out to be dominated by the mean rate of the transitions irrespective of their direction. All in all these findings allow one to probe collective dynamics by exploring transition kinetics of a tracer. Transition times and splitting probabilities can be measured locally by observing just one single potential well, instead of tracking particle trajectories through the whole system. We believe that the described effects could be directly measurable in confined crowded transport, for example, by tracking tagged particle motion in micropores, 51 or in recent experiments combining microfluidics with optical, 52–55 electric, 56 and/or magnetic 57–59 micromanipulation methods. Another option for an experimental test could be a recently observed two-component diffusion in nanopores 60 where the net drift of particles is achieved during adsorption of binary mixture inside a nanoporous material. 15



Acknowledgement Financial support by the Czech Science Foundation (project no. 17-06716S), the Deutsche ˇ Forschungsgemeinschaft (MA 1636/10-1), the MSMT (7AMB17DE014), and the DAAD (57336032) is gratefully acknowledged. We sincerely thank the members of the DFG Research Unit FOR 2692 for fruitful discussions.

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Graphical TOC Entry

Fast U(x)

Slow

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Counterintuitive Short Uphill Transitions in Single-File Diffusion - The

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