Tracer diffusion in crowded narrow channels

The single-file concept was introduced first in biophysical literature [68] to describe diffusion through pores in membranes, which are so narrow such that the initial order of particles is preserved at all times. In mid 60 s it was realised that in single-files the effect of interactions with the environment can be even stronger than a mere renormalisation of the diffusion coefficient—it was shown that the single-file constraint can change the very temporal evolution of the TP mean-squared displacement $\overline{X_t^2}$. Analysing the TP dynamics in a single-file of interacting diffusions, it was discovered in [69] that $\overline{X_t^2}$ obeys, in the asymptotic limit $t \to \infty$, a slower than diffusive law, $\overline{X_t^2} \sim \sqrt{t}$, i.e. $\overline{X_t^2}$ exhibits not a linear Stokes–Einstein but an anomalous sub-diffusive growth with time. The point is that in single-files—an extreme case of narrow channels—due to the constraint that the environment particles cannot bypass each other, in order to explore a distance X_t the TP has to involve in a cooperative motion  ∼$\rho X_t$ particles of the environment, with ρ being their mean density⁶. In consequence, the effective frictional force exerted by the medium on the TP also scales with the distance travelled by the TP, which entails a sub-diffusive motion⁷. In a way, this resembles (and in fact, it is linked on the level of the underlying mathematics) dynamics of a tagged bead in an infinitely long Rouse polymer chain; in order to explore progressively longer and longer spatial scales the tagged bead has to involve in motion more and more other beads of the chain [73].

This remarkable result has been subsequently re-derived in different settings by using a variety of analytical techniques [74–82] (see also recent [83, 84] for a more extensive review) to show that the exact leading asymptotic behaviour of $\overline{X_t^2}$ obeys

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{1} \overline{X_t^2} \underset{t\to\infty}{\sim} \frac{\left(1 - \rho\right) a^2}{\rho} \sqrt{\frac{2}{\pi}\frac{t}{\tau}}, \nonumber \end{align} \tag{ 1 }$$

where a is the lattice spacing and τ is the mean waiting time between jumps. When $\rho \to 1$, (i.e. the space available for diffusion shrinks), the right-hand-side of equation (1) vanishes, since the system becomes completely blocked. In turn, in the limit $\rho \to 0$ the right-hand-side of equation (1) diverges, meaning that a faster growth has to take place—in this limit, of course, one expects the standard Brownian motion result $\overline{X_t^2} \sim t$ to hold. For arbitrarily small, but finite ρ, the result in equation (1) will describe the ultimate long-time behaviour, while the diffusive law will appear as a transient.

The time dependence of equation (1) was also observed experimentally (see, e.g. [85–88] and also [83, 84]), which is not surprising given its universal nature. The TP dynamics in single-files still represents a challenging playground for testing different analytical techniques and probing other properties: convergence of the distribution of X_t to a Gaussian [78], explicit form of the distribution of X_t at arbitrary, not necessarily large times in dense systems [89], the TP dynamics in the presence of a slower-than-diffusive environment [90], large deviations properties [91–95] and emerging correlations [70, 96–100]. Surprisingly, some unexpected features of this seemingly exhaustively well-studied process keep on being revealed: it was shown recently [101] (see also [92, 102, 103]) that the preservation of the initial order implies, in fact, an infinite memory of the process on the specific initial conditions—in systems, in which the particles of the environment are initially out-of-equilibrium (also called a'quenched' initial condition), the growth of the mean-squared displacement of the TP proceed a factor of $\sqrt{2}$ slower, than in systems with an equilibrium (also called 'annealed' or uniform) initial distribution of the environment particles⁸. In [92] such a dichotomy in dynamics under annealed and quenched initial conditions has been also shown to persist for higher-order cumulants of the TP position, for which it becomes even more striking.

Before we pass from single-files to the behaviour on unbounded regular lattices of dimension d higher than 1, it seems relevant to focus on some geometrically 'intermediate' situation, in which a variety of anomalous diffusions emerge. Recently [105] studied the dynamics of a TP in the presence of hard-core environment particles on a comb-like lattice, i.e. a 1D backbone (see figure 3) connected at each site to a tooth—an infinite 1D lattice, with passages of particles between the teeth being allowed only along the backbone. Dynamics of a single isolated particle in such a system has been widely studied in the past as a toy model of dynamics in a geometrically disordered system [106]. It is well known that the mean-squared displacement of a single TP (in the absence of the environment particles) obeys, at sufficiently long times, $\overline{X_t^2} \sim \sqrt{t}$ [106, 107], i.e. the TP moves sub-diffusively along the x-axis due to progressively longer and longer excursions on the teeth. [105] thus analysed a natural extension of this model by adding stochastically moving environment particles and considering two situations: (a) the environment particles can move everywhere, i.e. both along the backbone and along the teeth, while the TP can go along the backbone only, and (b) both the TP and the environment particles are permitted to go everywhere.

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At the first glance, the case (a) seems to be very close to the situation discussed in [71, 72], i.e. a 1D lattice attached to a reservoir of particles. Hence, one may expect essentially the same 'diffusive' behaviour, i.e. $\overline{X_t^2} \sim D t$, where the diffusion coefficient D captures the combined effect of the geometry and dynamics. In reality, the behaviour appears to be more complicated. Bénichou et al [105] focused on densely populated systems and analysed the TP mean-squared displacement in the leading in the density $\rho_0 = 1 - \rho$ of vacancies order, in the limit $\rho_0 \to 0$. It was shown that, when the vacancies are initially uniformly spread across the comb with mean density $\rho_0$, the mean-squared displacement of the TP along the x-axis obeys, for times t sufficiently large but less than a certain large cross-over time $t_1 \sim 1/\rho_0^4 \ln^8(1/\rho_0)$:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{comba} \lim_{\rho_0 \to 0}\frac{\overline{X_t^2}}{\rho_0} = \frac{1}{2^{5/4} \Gamma(7/4)} t^{3/4} , \nonumber \end{align} \tag{ 2 }$$

where $\Gamma(x)$ is the gamma-function, which crosses over for t  >  t₁ to the diffusive behaviour of the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{combb} \lim_{t \to \infty}\frac{\overline{X_t^2}}{t} = \rho_0^2 \left(\ln\frac{1}{\rho_0}\right)^2 . \nonumber \end{align} \tag{ 3 }$$

The results in equations (2) and (3) reveal two interesting features: first, it is the existence of a transient regime with a sub-diffusive behaviour, which is more extended in time the denser the system is. Second, in the ultimate diffusive regime, the diffusion coefficient of the TP exhibits a rather unusual non-analytic dependence on the density of vacancies.

Bénichou et al [105] described an interesting out-of-equilibrium situation when the vacancies are all initially placed on the backbone only, with a linear density $\rho_0^{\rm (lin)} \ll 1$. Interestingly enough, in this case the mean-squared displacement of the TP grows sub-linearly

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{combc} \overline{X_t^2} = \frac{\rho_0^{\rm (lin)}}{2^{7/4} \Gamma(5/4)} t^{1/4} , \nonumber \end{align} \tag{ 4 }$$

for any (sufficiently large) time t, without any crossover to an ultimate diffusive regime. We note, as well, that all even cumulants of the TP displacement follow the dependences in equations (2) and (3) in case of a uniform distribution of vacancies on the comb, and the dependence in equation (4) in case of their placement on the backbone only, which signals that the distribution of the TP position is a Skellam distribution (see, [105] for more details). This means that the distribution converges to a Gaussian at long times, with an appropriately rescaled TP position.

Lastly, in the case (b) when both the environment particles and the TP can go along the teeth, the following dynamical scenario has been predicted and confirmed through an extensive numerical analysis [105]. For an extended time interval $0 \ll t \ll t^{(1)} \sim 1/\rho_0^2$, one has $\overline{X_t^2} \sim t^{3/4}$, in which regime the TP has not had enough time to explore any given tooth because its excursions were hindered by the environment particles. This regime is followed, on the time interval $t^{(1)} \ll t \ll t^{(2)}$ (precise form of t⁽²⁾ was not provided in [105]), by a rather unusual dependence $\overline{X_t^2} \sim t^{9/16}$, associated with the large-t tail of the corresponding distribution of the time spent by the TP on a given tooth. Ultimately, for $t \gg t^{(2)}$, one finds again $\overline{X_t^2} \sim t^{3/4}$. Note that the coincidence of the dynamical exponent 3/4 characterising the initial and the final stages is occasional; the underlying physics is completely different. Note, as well, that at all stages the growth of the TP mean-squared displacement proceeds faster than in case of a single isolated TP on an empty comb ($\overline{X_t^2} \sim t^{1/2}$). This is a consequence of interactions with the environment particles, which do not permit the TP to enter too often into the teeth and also to travel too far within each tooth.

On lattices of spatial dimension $d \geqslant 2$, the ultimate long-time dynamics of the TP is diffusive⁹, such that $\overline{X_t^2} = 2 d D t$. As we have already remarked, here the main issue is the calculation of the effective diffusion coefficient which embodies the full dependence on both the rates of the TP and of the environment particles, their density and the type of the lattice (if any), on which the evolution takes place. It was fairly well understood that calculation of the diffusion coefficient is a genuine many-body problem, which is unsolvable in the general case and one has to resort to either density expansions or some other approximations, verified by numerics. Numerous approaches have been developed and a large number of different results, both analytical and numerical, have been obtained (see, e.g. [111–121] for a few stray examples, [122] for some early review and [123] for a more recent summary). We focus below particularly on two approaches, proposed in [112] and [120, 121], respectively, which will be used in what follows for the analysis of the TP dynamics in narrow channels.

In [112], the self-diffusion constant of a tracer on regular lattices partially populated by identical hard-core particles has been analysed using an approximate approach, based on a perturbative expansion in powers of $\rho (1-\rho)$, which is exact at the two extrema of the volume fraction, $\rho=0$ and $\rho=1$. This approach gives, in particular, for an unbounded d-dimensional hyper-cubic lattice the following expression for the TP diffusion coefficient

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{2} D = \frac{(1 - \rho) a^2 }{2 d \tau} f(\rho, \omega) , \nonumber \end{align} \tag{ 5 }$$

where $1/\tau$ is the jump frequency of the TP, a is the lattice spacing, and $\omega = \tau^*/\tau$ with $1/\tau^*$ being the characteristic jump frequency of the environment particles. Lastly, $f(\rho, \omega)$ is the correlation factor which is explicitly given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f(\rho, \omega) = \frac{\left(\omega (1 - \rho) + 1 \right) \left(1 - \alpha\right)}{\omega (1 - \rho) + 1 - \alpha \left(1 + \omega (1 - 3 \rho)\right)} , \nonumber \end{align} \tag{ 6 }$$

with α being one of the Watson integrals [124]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{alpha} \alpha = \frac{1}{(2 \pi)^d} \int^{\pi}_{- \pi} {\rm d}k_1 \ldots \int^{\pi}_{-\pi} {\rm d}k_d \frac{\sin^2(k_1)}{\sum_{j=1}^d \left(1 - \cos(k_j)\right)} . \end{align} \tag{ 7 }$$

This latter factor relates D to the propagator of a special type of random walk on a d-dimensional hyper-cubic lattice. For d  =  1, one has $\newcommand{\e}{{\rm e}} \alpha \equiv 1$ and hence, D vanishes, as it should, since here we enter into the realm of a single-file diffusion. On comparing the analytical predictions against available at that time results of Monte Carlo simulations in d  =  3 systems, the authors demonstrated that their approximation gives a good interpolation formula in between of the two extrema. In [125, 126] and [127], the expression in equation (5) has been generalised, using a similar approach based on a decoupling of the TP-particle-particle correlation function into a product of two TP-particle correlation functions, for situations when the density ρ of the environment particles is not explicitly conserved (unlike, e.g. the case of a monolayer in a slit-pore, confined between two solid surfaces) but the particles undergo continuous exchanges with a reservoir maintained at a constant chemical potential. References [125, 126] focused specifically on such a situation in two dimensions, i.e. a monolayer on top of a solid surface exposed to a vapour phase, while [127] presented an explicit expression for the TP diffusion coefficient in three-dimensions, which case represents some generalised model of a dynamic percolation. The resulting expressions for the diffusion coefficient are quite cumbersome and we do not present them here, addressing an interested reader to the original papers [125–127]. We note parenthetically that in case when exchanges with the vapour are permitted, dynamics becomes diffusive and D is well-defined [71, 72] even in single-files, since the initial order is destroyed.

A different line of thought has been put forth in [120, 121], which studied the TP dynamics in a very densely populated system of hard-core particles, assuming that a continuous 'reshuffling' of the environment occurs due to the dynamics of the 'vacancies' (i.e. vacant sites, present on the lattice at mean density $\rho_0=1 - \rho$), rather than of the particles themselves. Focussing on such a 'vacancy-assisted' dynamical model on a 2D square lattice of a unit lattice spacing, the authors studied first the case of just a single vacancy, which moves in the discrete time n, $n=0, 1, 2, \ldots$, by exchanging its position, at each tick of the clock, with one of the neighbouring particles (including the TP, once it appears on the neighbouring site), chosen at random. In such a 2D model, any particle including the TP performs concerted random excursions and can, in principle, travel to infinity from its initial position. By counting all possible paths which bring the TP to position ${\bf X}$ at time moment n, the authors were able to calculate exactly the full probability distribution $P_n({\bf X})$ of finding the TP at this very site at time moment n. Surprisingly, this distribution appears to be non-Gaussian even in the limit $n \to \infty$ and the mean-squared displacement $\overline{{\bf X}_n^2}$ shows a striking sub-diffusive behaviour in the leading in n order [120]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{3} \overline{{\bf X}_n^2} = \frac{\ln(n)}{\pi (\pi - 1)} , \nonumber \end{align} \tag{ 8 }$$

with a very non-trivial numerical amplitude (which depends, of course, on the lattice structure).

Brummelhuis and Hilhorst [121], focused on the case of a very small density of vacancies. Discarding the events when any two vacancies appear at the adjacent sites (whose probability is of order of $\rho_0^2$), an analogous expression for $P_n({\bf X})$ has been derived, which does converge to a Gaussian in the limit $n \to \infty$. The TP dynamics in this case appears to be diffusive, since now the TP displacement is a sum of many independent events. In consequence, the mean-squared displacement, at sufficiently large n, obeys

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{4} \overline{{\bf X}_n^2} = \frac{\rho_0}{\pi - 1} n , \nonumber \end{align} \tag{ 9 }$$

with, again, a very non-trivial numerical amplitude, which depends on the lattice structure. We emphasise that this result is only valid in the linear in the density of vacancies $\rho_0$ order. There exist, of course, $\rho_0$-dependent corrections to the diffusion coefficient. Expanding the expression in equation (5) in Taylor series in powers of $\rho_0$ and retaining only the leading term in this expansion, one may straightforward verify that it coincides with the expression in equation (9).

The result in equation (9) has been subsequently generalised in [128] for the situations where the TP interacts with the environment particles in a different way, than the particles interact between themselves, which situation mimics, e.g. dynamics of an intruder (e.g. an Indium atom) in the close-packed upper most layer of atoms in a metal (e.g. ${\rm Cu}$), in the presence of one or a few naturally existing defects of packing—the vacancies [129, 130]. The expression derived in [128] revealed an interesting non-monotonic dependence of the self-diffusion coefficient on the strengths of these interactions. Vacancy-assisted dynamics on different types of lattices has been amply discussed in [131–136].

By assuming the validity of the Einstein relation, a generalisation of the expression in equation (9) for a hyper-cubic lattice of an arbitrary dimension d can be obtained from the expression for the TP mobility (see the supplementary information to [137]). This gives the following expression

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{5} D = \frac{\rho_0}{2 d} \frac{1 - \alpha}{1 + \alpha} , \nonumber \end{align} \tag{ 10 }$$

where α is defined in equation (7), $\tau^*= \tau=1$ and a  =  1. We note that equation (10) coincides with the leading term in the expansion in Taylor series in powers of $\rho_0$ of the expression in equation (5).

A model with a biased TP travelling in a lattice gas of particles with symmetric hopping probabilities has been studied in [150] in the particular limit when the force is infinitely large, such that the TP performs a totally directed random walk in the direction of the applied bias, constrained by hard-core interactions with the environment particles. It was shown in [150] that when the TP jumps instantaneously to the neighbouring lattice site on its right, as soon as it becomes vacant, and does not jump in the opposite direction, its mean displacement obeys at sufficiently long times

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{mean} \overline{X_t} = \gamma \, \sqrt{2 D_0 t} , \nonumber \end{align} \tag{ 11 }$$

where D₀ is the Stokes–Einstein diffusion coefficient a single isolated environment particle and the amplitude γ is defined implicitly as the solution of the following transcendental equation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \gamma I_{+}(\gamma) = \rho_0 , \,\,\, I_+(\gamma) = \sqrt{\frac{\pi}{2}} \exp\left(\frac{\gamma^2}{2}\right) \left(1 - {\rm erf}\left(\frac{\gamma}{\sqrt{2}}\right)\right) , \nonumber \end{align} \tag{ 12 }$$

with ${\rm erf}(x)$ being the error function. Remarkably, the TP does not move ballistically, i.e. does not have a constant velocity, but rather creeps. This happens because the displacement of the TP becomes effectively hindered by the environment particles; at each step to the right, the TP 'pumps' a vacancy to the left, such that the density of the environment particles in the phase in front of the TP effectively grows. In turn, there appears a sort of a traffic jam in front of the TP, which also grows in size in proportion to the distance X_t travelled by the TP. This means that the system is out of equilibrium (in contrast to the situations described in section 2.1) at any time, the density profile of the environment is a step-like function propagating to the right away of the TP such that more and more of the environment particles are entrained by the TP in a directed motion. Viewing X_t as a random process described by a Langevin equation, one may argue [150] that in this situation there is an effective frictional force exerted on the TP due to the appearance of the propagating traffic jam in front of it, which grows in proportion to X_t.

In [151, 152] a more complicated situation has been analysed when the force F exerted on the TP is finite, such that it may perform jumps in both directions¹⁰. It was shown in [151, 152] that the TP mean displacement still has a form in equation (11), but now the amplitude γ is determined by a more complicated transcendental equation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{A} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\, \left(\gamma I_+(\gamma) + \frac{{\rm e}^{-\beta F} - \rho_0}{1 - {\rm e}^{-\beta F}}\right)\left(\gamma I_-(\gamma) + \frac{1 - \rho_0 {\rm e}^{-\beta F} }{1 - {\rm e}^{-\beta F}}\right) = \frac{\rho_0^2 {\rm e}^{-\beta F}}{\left(1 - {\rm e}^{-\beta F}\right)^2} , \nonumber \end{align} \tag{ 13 }$$

where β is the reciprocal temperature and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle I_-(\gamma) = \sqrt{\frac{\pi}{2}} \exp\left(\frac{\gamma^2}{2}\right) \left(1 + {\rm erf}\left(\frac{\gamma}{\sqrt{2}}\right)\right) . \nonumber \end{align} \tag{ 14 }$$

References [151, 152] have also demonstrated that the density profiles of environment particles in front of and past the TP attain a stationary form in variable $x = (X - \overline{X_t})/\overline{X_t}$ characterised by a dense region in front of the TP and a depleted region in its wake. In consequence, in the laboratory frame, the environment particles progressively accumulate in front of the TP, and the size of the depleted region past the TP, which is devoid of particles, also grows progressively in time. Overall, the biased TP entrains in a directed motion the entire system, which is quiescent in the absence of the external bias. This is quite evident for the particles in front of the TP, but less trivial for the particles in its wake. The explanation is, however, rather simple and relies on one of the results presented in [78]; namely, it was shown that in case of an asymmetric layout of the environment particles around an unbiased TP, such that they are initially present past the TP and are absent in front of it, the mean displacement of the TP follows $\overline{X_t} = \sqrt{t \ln(t)}$, i.e. an unbiased TP moves faster than $\sqrt{t}$ due to an additional logarithmic factor, which stems from the pressure exerted on the TP by the particle phase behind it. This means that in case of a symmetric placement of the environment particles around the biased TP, once the TP goes away from the phase behind it, the closest to the TP particle sees the void space in front of it and starts to move as $\sqrt{t \ln(t)}$ catching eventually the TP, which moves as $\sqrt{t}$.

The results in equations (11) and (13) permit to make the following, conceptually important conclusion: define a generalised, time-dependent mobility $\mu_t$ of the TP in the presence of the force as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{mobility} \mu_t = \lim_{F \to 0} \frac{\overline{X_t}}{F t} , \nonumber \end{align} \tag{ 15 }$$

and the time-dependent diffusivity D_t in the absence of the external force as $D_t = \overline{X^2_t}/2 t$. Our aim is now to check if the Einstein relation $\mu_t = \beta D_t$ holds. Note that, first, for standardly defined mobility and the diffusion coefficient, which requires going to the limit $t \to \infty$, the Einstein relation trivially holds since 0  =  0. A legitimate and non-trivial question is if it holds as well for time-dependent $\mu_t$ and D_t. Calculating D_t from equation (1) and $\mu_t$—from equations (11) and (13), one realises that the Einstein relation holds exactly in the case when the initial distribution of the environment particles is random [151, 152], and does not hold when the latter is regular [101], which is a non-trivial result. The validity of the Einstein relation in single-file systems has been also investigated via molecular dynamics simulations in models of granular (inelastic) particles in [158, 159].

Within the last decade many other characteristic features of biased TP diffusion in single-file systems have been studied. In particular, the mean displacement of a biased TP has been analysed in case when the initial distribution of the environment particles has a shock-like profile: a higher density of the environment particles from the left from the TP than from the right of the TP, and the TP is subject to a constant force F pointing towards the higher density phase, (which mimics an effective attraction towards this phase) [160]. It was shown that $\overline{X_t}$ obeys equation (11) in which the amplitude γ is determined implicitly as the solution of a quite complicated transcendental equation (see, [160]). Interestingly enough, the TP not always moves in the direction of the force but only when the latter exceeds some threshold value F_c. For F  <  F_c, the TP moves against the force and the high density phase expands. When the force is exactly equal to this threshold value, $\gamma = 0$ such that the mean displacement vanishes; in this particular case, the mean-squared displacement is not zero, $\overline{X_t^2} \sim \sqrt{t}$ with a prefactor dependent on the densities in both phases (see, [160]).

In recent [161] dynamics of a pair of biased TPs in a single-file has been analysed in case when they experience action of the forces pointing in the opposite directions, such that the forces tend to separate the TPs. It was shown that in this situation, however, the TPs do indeed travel in the opposite directions only when the forces exceed some critical value; otherwise, the pair of TPs remains bounded.

The full distribution of the TP position as well as its cumulants have been calculated exactly for an arbitrary time in the leading order in the density of vacancies $\rho_0$ [89], upon an appropriate generalisation of the approach developed in [121]. The TP position distribution has been also determined in case of an arbitrary particles density within a certain decoupling approximation [162, 163], for the situation when exchanges of particles with a reservoir are permitted. It was shown that in the asymptotic limit $t \to \infty$ the distribution converges to a Gaussian. Interestingly enough, it was realised that in the presence of a bias the diffusion coefficient can be a non-monotonic function of the environment particles density ρ. Lastly, the emerging correlations in such systems have been analysed in [164, 165].

We close this subsection by noting that the models with a biased TP evolving in a single-file of particles with symmetric hopping probabilities appear also in different contexts (seemingly unrelated to the TP diffusion). In particular, they describe the dynamics of the front of a propagating precursor film which emanates from a spreading liquid droplet [166–170] or a time evolution of a boundary of a monolayer on a solid surface [171].

On unbounded lattices of spatial dimension $d \geqslant 2$, the biased TP attains ultimately a constant velocity $V$ along the direction of the force, whose dependence on system's parameters is rather non-trivial and will be discussed below. At the same time, the TP alters the environment, involving in a directional motion some of the environment particles and hence, drives the environment out of equilibrium—it is no longer homogeneous and some density profiles emerge. In the frame of reference moving with the TP, these density profiles attain a steady-state form meaning that, (in contrast to the case of single-files), here the entrainment is only partial, in the sense that upon an encounter with the TP an environment particle travels alongside the TP for some random time, leaving it afterwards and being replaced by another particle. The total amount of the entrained environment particles stays constant, on average, in time.

The emerging steady-state density profiles have been studied in detail in [125–127] for square and simple cubic lattices. It was realised that, in general, in front of the TP and in the direction perpendicular to the force, the approach of the perturbed (local) density to its average value is exponential, with the characteristic length dependent on the coordinates, such that they are asymmetric. In the wake of the TP, the form of these profiles depends essentially on whether one deals with the situation in which the number of the environment particles is explicitly conserved, or the system is exposed to some vapour phase and the environment particle may desorb from and re-adsorb into the system—in this latter case the particles number fluctuates in time around some average value. Then, the local density past the TP approaches the unperturbed value exponentially fast, but the characteristic length protrudes over much larger scales than in front of the TP. The situation is very different in the conserved number case. Here, the local density approaches its unperturbed value as a power-law, meaning that the environment remembers the passage of the TP over very large temporal and spatial scales. The approach of the local density $\rho(\lambda)$ to the mean density ρ obeys [125–127]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{dens} \rho - \rho(\lambda) \sim \frac{C}{\lambda^{(d+1)/2}} , \nonumber \end{align} \tag{ 16 }$$

where λ is the distance past the TP. The amplitude C in this decay law has been also calculated in [125–127].

We close this subsection with the following two remarks:

First, we note that such a spectacular behaviour of the density of the environment particles in the wake of a biased TP has been also seen in various continuous space settings (see, e.g. [138, 142, 143, 145, 172]), in granular media [173] and also in colloidal experiments (see, e.g. [139, 147]).

Second, we note that an interesting phenomenon may take place in situations where there is not a single biased TP, but there are a few or even a concentration of them. The point is that an out-of-equilibrium environment may mediate long-ranged mutual interactions between intruders, which are non-reciprocal and violate Newton's third law (see, e.g. [174, 175]). When several TPs are present, each of them will perturb the distribution of the environment particles in its vicinity and in order to minimise the micro-structural changes in the environment and to reduce the dissipation, they will start to move collectively. Stochastic pairing of two biased TPs has been observed in simulations in a model of a 2D lattice gas [176], in experiments in colloidal suspensions [177, 178], and also theoretically predicted for the relative motion of two TPs in a nearly-critical fluid mixture, due to emerging critical Casimir forces [179]. Formation of string-like clusters of TPs, or 'trains' of TPs, which reveals an emerging effective attraction between them has been evidenced for situations with a small concentration of TPs [180–182]. Lastly, in case when the concentration of the TPs is comparable to that of the environment particles, the TPs arrange themselves in lanes [183–187]. This self-organisation of the TPs resembles spontaneous lane formation in binary mixtures of oppositely charged colloids (see, e.g. [188]), in dusty complex plasmas [189] and also for such seemingly unrelated systems as pedestrian counter flows [190].

The functional dependence of the TP ultimate velocity $V$ on physical parameters has been studied in details within the last two decades. In the most general case, these parameters are the magnitude of the external force, the density of the environment, the rates of the particles exchanges with a vapour phase, and the jump rates of both the TP and of the environment particles. Several analytical approaches have been used in this analysis—an analytical technique based on a decoupling of correlations between the tracer particle and two given environment particles (see, [125–127]), and also a different approach based on an appropriate generalisation of the vacancy-assisted dynamical model of [120, 121], in which a bias exerted on the TP has been taken into account explicitly [191].

References [125–127] studied the functional dependence of $V$ on all physical parameters, including the exchange rates of the environment particles with the vapour phase (which define the value of the density ρ), in the general case of unequal jump rates of the TP and of the environment particles. The resulting expressions appear very cumbersome, and we address the reader to the original papers. Here we merely mention some features of the expressions obtained in [125–127]:

First, in the limit of very small forces, $V$ attains a physically meaningful form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Vel} V = \xi^{-1} F , \nonumber \end{align} \tag{ 17 }$$

which can be thought of as an analogue of the Stokes formula with ξ being the friction coefficient. This signifies that the frictional force exerted on the TP by the environment is viscous at small values of F. In turn, the friction coefficient can be decomposed into two contributions [125–127]: a mean-field one, proportional to an effective density of voids, and the second one which has a more complicated form and stems from the cooperative behaviour emerging in such a system—de-homogenisation of the environment by the TP and formation of some asymmetric density profiles of the environment particles around the TP, which we will discuss later on.

Second, it was observed in [125–127] that in the systems with an explicitly conserved particle density, the mobility of the TP exactly equals $\beta D$ with D defined in equation (5) meaning that the Einstein relation holds.

Third, we note that the expressions obtained in [125–127] have been re-examined recently in [192] and it was realised that for the environments which evolve on time scales larger than the one which defines the TP jump rates, the terminal velocity $V$ is a non-monotonic¹¹ function of the force F. The velocity first increases with an increase of F, as dictated by the linear response theory, reaches a maximal value and then, upon a further gradual increase of F, starts to decrease. In consequence, the differential mobility of the TP, defined beyond the linear response regime, becomes negative. In [192], a full phase chart has been presented indicating the region in the parameter space in which such a non-monotonic behaviour emerges. We note that the phenomenon of the so-called negative differential mobility is a direct consequence of an out of equilibrium dynamics. Various facets of this intriguing phenomenon have been studied in the past and it has been also observed in many realistic physical systems [34, 35, 193–203].

A different approach to calculation of the terminal velocity of a biased TP in very densely crowded environments has been pursued in [191], which generalised the analytical technique originally developed in [120, 121], over the case when the TP is subject to an external constant force F. Bénichou and Oshanin [191] first focused on the case of a single vacancy-mediated dynamics evolving in discrete time n and calculated exactly the mean displacement of a biased TP on an infinite square lattice to get

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{displacement} \overline{X}_n = \frac{1}{\pi} \frac{\sinh\left(\beta F/2\right)}{\left(2 \pi - 3\right) \cosh\left(\beta F/2\right) + 1} \ln(n) , \nonumber \end{align} \tag{ 18 }$$

i.e. the TP displaces anomalously slowly (logarithmically) with time. Accordingly, the time-dependent mobility $\mu_n$ (see equation (15)) of the biased TP obeys

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{mobb} \mu_n = \frac{\beta}{4 \pi (\pi - 1)} \frac{\ln(n)}{n} . \nonumber \end{align} \tag{ 19 }$$

Determining from equation (8) the effective diffusivity of the TP in the absence of the force, $D_n = \overline{X_n^2}/4 n$, one arrives at a surprising conclusion [191]: the Einstein relation $\mu_n = \beta D_n$ holds exactly even for such an anomalously confined diffusion! The full distribution function of the TP position has been also determined in [191] and it was shown that it does not converge to a Gaussian as $n \to \infty$, similarly to the case without an external bias [120].

Bénichou and Oshanin [191] extended this analysis over the case when vacancies are present on a square lattice at very small density $\rho_0$, to find that in the lowest order in the density of vacancies the mean displacement of the biased TP obeys

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{nnn} \overline{X}_n = \frac{\sinh\left(\beta F/2\right)}{\left(2 \pi - 3\right) \cosh\left(\beta F/2\right)+1} \rho_0 n , \nonumber \end{align} \tag{ 20 }$$

such that the TP now moves ballistically with a finite velocity $V = \overline{X}_n/n$. Inspecting the mobility of the TP and the diffusion coefficient of the TP in equation (9), one realises that the Einstein relation holds [191]. The distribution of X_n has also been determined in [191] and it was shown that the latter converges to a Gaussian as $n \to \infty$.

In more recent [137], the result in equation (20) has been generalised, using essentially the same technique as in [191], to calculate the mean velocity of a biased TP on a d-dimensional hypercubic lattice (including the simple cubic one, d  =  3) in the presence of a small density of vacancies. The result of [137] (see the supplementary materials to this paper) reads

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{a0} V = a_0 \rho_0 , \,\,\, a_0 = \frac{(1 - \alpha) \sinh\left(\beta F/2\right)}{\left(d (1 + \alpha)-1+\alpha\right) \cosh\left(\beta F/2\right)+1 - \alpha} , \nonumber \end{align} \tag{ 21 }$$

where the parameter α is defined in equation (7). For d  =  2, equation (21), multiplied by n, reduces to the expression in equation (20). For d  =  1, the parameter α becomes equal to 1, such that $V$ vanishes, as it should.

We concentrate on the time-evolution and the density dependence of the variance $\sigma^2_x$ of a biased TP displacement in the direction of the force F, $\sigma^2 = \overline{X_t^2} - \overline{X}_t^2$, on a 2D square and a 3D simple cubic lattices. This question has been raised only recently and several rather surprising results have been obtained. Note that in the absence of the force F, the variance simply becomes $\sigma^2_x = 2 d D t$, where the forms of the diffusion coefficient have been discussed above.

We start with the case of a square-lattice, densely populated by the environment particles, and a biased TP, and focus on the vacancy-assisted dynamics in discrete time n, which was put forth in [120, 121]. The first observation of a 'strange' behaviour of the variance $\sigma_x^2$ has been made in [204], in which it was predicted analytically and verified numerically that $\sigma_x^2 \sim n \ln(n)$. Surprisingly, it appeared that in a dense system the spread of fluctuations in the TP position is (weakly) super-diffusive. This question has been further inspected in [205] and the physical mechanism responsible for such a super-diffusive growth of fluctuations has been explained—it was shown that it emerges due to correlations between the successive jumps of the TP, which originate from interactions with a single vacancy returning to the TP position many times and carrying it in the direction of the force.

Deeper analysis of this model has been presented in [137], and it was shown that this super-diffusive regime is only a transient one which prevails up to times of order of $1/\rho_0^2$. Not only the leading term but also the correction terms to the leading behaviour have been calculated, to provide the following exact expression

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{var} \lim_{\rho_0 \to 0} \frac{\sigma_x^2}{\rho_0} &\sim a_0 n \Bigg[\frac{2 a_0}{\pi} \left(\ln(n) + \ln 8 + C -1 + \frac{\pi^2 (5 - 2 \pi)}{(2 \pi - 4)}\right)\nonumber\\&\quad+ \coth\left(\frac{\beta F}{2}\right) \Bigg] , \nonumber \end{align} \tag{ 22 }$$

where C is the Euler–Mascheroni constant, the parameter a₀ is defined in equation (21) and the omitted terms do not depend on n, i.e. are unimportant for sufficiently large n. Recalling that $a_0 \sim F$ for small forces, equation (22) implies that the coefficient in front of the leading super-diffusive term is proportional to F², meaning that super-diffusion emerges beyond the linear response. Note that using equation (21) at small F and equation (22) at F  =  0, the Einstein relation is shown to be satisfied.

For times greater than $1/\rho_0^2$, the weakly super-diffusive regime crosses over to a standard diffusive growth of the variance, which emerges due to renewal events—arrivals of other vacancies to the location of the TP—which fade out the memory about interactions with a single vacancy and make successive jumps of the TP uncorrelated. In this ultimate regime, one finds that in the leading in $\rho_0$ order the variance obeys [137]

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \lim_{n \to \infty} \frac{\sigma_x^2}{n} \sim&\,\, a_0 \rho_0 \Bigg[\coth\left(\frac{\beta F}{2}\right) +2 a_0 \left(\frac{2}{\pi} \ln\left(\frac{1}{a_0 \rho_0}\right) \right.\\&\left.+ \frac{1}{\pi} \ln 8 + \frac{\pi (5 - 2 \pi)}{2 \pi - 4} \right) \Bigg]. \nonumber \end{align*} \tag{ 23 }$$

Here, again, one observes that the effective diffusion coefficient acquires some additional terms which are quadratic with the force in the limit of small forces. Note that for moderate forces, however, these terms can become large due to the factor $\ln(1/\rho_0)$. The validity of the Einstein relation in this ultimate regime is again ensured by the term $a_0 n \coth(\beta F/2)$.

For the TP dynamics on a simple cubic lattice, densely populated by the environment particles, the growth of the variance of the TP displacement obeys at all time scales [137]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \lim_{\rho_0 \to 0} \frac{\sigma_x^2}{\rho_0} \sim a_0 n \left[\coth\left(\frac{\beta F}{2}\right) + A a_0\right] , \nonumber \end{align} \tag{ 24 }$$

where A is a numerical constant. Hence, the spread of fluctuations in three- (and actually, higher-) dimensions is diffusive but the effective diffusion coefficient contains, as well, quadratic with the force terms in the limit of small applied forces. The validity of the Einstein relation is again ensured by the term $a_0 n \coth(\beta F/2)$.

Lastly, we discuss the observations made in [206], which analysed the density-dependence of the variance of the TP displacement in a system of the environment particles evolving on a discrete lattice of an arbitrary dimension in continuous time t. The approach of [206] was based on a decoupling of the TP-particle-particle correlation function, similar to the one employed in [125–127], and the results were verified numerically and in the high- and low-density of particles limits, in which exact solutions are available [137, 207–209]. In particular, it was shown in [206] that in the ultimate regime ($t \to \infty$) the effective diffusion coefficient can be a non-monotonic function of the density, for forces which exceed a certain threshold value, well above the linear response regime, and is a monotonic function of ρ for F below the threshold. Moreover, in the case when the jump rate of the environment particles is substantially lower than the one characterising the jumps of the TP, this effective diffusion coefficient can be also a non-monotonic function of the force F—a behaviour we have already observed for the velocity of a biased TP.

Before we proceed to the discussion of the forms of $V$ in narrow channels, we briefly recall one of the results of [121], which concerns the form of the diffusion coefficient D_L for self-diffusion of an unbiased TP on an infinitely long strip of width L, densely populated by the environment particles. In [121], within the model with the vacancy-assisted dynamics, the following exact, in the first order in the density $\rho_0$ of vacancies, result has been presented

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{L} D_L = \frac{\rho_0}{4} \frac{1 - \alpha'}{1+ \alpha'} , \nonumber \end{align} \tag{ 25 }$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{alpha'} \alpha' = \frac{1}{4} \lim_{z \to 1^-} \left[P(0|0;z) - P(2 {\bf e}_x|0;z)\right] , \nonumber \end{align} \tag{ 26 }$$

with ${\bf e}_x$ being a unit vector in the positive x-direction and $P({\bf r}|0;z)$—the generating function associated with the propagator of a symmetric random walk on a 2D strip of width L, starting at the origin and ending at position ${\bf r} = (n_1, n_2)$,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{z} P({\bf r}|0;z) = \frac{1}{2 \pi L} \sum_{k=0}^{L-1} \int^{\pi}_{-\pi} {\rm d}q \frac{{\rm e}^{- {\rm i} n_1 q - 2 {\rm i} \pi k n_2/L}}{1 - z \left(\cos q + \cos(2 \pi k/L)\right)/2} . \nonumber \end{align} \tag{ 27 }$$

Note that $\alpha'$ here has essentially the same meaning as α in equation (7), except that now it is defined for confined geometries.

This situation has been re-examined in [137] in case of a TP biased by a constant external force, to show that for both 2D strips (d  =  2) and 3D capillaries (d  =  3) the velocity of the TP is given, in the same order in the density of vacancies, by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{V} \tilde{V} = \rho_0 a_0' , \,\,\, a_0' = \frac{(1 - \alpha') \sinh\left(\beta F/2\right)}{\left(1 + (2 d - 1) \alpha' \right) \cosh\left(\beta F/2\right) + 1 - \alpha'} , \nonumber \end{align} \tag{ 28 }$$

where $\alpha'$ is defined in equation (26) (with the factor 1/4 replaced by $1/(2 d)$) and with $P({\bf r}|0;z)$ given in equation (27) in case of 2D strips, and by a bit more complicated expression describing the propagator of a random walk in case of 3D capillaries (see [137] for more details). One may directly verify, upon comparing the diffusion coefficient defined in equation (25) and the mobility deduced from equation (28) with d  =  2, that the Einstein relation holds.

A surprising observation has been made subsequently in [213], in which it was realised that $\tilde{V}$ in equation (28) does not describe the true velocity of the biased TP in the limit $n \to \infty$, but only a constant velocity appearing at some transient stage, for times less than the cross over time $t_x \sim 1/\rho_0^2$ (which is however quite large when $\rho_0 \ll 1$). The point is that in the underlying derivation the limit $\rho_0$ is taken first and then the analysis focuses on the asymptotic behaviour in the limit $n \to \infty$. It appears, however, that due to some very subtle circumstances these limits do not commute and, in essence, the velocity $\tilde{V}$ in equation (28) corresponds to the temporal regime when the TP interacts with just a single vacancy. Recall that we have already encountered this phenomenon while describing the behaviour of the variance of the TP displacement on an infinite square lattice. The key difference consists in the fact that for a fixed small $\rho_0$ the random walk performed by any of the vacancies between two successive visits to the TP position is a biased random walk, in the frame of reference moving with the TP, due to the interactions of the TP with other vacancies. In consequence, to get the expression for the ultimate, terminal velocity $V$, one has to examine the general expressions taking the limit $n \to \infty$ first, and only then to concentrate on the leading behaviour in the limit $\rho_0 \to 0$. In doing so, it was shown in [213] that for $n \gg 1/\rho_0^2$ the true terminal velocity obeys

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{alpha''} V = \rho_0 a_0^{\prime\prime}, \,\,\, \frac{1}{a_0^{\prime\prime}} = \frac{1}{a'_0} + \frac{2 d}{1 - \alpha'} \frac{1}{L^{d - 1}} , \nonumber \end{align} \tag{ 29 }$$

where $a_0'$ is defined in equation (28) and $\alpha'$—in equation (26).

The exact expression in equation (29) permits to draw several important conclusions, which all were also confirmed in [213] by extensive Monte Carlo simulations:

• in strips and capillaries with a finite L the true terminal velocity is lower than the velocity $\tilde{V}$ at the transient stage;
• this effect does not exist on infinite lattices when $L \to \infty$, even for d  =  2 when the variance shows two distinct behaviours;
• the effect is more pronounced in strip-like geometry than in the capillaries;
• it requires quite long times (∼$1/\rho_0^2$) to observe the terminal velocity;
• the jump of the velocities $\tilde{V} - V \sim F^2$ for small forces F, meaning that this anomaly takes place beyond the linear response regime;
• the latter also ensures that the Einstein relation holds on both transient and terminal stages.

Lastly, the analysis in [213] culminated at the derivation of a complete expression for the time evolution of the TP mean displacement, which is valid at an arbitrary discrete time n:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{\overline{X}_n}{\rho_0 n} \sim g\left(\left(a_0^{\prime\prime}\right)^2 \rho_0^2 n\right) , \nonumber \end{align} \tag{ 30 }$$

with

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle g(\tau) &= a'_0 \Bigg[\frac{b}{b^2-1} \left({\rm erf}\left(\sqrt{\tau}\right) + \frac{{\rm e}^{-\tau}}{\sqrt{\pi \tau}}\right) + \frac{b}{2} \frac{b^2+1}{\left(b^2-1\right)^2} \frac{{\rm erf}\left(\sqrt{\tau}\right)}{\tau} \nonumber \\ &- \frac{1}{b^2 - 1} + \frac{1}{\tau} \left(\frac{b}{b^2-1}\right)^2 \left({\rm e}^{(b^2-1) \tau} {\rm erf}\left(b \sqrt{\tau}\right) -1\right) \Bigg] , \nonumber \end{align} \tag{ 31 }$$

where $b = a'_0/a_0^{\prime\prime} - 1$. This scaling function reproduces correctly both regimes and has been numerically verified in [213] for both strip-like and capillary-like geometries.

General force-velocity relation for a biased TP moving in crowded strip-like or capillary-like geometries, with an arbitrary density ρ of the environment particles and different characteristic jump times of the TP, τ, and of the environment particles, $\tau^*$, has been derived in [210] using the decoupling approximation. In [210], the terminal velocity has been obtained as an implicit solution of a rather complicated non-linear equation involving matrix determinants, which we do not present here. We merely depict in figure 4 the curves $V$ versus F for a particular case of strip-like geometries with the width L  =  3, for several densities ρ of the environment particles, fixed characteristic jump-time of the TP, $\tau =1$, and two different jump-times of the environment particles, $\tau^* = 1$ and $\tau^* = 10$. We observe that for the former case, when the characteristic jump-times of the TP and of the environment particles are equal to each other, the terminal velocity $V$ is a monotonic function of the force F, which was also noticed in [211]. Conversely, in case of a slowly varying environment, e.g. for $\tau^* = 10$, the velocity $V$ exhibits a strongly non-monotonic behaviour. For small F, a linear dependence $V \sim F$ is observed, as dictated by the linear response, then, $V$ reaches a peak value $V^*$ and then gradually decreases upon a further increase of F. This means that the negative differential mobility phenomenon takes place also in confinement.

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