A Continuity of h(z,t) across z=0

# Exact correlations in a single file system with a driven tracer

## Abstract

We study the effect of a single driven tracer particle in a bath of other particles performing the random average process on an infinite line using a stochastic hydrodynamics approach. We consider arbitrary fixed as well as random initial conditions and compute the two-point correlations. For quenched uniform and annealed steady state initial conditions we show that in the large time limit the fluctuations and the correlations of the positions of the particles grow subdiffusively as and have well defined scaling forms under proper rescaling of the labels. We compute the corresponding scaling functions exactly for these specific initial configurations and verify them numerically. We also consider a non translationally invariant initial condition with linearly increasing gaps where we show that the fluctuations and correlations grow superdiffusively as at large times.

47.60.-i
05.60.Cd
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02.50.Ey
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[letter-]rapline-mft-final_revised_nored Flow phenomena in quasi-one-dimensional systems Classical transport Stochastic processes

The motion of non-overtaking particles in narrow channels is known as single-file diffusion. In such one dimensional geometry the motion of any tagged particle (TP) is hemmed by its neighbors. The study of TP motion have been started independently by Harris and Jepsen [2, 1] in 1965. Since then there have been numerous theoretical [3, 5, 6, 4, 7, 12, 13, 9, 10, 8, 14, 15, 11, 16, 17, 18, 19] as well as experimental studies [20, 21, 22, 23, 24, 25, 26]. One interesting feature is that the variance of the TP grows subdiffusively () even when individual particles are diffusive and the prefactor of depends on the initial condition (IC) [1, 3, 5, 6, 4, 7, 13, 9, 10, 8, 14, 15, 16, 17, 18, 19]. Due to recent developments of experimental techniques like active microrheology and microfluidics [27, 28] there has been a resurgence of interest in studying locally driven interacting particle systems. In this paper we consider the case where the TP is driven externally and study its effect on the surrounding unbiased bath particles.

Single driven TPs (DTPs) in quiescent media have been used to probe rheological properties of complex media such as polymers [29, 30], granular media [31, 32] or colloidal crystals [33], and also to study directed cellular motion in crowded channels [34] or the active transport of vesicles in a crowded axon [35]. Some practical examples of DTPs are a charged impurity being driven by an applied electric field or a colloidal particle being pulled by an optical tweezer in the presence of other colloidal particles performing random motion.

On the theoretical side, situations have been considered where the surrounding medium is a Symmetric Simple Exclusion Process. In this context the effect of the DTP has been quantified in terms of both the tracer motion and the perturbation of the density profile [36, 37, 38, 39, 40] in the Steady State (SS). In particular it has been shown by using a decoupling approximation that both the mean [36, 37, 39] and the variance [37] of the position of the DTP grow as . The full distribution of the displacement of the DTP has recently been computed in the high density limit of bath particles [16]. However, all these studies always rely on some approximation, and mostly considered quantities related to the displacement of the TP.

Here we consider the motion of a DTP moving inside a pool of other particles performing the random average process (RAP) in one dimension. Using a stochastic hydrodynamic description for this model with a DTP, we compute the mean and the fluctuation of the position of not only the DTP but of all the particles in the large time limit exactly. In addition, we also compute the exact large time behavior of the correlation function between the positions of any two particles.

The RAP was introduced in [41]. It appears in many problems like force propagation in granular media [45, 43], in the porous medium equation [46], in models of mass transport [42, 43], voting systems [47] or wealth distribution [48] and in the generalised Hammersley process [49].

RAP Model: We consider an infinite number of particles labeled by integers on the infinite line (see fig. 1). We denote the position of the particle at time by . Each particle is allowed to move to the right or left by a random fraction of the space available until the nearest particle in the direction of hopping. The fraction is drawn from some distribution with moments . All particles hop towards right and left with rate except for the -th particle which moves asymmetrically, to the right with rate and to the left with rate . Without loss of generality we assume .

The ICs of the particles, more precisely the initial separations between successive particles are chosen from some arbitrary distribution . We are mainly interested in the following three ICs:

Quenched uniform (QU) initial conditions: At initial time the paticles are at fixed positions (’quenched’) with a constant spacing (’uniform’). We therefore consider , where is the particle density in the RAP.

Quenched linear (QL) initial conditions: For this case, the particles are arranged in such a way that the gaps between successive particles grows linearly with indices on both sides of the DTP ().

Steady state (SS) initial conditions: In the steady state initial condition, a configuration is chosen randomly from the steady state distribution obtained in absence of DTP. This distribution is explicitly given in terms of the initial gap profile by ,

 Gss[ω]=∫∞−∞dz [f(ω)−f(ν0)−f′(ν0)(ω−ν0)] (1)

with , . This expression of can be obtained by following the procedure detailed in [54, 55], which mainly consists in deriving an appropriate hydrodynamic description starting from a microscopic point of view.

The SS IC is the one expected to be realized in most experiments [29, 31, 32, 33], as the tracer particle is usually held and the rest of the fluid left untouched. Taking mesurements for the QU IC would require holding all the particles initially at fixed positions using either optical/magnetic tweezers or a periodic field, and then turning them off before starting the noisy dynamics. We here mainly focus on the difference between QU and SS ICs. The QL IC case is briefly discussed to demonstrate the possibility of superdiffusive behaviour.

Results: In the large time limit, we compute the mean, variance and correlations of the particles for arbitrary ICs. In particular, we show that for both QU and SS ICs, the mean and the two point correlation have the following scaling forms

 yk(T) = ν0√2μ1T Yb(k/√2μ1T), ckl(T) = ν20√2μ1T Cb(k/√2μ1T,l/√2μ1T) (2)

where is the bias and denotes average over stochastic evolution as well as over ICs. These scaling behaviors (2) have recently been observed in numerics and has been obtained in [51] using a microscopic point of view. Here we rederive it using a hydrodynamic approach and show that it is given by

 Yb(u)=u+b [e−u2/√π−|u|Erfc(|u|)], (3)

for both QU and SS ICs.

For the zero drive case () the scaling function has been computed in [44] for both ICs. In this translationally invariant case the correlation function depends only on the separation between the two particles. On the other hand, in presence of DTP the system is not translationally invariant and the correlation functions depend on the scaling variables and individually.

For the driven case an attempt to compute the scaling function for the QU IC has been made but for small drive strength and for the totally asymmetric choice (or ) [51]. In the latter case has been computed only for and  [51]. Here we compute these scaling functions over the full 2D plane exactly for arbitrary and and for arbitrary ICs. In particular, we show that the variance of the position of the DTP is given by

 c0,0(T)=ν20√2μ1T μ2√2 (√2−1)2√π(μ1−μ2)A(b), (4)

where is given by

 A(b)=⎧⎨⎩(√2+1)(1−b2)+12(1+b2),      QU(√2+1)(1−b2)+2+√22(1+b2),  SS. (5)

The equations (4), (5) match with previously known results for for both ICs [44], and in the QU case [51]. Note that the ratio of the prefactors in the QU and SS ICs depends on the drive and becomes for , as observed earlier [44, 13, 18].

To prove the above results, we map the RAP to a mass transfer problem [51]. The mass at site is equal to the the gap between the positions of the th and th particles in RAP for all . The DTP is mapped to a special link where mass transfer occurs with rate to the left and to the right while all the other links are symmetric. In terms of the gaps, the displacement of the -th particle at time is given by .

In the large length and time scales it seems convenient to consider a continuum description of the system. For that we coarse-grain the gap label to get a continuous variable as done in [10]. This allows us to describe the system in terms of the conserved mass density field and the current field , which are related via the continuity equation . The displacement of the th particle can be expressed as

 Xk(T)=xk(T)−xk(0)=∫k−∞dz [ω(z,T)−ω(z,0)]. (6)

Since the TP is driven, the region at its front will be more crowded than the region at its back, thus creating a lower gap density in front of it than at its back. This leads to the following discontinuity [51]

 q ω(z,t)|z→0−=p ω(z,t)|z→0+ . (7)

Since the mass transfer model falls in the class of ‘gradient type’ models, the typical (and also the average) gap density profile satisfies the diffusion equation [53, 52]

 ∂t¯¯¯ω=∂z[D(¯¯¯ω)∂z¯¯¯ω], (8)

where is the diffusivity [42]. To solve (8) we introduce the Green’s function that satisfies with the discontinuity conditions and and the delta initial condition . The Green’s function can be determined by standard methods and is given by

 Fb(z,z′,t)=e−(z−z′)22μ1t√2πμ1t−b sgn(z)e−(|z|+|z′|)22μ1t√2πμ1t, (9)

One can solve for satisfying (7) for arbitrary fixed IC to get [50]

 ¯¯¯ω(z,t)=∫∞−∞Fb(z,z′,t) ωin(z′) dz′, (10)

where is the sign function. Using both for QU and SS ICs, in the above equation we explicitly get with . Inserting this result in (6) we obtain (2)-(3).

For the QL IC case, where the particles are initially placed at positions one has . Using this in (10) and performing the integral one gets where . Once again inserting this result in (6) one obtains in contrast to behaviour: where and is the Heaviside theta function.

We now focus on computing the variances and the two point correlations. For that we need to consider the fluctuations of the density and current fields about their mean behaviors. According to the macroscopic fluctuation theory [55, 18], the fluctuations in these two fields can be described by adding Gaussian fluctuations to the current: where is the mobility and is a Gaussian white noise satisfying and . For the mass transfer model one can show  [42]. Due to the Gaussian nature of the noise, the joint probability of observing density and current profiles and different from the typical profiles is also Gaussian and is given by [55, 18]

 P[ω,j]≍exp[−∫T0dt∫∞−∞dz[j+(μ1/2)∂zω(z,t)]22σ(ω(z,t))2] (11)

where satisfy the continuity equation. Here the symbol represents logarithm equivalence.

In order to compute we look at the joint distribution of the displacements and . This distribution is completely characterized by the generating function

 μ(λk,λl)=⟨exp{λkXk(T)+λlXl(T)}⟩, (12)

where are Lagrange multipliers. From this quantity the two-particle connected correlations can be obtained as .

To proceed further we follow the technique developed by Krapivsky et al. [18] and compute using the joint distribution (34). We consider an ensemble of ICs drawn with probability . From (34) and (12) we see that can be expressed as a path integral

 μ(λk,λl)=∫D[ωin]e−G[ωin]∫ω|t=0=ωinD[ω,h]e−S[ω,h], (13)

where is an auxiliary field. We here stress the fact that the integrals over and have to be evaluated for a fixed IC before taking the average over initial configurations. The action is obtained via Martin-Siggia-Rose formalism [18],

 S[ω,h]=−λkXk(T)−λlXl(T)+∫T0∫∞−∞dz[h ∂tω +(μ1/2)(∂zω)(∂zh)−(σ(ω)/2)(∂zh)2], (14)

where we note that the integrand is singular at at all times. At large times the path integral over and can be computed by saddle point method to give , as it is dominated by optimal paths that depend on the initial configuration . In the final step one has to perform the remaining path integral over in (13) in the annealed case. This can again be carried out by saddle-point method.

We now have to find the optimal paths for arbitrary fixed IC . Using variational calculus we obtain that the optimal paths satisfy the following coupled differential equations [50]

 ∂tω∗(z,t)=(μ1/2)∂2zω∗−∂z(σ(ω∗) ∂zh∗), (15) ∂th∗(z,t)=−(μ1/2)∂2zh∗−(σ′(ω∗)/2) (∂zh∗)2, (16)

where . Writing for short, one can show that the solutions of (15) and (16) have to satisfy the discontinuity conditions

 q ω∗(z,t)|z→0−=p ω∗(z,t)|z→0+ , h∗(z,t)|0−0+=0 μ12[∂zω∗]0−0+=[σ(ω∗)∂zh∗]0−0+, [ρ∂zh∗]0−0+=0 (17)

and the initial/boundary conditions :

 ω∗(z,0) = ωin(z),  ω∗(z,t)||z|→∞=¯¯¯ω(z,t)|z|→∞, h∗(z,T) = λkΘ(k−z)+λlΘ(l−z). (18)

Hydrodynamic equations similar to (15)-(16) also appear in different other single file systems like Brownian motion models, exclusion processes [18]. These equations are normally non-linear and are usually hard to solve. In presence of a DTP, additional complications appear as the solutions generically become singular at the tracer position. However it turns out that for RAP these hydrodynamic equations can be solved perturbatively as we show in the following.

To obtain the two point correlations one only needs to know the cumulant generating function up to . Hence it is enough to solve (15)-(16) to . We therefore expand the paths in powers of and ,

 ω∗ = ¯¯¯ω+λkωk+λlωl+O(λ2), (19) h∗ = λkhk+λlhl+O(λ2). (20)

Inserting these expansions in (15)-(16) and equating terms of order and from both sides separately, we obtain equations for the fields and  [50],

 ∂tωϵ(z,t) = μ12∂2zωϵ−∂z[σ(¯¯¯ω) ∂zhϵ], (21) ∂thϵ(z,t) = −μ12∂2zhϵ, (22)

for with BCs

 ωϵ(z,0) = 0, ωϵ(z,t)||z|→∞→0, (23) and hϵ(z,T) = Θ(ϵ−z) (24)

obtained from (53). The discontinuities are obtained from (52) as :

 (1+b)ωϵ(0+,t)=(1−b) ωϵ(0−,t) , μ12[∂zωϵ]0−0+=[σ(¯¯¯ω)∂zhϵ]0−0+, (25) hϵ(z,t)|0−0+=0, (1+b)∂zhϵ|0−=(1−b)∂zhϵ|0+, (26)

where . We first solve (22) with conditions (24) and (26). Using a similar procedure as for the density profile, one can show that the solutions are given by

 hϵ(z,t) = 12erfc(z−ϵ√2μ1(T−t))+b2erfc(|z|+|ϵ|√2μ1(T−t)). (27)

Now that we have explicitly, we can solve (21). The solution is given by

 ωϵ(z,t) = −∫t0dτ∫∞−∞dz′Fb(z,z′,t−τ) (28) ×∂z′[σ(¯¯¯ω(z′,τ)) ∂z′hϵ(z′,τ)],

is given in (9). The action in (14) depends on the integrals like . Using and performing some manipulations, we obtain

 Φϵ(z,t)=∫t0dτ∫∞−∞dz′ σ(¯¯¯ω(z′,τ)) ∂z′hϵ(z′,τ) ∂z′hz(z′,τ), (29)

We can now insert and functions in (14) and simplify using eqs. (21)-(22) and find

 S[ω∗,h∗]=−12∫T0dτ∫∞−∞dz σ(¯¯¯ω) (λkfbk+λlfbl)2 −λk∫k−∞dz(¯¯¯ω−ωin)−λl∫l−∞dz(¯¯¯ω−ωin)+O(λ2), (30)

where and are given in (10) and (9). Once we have the optimal value of the action for given IC , we have to perform the average over the initial configurations by evaluating the remaining path integral over in (13).

To include all the ICs SS , QU and QL in a single analysis, we consider a general Gaussian distribution of , . The SS, QU and QL ICs are obtained by taking , and , respectively. In [50] we show that

 ck,l[χ,Σ]=∫T0dτ∫∞−∞dz σ(¯¯¯ω[χ](z,t))fbk(z,t)fbl(z,t) +∫∞−∞dz Σ(z)2Fbk(z,T)Fbl(z,T), with (31) Fbk(z,T)=12(erfc(|u−ϵ|√2μ1t)+b erfc(|ϵ|+|u|√2μ1t)).

The notation emphasizes that the average density profile depends on but not on as can be seen from (10). The first term comes from the time evolution only, while the second term, proportional to , comes from the variance of the initial configuration. In particular this expression shows that the variance for any choice of and and for any particle .

In general the integrals appearing in (31) can easily be evaluated numerically for arbitrary initial conditions after inserting the corresponding and . In some special cases further analytical simplifications are possible. For the QU and SS ICs the integrals in can be performed analytically and one can obtain the variance of the DTP as given in (4) and (5). In fig. 2 we compare these predictions with numerical simulations and find nice agreement. Similarly, for these two ICs in the totally asymmetric case , one finds the following expressions of the scaling functions defined in (2) after inserting and in (31) for SS and QU ICs, respectively and simplifying. Writing for short, we get

 CQU1(u,v)=a ∫10dττ∫∞−∞dy(1−sgn(y)erfc(y√1−τ))2× (32) (e−(y−u)2τ+sgn(y)e−(|y|+|u|)2τ)(e−(y−v)2τ+sgn(y)e−(|y|+|v|)2τ), CSS1(u,v)=CQU1(u,v)+πa2∫∞−∞dy [erfc(|y−u|)+erfc(|u|+|y|)] ×[erfc(|y−v|)+erfc(|v|+|y|)]. (33)

For the case we compare numerical results for and with the theory (32) and (33) in figs. 3 and 4 for both ICs. As expected, these expressions are coherent with the results found in [44] and [51] in the appropriate cases.

In fig. 3 we find that the variances of the positions of particles in the front of the DTP are smaller than at the back, because the particles in the front have less space than those at the back. This intuitively suggests that the variance should be maximum just behind the DTP. On the contrary, in fig. 3 we see that the maximum occurs at a finite distance behind the DTP. It seems like the driven motion of the TP creates a density perturbation moving forward which is accompanied by a corresponding vacancy perturbation moving backwards similar to what happens in Exclusion Processes [10].

Figure 4 displays the plot of the correlation (at time ) between the DTP and th particle as a function of the scaling variable . We notice that it has a vanishing slope for in the QU IC case implying that the particles at the front of the DTP are equally correlated up to a distance . This is another artifact of the crowding phenomenon at . On the other hand for SS ICs there is an extra contribution from the initial correlations that dominates at large times.

It is worth noting that the growth of the correlation holds only for the translationally invariant ICs like QU and SS. This however does not hold for non translationally invariant ICs like the QL IC for which one has and . In this case the DTP gets enough space on both sides and as a result the correlation grows superdiffusively at large times (see [50] for the proof).

In this work we consider a single-file system, the RAP, in presence of a locally driven particle which drives the whole system out of equilibrium. The drive manifests itself into discontinuities of the hydrodynamic fields at the DTP position. For this model we obtain the large-time two-particle correlation function for any IC exactly. The two-point correlation functions grow as for both the QU and SS ICs, similar to the annealed/quenched dichotomy in non-driven systems. This is in contrast with Exclusion Processes, where nonlinearities play an important role in the quenched case [56]. We also have looked at another specially designed quenched linear initial condition where the two point correlation functions grow as and have scaling forms under rescaling of particle labels by . These correlation functions will serve as a benchmark in analyzing the trajectories of a driven tracer particle in traffic flow, in active microrheology [27] and microfluidics [28] experiments. An extension of the method used here to a larger class of single file systems would be of great interest.

We acknowledge useful discussions with S. N. Majumdar and D. Mukamel. The support of the Israel Science Foundation (ISF) is gratefully acknowledged. AK would like to acknowledge the hospitality of the Weizmann Institute of Science where part of the work was done while he was visiting.

## Appendix A Continuity of h(z,t) across z=0

According to the Macroscopic Fluctuation Theory the joint probability of the density and current profiles and is given by

 P[ω(z,t),j(z,t)]≍e−F[ω(z,t),j(z,t)],  where, (34) F[ω,j]=∫T0dt∫∞−∞dz(j+(μ1/2)∂zω(z,t))22σ(ω(z,t))2, (35) with    ∂tω(z,t)=−∂zj(z,t). (36)

From (34) the probability of a given density profile can be obtained by optimizing over all the current profiles satisfying (36) and it is given by where From the explicit expression of in (35), the optimizing equation reads

 ∫∞−∞dz(jo(z,t)+D(ω(z,t))∂zω(z,t))σ(ω(z,t))=0, (37)

whose solutions can be written as where is an auxiliary function. Putting this optimized solution in , one finds that the auxiliary function is continuous across :

 h(z,t)|z→0+=h(z,t)|z→0−. (38)

## Appendix B Optimal field equations for a given fixed initial configuration ωin(z)

Here we derive the equations (LABEL:letter-opt-diff-eq-1) and (LABEL:letter-opt-diff-eq-2) of the main text. We start with the generating function as defined in (LABEL:letter-cumu-func) of the main text, where we do not yet perform the average over the initial condition:

 ⟨exp[λkXk(T)+λlXl(T)]⟩ωin=∫ω|t=0=ωinD[ω,h]e−S[ω,h], (39)

where the action, obtained via Martin-Siggia-Rose formalism, is given by

 S[ω,h]=−λkXk(T)−λlXl(T)+∫T0dt∫∞−∞dz(h∂tω+(μ1/2)(∂zω)(∂zh)−(σ(ω)/2)(∂zh)2), (40)

where

 Xk(T)=xk(T)−xk(0)=∫k−∞dz [ω(z,T)−ω(z,0)]. (41)

For large , the integral in (39) is dominated by the optimal paths that minimize the action and it is given by . To find out the optimal paths we consider small variations and of and and compute the variation of the action,

 δS = S[ω+ϕ, h+ψ]−S[ω, h], (42) = −∫∞−∞dz (λkδXkδω(z,0)+λl δXlδω(z,0)+h(z,0)) ϕ(z,0) −∫∞−∞dz (λkδXkδω(z,T)+λl δXlδω(z,T)−h(z,T)) ϕ(z,T) +∫T0dt∫∞−∞dz(ψ(z,t)∂tω(z,t)−ϕ(z,t)∂th(z,t)) −∫T0dt∫∞−∞dz(ϕ(z,t)σ′(ω(z,t)) (∂zh(z,t))2+2σ(ω(z,t)) ∂zh(z,t) ∂zψ(z,t))) +μ12∫T0dt∫∞−∞dz(∂zω(z,t) ∂zψ(z,t))+∂zh(z,t) ∂zϕ(z,t))) = ∫∞−∞dz (λk Θ(k−z)+λl Θ(l−z)−h(z,0)) ϕ(z,0) +∫∞−∞dz (−λ