# Optimum transport in systems with time-dependent drive and short-ranged
interactions

Deepsikha Das  Punyabrata Pradhan<EMAIL_ADDRESS>Sakuntala
Chatterjee<EMAIL_ADDRESS>Physics of Complex Systems, S.N.
Bose National Centre for Basic Sciences
Block-JD, Sector-III, Salt Lake, Kolkata 700106, India

###### Abstract

We study one-dimensional hardcore lattice gases, with nearest-neighbor
interactions, in the presence of an external potential barrier, that moves on
the periodic lattice with a constant speed. We investigate how the nature of
the interaction (attractive or repulsive) affects particle transport and
determine, using numerical simulations and mean-field calculations, the
conditions for an optimum transport in the system. Physically, the particle
current induced by the time-dependent potential is opposed by a diffusive
current generated by the density inhomogeneity (a traveling wave) built up in
the system, resulting in a current reversal, that crucially depends on the
speed of the barrier and particle-number density. Indeed the presence of
nearest-neighbor interaction has a significant impact on the current:
Repulsive interaction enhances the current, whereas attractive interaction
suppresses it considerably. Quite remarkably, when the number density is low,
the current increases with the strength of the repulsive interaction and the
maximum current is obtained for the strongest possible repulsion strength,
i.e., for the nearest-neighbor exclusion. However, at high density, very
strong repulsion makes particle movement difficult in an overcrowded
environment and, in that case, the maximal current is achieved for weaker
repulsive interaction strength.

††preprint: APS/123-QED

##  I Introduction

The ability to manipulate colloidal-particle motion in narrow channels using
time-varying optical potential has opened up new research avenues in driven
diffusive systems 1 ; 16 ; 18 ; 100 ; 13 . These studies have provided
valuable insights into a variety of important aspects of non equilibrium
systems, such as verification and applicability of fluctuation relations,
among other things. Experiments with colloidal particles driven by an optical
trap 14 ; 15  have previously established the validity of the fluctuation-
dissipation theorem, which predicts entropy production over finite time, and
that of a generalized Einstein relation 12 ; Additionally, the violation of
the second law of thermodynamics has been experimentally demonstrated for
small systems over short time scales 10 ; see Ref. 21  for review. Recently, a
particularly promising research direction that has received a lot of attention
is the characterization of particle transport in a periodically driven many-
particle system 21 ; 23 . These systems find application in a wide range of
situations. For example, stochastic pumps 19 ; 26 ; 27 ; 29 , in which the
time-varying external parameters drives the systems away from equilibrium, can
generate a directed particle flow; also consider the thermal ratchets 20 ; 53
; 30 , where non equilibrium fluctuations can induce a directed particle
motion. Indeed, much attention has been focused on, and significant progress
has been made, in understanding the underlying mechanism of directed flow in
thermal ratchets and molecular pumps 19 .

Notably, the characterization of particle transport in time-varying external
potential is important also in the context of driven fluids in confined
geometry, leading to the identification of number of unexpected consequences,
such as negative differential resistance and absolute negative mobility, etc.,
among others 5 ; 6 ; 2 ; 61 ; 62 ; 64 ; 66 ; 67 . In the past, particle
transport in colloidal suspensions in narrow channels have motivated studies
of noninteracting particles, driven by a moving potential barrier, using
dynamic density functional theory 5 ; 6 . Eventually, several many-particle
models were also put forth in an effort to theoretically understand the role
of hardcore interactions in these systems 31 . One particularly important
question in this context is whether the system can support a nonzero dc (time-
averaged) current when it is driven by a time-periodic driving force. Although
the presence of an external forcing would typically suggest the presence of a
current in the system, the periodic nature of the driving however means that
the net force acting on the system over a time period is zero. In that case,
do such systems still carry a nonzero dc current? If so, in what direction
does the current flow? Another intriguing question is whether it is possible
to optimize the particle current by tuning various control parameters.

In order to address the above issues, a series of works 31 ; 32 ; 35
considered the paradigmatic models of simple exclusion processes 37 , in which
the interaction among the particles was assumed to be of the simplest possible
form, i.e., hardcore exclusion. The motion of the particles were described on
a lattice where particles hop from one site to a neighboring unoccupied site;
in that case, the periodic external potential was simply represented by space
and time dependent hopping rates. Depending on whether the time-varying
hopping rates were present only on particular sites or were present throughout
the system, it was shown through numerical simulations and a perturbative
approach that the dc current flowing through such a system could either vanish
(inversely with system size) or have a finite value. Furthermore, the dc
current was found to exhibit non-monotonic dependence on the time-period of
the drive. Several interesting features such as current reversal and system-
size dependent transport were observed 35  in the case when time-varying
potential maintains a position-dependent phase relation among sites, that
results in a nonzero dc current.

Subsequently, in another study of a many-particle lattice model 33 , our group
had developed a new and simple method of modeling a periodically moving drive
in a system of hardcore particles diffusing on a ring. Motivated by moving
trap or barrier used in experiments with particles in an optical potential, we
studied a system with a “defect” site, which had a hopping rate different from
the rest of the system. Then, in the non equilibrium setting, the defect site
was considered to move on the lattice with a speed $v$, and, for a one
dimensional ring of $L$ sites, to complete one cycle after a time period
$L/v$. Using numerical simulation and a mean-field theory, we observed that,
in the time-periodic steady state, a density inhomogeneity is created around
the defect, resulting in a dc current in the system that scales as $1/L$. The
direction and magnitude of the dc current was controlled by tuning the defect
speed, particle density and the bulk diffusivity of particles. Moreover, in
the presence of multiple defect sites 34 , an interesting collective behavior
was observed when the defect sites were close enough so that their respective
density patterns generated by each of the defects overlap with each other.
Interestingly, reversal of current has also been observed in a slightly
different set up 300  in the context of a single particle, which diffuses in a
two-dimensional channel of varying width and is driven by a force having a
random orientation across the channel; in this case, the current reversal
happens by tuning both the transverse and the longitudinal drive.

So far, in the previous studies of many-particle lattice models, the only type
of the interaction considered between the particles was the hardcore
exclusion. However, in real systems, particles can also experience short-
ranged attraction or repulsion and the interplay between external driving and
inter-particle interactions are expected to give rise to nontrivial effects.
In order to investigate this scenario, in the present work we consider a many-
particle lattice model in which hardcore particles diffuse and interact via
nearest-neighbor attractive or repulsive potential. In other words, in
addition to the hardcore exclusion, a particle, in the case of repulsive
(attractive) interaction, now prefers to have its neighboring site empty
(occupied). Here we are primarily interested in exploring how the strength of
the interaction potential affects the particle current in the system. Does the
system still supports current reversal and, if so, how are the transport
characteristics affected by many-particle interactions? Is there an optimum
interaction strength for which magnitude of the current in either direction is
largest?

In this paper, by performing Monte Carlo simulations and using a modified
mean-field theory, we have determined the condition of optimal transport in
the system and studied how attractive or repulsive interaction among the
particles affect the transport. We show that a moving defect always induces
current in the negative direction, i.e., along the direction opposite to the
defect movement. But, due to the density inhomogeneity produced by the defect
movement, the diffusive current in the system flows in the positive direction.
As a result, when the bulk diffusion in the system is negligibly small, we
find current in the negative direction. However, as the bulk diffusion becomes
stronger, the current changes sign and becomes positive. By varying the defect
speed, particle density and the interaction strength, we determine the
parameter regime, that yields the optimum current in the system in either
direction. It turns out that an attractive interaction among the particles
hinders transport, while a repulsive interaction enhances it. For small
particle density, current is largest when the strength of the repulsive
interaction assumes its highest possible value. However, for large particle
density, the system is overcrowded and a very strong repulsion indeed blocks
certain transitions and consequently reduces the current. In that case, the
optimum transport is obtained when the interaction strength lies somewhat
below the largest possible value. However, unlike repulsive interaction and
irrespective of defect speed and bulk density, the current decreases
monotonically with the attractive interaction strength.

The organization of the paper is as follows: We describe the model in section
II. In Section III, we describe the simplest case where dynamics in the bulk
of the system is absent (the case where particle speed is larger compared to
the corresponding rate of bulk diffusion). Analytical formalism for this
particular situation is presented in subsections III A and III B while results
are shown and discussed in subsection III C. In section IV, we discuss the
case when bulk dynamics is also compared to the other rates in the system. Our
conclusions are presented in section V.

##  II The Model

We consider paradigmatic models of exclusion processes involving hardcore
particles with nearest-neighbor interaction 43 . We incorporate the
periodically moving external potential barrier simply as a set of moving
“defects” 33 ; 34 , each of them resides at a site for duration $\tau$ before
moving to the right. The energy function for the system can be written as

$H=-\dfrac{J}{2}\sum_{\begin{subarray}{c}i,j\\\
<i,j>\end{subarray}}\eta^{\\{\alpha_{k}\\}}_{i}\eta^{\\{\alpha_{k}\\}}_{j}+\sum_{i}\eta^{\\{\alpha_{k}\\}}_{i}V_{i}$
(1)

where, $\eta^{\\{\alpha_{k}\\}}_{i}$, $\eta^{\\{\alpha_{k}\\}}_{j}$ denote
occupancy of sites which can take values 0 or 1. The indices
$\\{\alpha_{k}\\}\equiv\\{\alpha_{1},\alpha_{2},\dots,\alpha_{N}\\}$ are a set
of $N$ elements with the $k$-th element, $\alpha_{k}$, denoting the position
of the $k$-th defect and $<i,j>$ denotes that sites $i$ and $j$ are the
nearest neighbors, A site is called a defect site when a potential barrier is
present there and it is called a bulk site otherwise. The potential at site
$i$ is $V_{i}=\sum_{k}V_{0}\delta_{i,\alpha_{k}}$, where $V_{0}$ represents
height of the onsite potential barriers, and $J$ denotes the interaction
strength that can vary in the range $-\infty$ to $+\infty$. A mapping 44  can
be performed from $J$ to a dimensionless parameter $\epsilon$, as given below

$\displaystyle e^{-\beta J}=\dfrac{(1+\epsilon)}{(1-\epsilon)}$ (2)

such that $|\epsilon|\leq 1$ and thus $0\leq e^{-\beta J}<\infty$. Note that,
for a left (right) hopping to take place, the departure site has to be
occupied by a particle and its immediate left (right) site has to be empty.
Therefore, particle hopping towards left (right) can happen in four possible
ways and all possible transition rates 44  for left and right hops are written
as following:

$\begin{array}[]{ccl}0010\xrightleftharpoons[ce^{-\beta V_{i}}]{c}0100\\\
1010\xrightleftharpoons[c(1+\epsilon)e^{-\beta V_{i}}]{c(1-\epsilon)}1100\\\
0011\xrightleftharpoons[c(1-\epsilon)e^{-\beta V_{i}}]{c(1+\epsilon)}0101\\\
1011\xrightleftharpoons[ce^{-\beta V_{i}}]{c}1101\end{array}$

$\begin{array}[]{ccl}0100\xrightleftharpoons[ce^{-\beta V_{i}}]{c}0010\\\
0101\xrightleftharpoons[c(1+\epsilon)e^{-\beta V_{i}}]{c(1-\epsilon)}0011\\\
1100\xrightleftharpoons[c(1-\epsilon)e^{-\beta V_{i}}]{c(1+\epsilon)}1010\\\
1101\xrightleftharpoons[ce^{-\beta V_{i}}]{c}1011\end{array}$ (3)

with $\beta V_{0}=\ln{(p/r)}$, where $\beta$ is the inverse of temperature and
(i) $c=p/2$ when the departure site is a defect site and a particle hops out
of it, (ii) $c=q/2$ when the departure site is a bulk site, (iii) $c=r/2$ when
the departure site is a neighbor of a defect and the destination is a defect
site. The transition rates and corresponding reverse rates follow detailed
balance condition for defect velocity $v=0$. For simplicity, in our study, we
have considered only a single defect, which represents an infinite potential
barrier, periodically moving over a lattice of length $L$ with speed $v$.
Consequently $r=0$ and $p=1$ are maintained throughout the paper.

Figure 1: Schematic diagram of the model. Red solid (empty) ellipses represent
occupied (empty) defect sites while the blue solid (empty) circles are
occupied (empty) bulk sites. In one time step a particle can jump to one of
its neighboring site provided the destination site is empty. The transition
rates depend on the local configurations around the departure site, as
specified in Eq. (3).

##  III $p=1$, $r=0$, $q=0$: No dynamics in the bulk

###  III.1 Analytical formalism for a periodically moving defect

In this section, we consider the simplest case when the inverse hopping rate
is much larger compared to the typical residence time scale of the defect. In
that case, we assume the bulk hopping rate $q=0$, i.e., dynamics in the bulk
is completely frozen. A particle can hop during an infinitesimal time interval
$dt$ only if its position coincides with the position of the defect denoted by
$\alpha$. Starting from an initial configuration, the system reaches a time
periodic steady state, given sufficient time has passed. The density profile
of the system has a form of a traveling wave moving over the lattice with the
same speed $v$ as that of the moving defect. The defect spends time $\tau$ at
a particular site before moving on to the next site where $\tau=1/v$ is the
residence time of the defect. When the defect spends one Monte-Carlo step on
each lattice site, $v$ is measured as $1$. We have measured density profile at
time-steps $t=n\tau$ just before the defect moves on to the next site, after
spending time $\tau$ at the previous site, with $n=0,1,2,....,\infty$. For
time $t=n\tau$ we write the discrete time evolution equation 33  for density
$\rho_{i}^{(\alpha)}(t)=\langle\eta_{i}^{(\alpha)}(t)\rangle$,

$\langle\rho^{(\alpha+1)}(t+\tau)|=\langle\rho^{(\alpha)}(t)|W^{(\alpha+1)}$
(4)

Here,
$\langle\rho^{(\alpha)}(t))|\equiv\\{\rho_{1}^{(\alpha)}(t),....,\rho_{i}^{(\alpha)}(t),....,\rho_{L}^{(\alpha)}(t)\\}$
is a row vector of length $L$, its $i$-th element being
$\rho_{i}^{(\alpha)}(t)$ with $\alpha$ denoting the position of the defect.
$W^{(\alpha+1)}$ is the transition matrix with the defect site at $\alpha$ \+
1. Its structure depends on the position of the defect site. For example when
$\alpha+1=1,2$, respective transition matrices can be written as,
$W^{(1)}=\begin{bmatrix}1-a_{+}-a_{-}&a_{+}&0&\dots&0&a_{-}\\\
0&1&0&0&\dots&0\\\ \dots&\dots&\dots&\dots&\dots&\dots\\\
\dots&\dots&\dots&\dots&\dots&\dots\\\ 0&\dots&0&0&1&0\\\
0&0&\dots&0&0&1\end{bmatrix}$ $W^{(2)}=\begin{bmatrix}1&0&0&\dots&0&0\\\
a_{-}&(1-a_{+}-a_{-})&a_{+}&0&\dots&0\\\ 0&0&1&0&\dots&0\\\
\dots&\dots&\dots&\dots&\dots&\dots\\\ 0&\dots&0&0&1&0\\\
0&0&\dots&0&0&1\end{bmatrix}$ (5)

The $i,j$-th element of the transition matrix 33  can be written as,

$\displaystyle W_{ij}^{\alpha+1}$
$\displaystyle=1-a_{+}-a_{-}\quad\quad\text{for}\quad\quad i=j=\alpha+1$
$\displaystyle W_{ij}^{\alpha+1}$
$\displaystyle=a_{-}\quad\quad\text{for}\quad\quad i=j+1=\alpha+1$
$\displaystyle W_{ij}^{\alpha+1}$
$\displaystyle=a_{+}\quad\quad\text{for}\quad\quad i=j-1=\alpha+1$ (6)
$\displaystyle W_{ij}^{\alpha+1}$
$\displaystyle=1\quad\quad\text{for}\quad\quad i=j\neq\alpha+1$ $\displaystyle
W_{ij}^{\alpha+1}$ $\displaystyle=0\quad\quad\text{for}\quad\quad
i\neq\alpha+1,\,i\neq j$

Here $a_{\pm}$ are the conditional probabilities that, given the defect site
is occupied, particle from the defect site moves to its unoccupied right(left)
neighboring site during the residence time $\tau$. Starting from microscopic
dynamics we can have their expressions as following:

$\displaystyle
a_{+}=\sum_{m=1}^{6}{\mathcal{C}}_{m}^{+}\omega_{m}^{+}\quad,\quad$
$\displaystyle a_{-}=\sum_{n=1}^{6}{\mathcal{C}}_{n}^{-}\omega_{n}^{-}$ (7)

where ${\mathcal{C}}_{m}^{+}$, ${\mathcal{C}}_{n}^{-}$ are the conditional
probabilities of different local configurations favorable for right and left
hopping respectively during the residence time $\tau$, given the defect site
is occupied. $\omega_{m}^{+}$, $\omega_{n}^{-}$ denote transition
probabilities for right and left hopping respectively from an occupied defect
site during $\tau$. For example,

${\mathcal{C}}_{1}^{+}=\text{Prob}.(00\hat{1}01|\eta^{(\alpha)}_{\alpha+1}=1)=\dfrac{\bigg{\langle}(1-\eta^{(\alpha)}_{\alpha-1})(1-\eta^{(\alpha)}_{\alpha})\eta^{(\alpha)}_{\alpha+1}(1-\eta^{(\alpha)}_{\alpha+2})\eta^{(\alpha)}_{\alpha+3}\bigg{\rangle}}{\langle\eta^{(\alpha)}_{\alpha+1}\rangle}$
(8)

${\mathcal{C}}_{1}^{-}=\text{Prob}.(10\hat{1}00|\eta^{(\alpha)}_{\alpha+1}=1)=\dfrac{\bigg{\langle}\eta^{(\alpha)}_{\alpha-1}(1-\eta^{(\alpha)}_{\alpha})\eta^{(\alpha)}_{\alpha+1}(1-\eta^{(\alpha)}_{\alpha+2})(1-\eta^{(\alpha)}_{\alpha+3})\bigg{\rangle}}{\langle\eta^{(\alpha)}_{\alpha+1}\rangle}$
(9)

$\omega_{1}^{+}=\omega_{1}^{-}=\dfrac{1-\epsilon}{2-\epsilon}\biggl{(}1-e^{-(2-\epsilon)/4v}\biggr{)}$
(10)

where ${\mathcal{C}}_{1}^{+}$ and ${\mathcal{C}}_{1}^{-}$ represent the
conditional probabilities for local configurations $00\hat{1}01$ and
$10\hat{1}00$ respectively given the defect site is occupied. $\omega_{1}^{+}$
and $\omega_{1}^{-}$ denote the transition probabilities corresponding to
configuration $00\hat{1}01$ and $10\hat{1}00$ respectively during a residence
time $\tau$ with $\hat{1}$($\hat{0}$) denoting an occupied (unoccupied) defect
site (details are given in Appendix). Due to the time periodic structure of
the steady state, the density profile comes back to itself each time the
defect moves across the ring and completes a cycle. So,
$W^{(\alpha+1)}...W^{(L)}W^{(1)}...W^{(\alpha-1)}W^{(\alpha)}$ has an
eigenvector $\langle\rho_{st}^{(\alpha)}|$, with eigenvalue unity. The steady
state density 33  at $i$-th site satisfies,

$\rho_{st,i}^{(\alpha+1)}=\rho_{st,i-1}^{(\alpha)}$ (11)

which follows from the time periodic structure of the steady state density and
from Eq. (4). To solve for the density profile in a time periodic steady
state, we find that at the time of measurement, the defect site $\alpha$
registers lower density compared to the bulk as for $r$=0 particles can’t hop
into the defect site. Rather they can only hop out of the defect site. For
$q$=0 the neighboring sites $(\alpha\pm 1)$ can only receive particles from
the defect site without any loss. The site $(\alpha+1)$ thus has a higher
density compared to that at the bulk. On the other hand, the site $(\alpha-1)$
which was previously occupied by the defect and has already registered lower
density, could only receive particle from the defect site $\alpha$ and its
density goes back to the bulk level. Therefore regarding the structure of the
density profile as a function of position, we formulate an ansatz 33  in the
form of a traveling density wave which moves with the defect $\alpha$.

$\displaystyle\rho_{st,i}^{(\alpha)}=$
$\displaystyle\rho_{-}\quad\text{for}\quad i=\alpha$
$\displaystyle\rho_{st,i}^{(\alpha)}=$
$\displaystyle\rho_{+}\quad\text{for}\quad i=\alpha+1$ (12)
$\displaystyle\rho_{st,i}^{(\alpha)}=$
$\displaystyle\rho\quad\quad\text{otherwise}$

The ansatz can be used in Eqs. 4 and 11, to obtain Eqs. 13, 14.

$\rho_{+}a_{-}+\rho_{-}=\rho_{b}$ (13) $\rho_{+}a_{+}+\rho_{b}=\rho_{+}$ (14)

which can be solved by using particle number conservation
$\rho_{+}+\rho_{-}+(L-2)\rho_{b}=L\rho$ to get the exact densities,

$\rho_{b}\quad=\quad\dfrac{(1-a_{+})L}{2-a_{+}-a_{-}+(1-a_{+})(L-2)}\rho\quad\simeq\quad\rho$
(15) $\displaystyle\rho_{+}\quad\simeq\quad$
$\displaystyle\dfrac{1}{1-a_{+}}\rho$ (16)
$\displaystyle\rho_{-}\quad\simeq\quad$
$\displaystyle\dfrac{1-a_{+}-a_{-}}{1-a_{+}}\rho$ (17)

as $L>>1$. From Eqs. 16 and 17 it is evident that $\rho_{+}>\rho$ and
$\rho_{-}<\rho$ i.e a peak and a trough are formed in front of the defect site
and at the defect site respectively.

The trough and the peak present in the density profile are different in size
which will result in a non-zero particle current in general. Contribution to
particle current comes from only two bonds adjacent to the defect site as no
hopping takes place across any other bond for $q=0$. Particle current consists
of two components $\text{$J$}_{+}$ and $\text{$J$}_{-}$, defined to be the
time rate of rightward and leftward movement of particles respectively, from
the defect site. The total current is the algebraic sum of them. The defect
visits a particular site with rate $v/L$. Thus the expression for particle
current can be written as

$J=J_{+}+J_{-}=\dfrac{v}{L}(\langle\eta^{(\alpha)}_{\alpha+1}\rangle
a_{+}-\langle\eta^{(\alpha)}_{\alpha+1}\rangle a_{-})$ (18)

which can be written in terms of $\rho_{\pm}$ from Eq. 16 and 17 as

$\displaystyle
J=\dfrac{v}{L}((\rho_{+}-\rho)+(\rho_{-}-\rho))=\dfrac{v}{L}(\rho_{+}+\rho_{-}-2\rho)$
(19)

where

$J_{+}=\dfrac{v}{L}(\langle\eta^{(\alpha)}_{\alpha+1}\rangle
a_{+})=\dfrac{v}{L}(\rho_{+}-\rho)$ (20)
$J_{-}=-\dfrac{v}{L}\langle\eta^{\alpha}_{\alpha+1}\rangle
a_{-}=\dfrac{v}{L}(\rho_{-}-\rho)$ (21)

###  III.2 Mean-field theory

The exact expression for $a_{+}$ and $a_{-}$ are given in Eq. 7 and in Eq. A-1
to A-17. To write $a_{+}$ and $a_{-}$ as explicit functions of $\epsilon$,
$\rho$ and $v$, we use mean-field approximation, where many-point correlations
${\mathcal{C}}_{m}^{+}$, ${\mathcal{C}}_{n}^{-}$ (for m, n = 1, 2,…, 6) are
assumed to be factorized. Thus we have,

$\displaystyle{\mathcal{C}}_{1}^{+}={\mathcal{C}}_{1}^{-}=\rho(1-\rho_{-})(1-\rho)^{2}$
(22)

etc. and

$\displaystyle
a_{+}=(1-\rho)\Biggl{[}(1-\rho)(1-\rho_{-})\rho\omega_{1}^{+}+\rho^{2}(1-\rho_{-})\omega_{2}^{+}+(1-\rho)^{2}(1-\rho_{-})\omega_{3}^{+}$
(23)
$\displaystyle+\rho(1-\rho_{-})(1-\rho)\omega_{4}^{+}+\rho_{-}(1-\rho)\omega_{5}^{+}+\rho_{-}\rho\omega_{6}^{+}\Biggr{]}$

$\displaystyle
a_{-}=(1-\rho_{-})\Biggl{[}\rho(1-\rho)^{2}\omega_{1}^{-}+\rho^{2}(1-\rho)\omega_{2}^{-}+(1-\rho)^{3}\omega_{3}^{-}$
(24)
$\displaystyle+\rho(1-\rho)^{2}\omega_{4}^{-}+\rho(1-\rho)\omega_{5}^{-}+\rho^{2}\omega_{6}^{-}\Biggr{]}$

Combining Eqs. 16, 17 we have the following form.

$(\rho_{-}-\rho)(1-a_{+})+a_{-}\rho=0$ (25)

If expressions for $a_{+}$ and $a_{-}$ from Eqs. 23, 24 are put into Eq. 25 we
will obtain the following quadratic equation for $\rho_{-}$,

$\displaystyle(\rho_{-}-\rho)[1-(1-\rho)\\{(1-\rho)(1-\rho_{-})\rho\omega_{1}^{+}+\rho^{2}(1-\rho_{-})\omega_{2}^{+}+(1-\rho)^{2}(1-\rho_{-})\omega_{3}^{+}$
(26)
$\displaystyle+\rho(1-\rho_{-})(1-\rho)\omega_{4}^{+}+\rho_{-}(1-\rho)\omega_{5}^{+}+\rho_{-}\rho\omega_{6}^{+}\\}]+\rho(1-\rho_{-})[\rho(1-\rho)^{2}\omega_{1}^{-}$
$\displaystyle+\rho^{2}(1-\rho)\omega_{2}^{-}+(1-\rho)^{3}\omega_{3}^{-}+\rho(1-\rho)^{2}\omega_{4}^{-}+\rho(1-\rho)\omega_{5}^{-}+\rho^{2}\omega_{6}^{-}]=0$

Solving Eq. 26 in Mathematica, retaining the physically acceptable solution
(not larger than unity) we have obtained $\rho_{+}$ and $\rho_{-}$ against
$\epsilon$.

Then meanfield expression for current can be written using Eqs. 23, 24 and
Eqs.A-1 to A-17, as

$\displaystyle\text{$J$}=\dfrac{v}{L}\rho_{+}(\rho_{-}-\rho)\\{(1-\rho)(1-e^{-(1+\epsilon)/4v})+\rho(1-e^{-1/4v})\\}$
(27)

In the limit of small and large $\rho$, the solutions for $\rho_{\pm}$ and
hence for particle current $J$ take simple forms as we write $a_{\pm}$ in
leading order of $\rho$ and $(1-\rho)$ respectively.

####  III.2.1 Small-density approximation

For small $\rho$, we retain leading order terms in $\rho$ in its functions. We
obtain,

$a_{+}\approx\rho_{-}[\rho(3\omega_{3}-\omega_{1}-\omega_{4}-2\omega_{5}+\omega_{6})-(\omega_{3}-\omega_{5})]+\rho(\omega_{1}-3\omega_{3}+\omega_{4})+\omega_{3}$
(28)

$a_{-}\approx\rho_{-}[\rho(3\omega_{3}-\omega_{1}-\omega_{4}-\omega_{5})-\omega_{3}]+\rho(\omega_{1}-3\omega_{3}+\omega_{4}+\omega_{5})+\omega_{3}$
(29)

Substituting Eqs. 28 and 29 into Eq. 25 we have,

$\rho_{-}=\dfrac{1-2\omega_{3}}{1-\omega_{3}}\rho$ (30)

$\rho_{+}=\dfrac{\rho}{1-\omega_{3}}$ (31)

Scaled current $JL$ takes the form,

$JL=\dfrac{v\omega_{5}(1-2\omega_{3})\rho^{2}}{(1-\omega_{3})^{2}}$ (32)

####  III.2.2 Large-density approximation

For large $\rho$, we retain terms in leading order of $(1-\rho)$. We obtain,

$a_{+}\approx\omega_{6}(1-\rho)+(\omega_{2}-\omega_{6})(1-\rho)(1-\rho_{-})$
(33)

$a_{-}\approx\omega_{6}(1-\rho_{-})+(\omega_{2}+\omega_{5}-2\omega_{6})(1-\rho)(1-\rho_{-})$
(34)

Substituting Eqs. 33 and 34 into Eq. 25 we have,

$\rho_{-}=1-\dfrac{(1-2\omega_{2}-2\omega_{5}+3\omega_{6})}{(1-\omega_{2}-\omega_{5}+\omega_{6})^{2}}(1-\rho)$
(35)

$\rho_{+}=1-(1-\omega_{6})(1-\rho)$ (36)

Scaled current $JL$ takes the form,

$JL=\dfrac{v\omega_{6}}{\omega_{6}-1}(1-\rho)$ (37)

.

###  III.3 Simulation results and comparison with mean-field theory

We present here numerical results obtained from simulation along with
analytical calculations from mean-field theory. We have used system size
$L=512$ throughout. We have studied variation of density peak $\rho_{+}$,
density trough $\rho_{-}$ and particle current $J$ against bulk density
$\rho$, interaction strength $\epsilon$ and defect velocity $v$. For $q=0$, it
has been observed that magnitude of particle current is maximum around
$v=0.16$. Since we are interested in optimum particle transport, $\rho$ and
$\epsilon$ dependence of all the quantities have been studied for $v=0.16$.
Variation against $\epsilon$ and $v$ have been studied for two different
densities, $\rho=0.29$ and 0.75, well below and above $\rho=0.5$ while
variation against $\rho$ and $v$ are carried out for $\epsilon=0.6$ and
$-0.6$, corresponding to repulsive and attractive interaction respectively.

From density profiles depicted in Figs. 2 and 2 and from variation of
$\rho_{\pm}$ vs $\rho$ depicted in Fig. 3, it can be observed that, density
peak and trough become more pronounced in case of repulsive interaction.
Variation of $\rho_{-}$ with $\rho$ is stronger compared to that of
$\rho_{+}$. Such a behavior is also supported by mean-field theory.

(a) $\rho$=0.29

(b) $\rho=0.75$

Figure 2: Particle density profile $\rho(x)$ where $x$ denotes the distance
from the defect site. For all interactions the defect site has a density
trough and its right neighbor has a peak. For attractive interaction the
trough and peak are relatively shallower.  Figure 3: $\rho_{\pm}$ are plotted
against bulk density $\rho$. Mean-field results are presented by solid
($\epsilon=-0.6$), dashed ($\epsilon=0$) and dotted ($\epsilon=0.6$) lines.
$\rho_{+}$ shows a weaker dependence on $\rho$ compared to $\rho_{-}$ . For
attractive interaction $\rho_{+}(\rho_{-})$ is noticeably smaller (greater)
than that for the hardcore and repulsive interactions. Mean-field results show
good agreement for $\epsilon\geq 0$ but for $\epsilon<0$ quantitative
deviation from numerical data is observed.

In Fig. 4, we find that as $\epsilon$ increases from negative to positive
values, the differences $(\rho_{+}-\rho)$ and $(\rho-\rho_{-})$ also increase.
The variation in both these quantities against $\epsilon$ are non-monotonic
with a peak at large positive $\epsilon$ values. Our mean-field calculation
captures this non-monotonic behavior but does not provide good quantitative
agreement with the numerical data.

(a) $\rho=0.29$

(b) $\rho=0.75$

Figure 4: Size of the density peak and trough, ($\rho_{+}-\rho$) and
($\rho-\rho_{-}$) are plotted against $\epsilon$ in panels (a) and (b). Mean-
field results are presented by solid and dotted lines respectively. It is
evident from both the panels that depth of the trough is always greater than
height of the peak. Non-monotonic variation against $\epsilon$ can be seen in
both the quantities which is more pronounced for ($\rho-\rho_{-}$) with a
maximum at a large positive $\epsilon$. Mean-field theory can qualitatively
capture such behavior in the repulsive region while it fails in the region of
attractive interaction.

In Figs. 5 and 5 we show the variation of $\rho_{\pm}$ with $v$ for two
different $\rho$ values. For small $v$ a particle can almost always hop out of
the defect site but as $v$ increases such a transition may not always be
possible because of short residence time of the defect 33 , 34 . Therefore
$\rho_{-}$ ($\rho_{+}$) increases (decreases) with $v$, finally saturating to
$\rho$ for very large $v$. As $\epsilon$ increases from negative to positive
values, for all $v$, $\rho_{-}$ becomes systematically lower and $\rho_{+}$
becomes higher, consistent with what we have shown in Fig. 3.

(a) $\rho=0.29$

(b) $\rho=0.75$

Figure 5: $\rho_{\pm}$ are plotted against defect velocity $v$. Mean-field
results are presented by solid ($\epsilon=-0.6$), dotted ($\epsilon=0$) and
dashed ($\epsilon=0.6$) lines which qualitatively capture the variation. For
large $v$ both these quantities approach $\rho$ while for small $v$ they show
weak variation. Comparing the data for different $\epsilon$ values show that
for all $v$ repulsive interaction causes highest (lowest)
$\rho_{+}(\rho_{-})$.

In Fig. 6 the variation of scaled particle current $JL$ with $\epsilon$ has
been shown for different $\rho$ values. At $\epsilon=-1$ because of strong
attractive interaction among the particles, the system supports one single
cluster containing all the particles. Therefore current vanishes in this
limit. As $\epsilon$ increases, around the defect site a density profile
consisting of peak and trough as shown in Figs. 2, 2 is formed and current
becomes non-zero. As $\epsilon$ increases further the density peak and trough
become more pronounced (as shown in Fig 4) resulting in larger current
magnitude. However, for positive $\epsilon$, current shows qualitatively
different variations for small and large $\rho$. For small $\rho$ values,
current remains almost constant with $\epsilon$ before showing a mild increase
near $\epsilon=1$.

Figure 6: scaled current are plotted against epsilon along with mean-field
results (presented by dotted line ($\rho=0.2$), short-dashed line
($\rho=0.29$), dot-dashed line ($\rho=0.4$), solid line ($\rho=0.75$) and
dashed line ($\rho=0.85$). Current vanishes at $\epsilon=-1$ and remains
negative elsewhere. For small and intermediate $\rho$ current is largest for
$\epsilon=1$, while for large $\rho$ it shows a peak at a slightly smaller
$\epsilon$ value.

This behavior can be explained from our data in Fig. 4, where the difference
between the two curves (red square and blue diamond), which represents the
asymmetry between the sizes of density peak and trough, remains unchanged for
a significant range of positive $\epsilon$ and increases when $\epsilon$ is
close to $1$. Note that this asymmetry is directly related to the current as
shown in Eq. 19. For large $\rho$, on the other hand, current shows a peak at
$\epsilon\simeq 0.7$ and decreases beyond that. This behavior is consistent
with our data in Fig. 4, where the two curves are seen farthest apart at that
particular $\epsilon$. Note that mean-field theory can qualitatively capture
the peak in current for large $\rho$ but for small $\rho$ it is unable to
reproduce the upswing shown by our data near $\epsilon=1$. We find similar
disagreement in Fig. 4 as well where mean-field theory hardly captures the
variation of trough size.

Fig. 7 shows the plot of scaled current $JL$ vs bulk density $\rho$ for
various $\epsilon$ values. In the limit $\rho\to 0$ and $\rho\to 1$ current
vanishes for all $\epsilon$ as expected. We have been able to analytically
show (see Eqs. 32, 37) that in the small density limit current $\sim\rho^{2}$,
while in the large density limit current $\sim(1-\rho)$. This limiting
behavior agrees reasonably well with our numerical data.

Figure 7: Scaled current vs density for various interaction strengths.
Discrete points show simulation data and lines show mean-field calculations.
We have used line-styles dotted ($\epsilon=-0.9$), short-dashed
($\epsilon=-0.5$), dashed ($\epsilon=0$), solid ($\epsilon=0.5$), dot-dashed
($\epsilon=0.97$) and a dot-dot-dashed line ($\epsilon=1$). For all epsilon
values, current shows non-monotonic variation with density. For attractive
interaction and moderate repulsive interaction, current shows a single peak at
a density $>1/2$. However, as repulsion becomes stronger, current shows two
peaks, separated by a minimum at $\rho=1/2$. Although mean field theory fails
to capture the double peak, we offer an alternative simple explanation in the
text.

From Fig. 7, we see that, for an intermediate density $\rho^{\ast}$, the
current shows a maximum. When $\epsilon$ takes large negative value, the
overall magnitude of the current is low because of strong attractive
interaction among the particles. As the particle attraction weakens, the
current also becomes larger and the peak at $\rho^{\ast}$ gets higher. Our
mean-field results successfully capture this trend although $\rho^{\ast}$
shows dependence on $\epsilon$ unlike a nearly constant $\rho^{\ast}\simeq
0.75$ obtained from numerics for all $\epsilon$. However, as $\epsilon$
changes sign and becomes positive the repulsive interaction does not favor
successive occupied sites. This gives rise to a special point at $\rho=0.5$
and $\epsilon=1$ when the configuration with alternate sites occupied by
particles is the only allowed configuration. No transitions are possible from
this configuration and hence current vanishes. This is verified from our
numerics where current sharply becomes zero at $\rho=0.5$ for $\epsilon=1$.
This generates another peak in current at a lower density $\rho<0.5$. However,
even as $\epsilon$ falls slightly below unity, this effect weakens and the
zero of current at half -filled density is replaced by a mild minimum.
Unfortunately, our mean-field calculations are unable to capture this effect
and predicts a single peak for current for all $\epsilon$.

Fig. 8 depicts variation of current with defect velocity $v$ for different
interaction strength and two different $\rho$ values. In all cases $v\to 0$
corresponds to the equilibrium limit when current vanishes. For very large $v$
the defect movement becomes too fast for the particles to respond and current
vanishes here too. An intermediate $v$ therefore maximizes the current which
can be seen both from our numerical data and mean-field calculations. As
$\epsilon$ increases from negative to positive values the peak current
increases monotonically for $\rho=0.29$ (Fig. 8), while for $\rho=0.75$ the
peak current shows a non-monotonic variation for positive $\epsilon$ (Fig. 8).
This is consistent with the variation observed in Fig. 6.

(a) $\rho=0.29$

(b) $\rho=0.75$

Figure 8: Scaled current $JL$ plotted against defect velocity $v$ along with
mean-field results (represented by dotted ($\epsilon=-0.6$), short-dashed
($\epsilon=0$), solid ($\epsilon=0.6$) and dot-dashed line ($\epsilon=1$)) in
panels (a) and (b). Current vanishes in the small $v$ and large $v$ limit and
shows a peak in between. The peak height increases as interaction changes from
attraction to repulsion. Largest peak is obtained for a large positive value
of $\epsilon$. Mean field theory explains the numerical data qualitatively.

In Figs. 6 to 8, we have plotted current as a function of one of the three
variables $\rho$, $\epsilon$ and $v$, keeping other two constant. To
understand the condition of optimum transport, we need to identify how $\rho$,
$\epsilon$ and $v$ should be chosen such that the current in the system is
maximum. To this end, we present heat-maps in Fig. 9 where we simultaneously
vary $\epsilon$ and $v$ for fixed $\rho$. Our numerical data are presented in
panels (a), (b) and our mean-field calculations appear in panels (c), (d) in
Fig. 9. These plots clearly show repulsive interaction facilitates particle
transport. For smaller density current always increases as $\epsilon$
increases and largest current is obtained at $\epsilon=1$. For larger density
on the other hand, very strong repulsion makes certain transitions
energetically unfavorable. This hinders particle transport. Therefore in this
case optimum transport is obtained at an intermediate $\epsilon$ value. Mean-
field calculations manage to reproduce this optimality correctly in Fig. 9,
but do not work so well in Fig. 9. Note that the scale used for low density is
widely different from that in the high density. This means when the density is
low, a condition for optimum transport can be derived but the current is far
smaller than optimum regime for high density. This is seen more clearly in
Fig. 10.

(a) $\rho=0.29$

(b) $\rho=0.75$

(c) $\rho=0.29$

(d) $\rho=0.75$

Figure 9: Scaled particle current $JL$ is plotted against $\epsilon$ and $v$.
Panels (a), (b) represent numerical data while panels (c), (d) show mean-field
results. The heat-maps help to trace out the region of $\epsilon$ and $v$
corresponding to the optimum transport in the system. Panel (a) shows that for
small density, current is maximum for strongest repulsion $\epsilon=1$, while
panel (b) shows that for large $\rho$ a positive $\epsilon<1$ optimizes the
transport. Note however, the scales chosen for left and right panels which
show magnitude of the optimum current is much larger when $\rho$ is large.
Mean- field results work reasonably well for large $\rho$, but fail to capture
the optimum transport regime for small $\rho$.

Fig. 10 depicts the heat-maps for current when $\epsilon$ and $\rho$ are
varied keeping $v$ constant. Apart from the usual choice of $v=0.16$, we have
also presented data for $v=1$ here. These plots show to obtain the optimum
transport, $\rho$ needs to be sufficiently high. Since in the high density
regime an intermediate strength of repulsive interaction gives maximum
current, the optimum transport happens away from $\epsilon=1$. Note that even
in this figure the actual value of maximum current is far higher for $v=0.16$
compared to $v=1$ case.

(a) $v=0.16$

(b) $v=1.0$

(c) $v=0.16$

(d) $v=1.0$

Figure 10: Numerical results for particle current $JL$ is plotted against
$\epsilon$ and $\rho$ in panel (a) and (b) while mean-field results are
represented in panel (c) and (d). The heat-maps trace out the optimum region
for particle current in $\epsilon-\rho$ plane. The region corresponds to large
$\rho$ and a high positive $\epsilon$. Magnitude of optimum current is higher
in left panels where intermediate $v$ value is used.

##  IV Nonzero bulk-hopping rate: $p=1$, $r=0$, $q\neq 0$

In the previous section, we had considered the case when the only possible
transition in the system is particle hopping out of the defect site. In the
present section we consider $q\neq 0$ which allows movement of particles in
the bulk of the system. We are interested to find out how this bulk dynamics
affects the current. As expected, for very small $q$ our results are similar
to what we had presented in the previous section. But as $q$ increases, there
is a significant effect on the current. We argue below that the bulk dynamics
is expected to make a positive contribution to the current. Note that for
$q\neq 0$ the density profile remains homogeneous far from the defect site and
therefore the non-vanishing contribution to current comes from the dynamics
around the defect site. For small enough $q$ the density profile remains
qualitatively similar to our plot in Fig. 2. A diffusive current will flow
between the site with density $\rho_{-}$ and its left neighbor with density
$\rho$. Since $\rho_{-}<\rho$, this current will be in the positive direction.
Therefore, inclusion of bulk dynamics adds a positive component to the system
current. We do not have a mean-field theory for this case to support the
numerical data. We have considered only small and moderate $q$ in our study.

In Fig. 11 we plot $JL$ vs $\epsilon$ for fixed $v$ and two different $\rho$
values. For $q=0.05$ the behavior is very similar to the trend observed in
Fig. 6. As $q$ increases $JL$ becomes more positive as explained above. For
small density $JL$ even reverses sign for $q=0.5$. For large density since
$JL$ starts from large negative values for $q=0$ it remains negative even when
$q=0.5$ although its magnitude decreases because of larger positive
contribution coming from bulk dynamics. Similar trends are observed in the
variation of $JL$ vs $\rho$ (Fig. 12)or $JL$ vs $v$ (Fig. 13).

(a) $\rho=0.29$

(b) $\rho=0.75$

Figure 11: Scaled current $JL$ is plotted against epsilon for different $q$ in
panel (a) and (b). Current is larger for repulsive interaction compared to
attractive ones. At low density and at a moderate $q$, current shows reversal
of sign as $\epsilon$ increases while for large density it remains negative
for all $q$.

(a) $\epsilon=-0.5$

(b) $\epsilon=1.0$

Figure 12: Variation of scaled current $JL$ with density $\rho$ for different
$q$ values. For small $q$, current shows similar behavior as in $q=0$ case
both for attractive and repulsive interaction. For moderate $q$ values current
shows a positive peak at small density and a negative peak at large density.
For $\epsilon=1$ current crosses zero exactly at $\rho=0.5$.

(a) $q=0.1$

(b) $q=0.5$

Figure 13: Scaled current $JL$ is plotted against defect velocity $v$ for
different $\epsilon$. For small $q$ current remains almost flat at small $v$
and reverses its direction at an intermediate $v$, while for moderate $q$ it
remains positive throughout for all $\epsilon$.

To identify the parameter regime for optimal transport we show the heatmap in
Fig.14. We have four relevant parameters here: $\epsilon,v,\rho$ and $q$. For
a fixed $\rho$ we show the variation of current in $\epsilon-v$ plane for two
different $q$ values. For $q=0.1$ current has both positive and negative
peaks, i.e optimum current can flow in the same direction of defect movement
or in the opposite direction. This can be clearly seen from Fig. 14. For
$q=0.5$ however, only positive current is possible and optimum transport
always happens in the direction of defect movement.

(a) $q=0.1$

(b) $q=0.5$

Figure 14: Numerical results for current $JL$ is plotted against $\epsilon$
and $v$ at $\rho=0.29$. For small $q$, a positive and a negative peak in the
variation of current can be observed from panel (a) while panel (b) shows that
there exists a single positive peak in its variation. Such peaks occur at
large positive $\epsilon$.

##  V Summary and concluding remarks

In this paper, we have studied a class of stochastic lattice gases of hardcore
particles with nearest-neighbor interaction, where the system is driven by a
localized potential barrier (referred to as a “defect”) moving on a ring. We
find that the inter-particle interaction is crucial in controlling particle
transport in the system: In the presence of an attractive interaction, the
time-averaged dc current decreases, whereas a repulsive interaction increases
the current significantly, thus resulting in an interaction-dominated regime
of particle transport. The moving potential barrier creates a traveling
density inhomogeneity, which generates a current in the negative direction,
i.e., in the direction opposite to its movement, while the bulk diffusion
generates a current in the direction along the barrier movement. As a result,
when the bulk hopping (diffusion) rate vanishes, i.e., when $q=0$, the
particle current is always negative and shows a negative peak as the barrier
movement speed $v$ and bulk density $\rho$ are varied. Quite remarkably, the
negative peak in the current is further enhanced when a strong repulsive
interaction is present among the particles. On the other hand, for the bulk
hopping rate $q\neq 0$, as defect speed $v$, bulk density $\rho$ and
interaction strength $\epsilon$ are varied, the particle current shows both
positive and negative peaks, which are due to the competition between the
positive contribution from bulk diffusion and the negative contribution from
the defect movement; however the extent of variation is weaker in this case
compared to that for $q=0$. We have been able to identify the precise
parameter regime for an optimum transport, which indeed maximizes the
magnitude of the current. In the case of attractive interaction, a particle
prefers to have its nearest neighbor occupied, giving rise to particle
clustering. The contribution in the current from the transitions which cause
fragmentation of the clusters decreases as the strength of attractive
interaction increases, thus resulting in a decreased current. Indeed, unlike
repulsive interaction, the current decreases monotonically with the attractive
interaction strength, irrespective of defect speed and bulk density. To
theoretically understand the above results, we perform a modified mean-field
calculation, which - for repulsive interaction, high particle density and
negligible bulk diffusion - agrees reasonably well with simulations. Strong
attractive interaction causes particle clustering, leading to strong spatial
correlations in the system, and the mean-field theory in that case does not
work well. Also, for large bulk diffusion, our mean-field theory does not show
satisfactory agreement with simulations, again due to the built up of quite
strong spatial correlations in the system.

The role of inter-particle interactions in controlling particle transport in
the presence of a time-periodic drive can be tested in experiments. A periodic
potential energy landscape can be created by superimposing external rotating
magnetic field on local periodic arrangement of micro-magnets 70 . With the
help of this periodic potential, micron-size super paramagnetic beads can be
separated from a complex mixture by transporting the beads across a substrate.
Then, by tuning the rotational frequency of the external field, the mobility
of a specific type of beads can be significantly reduced. the interaction
among the paramagnetic colloidal particles can be directly tuned using a
modulated ratchet potential 71 . In a system of paramagnetic particles
dispersed in water, driven across a striped patterned magnetic garnet film, an
external rotating magnetic field induces a periodic potential energy landscape
and causes directed motion of the particles. Interestingly, by varying the
ellipticity of the rotating magnetic field, the inter-particle interaction can
be changed from attractive to repulsive. Our conclusions can be tested in this
kind of a setup.

Throughout this work, we have considered a class of interacting many-particle
models, albeit only on a lattice where particles hop in discrete steps, and it
would be quite interesting to investigate particle transport in a continuum.
Indeed, in the past, there has already been some progress in this direction
where the directed particle transport in continuum was found to be crucially
dependent on the precise protocols of the external drive under consideration.
For example, a sinusoidally varying traveling wave potential is known to
generate a current always in the direction of the traveling wave for a system
of particles diffusing on a one-dimensional ring 200 . However, in a previous
work, from our group it was demonstrated, using numerical simulations, that a
moving potential barrier can in fact generate current in either direction,
depending on whether the potential moves uniformly or in discrete jumps 201 .
Interestingly, some recent studies have reported multiple current reversal for
Brownian particles in the presence of a traveling wave potential 202 ; 203 ;
204 . In a slightly different context, Ref. 72  numerically investigated the
effect of interaction on particle transport in asymmetric channels and
observed that, depending on the frequency of the external periodic drive, it
is possible to enhance transport by tuning the interaction potential. For
single-file diffusion of colloidal particles in an external time-varying force
field, various types of interactions such as Weeks-Chandler-Andersen, Yukawa,
and super paramagnetic potentials were considered 73 , and anomalous transport
was observed. Indeed, theoretical understanding of transport in continuum,
with such realistic potentials and in the presence of a time-periodic drive,
will be of significant interest in the context of obtaining the most efficient
directed flow. However, it is worth mentioning here that analytical
calculations in such a many-particle continuum models is quite challenging. In
this scenario, theoretical studies of lattice models such as those presented
here are quite relevant and useful, particularly in terms of analytically
calculating the transport properties of these systems, and could initiate
further research in this direction.

## Appendix : Calculation of $a_{\pm}$ for $q=r=0$

In the main text, $a_{+}(a_{-})$ is defined as the conditional probability
that given the defect site is occupied, a particle hops from the defect site
to its empty right (left) neighbor site during the residence time $\tau$ of
the defect at a single site. Eq. (7) provides a formal mathematical definition
for $a_{\pm}$. In this appendix we outline the calculation for
$\omega^{\pm}_{i}$ with $i=1,2...,6$ as explicit functions of $\epsilon$ and
$v$.

Let $\hat{1}$ ($\hat{0}$) denote an occupied (empty) defect site. When a
particle hops rightward from the defect site, there are six possible local
configurations, which are : $00\hat{1}01$, $10\hat{1}01$, $00\hat{1}00$,
$10\hat{1}00$, $1\hat{1}00$, $1\hat{1}01$. We number them as $i=1,2,...,6$.
Similarly, for leftward hopping possibilities are: $10\hat{1}00$,
$10\hat{1}01$, $00\hat{1}00$, $00\hat{1}01$, $00\hat{1}1$ and $10\hat{1}1$.
For a system of size $L$ we divide one Monte Carlo step in $L$ time-intervals
of length $dt=1/L$, where $L\gg 1$. For a specific local configuration $i$,
$\omega^{+}_{i}(\omega^{-}_{i})$ is defined as the probability that a particle
hops from the defect site to its right (left) neighboring site during $\tau$.
The configurations are numbered in such a way, that
$\omega^{+}_{i}=\omega^{-}_{i}=\omega_{i}$. Below we discuss only the
rightward hopping events, which can be easily generalized for leftward hopping
as well.

### Calculation for $\omega_{1}=\omega(00\hat{1}01)$

For the local configuration $00\hat{1}01$ the probability that the particle
hopping event takes place during the first infinitesimal time step $dt$ is
given by $(1-\epsilon)pdt/4$ (see Fig. 1). Probability that no hopping takes
place in this interval is

$\biggl{(}1-\dfrac{pdt(1-\epsilon)}{4}-\dfrac{pdt}{4}\biggr{)}$ (A-1)

which includes the possibilities both leftward and rightward hopping attempts
was unsuccessful. The probability that the hopping event takes place after
time $2dt$ is therefore,

$\biggl{(}1-\dfrac{pdt(1-\epsilon)}{4}-\dfrac{pdt}{4}\biggr{)}\biggl{(}\dfrac{(1-\epsilon)pdt}{4}\biggr{)}.$
(A-2)

Similarly the probability that it takes place at time $3dt$ is

$\biggl{(}1-\dfrac{pdt(1-\epsilon)}{4}-\dfrac{pdt}{4}\biggr{)}^{2}\biggl{(}\dfrac{(1-\epsilon)pdt}{4}\biggr{)}$
(A-3)

and so on. So the probability $\omega_{1}$ that the hopping happens in any of
the $\tau/dt$ time steps is

$\dfrac{(1-\epsilon)pdt}{4}\biggl{[}1+\biggl{(}1-\dfrac{(2-\epsilon)pdt}{4}\biggr{)}+\biggl{(}1-\dfrac{(2-\epsilon)pdt}{4}\biggr{)}^{2}+.....+\biggl{(}1-\dfrac{(2-\epsilon)pdt}{4}\biggr{)}^{(\tau/dt)-1}\biggr{)}\biggr{]}$
$\displaystyle=\dfrac{(1-\epsilon)pdt}{4}\biggl{(}\dfrac{1-(1-(2-\epsilon)pdt/4)^{\tau/dt}}{1-(1-(2-\epsilon)pdt/4)}\biggr{)}=\dfrac{1-\epsilon}{2-\epsilon}\biggl{(}1-e^{-(2-\epsilon)/4v}\biggr{)}$
(A-4)

where we have used $\tau=1/v$ and $dt\to 0$.

### Results for remaining $\omega$

Following similar steps as outlined above, expressions for all other $\omega$
can be derived. We directly present the final results here

$\displaystyle\omega_{2}=\omega(10\hat{1}01)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\biggl{(}1-e^{-p(1-\epsilon)/2v}\biggr{)}$
$\displaystyle\omega_{3}=\omega(00\hat{1}00)$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\biggl{(}1-e^{-p/2v}\biggr{)}$
$\displaystyle\omega_{4}=\omega(10\hat{1}00)$ $\displaystyle=$
$\displaystyle\dfrac{1}{2-\epsilon}\biggl{(}1-e^{-p(2-\epsilon)/4v}\biggr{)}$
(A-5) $\displaystyle\omega_{5}=\omega(1\hat{1}00)$ $\displaystyle=$
$\displaystyle\biggl{(}1-e^{-p(1+\epsilon)/4v}\biggr{)}$
$\displaystyle\omega_{6}=\omega(1\hat{1}01)$ $\displaystyle=$
$\displaystyle\biggl{(}1-e^{-p/4v}\biggr{)}$

### Expressions for ${\mathcal{C}}_{i}^{\pm}$

We provide the formal definitions for ${\mathcal{C}}_{i}^{\pm}$ below. These
denote the conditional probability of a specific local configuration, given
that the defect site is occupied.

${\mathcal{C}}_{1}^{+}=\text{Prob}.(00\hat{1}01|\hat{1})=\dfrac{\bigg{\langle}(1-\eta^{(\alpha)}_{\alpha-1})(1-\eta^{(\alpha)}_{\alpha})\eta^{(\alpha)}_{\alpha+1}(1-\eta^{(\alpha)}_{\alpha+2})\eta^{(\alpha)}_{\alpha+3}\bigg{\rangle}}{\langle\eta^{(\alpha)}_{\alpha+1}\rangle}$
(A-6)

${\mathcal{C}}_{2}^{+}=\text{Prob}.(10\hat{1}01|\hat{1})=\dfrac{\bigg{\langle}\eta^{(\alpha)}_{\alpha-1}(1-\eta^{(\alpha)}_{\alpha})\eta^{(\alpha)}_{\alpha+1}(1-\eta^{(\alpha)}_{\alpha+2})\eta^{(\alpha)}_{\alpha+3}\bigg{\rangle}}{\langle\eta^{(\alpha)}_{\alpha+1}\rangle}$
(A-7)

${\mathcal{C}}_{3}^{+}=\text{Prob}.(00\hat{1}00|\hat{1})=\dfrac{\bigg{\langle}(1-\eta^{(\alpha)}_{\alpha-1})(1-\eta^{(\alpha)}_{\alpha})\eta^{(\alpha)}_{\alpha+1}(1-\eta^{(\alpha)}_{\alpha+2})(1-\eta^{(\alpha)}_{\alpha+3})\bigg{\rangle}}{\langle\eta^{(\alpha)}_{\alpha+1}\rangle}$
(A-8)

${\mathcal{C}}_{4}^{+}=\text{Prob}.(10\hat{1}00|\hat{1})=\dfrac{\bigg{\langle}\eta^{(\alpha)}_{\alpha-1}(1-\eta^{(\alpha)}_{\alpha})\eta^{(\alpha)}_{\alpha+1}(1-\eta^{(\alpha)}_{\alpha+2})(1-\eta^{(\alpha)}_{\alpha+3})\bigg{\rangle}}{\langle\eta^{(\alpha)}_{\alpha+1}\rangle}$
(A-9)

${\mathcal{C}}_{5}^{+}=\text{Prob}.(1\hat{1}00|\hat{1})=\dfrac{\bigg{\langle}\eta^{(\alpha)}_{\alpha}\eta^{(\alpha)}_{\alpha+1}(1-\eta^{(\alpha)}_{\alpha+2})(1-\eta^{(\alpha)}_{\alpha+3})\bigg{\rangle}}{\langle\eta^{(\alpha)}_{\alpha+1}\rangle}$
(A-10)

${\mathcal{C}}_{6}^{+}=\text{Prob}.(1\hat{1}01|\hat{1})=\dfrac{\bigg{\langle}\eta^{(\alpha)}_{\alpha}\eta^{(\alpha)}_{\alpha+1}(1-\eta^{(\alpha)}_{\alpha+2})\eta^{(\alpha)}_{\alpha+3}\bigg{\rangle}}{\langle\eta^{(\alpha)}_{\alpha+1}\rangle}$
(A-11)

${\mathcal{C}}_{1}^{-}=\text{Prob}.(10\hat{1}00|\hat{1})=\dfrac{\bigg{\langle}\eta^{(\alpha)}_{\alpha-1}(1-\eta^{(\alpha)}_{\alpha})\eta^{(\alpha)}_{\alpha+1}(1-\eta^{(\alpha)}_{\alpha+2})(1-\eta^{(\alpha)}_{\alpha+3})\bigg{\rangle}}{\langle\eta^{(\alpha)}_{\alpha+1}\rangle}$
(A-12)

${\mathcal{C}}_{2}^{-}=\text{Prob}.(10\hat{1}01|\hat{1})=\dfrac{\bigg{\langle}\eta^{(\alpha)}_{\alpha-1}(1-\eta^{(\alpha)}_{\alpha})\eta^{(\alpha)}_{\alpha+1}(1-\eta^{(\alpha)}_{\alpha+2})\eta^{(\alpha)}_{\alpha+3}\bigg{\rangle}}{\langle\eta^{(\alpha)}_{\alpha+1}\rangle}$
(A-13)

${\mathcal{C}}_{3}^{-}=\text{Prob}.(00\hat{1}00|\hat{1})=\dfrac{\bigg{\langle}(1-\eta^{(\alpha)}_{\alpha-1})(1-\eta^{(\alpha)}_{\alpha})\eta^{(\alpha)}_{\alpha+1}(1-\eta^{(\alpha)}_{\alpha+2})(1-\eta^{(\alpha)}_{\alpha+3})\bigg{\rangle}}{\langle\eta^{(\alpha)}_{\alpha+1}\rangle}$
(A-14)

${\mathcal{C}}_{4}^{-}=\text{Prob}.(00\hat{1}01|\hat{1})=\dfrac{\bigg{\langle}(1-\eta^{(\alpha)}_{\alpha-1})(1-\eta^{(\alpha)}_{\alpha})\eta^{(\alpha)}_{\alpha+1}(1-\eta^{(\alpha)}_{\alpha+2})\eta^{(\alpha)}_{\alpha+3}\bigg{\rangle}}{\langle\eta^{(\alpha)}_{\alpha+1}\rangle}$
(A-15)

${\mathcal{C}}_{5}^{-}=\text{Prob}.(00\hat{1}1|\hat{1})=\dfrac{\bigg{\langle}(1-\eta^{(\alpha)}_{\alpha-1})(1-\eta^{(\alpha)}_{\alpha})\eta^{(\alpha)}_{\alpha+1}\eta^{(\alpha)}_{\alpha+2}\bigg{\rangle}}{\langle\eta^{(\alpha)}_{\alpha+1}\rangle}$
(A-16)

${\mathcal{C}}_{6}^{-}=\text{Prob}.(10\hat{1}1|\hat{1})=\dfrac{\bigg{\langle}\eta^{(\alpha)}_{\alpha-1}(1-\eta^{(\alpha)}_{\alpha})\eta^{(\alpha)}_{\alpha+1}\eta^{(\alpha)}_{\alpha+2}\bigg{\rangle}}{\langle\eta^{(\alpha)}_{\alpha+1}\rangle}$
(A-17)

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