# Sloppy Gear Mechanism for Coupled Stochastic Transportation: from anti-
equilibrium flow to infinite selectivity

Chase Slowey  Zhiyue Lu<EMAIL_ADDRESS>Department of Chemistry, University
of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3290

###### Abstract

Non-equilibrium transportation of particles through a restricted space (such
as porous media or narrow channels) significantly differs from free space.
With a simple model of two types of particles competing to transport via a
passive single-lane channel connecting two chemical baths, we find two
dynamical modes of transportation—the dud mode and the ratchet mode. At the
ratchet mode, the gradient flow of one type of particle forces the other into
a transient anti-equilibrium flow against its gradient. At the dud mode, the
two types of particles both flow according to their gradients. This counter-
intuitive non-equilibrium effect can be explained by a sloppy gear mechanism.
In addition to the anti-gradient flow observed in the ratchet mode, we find
that the non-equilibrium sloppy gear mechanism suppresses the flow of one
particle while allowing the flow for the other. This mechanism provides a
general theoretical framework to explain and design the selective
transportation of particles via passive channels.

Stochastic Thermodynamics $|$ Selective Transportation $|$ Non-equilibrium
Statistical Mechanics

††preprint: APS/123-QED

##  I Introduction

Systems far from equilibrium are ubiquitous in chemistry, physics and living
systems. Their dynamics are typically too far from equilibrium and can not be
described by linear response theory or mean-field approaches. Intuitions from
near-equilibrium processes can be misleading when directly applied to such
non-equilibrium processes. Examples of surprising and even sometimes counter-
intuitive non-equilibrium phenomenons includes stochastic pumps [1], kinetic
proofreading [2], chemical reaction oscillations (B-Z reactions) [3], self-
organized Rayleigh-Benard patterns [4, 5], and the Mpemba effect [6, 7]. The
development of stochastic thermodynamics [8] provides us with a powerful set
of tools to study dissipative processes far from equilibrium. Here, using a
simple kinetic model, we present a counter-intuitive anti-equilibrium
phenomenon in the passive transportation of particles through a narrow tube.

Stochastic transportation of particles via a narrow tube can be well described
by the famous set of simple models of exclusion process. [9] These models
consider a narrow tube as an array of one-dimensional lattice sites in which
each site can only be occupied by up to one particle. Particles excludes each
other via hard-sphere repulsion and transport through the tube via stochastic
hopping between neighboring sites. If the hopping is biased towards one
direction, the model becomes the asymmetric exclusion process (or totally
asymmetric exclusion process when hopping is only allowed for one direction).
[10, 11, 12, 13, 14] These models have successfully captured the essential
physical nature of a single type of particle transporting via a restricted
space and have demonstrated interesting dynamical phase transitions: when the
density of particles in a lattice increase above a threshold, kinetic jamming
takes place resulting in a sharp decrease in the transportation rate.[11, 15,
16, 10, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]

Non-equilibrium transportation can be more complicated if more than one type
of particle transport via the same narrow tube. In this situation, the flows
of different types of particles can be strongly coupled to each other due to
the confinement and demonstrate various counter-intuitive transportation
properties. In this work, we report two types of unexpected transportation
effects: _anti-equilibrium flow_ and _coupling-induced selective
transportation_.

We build upon existing single-species exclusion process models to study the
competitive/cooperative transportation of two types of particles (A and B)
across a narrow tube. We model the tube as a symmetric exclusion lattice array
of sites coupled to two chemical baths consisting of mixtures of particles A
and B. The tube itself is passive, as the internal transportation of particles
satisfies detailed balance conditions without an active source of energy, but
is driven away from equilibrium by the concentration gradients between the two
chemical baths. We find that under certain conditions (i.e., varying energy
landscapes and concentration gradients for A and B) the tube can generate a
counter-equilibrium flow, i.e., one type of particle is temporarily
transported from a low concentration bath to a high concentration bath. This
counter-intuitive process is not violating the second law of thermodynamics
since it is achieved at the cost of entropy production of the other type of
particle. This interesting phenomenon indicates that when two types of
particles are present, a simple tube acts like a chemical engine (entropy
rectifier) and its thermodynamics efficiency analysis resembles those found in
the modern models of Maxwell’s demons. [27, 28, 29, 30]. This counter-
intuitive effect can be explained by the strong coupling between the flows of
two types of particles, intuitively presented in this work as a _sloppy gear
mechanism_. Moreover, this sloppy gear mechanism demonstrates a surprising
selective transportation in tubes. The coupling between the flows of the two
species itself could generate a bias, allowing only one type of particle to
transport via the tube while blocking the flow of the other, even when the two
types of particles are indistinguishable in terms of their interaction with
the tube.

The paper is structured as follows. We first introduce a minimal Markov model
of two types of particles transporting through a single-track tube, whose non-
equilibrium kinetics can be exactly solved by the master equation. In a simple
scenario without tuning parameters, we demonstrate a generic anti-equilibrium
flow effect where particles appear to spontaneously travel against their
gradient (illustrated by an exactly solved phase diagram of ratchet-dud
transitions). The phenomenon is explained by an intuitive sloppy gear
mechanism. Using the sloppy gear mechanism, we discover that the kinetic
coupling between two types of particles (two gears) can be used to predict and
optimize the design of highly selective transportation in rather simple tubes.
In the end, we treat the tube as a chemical engine and analyze its
thermodynamic efficiency.

##  II General kinetic model

Let us describe a general regime of particle transportation via a narrow tube.
Consider a 1-D tube with $n$ lattice sites connecting two large chemical
baths. These chemical baths consist of mixtures of two types of solute
particles (A and B). Both types of particles can enter and exit the edge sites
of the tube. We assume that the particles interact with each other via hard-
sphere repulsion. Similar to a symmetric exclusion process model, [31, 32, 33]
each particle can hop from one lattice site to an empty neighboring site, and
each site can hold up to one particle due to the hard-sphere repulsion. The
hopping rates of each particle obeys the Arrhenius law [34], obtained from the
energy landscape shown in Fig. 1.

In the general model, we assume that the two types of particles can have
distinct binding affinities with the lattice site ($E_{s,X}$), distinct energy
barriers for hopping between sites ($E_{b,X}$), and distinct solvation free
energies ($F_{sol,X}$) where ‘X’ refers to the particle type A or B. However,
for simplicity, we assume that the solvent in both baths are identical and
that the $n$ sites of the tube are identical (homogeneous tube). The two
chemical baths with different concentrations of particles (different chemical
potentials) can lead to a net transportation of particles through the tube. We
denote the chemical potential as $\mu_{X,Z}=\beta^{-1}\ln{[X]_{Z}}$, where
$\beta$ is the inverse temperature and $[X]_{Z}$ is the concentration of
particle ‘X’ (i.e., A or B) in bath ‘Z’, where ‘Z’ can represent either the
left bath (‘L’) or the right bath (‘R’). The energy that a single particle A
(or B) experiences at different locations within or outside the channel is
plotted by the energy landscape in Fig. 1(b).

Figure 1: (a) A 5-site single-lane tube connecting two chemical baths of
particles A (red) and B (blue). (b) Energy landscape of a single particle. The
red dashed line represents the energy landscapes that a particle A experiences
at different locations. The 5 energy wells describes the binding energy
(affinity) between A and the tube’s 5 binding site, $E_{s,A}$. The energy
hills $E_{b,A}$ represents the activation barrier for particle A to hop from
one site to another, the solvation free energy of a single particle A is
represented by $F_{sol,A}$ reflecting As binding affinity with the solvent.
For particle B, the landscape is shown as a blue solid line. In the simplest
scenario (inert bare-tube), both A and B landscapes are totally flat, when we
refer to the tube as an inert tube.

At any given concentrations (chemical potentials) the non-equilibrium
transportation rate of particles can be calculated by solving for the
probability of particle-occupancy configurations (micro-states) of the tube.
There are three states for each site: empty, occupied by an A, or occupied by
a B. Then, for a 1-D tube that has $n$ sites, there are $N=3^{n}$ possible
configurations (i.e., micro-states) of the tube. For a 5-site tube, there are
$3^{5}=243$ possible micro-states including, but not limited to, (A A A A A),
(A $\\_$ A $\\_$ B), and ($\\_~{}\\_~{}\\_~{}\\_~{}\\_$), where “$\\_$”
denotes an empty site. The probability over $N$ micro-states is denoted by a
$N$-vector $\vec{p}(t)$ which evolves according to the master equation:

$\frac{d\vec{p}(t)}{dt}=\hat{R}\cdot\vec{p}(t)$ (1)

where $\hat{R}$ is a $N\times N$ rate matrix whose off-diagonal element,
$R_{ik}$, represents the probability transition rate from micro-state ‘$k$’ to
‘$i$’. These transitions are among three different types: One particle
$X=A~{}{\rm or}~{}B$ hops from one site to an empty neighboring site with rate

$R_{ik}=\exp[{-\beta(E_{b,X}-E_{s,X})}]~{};$ (2)

One particle ‘X’ enters an empty left/right edge-site from the left/right
chemical bath with rate

$R_{ik}=[X]_{L/R}\cdot\exp[-\beta(E_{b,X}-F_{sol,X})]~{};$ (3)

One particle of type ‘X’ exits the tube from the left/right and enters the
left/right chemical bath with rate

$R_{ik}=\exp[-\beta(E_{b,X}-E_{s,X})]~{}.$ (4)

As illustrated in Fig. 1, $E_{b,X}$ is the barrier energy for the nearest
neighbor site hopping for particle ‘X’ hop, which is assumed to be uniform
throughout the entire tube. For convenience, we employ the unit system such
that the inverse temperature is unity, $\beta=(K_{B}T)^{-1}=1$.

Separation of time-scale: If one considers two well-mixed large baths
connected by a microscopic tube, we can separately treat the dynamics within
the tube and the dynamics of the baths. On the one hand, the particles enter,
leave, and hop within the tube erratically, which occurs at a very fast time
scale. On the other hand, the concentrations of the baths are changed by the
tube extremely slowly due to the big size of the bath. As a consequence, at
any given time we can assume that the tube reaches a non-equilibrium steady
state (NESS) with stationary bath concentrations. As a result, if one obtains
$J_{X}([\text{A}]_{R},[\text{B}]_{R})$, the tube’s NESS transportation rate of
particle ‘X’ at the pseudo-stationary concentrations of the baths
($[\text{A}]_{R},[\text{B}]_{R}$), then one can use such NESS rates to predict
the slow change of baths concentrations at the longer time scale. The slow
dynamics of the bath concentration follows the following ordinary differential
equations (ODEs)

$\dot{[X]}_{R}=J_{X}([\text{A}]_{R},[\text{B}]_{R})$ (5)

where $\dot{[X]}_{R}$ is the time derivative of the concentration of ‘X’ (A or
B) in the right bath, and $J_{X}([\text{A}]_{R},[\text{B}]_{R})$ is calculated
in Eq. 8. Notice that due to the conservation of materials, we only need to
specify $[\text{A}]_{R},[\text{B}]_{R}$ to determine the concentration of both
particles in the left bath: $[\text{A}]_{R}+[\text{A}]_{L}=$ const. and
$[\text{B}]_{R}+[\text{B}]_{L}=$ const. The ODEs are later illustrated by
2-dim vector fields (normalized) in Fig 2.

NESS transportation rates: To obtain the NESS transportation behavior of the
tube we solve for the NESS probability distribution $\vec{p}^{ss}$ of the
tube’s micro-states, as the null-space of rate matrix:

$\frac{d\vec{p}^{\,ss}}{dt}=\hat{R}\cdot\vec{p}^{\,ss}=0$ (6)

Notice that the probability transition rate matrix, $\hat{R}$, is
parameterized by the pseudo-stationary concentrations of A and B in both
baths.

Using the NESS probability distribution, $\vec{p}^{\,ss}$, solved for in Eq. 6
with the pseudo-stationary bath concentrations, we can compute the steady
state probability transition rates $j^{ss}_{ik}$ for the tube’s configuration
change from micro-state $k$ to $i$:

$j^{ss}_{ik}=R_{ik}p^{ss}_{k}$ (7)

where $p^{ss}_{k}$ is the steady-state probability of the tube at micro-state
$k$. Thus, at the steady state of given bath concentrations, the tube is able
to facilitate a net transportation of particle type ‘X’ through the tube. This
transportation rate can be computed by first selecting the micro-state
transitions that correspond to a particle of type ‘X’ entering the right bath
(positive event) and the micro-state transitions that correspond to a particle
of type ‘X’ entering the tube from the right bath (negative event). Then by
summing the detailed probability currents for all of the events that
contributed to the net transportation of particle ‘X’, we can obtain the
particle flow rate (transportation rate) $J_{X}$ for particles of type ‘X’ at
the NESS for any given bath concentrations:

$J_{X}=\sum_{i,k}j^{ss}_{ik}\cdot T^{X}_{ik}$ (8)

where $T^{X}_{ik}$ is a current indicator, $T^{X}_{ik}=1$ for positive events,
$T^{X}_{ik}=-1$ for negative events, and $T^{X}_{ik}=0$ for transitions that
are not associated with particle ‘X’ leaving/entering the right bath.

Simplest model: parameter-free inert bare-tube. The general model described
above allows us to calculate non-equilibrium coupled transportation of two
types of particles via a narrow $n$-site tube given parameters to tune the
interaction between the particle and sites and between particles and the
solvent. We emphasize that the novel transportation effects described in this
paper can be demonstrated by a simplest parameter-free scenario. In the
simplest _parameter-free model_ , we set
$F_{sol,A}=E_{s,A}=E_{b,A}=F_{sol,B}=E_{s,B}=E_{b,B}=0$ to represent an _inert
bare-tube_ with flat energy landscapes. In this case, both types of particles
are “indistinguishable” to the tube as they interact with the tube’s sites
identically. Also, since the hopping energy barriers are set to 0, the whole
tube can be considered as a smooth bare-tube. Also, due to $F_{sol,X}=0$, the
two types of particles are “indistinguishable” in terms of their interaction
with the solvent.

General model and future extensions: The presented model is more general than
a bare-tube and thus can be used to study of transportation effects by tuning
shape of the energy landscapes. In this case, the two types of particles can
assume different interaction intensity with the tube: $E_{s,A}\neq E_{s,B}$,
and $E_{b,A}\neq E_{b,B}$, the two types of particles can also have distinct
interaction with the solvent: $F_{sol,A}\neq F_{sol,B}$. This general model
allows us to design tubes that can facilitate optimized selectivity into the
transportation. For future work, this model may be extended to account for the
presence of external fields by tilting the energy landscapes, allowing for the
examination of charged particles transporting across the tube under an
external electrical field. In the tilted case, the model is akin to asymmetric
exclusion processes [35] with two competing types of particles. Such an
extension may be a step toward describing transportation of charged particles
such as ions under external electrical field. However, the tilted energy
landscape and charge-charge interactions are beyond the scope of this
manuscript.

##  III Dud-Ratchet Transition

###  III.1 Anti-equilibrium flow via inert bare-tube

Here we demonstrate that an anti-equilibrium flow can be observed via narrow
tubes. First let us examine the simplest case – an inert 5-site bare-tube,
where the two types of particles, A and B, are equally inert to the tube’s
binding sites and inert to the solvents. 111Notice that here we choose a
5-site tube for illustrative purposes. We demonstrate this effect in a short
tube to avoid confusing this novel effect with the dynamical phase transitions
discovered in the TASEP or ASEP models where the transitions occurs at the
long-tube limit. In this case, the energy landscapes for both particle A and
particle B are perfectly flat horizontal lines (simpler than that in Fig. 1).
Specifically in the simple model we set $E_{b,X}=F_{sol,X}=E_{s,X}=0$ for both
types of particles. Under this assumption, by numerically solving for Eq. 8,
the NESS current of particle ‘X’, $J_{X}([\text{A}]_{R},[\text{B}]_{R})$, of a
5-site tube at all possible combinations of concentrations under the
restriction that $[\text{A}]_{R}+[\text{A}]_{L}=1000$ and
$[\text{B}]_{R}+[\text{B}]_{L}=1000$, we can represent the concentration
evolution by a normalized vector field,

$\vec{f}^{\,ss}=\frac{(\dot{[\text{B}]}_{R},\dot{[\text{A}]}_{R})}{|(\dot{[\text{B}]}_{R},\dot{[\text{A}]}_{R})|}=\frac{(J_{B},J_{A})}{|(J_{B},J_{A})|}$
(9)

which is a function of the baths’ concentrations, illustrated as unit arrows
Fig. 2(a).

In this simplest parameter-free model, one can identify a transient anti-
equilibrium transportation. Starting at an initial concentrations
$[\text{B}]_{R}=[\text{B}]_{L}=500$, $[\text{A}]_{R}=0$, and
$[\text{A}]_{L}=1000$, one can observe that its relaxation toward final
equilibrium is non-monotonic. The combined relaxation of A and B is
illustrated in Fig. 2(a) by the red trajectory tracing the vector field
starting from (500,0) and ending at (500,500). The anti-equilibrium flow for B
is found at the begining of the relaxation: even though B particles are
initiated at thermal equilibrium ($[\text{B}]_{R}=[\text{B}]_{L}=500$), a
spontaneous anti-equilibrium flow of B toward the right bath is observed Fig.
2(a). Following the trajectory in the concentration space, this anti-
equilibrium flow of B starting at (500,0) temporarily increases
$[\text{B}]_{R}$ to around $570$ before eventually relaxing to the ultimate
thermal equilibrium
($[\text{A}]_{R}=[\text{A}]_{L}=[\text{B}]_{R}=[\text{B}]_{L}=500$). This
anti-equilibrium flow of B appears to violate the second law of
thermodynamics, as the entropy of particle B spontaneously decreases. However,
this anti-equilibrium transportation is driven by the spontaneous process of
the coupled flow of particle A and the total entropy of the system remains
increasing in time. This counter-intuitive dynamics indicate that even a
simple tube is able to function as a chemical engine, rectifying the chemical
potential gradient of particle A to replenish the chemical potential gradient
for particle B, which will be discussed in detail in Sec. V. In this section,
we focus on two distinct operating modes of the tube. Notice that the anti-
equilibrium flow, even though is demonstrated first by a simple bare-tube, is
a general effect for more complicated energy landscapes. Below we provide a
general description and a kinetic physical explanation for this effect.

Figure 2: Concentrations’ relaxation for both A and B ($J_{A}$ and $J_{B}$)
illustrated by normalized vector field $\vec{f}^{\,ss}$ for an inert tube
(flat energy landscapes). Each vector represents the normalized NESS flow
(normalized rate of concentration change of $[\text{B}]_{R}$ and
$[\text{A}]_{R}$) at the given bath condition. (a) Two distinct modes of
transportation can be found in the system: a ratchet mode (yellow) and dud
mode (blue). At the ratchet mode, the rate vector points away from the final
equilibrium (500,500), indicating an anti-equilibrium flow. As a consequence
of the ratchet mode, if the bath starts at (500,0), it follows the red line,
first increasing the concentration gradient for particle B before final
equilibrium is reached, contrary to equilibrium intuition where both
concentrations monotonically relax to (500,500). (b) On the same vector field,
we illustrate the contour lines for the tube’s selectivity ($\xi$). Shaded
regions indicate where the tube is highly selective $\xi\leq 0.1$. Arrows in
both figures are colored according to the selectivity $\xi$.

###  III.2 Sloppy gear mechanism

Using the vector field in Fig. 2(a), we identify two distinct kinetic modes of
transportation—the _ratchet mode_ and the _dud mode_. At a given concentration
(i.e., one point in Fig. 2(a)), if one type of particle is transported against
its own concentration gradient (i.e., $\vec{f}^{\,ss}$’s projection on that
particle’s axis points away from the center), the tube facilities an anti-
equilibrium flow and the tube is in the ratchet mode (yellow shade). If both
types of particles are transported down their concentration gradient (i.e.,
$\vec{f}^{\,ss}$’s projections to the both axis both point toward the center),
the tube is in the dud mode (blue shade). The ratchet mode, where one type of
particle is transported against its own gradient can be explained by a non-
equilibrium kinetic mechanism—the _sloppy gear mechanism_ , illustrated in
Fig. 3 and detailed below.

The sloppy gear mechanism explains the counter-intuitive ratchet mode by the
stochastic coupling between the particle flows of A and B. Fig. 3associates
the transportation behavior with the stochastic dynamics of the multi-site
particle occupancy configurations, e.g., ($~{}\\_$ B A A B B A) sketched for a
7-site tube in Fig. 3. The site number 7 is chosen arbitrarily for
illustrative purposes. Suppose the gradient-driven-flows of particles A and B
can be loosely visualized as two imaginary sloppy gears, A and B. The
imaginary teeth of each gear represent the particles A or B within the tube.
The two gears are loosely coupled since both the number of gear teeth in the
tube and the positions of the teeth are stochastic. However, due to the
exclusion between particles, the teeth of the gear (particles) cannot bypass
each other, resulting in a gear-gear coupling. Consider
$[\text{A}]_{L}>[\text{A}]_{R}$ and $[\text{B}]_{L}<[\text{B}]_{R}$, then the
imaginary gear teeth of A (or B) favors motion to the right (or left). Then,
the two imaginary gears push against each other via teeth-teeth exclusion
interactions. When there are enough teeth in the tube for both particles, the
motion of the two imaginary gears are strongly coupled (i.e., the motion of
two types of particles are strongly correlated), and the flow is dominated by
the gear with the “stronger” driving force. In this scenario, the stronger
gear drives the other against gradient force, resulting in a ratchet mode of
transportation, where one type of particle flows against its own gradient.
However, when the number of teeth in the tube is low, the two imaginary gears
are too sloppy to slip against each other, allowing for both types of
particles to transport down their gradients, as seen in the dud mode. In the
dud mode, the correlation between the flow of the two types of particles is
not strong enough to invert the flow’s direction but is still present as
discussed in the section of selectivity.

Figure 3: Cartoon illustration of the proposed sloppy gear mechanism. Each
particle, as it enters the tube, mimics a tooth of an imagined gear (A in red
or B in blue). The gear teeth are loose as their relative positions can change
due to allowed particle hopping. Both A and B imaginary gears are driven by
their concentration gradients ($\Delta\mu_{A}$ and $\Delta\mu_{B}$). Due to
the stochastic nature of the particles in the tube, the gear teeth are erratic
and sloppy and gears may slip past each other. The longer the tube, and the
more particles are found in the tube, the stronger the coupling between two
imaginary gears. The strong coupling between two gears results in the ratchet
mode and can lead to a highly selective transportation of particles.

The two kinetic modes of operations discussed above are not restricted to
inert bare-tubes (flat energy landscapes). Let us consider a general case
where the two types of particles have different interaction affinities to the
tube’s sites and different solvation free energies in the solvent: We
arbitrarily choose the energy landscape such that $E_{s,B}=F_{sol,B}=-4.0$,
$E_{b,B}=-1.2$, $E_{b,A}=-2.2$, while allowing for As interaction with the
tube and the bath to vary. Thus $E_{s,A}$ and $F_{sol,A}$ are the only two
free parameters from which we obtain a 2-dim phase diagram. We numerically
solve for the NESS currents of the specified 5-site tube connecting two baths
with fixed concentrations, $[\text{A}]_{L}=1$,
$[\text{A}]_{R}=[\text{B}]_{L}=1000$, and $[\text{B}]_{R}=100$, and confirm
that we can find both the ratchet mode and the dud mode in the
$E_{s,A}$–$F_{sol,A}$ diagram, illustrated in Fig. 4(a).

Notice that the ratchet mode where particle A inverts the flow of particle B
cannot be simply explained merely by the particle-tube interaction ($E_{s,A}$)
nor the particle-solvent interaction ($F_{sol,A}$). Otherwise, one should not
observe this effect in the inert bare-tube. However, the sloppy gear mechanism
(Fig. 3), is a non-equilibrium kinetic theory, which explains the anti-
equilibrium transportation based on the kinetic coupling between the two types
of particles within the narrow tube. To correctly understand the roles played
by the particle-tube interaction ($E_{s,A}$) and particle-solvent interaction
($F_{sol,A}$) in the transportation effect, one should focus their impact on
the stochastic occupancy configurations (micro-states) of the tube. For
example, altering the energy landscape non-trivially changes the typical
“teeth arrangements” of the two sloppy gears, which results in a complex non-
monotonic phase boundary between the ratchet mode and the dud mode shown in
Fig. 4(a).

Figure 4: At a a static concentration $[\text{A}]_{L}=1$,
$[\text{A}]_{R}=[\text{B}]_{L}=1000$, and $[\text{B}]_{R}=100$, NESS behavior
of a non-inert tube with non-flat energy landscape ($E_{s,B}=F_{sol,B}=-4.0$,
$E_{b,B}=-1.2$, $E_{b,A}=-2.2$. (a) The ratchet and dud modes of
transportation can be achievable by tuning the interaction energy ($E_{s,A}$)
and solvation free energy ($F_{sol,A}$) for A particle. In the ratchet mode B
particles are forced to go against their gradient due to the flow of A. (b)
Illustration of tube’s selectivity ($\xi$). The shaded regions ($\xi\leq 0.1$)
indicate where the selective tube strongly favors the current of A and
suppresses the current of B. Notice that selective transportation $\xi\leq
0.1$ can be achieved beyond the ratchet mode.

##  IV Selective Transportation

###  IV.1 Sloppy gear induced selectivity

The sloppy gear mechanism reveals that the non-equilibrium coupling between
the flows of two types of particles can result in selective transportation.
Due to the sloppy gear teeth coupling (particle-particle exclusion), one type
of particle’s flow (the dominant imaginary gear’s teeth) can suppress the flow
of the other, resulting in the tube strongly favoring the transportation of
one type of particle while blocking the flow of the other. Here we name the
type of particle whose transportation current is suppressed the “unfavored”
type, and the other type, “favored”. Then we can denote the selectivity of the
tube by the absolute value of the transportation current ratio between the
unfavored (‘uf’) particle and the favored (‘f’) particle:

$\xi=\left|\frac{J_{\text{uf}}}{J_{\text{f}}}\right|$ (10)

where the current $J_{\text{uf}}$ and $J_{\text{f}}$ refers to the NESS
currents of the unfavored type of particle and favored type of particle, as
defined in Eqn. 8. For illustrative purposes, let us define a threshold for
“high selectivity” as $\xi\leq 0.1$. This allows us to illustrate the highly
selective range of parameters as shaded areas on the selectivity contour
diagrams where we have shown the contour plots of $\xi$ in Fig. 2(b) and Fig.
4(b). Notice that the tube can be _infinitely selective_ ($\xi=0$) at the
phase boundary between the ratchet mode and the dud mode, as the NESS current
of the unfavored particle is totally stalled by the sloppy gear interaction
with the favored particle flow. Also notice that the tube is highly selective
not only in the ratchet phase (see shaded area in Fig. 4b).

It is worth to mention that the selective transportation is a non-equilibrium
effect caused by the coupling between the two particle flows (sloppy gear
mechanism). Thus it cannot be explained by intuition from equilibrium
thermodynamics. Intuitively, the selectivity of transportation of particles
can be determined by the particles’ interaction with tube’s binding sites
($E_{s,X}$), particle-solvent interactions ($F_{sol,X}$), and the hopping
barrier between neighboring sites ($E_{b,X}$). However, as shown in Fig. 2(b),
even when the two types of particles are physically and chemically
indistinguishable in terms of their interactions with the tube and the
solvents, the coupling between their flows can result in an infinitely high
selectivity, where one type of particle with a stronger chemical potential
gradient can dominate the tube while suppressing the flow of the other type to
0. In this example, the selectivity is achieved purely by the coupling between
the non-equilibrium flows. Thus, in the simplest model, the equilibrium
distinctions between particles A and B are totally removed from the discussion
of selective transportation (e.g., at equilibrium, particle A and B are
indistinguishable to the tube and to the baths, yet the tube can still
demonstrate selective transportation).

This sloppy gear mechanism for tube selectivity provides a novel non-
equilibrium perspective to understand and optimize selective transportation in
narrow tubes. Rather than focusing on the equilibrium distinctions between A
and B (e.g., $E_{s,X}$, $E_{b,X}$, and $F_{sol,X}$), selective transportation
can be understood by flow coupling between the two species. The strength of
the flow for each species are impacted by the kinetic rates of particle-
hopping, which are influenced by concentration gradients, temperature,
$E_{s,X}$, $E_{b,X}$, $F_{sol,X}$, and the kinetic micro-state of the tube.
Below, we demonstrate that the sloppy gear mechanism can be used to design
highly selective transportation tubes.

###  IV.2 Optimal design of selectivity

Above discussion have demonstrated that even an inert bare-tube can
selectively transport particles, which is illustrated by Fig. 2(b), where
$E_{b,A}=F_{sol,A}=E_{s,A}=0$ and $E_{b,B}=F_{sol,B}=E_{s,B}=0$. However, the
highly selective regions are relatively small compared to the whole phase
diagram, and it is obviously not optimized in terms of selectivity. Here we
explore the optimal design of selective transportation for the general model
of a narrow tube (i.e., with non-flat energy landscapes caused by chemical
decorations at each site). Specifically, let us consider the particle–site
interaction and particle–solvent interactions and study their effect on both
the current and the selectivity. In contrast to Fig. 2(b) obtained for an
inert bare-tube, we observe enhanced selectivity as we slightly alter the
bare-tube into realistic tubes (see the larger area of highly selective region
in Fig. 5). Here we numerically calculate the selectivity contour for modified
inert tubes (modifying from $E_{b,X}=F_{sol,X}=E_{s,X}=0$ for both ‘X’ = A and
B) with the only modification that (1) $E_{s,A}=-1$, resulting in an enhanced
selectivity favoring the flow of A shown in Fig. 5(a), or the only
modification that (2) $F_{sol,A}=-1$, resulting an enhanced selectivity
favoring the flow of B shown in Fig. 5(b).

As a summary, the sloppy gear mechanism and the kinetic analysis provides us
with a guiding principle toward designing selective transportation in rather
simple models of narrow tubes. We find that both $E_{s,A}$, the particle’s
interaction with the tube and $F_{sol,A}$, the particle’s interaction with the
solvent can significantly and non-monotonically impact the selectivity. By
comparing Fig. 5(a) and (b), we notice that altering $F_{sol,A}$, particle-
bath interaction can more prominently impact the selectivity of the tube.
Moreover, through a simple kinetic argument, we expect that altering
$F_{sol,A}$ is advantageous over altering $E_{s,A}$ toward designing a _low-
resistance selective tube_ : reducing $E_{s,A}$ can indeed make the sloppy
gear A dominant over the gear B, and favor the flow of particle A over B.
However, decreasing $E_{s,A}$ can significantly reduce the stochastic hopping
rate of particle A (see Eq. 2), and thus also reduce the transportation
current of particle A. In contrast, via increasing $F_{sol,A}$ or decreasing
$F_{sol,B}$, we can achieve the same enhancement favoring the flow of particle
A because it increases the probability for particle As to occupy the tube,
thus making the ‘gear’ A the dominant gear. Moreover, this enhancement is not
at the cost of slowing down the particle hopping rates for either A or B
because it does not impact Eq. 2. Thus if one intends to enhance the
selectivity without impeding the flow rate of the favored particle, the design
should lean more toward altering $F_{sol,A}$ rather than altering $E_{s,A}$.
222One may argue that similar optimized performance of selective
transportation of A could be achieved via increasing $E_{s,B}$,
i.e.,introducing a strong repulsion between the particle B and the tube.
However, this solution is trivial as it is simply requires that one find a
tube that is simply not permeable to particle B.

Figure 5: Comparison of the selective transportation for two types of
landscapes. Shown side-by-side are normalized vector fields $\vec{f}^{\,ss}$’s
for modified inert tubes with two different alternations: (a) $E_{s,A}=-1$ and
(b) $F_{sol,A}=-1$. Shaded regions indicate high selectivity $\xi\leq 0.1$.
Compared to a purely inert tube (Fig. 2(b)), both alternations results in an
increased selectivity. Notice that the high selectivity region is larger in
(b) than in (a), indicating the selectivity is more sensitive to particles’
difference in solvation free energies.

##  V Entropic Efficiency

At the ratchet mode, the tube can be considered as a chemical engine,
rectifying chemical energy of one type of particle and using it to increase
the chemical potential difference in another type of particle. Here let us
examine the thermodynamic entropy production. For simplicity, let us consider
a ratchet mode where A pushes ’B’ and thus forces B to an anti-equilibrium
flow against its own gradient. As a consequence, the total entropy of particle
B decreases over time. (Similar arguments can be made for the other ratchet
mode where particle B pushes A.) Thus, we have a mechanism to harness the
spontaneous increased entropy of particle A and use it to reduce the entropy
for particle B. At the NESS, the rate of entropy decrease of the particle B
can be computed as

$\dot{S}_{B}=J_{B}\ln{\frac{[\text{B}]_{L}}{[\text{B}]_{R}}}$ (11)

The second law of thermodynamics requires that the total entropy for both
particle A and B must not decrease in time:

$\dot{S}_{tot}=\dot{S}_{A}+\dot{S}_{B}\geq 0$ (12)

Thus, we characterize the efficiency of this chemical engine (in the ratchet
mode) by the steady-state entropy efficiency which must not exceed $1$
according to the second law of thermodynamics:

$\eta_{B}=\frac{-\dot{S}_{B}}{\dot{S}_{A}}=-\frac{J_{B}}{J_{A}}\cdot\frac{\mu_{B,R}-\mu_{B,L}}{\mu_{A,R}-\mu_{A,L}}\leq
1$ (13)

If one chooses to study the ratchet effect where particle B pushes A, then
efficiency should be defined as $\eta_{A}=\frac{-\dot{S}_{A}}{\dot{S}_{B}}$.
Our result confirms that the entropic efficiency is always below $1$.

Figure 6: Ratchet efficiency, $\eta$, for inert tubes (flat energy landscapes)
of different length: (a) 3-site tube and (b) 5-site tube. In agreement with
intuition from the sloppy gear mechanism, the longer the tube, the larger the
ratchet mode region. Also, the longer the tube, the higher the maximum
efficiency, as the two gears are less likely to slip through.

Here we briefly discuss the efficiency of the ratchet modes obtained at
different tube lengths, see Fig. 6. Our results indicate that the ratchet
effect is more prominent in longer tubes, accompanied by a higher maximum
entropic efficiency. This can be explained by the sloppy gear mechanism: As
the tube becomes longer the two imaginary gears have more possible teeth
biting with each other and thus strengthen the kinetic coupling between the
flows of particles A and B, reducing the possibility for the two gears to slip
by one another (suppression of the dud mode). The longer the tube, the greater
the coupling between the flows for particles A and B. It should be noted that,
as expected, the efficiency of the ratchet effect reaches its maximum near the
“linear-response regime” where concentration gradients for both particle types
is near zero. Similar efficiency maps with an engine mode and dud mode have
been found in various designs of stochastic information engines or modern
variations of Maxwell’s demons. [28, 29, 30]

##  VI Conclusion

This work demonstrates a generic anti-equilibrium transportation phenomenon
via a simple SEP-like model describing two types of particles transporting
through a purely passive narrow tube. The anti-equilibrium transportation
(demonstrated in a bare-tube and then more complex tubes) can be explained by
an intuitive sloppy gear mechanism and this general effect can occur without
carefully tuning the interactions between the particles and the tube/baths.
Moreover, this effect sheds light on a possible mechanism to explain the
selective transportation via narrow tubes. By using this simple model we seek
the optimal design of selective transportation via narrow tubes. Although both
the particle-tube interaction $E_{s,X}$ and the particle-bath interaction
$F_{sol,X}$ alter the selectivity of the tube, their influences on the
performance of the selective tube are not equivalent: Intuitively, decreasing
$E_{s,A}$ or increasing $F_{sol,A}$ would both favor the transportation of
particle $A$ over $B$. However, decreasing $E_{s,A}$ reduces the Arrhenius
transportation rate and comes with a cost of impeding the transportation rate
of $A$. Whereas increasing $F_{sol,A}$ (or decreasing $F_{sol,B}$) does not
impede the transportation rate of $A$ within the tube. Thus the optimal design
of the selective tube should be achieved via engineering the particle-
solvation interaction.

From a thermodynamic perspective, the tube functioning at the ratchet mode
behaves like a free energy transducer, harnessing the free energy decrease
from one type of particle (that transports down the concentration gradient)
and storing it into the free energy of the other type of particle (increasing
concentration gradient). The analysis of the entropic efficiency shows that
the longer the tube, the more efficient the free energy transduction.

##  VII Acknowledgments

This work is inspired from discussion with Michael Zwolak, Subin Sahu, at the
workshop “Information Engines at the Frontiers of Nanoscale Thermodynamics” at
TSRC, in 2019. The authors also appreciate Max Berkowitz, Hong Qian, Chris
Jarzynski, Sa Hoon Min, and Yueheng Lan for useful discussions. This work is
supported by the University start-up fund at UNC-Chapel Hill. We would like to
thank the University of North Carolina at Chapel Hill and the Research
Computing group for providing computational resources and support that have
contributed to these research results.

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  * Note [1] Notice that here we choose a 5-site tube for illustrative purposes. We demonstrate this effect in a short tube to avoid confusing this novel effect with the dynamical phase transitions discovered in the TASEP or ASEP models where the transitions occurs at the long-tube limit.
  * Note [2] One may argue that similar optimized performance of selective transportation of A could be achieved via increasing $E_{s,B}$, i.e.,introducing a strong repulsion between the particle B and the tube. However, this solution is trivial as it is simply requires that one find a tube that is simply not permeable to particle B.