††thanks: Corresponding author<EMAIL_ADDRESS>Corresponding author<EMAIL_ADDRESS>
# Inertial effects on rectification and diffusion of active Brownian particles
in an asymmetric channel

Narender Khatri  Chemical Physics Theory Group, Department of Chemistry,
University of Toronto, Toronto, Ontario M5S 3H6, Canada  Raymond Kapral
Chemical Physics Theory Group, Department of Chemistry, University of Toronto,
Toronto, Ontario M5S 3H6, Canada

###### Abstract

Micro- and nano-swimmers moving in a fluid solvent confined by structures that
produce entropic barriers are often described by overdamped active Brownian
particle dynamics, where viscous effects are large and inertia plays no role.
However, inertial effects should be considered for confined swimmers moving in
media where viscous effects are no longer dominant. Here, we study how inertia
affects the rectification and diffusion of self-propelled particles in a two-
dimensional asymmetric channel. We show that most of the particles accumulate
at the channel walls as the masses of the particles increase. Furthermore, the
average particle velocity has a maximum as a function of the mass, indicating
that particles with an optimal mass $M^{*}_{\rm op}$ can be sorted from a
mixture with particles of other masses. In particular, we find that the
effective diffusion coefficient exhibits an enhanced diffusion peak as a
function of the mass, which is a signature of the accumulation of most of the
particles at the channel walls. The dependence of $M^{*}_{\rm op}$ on the
rotational diffusion rate, self-propulsion force, aspect ratio of the channel,
and active torque is also determined. The results of this study could
stimulate the development of strategies for controlling the diffusion of self-
propelled particles in entropic ratchet systems.

##  I Introduction

Many biological microorganisms, as well as artificial active particles, take
free energy from their environments and convert it under nonequilibrium
conditions into persistent motion. The mechanisms that underlie such active
motion and the dynamical properties of these systems are diverse and have been
studied extensively [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].

For the most part, the biological and synthetic active agents mentioned above
have micrometer or sub-micrometer dimensions and move in viscous environments
under conditions where inertia does not play an important role. In such
circumstances, the active dynamics is often described by the overdamped
Langevin or continuum models that neglect inertia. Inertia cannot always be
neglected, and an increasing body of research [16] considers the effects of
inertia on active particle motion and describes the new phenomena that arise
as a result of its inclusion. While the systems where inertial effects are
important are diverse, some examples include systems that support a
temperature gradient across coexisting phases [17], vibrobots [18, 19, 20],
active particle motion in low-density media, such as gases [21, 22, 23],
plasmas [24, 25, 26], superfluids [27], and active aerosols [28], etc. Such
inertia-dominated active particles are termed micro- and nano-flyers rather
than swimmers [16].

The rectification of artificial active particles in confined environments in
the absence of external forces has attracted interest [29, 30, 31, 32, 33,
34]. Geometrical confinement controls the volume of phase space that is
accessible to the active particles, resulting in entropic barriers that
significantly influence their transport properties [35, 36, 37, 38, 39, 40,
41, 42]. As well, confined environments possessing spatial ratchet asymmetry
give rise to an entropic ratchet potential that can induce active directed
transport in the system [29, 33].

We investigate the underdamped dynamics of self-propelled particles confined
by a two-dimensional asymmetric channel. We use a minimal underdamped Langevin
model for the dynamics of the self-propelled particles that accounts for
inertia. The collisional dynamics of particles with the channel walls are
modeled by sliding-reflecting boundary conditions [33, 29, 43]. We focus on
how inertial effects influence the rectification and diffusion of active
particles in the asymmetric channel.

The article is organized as follows: in Sec. II, we introduce the underdamped
Langevin model used to describe the dynamics of the active particles in the
two-dimensional asymmetric channel. Section III discusses the effects of
inertia on the spatial distribution of particles, while Sec. IV presents
results on the rectification and effective diffusion in the channel. The main
conclusions of the article are given in Sec. V.

##  II Model

Figure 1: Schematic illustration of an active (self-propelled) Brownian
particle of mass $M$ and moment of inertia $I$ confined in a two-dimensional
triangular channel with periodicity $L$. The active force
$\bm{F_{0}}=F_{0}\bm{\hat{n}}$, angle $\theta$, active torque $T_{0}$, local
width of the channel $2\,{\rm w}(x)$, and local length of a cell of the
channel $\Delta(y)$ are indicated. The shape of the channel structure is
prescribed by Eq. (2). The particle cannot penetrate through the channel
walls, which are considered rigid; however, the particle is free to rotate and
slide along the walls.

We consider a system comprising active particles dilutely dispersed in a
dissipative medium and confined to a two-dimensional asymmetric channel with
periodicity $L$ (see Fig. 1). The active particle density is assumed to be
sufficiently small so that direct interactions among the particles may be
neglected. As in other studies [16], a minimal underdamped Langevin model is
used to describe the active dynamics under conditions where inertia is
important. The coupled Langevin equations for an active particle with position
$\bm{r}=(x,y)$, orientation $\bm{\hat{n}}=(\cos{\theta},\sin{\theta})$, mass
$M$, and moment of inertia $I$ read

$\displaystyle M\ddot{\bm{r}}(t)$
$\displaystyle=-\gamma_{t}\dot{\bm{r}}(t)+F_{0}\bm{\hat{n}}(t)+\gamma_{t}\sqrt{2D_{t}}~{}\bm{\xi}(t),$
(1) $\displaystyle I\ddot{\theta}(t)$
$\displaystyle=-\gamma_{r}\dot{\theta}(t)+T_{0}+\gamma_{r}\sqrt{2D_{r}}~{}\zeta(t).$

Here $\gamma_{t}$ and $\gamma_{r}$ are the translational and rotational
friction coefficients, $D_{t}$ and $D_{r}$ are the translational and
rotational diffusion constants, and $F_{0}$ and $T_{0}$ are the active force
and torque, respectively. The random variables $\bm{\xi}(t)$ and $\zeta(t)$
are the Gaussian white noise terms with zero mean and unit variances given by
$\langle\bm{\xi}(t)\otimes\bm{\xi}(t^{\prime})\rangle=\delta(t-t^{\prime})\bm{\mathrm{1}}$
and $\langle\zeta(t)\zeta(t^{\prime})\rangle=\delta(t-t^{\prime})$,
respectively, where $\bm{\mathrm{1}}$ is the unit matrix. Such a set of
coupled Langevin equations can serve as models for a variety of active
systems, including systems subject to athermal noise [44, 45, 46, 47], if
$\\{\gamma_{t},\gamma_{r},D_{t},D_{r}\\}$ are regarded as independent
parameters, and $D_{t}$ and $D_{r}$ control the strength of noise terms. For
systems with thermal noises that satisfy the fluctuation-dissipation relation,
the Einstein relations apply, $D_{t}=k_{B}T/\gamma_{t}$ and
$D_{r}=k_{B}T/\gamma_{r}$. The set of coupled Langevin equations (1) is a
simplified version of a more general set of coupled Langevin equations that
applies to asymmetric particles and accounts for translation-rotation coupling
[48]. The set of general underdamped coupled Langevin equations for
chemically-active colloids has been derived using fluctuating
chemohydrodynamics [49] and molecular theory [50], where expressions for the
active force and torque are given.

The two-dimensional asymmetric and spatially periodic channel shown in Fig. 1
is specified as follows: for the upper wall, we have

${\rm w}_{u}(x)=\begin{cases}{\rm w}_{\mathrm{min}},&x=0\\\ {\rm
w}_{\mathrm{max}}-({\rm w}_{\mathrm{max}}-{\rm
w}_{\mathrm{min}})\frac{x}{L},&0<x\leq L,\end{cases}$ (2)

where $L$ is the periodicity of the channel, and ${\rm w}_{\mathrm{min}}$ and
${\rm w}_{\mathrm{max}}$ refer to the minimum and maximum half-widths of the
channel, respectively. The dimensionless aspect ratio of the channel is
$\epsilon={\rm w}_{\mathrm{min}}/{\rm w}_{\mathrm{max}}$, where we set ${\rm
w}_{\mathrm{max}}=1$ throughout the work. The local length of a cell of the
channel along $x$ for a given $y$ is $\Delta(y)=({\rm
w}_{\mathrm{max}}-|y|)L/({\rm w}_{\mathrm{max}}-{\rm w}_{\mathrm{min}})$, for
${\rm w}_{\mathrm{min}}<|y|\leq{\rm w}_{\mathrm{max}}$, and $\Delta(y)=L$, for
$-{\rm w}_{\mathrm{min}}\leq y\leq{\rm w}_{\mathrm{min}}$. For a given value
of $x$, $y$ is bounded between the lower and upper walls. Due to the symmetry
about the principal axis of the channel, the lower wall is described by ${\rm
w}_{l}(x)=-{\rm w}_{u}(x)$. Consequently, $2\,{\rm w}(x)={\rm w}_{u}(x)-{\rm
w}_{l}(x)$ corresponds to the local width of the channel.

When a particle encounters a channel wall, it is elastically reflected [33,
51, 52, 43], and its orientation $\theta$ is unchanged during the collision
(sliding-reflecting boundary conditions [29, 43, 53]). So, the particle slides
along the channel walls until a fluctuation in the orientation vector
$\hat{\bm{n}}$ causes it to change so that it may move away from the wall.
While we restrict our considerations to this previously-used boundary
condition, other boundary conditions that depend on the scale of the wall
roughness may be applied. For example, if the particle interacts with the
channel walls through rough sphere collisions, the orientation of the particle
will change through this collision mechanism, as well as through interactions
with the solvent. Consequently, the primary effects on the particle dynamics
will likely be more rapid reorientation, decreased residence time near the
wall, and reduced rectification.

For translational motion described by Eqs. (1), inertial effects dominate
frictional effects for times $t\ll M/\gamma_{t}\equiv\tau_{v}$, while for
orientational motion, they are important for times $t\ll
I/\gamma_{r}\equiv\tau_{\omega}$, where $\tau_{v}$ and $\tau_{\omega}$ are the
characteristic times for linear and angular velocity relaxation, respectively.
For active particles in a bulk medium, these times may be compared to the
reorientation time $\tau_{r}=1/D_{r}$ that determines the time scale on which
orientation vector $\hat{\bm{n}}$ decays, and the time $\tau_{a}=2a/v_{0}$ it
takes a particle with speed $v_{0}=F_{0}/\gamma_{t}$ to move a distance equal
to the particle diameter $2a$.

For the present study, where the dynamics takes place in a periodic channel,
we are interested in the effects of inertia on time scales that reflect the
motion on the length scale $L$ of the periodic channel. We define the
characteristic diffusion time $\tau=L^{2}/D_{t}$, which gauges the time the
particle takes to diffuse one period of the channel length. For systems with
thermal noise, this characteristic diffusion time is related to $\tau_{v}$ by
$\tau_{v}/\tau=\big{(}\tau_{v}/\tau_{\rm th}\big{)}^{2}$, where $\tau_{\rm
th}=L/v_{\rm th}$ with $v_{\rm th}=\sqrt{k_{B}T/M}$ the thermal speed.

Given these considerations, we use a dimensionless description where lengths
are scaled by the periodicity of the channel $L$, $\bm{r}^{\prime}=\bm{r}/L$,
and time by $\tau$, $t^{\prime}=t/\tau$, analogous to that in Refs. [54, 39],
so that Eqs. (1) read,

$\displaystyle M^{*}\ddot{\bm{r}}(t)$
$\displaystyle=-{\dot{\bm{r}}}(t)+f_{0}\bm{\hat{n}}(t)+\sqrt{2}~{}\bm{\xi}(t),$
(3) $\displaystyle I^{*}\ddot{\theta}(t)$
$\displaystyle=-\dot{\theta}(t)+t_{0}+\sqrt{2\alpha}~{}\zeta(t),$

and we dispensed with the primes in writing this coupled set of equations. The
dimensionless mass and moment of inertia are given by
$M^{*}=\tau_{v}/\tau=MD_{t}/(\gamma_{t}L^{2})$ and
$I^{*}=\tau_{\omega}/\tau=ID_{t}/(\gamma_{r}L^{2})$, respectively. In these
variables, the dimensionless mass $M^{*}$ depends on physical mass $M$ as well
as $D_{t}$, $\gamma_{t}$, and $L$. Similarly, $I^{*}$ depends on the moment of
inertia $I$ as well as $D_{t}$, $\gamma_{r}$, and $L$. The dimensionless
active force and torque are $f_{0}=F_{0}L/(D_{t}\gamma_{t})$ and
$t_{0}=T_{0}L^{2}/(D_{t}\gamma_{r})$, respectively, and the parameter
$\alpha=D_{r}\tau$. Other choices of dimensionless units could have been used
[16].

In the following sections, we present results for the average velocity,
effective diffusion coefficient, and other properties obtained from
simulations of the coupled Langevin equations (3) in the channel [55]. At
$t=0$, the particles are uniformly distributed with random orientations in a
periodic cell of the channel located between $x=0$ and $x=1$. The results are
obtained from averages over $10^{4}$ stochastic trajectories.

##  III Spatial distribution

Figure 2: The steady state distribution of particles, mapped into a single
cell of the channel, is depicted in (a)-(c) for different values of $M^{*}$.
The corresponding probability densities $P_{st}(y)$ along the $y$ direction
are depicted in (d). The parameters are:
$I^{*}=0.001,f_{0}=5,t_{0}=0,\alpha=0.1$, and $\epsilon=0.1$.

In order to analyze the effects of inertia on the rectification and diffusion
of self-propelled particles in an asymmetric channel, we first consider the
spatial distribution of particles mapped onto a single cell of the channel.
Taking $P(x,y)$ to be the probability density of finding a particle at $(x,y)$
in the cell, we define the probability density to find a particle at $y$ per
unit local cell length $\Delta(y)$ at this $y$-value:
$P(y)=\int_{0}^{\Delta(y)}dx\;P(x,y)/\Delta(y)$. Figure 2 shows the steady
state distribution of particles and the corresponding probability density
$P_{st}(y)$ along the $y$ direction for different values of $M^{*}$. For
$M^{*}\to 0$ in the overdamped limit, the distribution of particles is
inhomogeneous, and most of the particles tend to accumulate near the left
corners of the cell (see Fig. 2(a) and the $P_{st}(y)$ plots in (d)). The
inhomogeneous distribution of particles can be ascribed to the active motion
due to broken detailed balance and the presence of the spatial asymmetry
imposed by the shape of the channel [29, 33]. It is interesting to see that on
increasing $M^{*}$ further, most of the particles quickly accumulate at the
channel walls; while the rest of the particles adopt the shape of a funnel
about the principal axis of the channel at the middle region due to narrow
bottleneck openings (see Fig. 2(b)). Most of the particles accumulate at the
left corners of the cell, and the distribution of particles is symmetric about
the principal axis of the channel (see the $P_{st}(y)$ plots, especially the
curve for $M^{*}=1$). Such an observation provides evidence that the
rectification of particles in an asymmetric channel can be enhanced by
increasing $M^{*}$. For strongly underdamped situations where
$M^{*}\to\infty$, the qualitative behavior of the distribution of particles
and the corresponding $P_{st}(y)$ are very similar; however, the rectification
and diffusion of particles approach zero because inertia dominates the self-
propulsion mechanism [56].

Figure 3: The time evolution of the distribution of particles with $M^{*}=1$
is shown in (a)-(c) at increasing times, $t=0,0.5,1$, respectively. The
probability density $P(y)$ versus $y$ is plotted in (d). The parameters are:
$I^{*}=0.001,f_{0}=5,t_{0}=0,\alpha=0.1$, and $\epsilon=0.1$.

The probability density evolution to its steady state distribution is shown in
Fig. 3 for a system with $M^{*}=1$ and parameters as in Fig. 2. At $t=0$,
particles are distributed uniformly inside the periodic cell consistent with
thermodynamic equilibrium [51, 57, 35, 36, 37]. As time increases to $t=0.5$,
the distribution becomes inhomogeneous: particles near the channel walls
quickly begin accumulating there, and the funnel-shaped distribution of
particles in the middle region of the cell starts to develop. At a yet later
time $t=1$, the funnel-shaped distribution sharpens, and particles have
continued to accumulate at the walls. The distribution at this time closely
resembles the steady state distribution shown in Fig. 2(b). The probability of
accumulation at the left corners of the cell is much higher than in other
regions, which is reflected in the structure of the probability density $P(y)$
in Fig. 3(d).

##  IV Rectification and effective diffusion

Figure 4: Average velocity $v$ as a function of $M^{*}$ is shown in (a) for
different values of the self-propulsion force $f_{0}$. The corresponding
effective diffusion coefficient $D_{\rm eff}$ is shown in (b). The inset plots
$D_{\rm eff}$ versus $M^{*}$ for $f_{0}=1$ on an expanded scale to show its
structure. Here and below, the solid lines are guides to the eye, and the
statistical errors for $v$ and $D_{\rm eff}$ are smaller than the symbol
sizes. The other parameters are: $I^{*}=0.001,t_{0}=0,\alpha=0.1$, and
$\epsilon=0.1$.

Figure 4 shows the dependence of the average velocity $v$ and effective
diffusion coefficient $D_{\rm eff}$ on $M^{*}$ for different values of the
self-propulsion force $f_{0}$. We observe that the particles exhibit
rectification ($v\neq 0$) in the positive $x$ direction; $v$ is positive due
to the chosen shape of the channel. If the shape of the channel were inverted
with respect to the $y$ axis, then the magnitude of rectification would remain
the same, but $v$ would lie in the negative $x$ direction. In the small
$M^{*}$ limit, the channel asymmetry affects the particle dynamics more
strongly since most of the particles accumulate at the channel walls (see
Figs. 2 and 3), resulting in an increase in $v$ with $M^{*}$. In the strongly
underdamped limit, when $M^{*}\to\infty$, inertia dominates self-propulsion
resulting in a rapid decay of $v$ and $D_{\rm eff}$ with $M^{*}$. Therefore,
as one might expect, $v$ has a peak at an optimal mass $M_{\rm op}^{*}$; thus,
in a mixture, rectified particles with $M_{\rm op}^{*}$ will have a higher
speed compared to particles with other $M^{*}$ values. The peak is more
pronounced as $f_{0}$ increases, and the value of the optimal mass can be
controlled by changing $f_{0}$. For the effective diffusion coefficient, we
see that $D_{\rm eff}$ initially decreases with increasing $M^{*}$, but on
increasing $M^{*}$ further, $D_{\rm eff}$ exhibits an enhanced diffusion peak,
which is a signature of the accumulation of most of the particles at the
channel walls [43]. Note that the two maxima, in $v$ and in the enhanced
diffusion peak, do not occur at the same $M^{*}$ value. As expected, $D_{\rm
eff}$ increases monotonically with $f_{0}$. As $f_{0}\to 0$, the motion of
self-propelled particles approaches that of passive Brownian motion; thus, $v$
will tend to zero, and the enhanced diffusion peak will no longer be present
(see the inset of Fig. 4(b)).

Figure 5: Average velocity $v$ and effective diffusion coefficient $D_{\rm
eff}$ as a function of $M^{*}$ are plotted in (a) and (b), respectively, for
different values of the scaled rotational diffusion rate $\alpha$. The inset
in (b) shows $D_{\rm eff}$ versus $M^{*}$ for $\alpha=1$ and $\alpha=10$ on an
expanded scale. The other parameters are: $I^{*}=0.001,t_{0}=0,f_{0}=5$, and
$\epsilon=0.1$.

Figure 6: Average velocity $v$ and effective diffusion coefficient $D_{\rm
eff}$ as a function of $M^{*}$ are plotted in (a) and (b), respectively, for
different values of the active torque $t_{0}$. The inset in (b) shows $D_{\rm
eff}$ versus $M^{*}$ for $t_{0}=1$ and $t_{0}=10$ on an expanded scale. The
parameters are: $I^{*}=0.001,\alpha=0.1,f_{0}=5$, and $\epsilon=0.1$.

The variation of $v$ and $D_{\rm eff}$ with $M^{*}$ for different values of
the rotational diffusion rate $\alpha$ is shown in Fig. 5. The qualitative
trends for different $\alpha$ are the same as those described above for
different $f_{0}$; however, the peak is more pronounced as $\alpha$ decreases,
and now $M_{\rm op}^{*}$ can be changed by tuning $\alpha$. As expected,
$D_{\rm eff}$ decreases monotonically with increasing $\alpha$. In particular,
as $\alpha\to\infty$ where reorientation is rapid, the self-propelled motion
tends to passive Brownian motion, $v$ tends to zero, and the enhanced
diffusion peak vanishes (see the inset of Fig. 5(b)).

Figure 6 shows $v$ and $D_{\rm eff}$ versus $M^{*}$ for different values of
the active torque $t_{0}$. The distribution of particles along the $y$
direction does become asymmetric with respect to the principal axis of the
channel when $t_{0}\neq 0$, as reported in Refs. [43, 58, 59]. For $t_{0}>0$,
the density of particles will be higher above the principal axis of the
channel; whereas, for $t_{0}<0$, the same density increase will occur in the
regime below the principal axis of the channel. However, our channel is
symmetric about its principal axis, i.e., it has top-down symmetry, and we
have verified that $v$ and $D_{\rm eff}$ are independent of the sign of the
active torque $t_{0}$. Again one observes that the qualitative trends are the
same as those discussed above, and the values of $M_{\rm op}^{*}$ can also be
changed by changes in the active torque $t_{0}$.

Finally, we point out that the qualitative behavior of $v$ and $D_{\rm eff}$
versus $M^{*}$ remains the same for different values of the moment of inertia
$I^{*}$ and aspect ratio of the channel $\epsilon$. In addition, $M_{\rm
op}^{*}$ is found to be independent of $I^{*}$, and it is only weakly
dependent on $\epsilon$.

Figure 7: Dependence of $M^{*}_{\rm op}$ on the rotational diffusion rate
$\alpha$ (a), self-propulsion force $f_{0}$ (b), aspect ratio of the channel
$\epsilon$ (c), and active torque $t_{0}$ (d). The moment of inertia
$I^{*}=0.001$.

The dependence of $M^{*}_{\rm op}$ on the rotational diffusion rate $\alpha$,
self-propulsion force $f_{0}$, aspect ratio of the channel $\epsilon$, and
active torque $t_{0}$ is shown in Fig. 7. For fixed values of $f_{0}$,
$\epsilon$, and $t_{0}$, $M^{*}_{\rm op}$ decreases monotonically with
$\alpha$ (panel (a)). From panel (b), for fixed values of $\alpha$,
$\epsilon$, and $t_{0}$, $M^{*}_{\rm op}$ varies nonmonotonically with $f_{0}$
with the appearance of a peak. We observe that $M^{*}_{\rm op}$ is weakly
dependent on $\epsilon$ (panel (c), see change in the ordinate scale); for
fixed values of $\alpha$, $f_{0}$, and $t_{0}$, $M^{*}_{\rm op}$ has a
minimum. Lastly, from panel (d), $M^{*}_{\rm op}$ monotonically decreases with
$t_{0}$ for fixed values of $\alpha$, $f_{0}$, and $\epsilon$.

##  V Remarks and Conclusion

This study of the rectification and diffusion of self-propelled particles in a
two-dimensional asymmetric channel showed that the inclusion of inertia leads
to several distinctive features. In particular, most of the particles
accumulate at the channel walls with increasing particle mass, while the
remaining particles are distributed in a funnel-shaped region about the
principal axis of the channel. This effect leads to enhanced rectification of
heavier particles. The presence of a maximum in the effective diffusion
coefficient as a function of the mass is also a consequence of the
accumulation of most of the particles at the channel walls. Furthermore, for
various parameter values, the average particle velocity has a maximum as a
function of the mass, indicating that particles with an optimal mass
$M^{*}_{\rm op}$ drift faster than other particles; hence, they can be sorted
from a mixture with particles of different masses.

While the Langevin model (1) can describe a wide variety of physical systems
whose active agents are powered by various mechanisms and are subject to
either thermal or athermal noise [16], it is instructive to discuss possible
experimental realizations of the rectification effects described above. The
asymmetric channels we considered can be constructed by microprinting on a
substrate, and the effects of inertia can be determined from measurements of
the average velocity and effective diffusion coefficient [60, 61, 62]. From
the results presented in the text, one can see that the effects of inertia
described above will manifest themselves only for certain values of the system
parameters, in particular, the particle mass, friction coefficients, diffusion
constants, self-propulsion force, solvent viscosity, etc. The ability to
control all of these parameters within desirable ranges places limits on the
physical systems.

A class of systems that may be of interest in this context are aerosols [28],
where diffusiophoresis has been used to separate micrometer-scale particles
[63, 64]. As an example, consider identically-sized active particles with
radii $a\sim 200~{}\mathrm{nm}$, mass $M\sim 10^{-15}~{}\mathrm{kg}$, moment
of inertia $I\sim 10^{-31}~{}\mathrm{kg~{}m^{2}}$ in air at room temperature
and pressure $p=10^{4}-10^{5}~{}\rm{Pa}$. The viscosity is given by $\eta\sim
10^{-5}~{}\mathrm{kg/(m~{}s)}$, independent of pressure, with translational
and rotational friction coefficients, $\gamma_{t}\sim
10^{-11}~{}\mathrm{kg/s}$ and $\gamma_{r}\sim
10^{-24}~{}\mathrm{kg~{}m^{2}/s}$, respectively. The active force lies in the
range $F_{0}\sim 0.1-1~{}\mathrm{pN}$, and the active torque is taken to be
zero. The translational and rotational noise strengths are then determined by
the values of $D_{t}$ and $D_{r}$, respectively. The ratchet channel
parameters are $L=10~{}\mu{\rm m}$, ${\rm w}_{\mathrm{max}}=10~{}\mu{\rm m}$,
and $\epsilon=0.1$.

For thermal noise, the Einstein relations hold, and
$D_{t}=k_{B}T/\gamma_{t}\sim 10^{-9}~{}\mathrm{m^{2}/s}$ and
$D_{r}=k_{B}T/\gamma_{r}\sim 10^{3}~{}\mathrm{s^{-1}}$. In the dimensionless
units introduced earlier, we have $\tau\sim 0.1~{}{\rm s}$, $M^{*}\sim
10^{-3}$, $I^{*}\sim 10^{-6}$, $f_{0}\sim 10^{2}-10^{3}$, and $\alpha\sim
10^{2}$. From these parameter values, we can see that the regime where
inertial effects play a role cannot be accessed.

For athermal noise, take $D_{t}\sim 10^{-8}-10^{-6}~{}\mathrm{m^{2}/s}$ and
$D_{r}\sim 10^{3}~{}\mathrm{s^{-1}}$. We then have $\tau\sim
10^{-2}-10^{-4}~{}{\rm s}$, $M^{*}\sim 0.01-1$, $I^{*}\sim 10^{-5}-10^{-3}$,
$f_{0}\sim 100-0.1$, and $\alpha\sim 10-0.1$. The persistence length and
Péclet number are given by $l_{p}=v_{0}/D_{r}\sim 10-100~{}\mu{\rm m}$ and
$Pe=v_{0}/\sqrt{D_{t}D_{r}}\sim 0.3-30$, respectively. Under these conditions,
the interesting regime where inertial effects lead to an optimal mass can be
reached, provided the system is driven by external noise sources.

These results could stimulate the development of strategies for controlling
the diffusion of active particles in entropic ratchet systems. Moreover, since
the rectification of particles strongly depends on their mass, the model could
be used to design lab-on-a-chip devices and artificial channels for the mass-
based separation of particles.

##  VI Acknowledgment

This work was supported in part by the Natural Sciences and Engineering
Research Council (NSERC) of Canada and Compute Canada (www.computecanada.ca).

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