# Active transport of a passive colloid in a bath of run-and-tumble particles

Tanumoy Dhar  David Saintillan<EMAIL_ADDRESS>Department of Mechanical and
Aerospace Engineering, University of California San Diego, 9500 Gilman Drive,
La Jolla, CA 92093, USA

###### Abstract

The dispersion of a passive colloid immersed in a bath of non-interacting and
non-Brownian run-and-tumble microswimmers in two dimensions is analyzed using
stochastic simulations and an asymptotic theory, both based on a minimal model
of swimmer-colloid collisions characterized solely by frictionless steric
interactions. We estimate the effective long-time diffusivity $\mathcal{D}$ of
the suspended colloid resulting from its interaction with the active bath, and
elucidate its dependence on the level of activity (persistence length of
swimmer trajectories), the mobility ratio of the colloid to a swimmer, and the
number density of swimmers in the bath. We also propose a semi-analytical
model for the colloid diffusivity in terms of the variance and correlation
time of the net fluctuating active force on the colloid resulting from swimmer
collisions. Quantitative agreement is found between numerical simulations and
analytical results in the experimentally-relevant regime of low swimmer
density, low mobility ratios, and high activity.

##  I Introduction

Systems driven out of equilibrium are ubiquitous, and underlying active
fluctuations play an important role in governing their behavior. One such
example is the ability of active microswimmers to transport colloids [1, 2, 3,
4] and perform mechanical work [5, 6, 7, 8]. The sole propulsion of individual
microswimmers can lead to novel transport characteristics, such as enhanced
diffusion [9, 10, 11, 12], where the long-time diffusivity of suspended
passive particles in bacterial suspensions far exceeds its Brownian
counterpart. Seminal experiments by Wu and Libchaber [1] first highlighted
this phenomenon and uncovered a transition from short-time directed motion
(non-Gaussian and super-diffusive) to long-time diffusive (Gaussian) motion. A
succession of subsequent experiments [9, 13, 3, 14, 15, 12, 16, 17, 18]
further probed this effect, and for instance examined the role of the
concentration of bacteria and the effect of the size ratio of the colloid to
the bacteria. The type of microswimmer used was also shown to affect the
dynamics of the colloids suspended in a bath: for example, certain algae (C.
reinhardtii) drive oscillatory flows [19, 20] resulting in loop-like colloid
trajectories [21], an effect that can be rationalized using far-field
hydrodynamics of this swimming organism [22, 23, 24]. An extensive amount of
theoretical [3, 12, 25, 26, 27, 28, 29, 30] and numerical works [31, 2, 10,
32, 26] were performed over the last two decades, and several models were
proposed to elucidate the respective roles of steric [27, 18] vs hydrodynamic
[10, 32, 26, 33] effects in driving colloidal diffusion. Recent experiments
have shown that bacteria [34, 4] and other self-propelled particles [35, 34,
36] can undergo direct collisions with colloidal clusters, substantiating the
significance of steric interactions relative to hydrodynamic interactions.
However, modeling endeavors focused on predicting colloid dispersion in active
suspensions have encountered various limitations, primarily attributable to
the challenges involved in quantifying the scattering dynamics between the
active particle and the colloid. Recently, Lagarde  _et al._ [18] put forth a
collision model that treats bacteria-colloid interactions as purely repulsive.
The present work is inspired by the scattering models of Jakuszeit  _et al._
[37] and Saintillan [38], used to study the dispersion of an active particle
in porous media. Regardless of the nature of interactions, the anomalous
short-time dynamics exhibited by colloids suspended in bacterial baths are
always found to map to Gaussian statistics at long times. The quantitative
understanding of this microscopic enhancement of the long-time diffusivity of
passive colloids in active baths remains, however, incomplete, even in the
seemingly simple case of steric interactions. Previous works [3, 15, 12, 26]
have provided phenomenological arguments for such diffusion. Yet, an
analytical model relating the diffusion coefficient to system parameters based
on first principles is still lacking for baths of run-and-tumble particles.

To address this gap, we propose a minimal model incorporating swimmer-colloid
collisions in two dimensions (where the majority of the experiments [1, 14, 3,
39, 18, 4] are performed) to effectively capture the effect of system
parameter variations on the dispersion. The proposed model relies on a
collision-resolution algorithm that solves for constraint forces arising from
swimmer-colloid contacts at every instant in time. We employ stochastic
simulations to estimate the effective diffusivity $\mathcal{D}$ of the
suspended colloid, highlighting its dependence on the level of activity or
Péclet number $Pe$, mobility ratio $\mu$ of the colloid to a swimmer, and
number density $\rho$ of swimmers in the active bath. These system parameters
are defined more precisely later. We also posit a semi-analytical model for
the colloid diffusivity in terms of the variance and correlation time of the
fluctuating active force resulting from swimmer collisions. Our theoretical
predictions agree well with particle simulations in the limit of low swimmer
density, low mobility ratio and high Péclet number, and provide a simple
analytical framework for the description of long-time colloidal dispersion in
baths of run-and-tumble microswimmers.

The paper is organized as follows. In Section II, we present the swimmer-
colloid collision model, the dimensionless governing equations, and the
computational framework. We present results from stochastic simulations in
Section III, where we highlight the dependence of the long-time colloid
diffusivity on the system parameters. In Section IV, we propose an asymptotic
theory valid for low mobility ratios $(\mu\longrightarrow 0)$. This theory
relates the diffusivity to the variance and correlation time of the
fluctuating force on a fixed colloid subject to collisions with run-and-tumble
microswimmers, and involves the determination of the mean number of swimmers
in contact with the colloid. The proposed semi-analytical theory is shown to
match well with the numerical simulation results in the limit of large
colloids and long swimmer persistence lengths. We conclude in Section V.

##  II Problem formulation

###  II.1 Swimmer-colloid collision model

We analyze the dispersion of a passive colloid suspended in a dilute
collection of non-interacting self-propelled particles in two dimensions using
a minimal model for swimmer-colloid collisions. Swimmers ($i=1,\dots,N$), with
positions $\bm{r}_{i}$ and orientations $\bm{p}_{i}$ (where $|\bm{p}_{i}|=1$),
are modeled as point particles and perform a run-and-tumble motion [40, 41]
away from the colloid: straight swimming runs of duration $\tau$ with constant
velocity $u$ along the unit director $\bm{p}_{i}$ are interrupted with
instantaneous reorientation events or tumbles (see Fig. 1b where the swimmer
with orientation $\bm{p}_{1}(t)$ performs a tumble at a point denoted by
$T_{1}$). The time $\tau$ between consecutive tumbles is an exponentially
distributed random variate with mean $\bar{\tau}$ and cumulative probability
distribution function $\Sigma(\tau)=1-\exp(-\tau/\bar{\tau})$ for $\tau\geq
0$. The choice of an exponential distribution to model the tumbling rate of
the Escherichia coli bacterium [42, 43, 44] is an assumption and has been
widely used in previous theoretical studies [45, 46]. In the present work, we
assume that the swimming velocity $u$, orientation $\bm{p}_{i}$ and run time
$\tau$ of the swimmer remain independent of its spatial proximity to the
colloid, which serves as a convenient approximation for computational
simulations. In free space, far from the colloid, a swimmer performs unimpeded
run-and-tumble motion. During a run away from the colloid, the swimmer
dynamics simply follows:

$\dot{\bm{r}}_{i}=u\bm{p}_{i}.$ (1)

A steric collision occurs whenever a swimmer comes in contact with the surface
of the colloid, assumed to be of circular shape with radius $A$. The geometry
of a collision is illustrated in the schematic of Fig. 1a. We choose a
Cartesian $(x,y)$ coordinate system that is fixed in the active bath, where
$\bm{r}_{i}$ and $\bm{R}$ are the instantaneous swimmer and colloid positions,
respectively. The current orientation of the swimmer is
$\bm{p}_{i}=(\cos\theta_{i},\sin\theta_{i})$, where the angle $\theta_{i}$
during a particular run is uniformly distributed in the interval $[0,2\pi)$.
The wall-normal unit vector $\bm{q}_{i}$ at the point of contact $C$ is
directed from the colloid into the active bath. The angle
$\alpha_{i}\in[-\pi,\pi]$ in Fig. 1a captures the projection of $\bm{p}_{i}$
on $\bm{q}_{i}$ and is henceforth called contact angle. Note that $\alpha_{i}$
changes with time over the course of a run as the swimmer slides over the
colloid surface. Symmetry with respect to the direction $\bm{q}_{i}$ dictates
that the probability density function of $\alpha_{i}$ be an even function.
Finally, the angle $\beta_{i}\in[0,2\pi)$ captures the angular position of the
swimmer on the colloid surface with respect to the $(X,Y)$ coordinate system.

We model swimmer-colloid interactions as steric frictionless collisions: When
a swimmer comes in contact with the colloid, its relative velocity with
respect to the colloid vanishes in the normal direction, effectively
establishing a no-penetration boundary condition. By assumption, the swimmer
orientation remains unchanged for the duration of the current run, and it
simply undergoes sliding motion along the colloid surface until it either
swims or tumbles away. If the contact angle reaches $\pm\pi/2$ before the end
of the run, the swimmer escapes from the colloid surface and completes its run
in a straight line in the bulk (see Fig. 1b where swimmer 2 comes in contact
with the colloid at point $C_{2}$ and escapes at point $E_{2}$ where
$\bm{p}_{2}$ becomes tangent to the surface). If, on the other hand, the
swimmer concludes its run on the surface of the colloid, it will perform its
next tumble on the surface. If its new orientation points into the bulk, it
will immediately leave the colloid surface (see swimmer 3 in Fig. 1b). If its
new orientation points toward the colloid, it will start the next run sliding
on the colloid surface, until it either escapes or tumbles again.

Figure 1: Problem definition: (a) Schematic diagram of a swimmer in contact
with the colloid and associated nomenclature (see main text for details); (b)
Representative swimmer dynamics, where sections of a run performed in the bulk
or on the colloid surface are shown in green and yellow, respectively, and
where tumbles are depicted as red dots. As a swimmer slides on the colloid
surface, it can either escape by swimming away tangentially (swimmer 2), or
finish its run and perform its next tumble on the surface (swimmer 3).

During the interaction between the swimmer and the colloid, both entities
apply contact forces $\pm F_{i}\bm{q}_{i}$ of equal magnitude but in opposite
directions. As the swimmer slides on the surface of the colloid, the contact
force magnitude changes with the swimmer’s evolving position and orientation.
Tangential resistance to sliding is not considered, resulting in a contact
force directed solely along the normal direction. The swimmer’s velocity is
modified as a result of the collision:

$\dot{\bm{r}}_{i}=u\bm{p}_{i}+mF_{i}\bm{q}_{i},$ (2)

where $m$ is the mobility of the swimmer, assumed to be isotropic for
simplicity. The dynamics of the colloid is driven by the net contact force
resulting from the collisions with any swimmers currently on its surface and
follows

$\dot{\bm{R}}=-M\sum_{i=1}^{c}F_{i}\bm{q}_{i},$ (3)

where $M$ is the mobility of the colloid. Here, $c(t)$ is the number of
swimmers in contact with the colloid at any given instant in time. As we
explain in detail in Section II.3, the contact forces $F_{i}$ are determined
numerically to satisfy the no-penetration condition:

$(\dot{\bm{r}}_{i}-\dot{\bm{R}})\cdot\bm{q}_{i}=0.$ (4)

We also introduce $\zeta=1/m$ and $Z=1/M$ as the friction coefficients of a
swimmer and of the colloid, respectively. Throughout the remainder of the
present work, we choose the convention of employing uppercase letters to
represent physical variables associated with the passive colloid, and
lowercase letters for those associated with a swimmer.

We underscore that the minimal model presented herein provides a simple
framework for understanding colloidal dispersion in active baths, but also
relies on strong simplifying assumptions: in the context of experiments,
short-ranged hydrodynamic effects (lubrication [47, 48, 49] and alignment
torques [50, 51]) experienced by the swimmers can lead to scattering at angles
that are often non-tangent to the colloid surface [52]. Experiments
demonstrate that the tumbling rate $1/\tau$ is reduced for certain swimmers
such as E. coli near boundaries, which effectively acts towards trapping the
swimmer at the walls [53, 54]. Finally, the present model does not capture the
convex trajectory of the swimmer around the colloid predicted by models that
account for swimmer-colloid hydrodynamic interactions [52, 55], but is
supported by the recent experimental observations of Lagarde  _et al._ [18].

###  II.2 Dimensionless governing equations

The governing equations are non-dimensionalized using length scale $A$ and
time scale $A/u$. Contact forces are made dimensionless using the swim force
[56, 57, 58] $F^{\text{swim}}=\zeta u$, which is the force one needs to apply
to a swimmer to fix its position in space. In free space, away from the
colloid, the dynamics of a swimmer during its run simply follows

$\bm{\dot{r}}_{i}=\bm{p}_{i}.$ (5)

At the colloid surface, it becomes:

$\dot{\bm{r}_{i}}=\bm{p}_{i}+F_{i}\bm{q}_{i}.$ (6)

The evolution equation for the position of the colloid reads:

$\dot{\bm{R}}=-\mu\sum_{i=1}^{c}F_{i}\bm{q}_{i},$ (7)

where the mobility ratio $\mu$ is defined as

$\mu=\frac{M}{m}=\frac{\zeta}{Z}.$ (8)

In addition to the mobility ratio, two dimensionless parameters govern the
problem. The Péclet number, defined as $Pe=u\bar{\tau}/A$, compares the mean
run time $\bar{\tau}$ to the time $A/u$ for a swimmer to travel a distance of
one colloid radius, and is a dimensionless measure of the persistence of
swimmer trajectories. Note that $Pe$ does not appear explicitly in the
governing equations, but defines the mean of the exponential distribution used
to model the swimmer run times. Finally, the number density $\rho$ of swimmers
is defined as the ratio of the total number of swimmers $\nu$ in the square
domain of length $L$ to the area $L^{2}-\pi A^{2}$ available to the swimmers.
In dimensionless variables,

$\rho=\frac{\nu A^{2}}{L^{2}-\pi A^{2}}.$ (9)

###  II.3 Numerical method

We implement a contact-force resolution-based computational algorithm for
simulating the dynamics of a single passive mobile colloid in a suspension of
non-interacting run-and-tumble particles in two dimensions. The simulations
are performed inside a periodic square domain of linear dimension $L$ chosen
to be much larger than the colloid size and typical run length
($L\gg\mathrm{max}\\{Pe,1\\}$ in dimensionless variables), which $\nu$
swimmers are distributed at random. The values of $L$ and $\nu$ are chosen to
achieve the desired number density $\rho$ as defined in eqn (9). We discretize
time with a fixed time step $\Delta t\ll\mathrm{min}\\{Pe,1\\}$ and use time-
marching to advance the positions of the swimmers and colloid.

Far from the colloid, a swimmer trajectory simply evolves as
$\bm{r}_{i}(t+\Delta t)=\bm{r}_{i}(t)+\bm{p}_{i}\Delta t$ over the course of
one time step. During any given time step, a number $c(t)$ of swimmers will
collide with the colloid. The positions of these swimmers are updated as

$\bm{r}_{i}(t+\Delta t)=\bm{r}_{i}(t)+[\bm{p}_{i}+F_{i}(t)\bm{q}_{i}]\Delta
t,$ (10)

where the normal contact force $F_{i}$, taken to be constant over one time
step, must be determined. During that same time step, the position $\bm{R}$ of
the colloid is evolved as

$\bm{R}(t+\Delta t)=\bm{R}(t)-\mu\sum_{i=1}^{c(t)}F_{i}(t)\bm{q}_{i}\Delta t,$
(11)

which couples all the contact forces. To solve for the magnitude of these
forces, we apply the no-penetration condition of eqn (4) at the location of
each contact. This leads to the following set of coupled linear equations:

$(\bm{\mathscr{I}}+\mu\bm{\mathscr{A}})\cdot\bm{\mathscr{F}}=-\bm{\mathscr{B}},$
(12)

where, $\bm{\mathscr{F}}=\\{F_{1},F_{2}...F_{c}\\}$ is a vector of length
$c(t)$ containing the unknown contact forces, and $\bm{\mathscr{I}}$ is the
identity matrix. The matrix $\bm{\mathscr{A}}$ and the vector
$\bm{\mathscr{B}}$ are geometric quantities that depend on the current
configuration of the swimmers on the colloid surface:

$\displaystyle\begin{split}\bm{\mathscr{A}}=\begin{Bmatrix}\bm{q}_{1}\cdot\bm{q}_{1}&\bm{q}_{1}\cdot\bm{q}_{2}&\bm{q}_{1}\cdot\bm{q}_{3}&...&\bm{q}_{1}\cdot\bm{q}_{c}\\\
\bm{q}_{2}\cdot\bm{q}_{1}&\bm{q}_{2}\cdot\bm{q}_{2}&\bm{q}_{2}\cdot\bm{q}_{3}&...&\bm{q}_{2}\cdot\bm{q}_{c}\\\
\bm{q}_{3}\cdot\bm{q}_{1}&\bm{q}_{3}\cdot\bm{q}_{2}&\bm{q}_{3}\cdot\bm{q}_{3}&...&\bm{q}_{3}\cdot\bm{q}_{c}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
\bm{q}_{c}\cdot\bm{q}_{1}&\bm{q}_{c}\cdot\bm{q}_{2}&\bm{q}_{c}\cdot\bm{q}_{3}&...&\bm{q}_{c}\cdot\bm{q}_{c}\end{Bmatrix},\\\
\bm{\mathscr{B}}=\begin{Bmatrix}\bm{p}_{1}\cdot\bm{q}_{1}\\\
\bm{p}_{2}\cdot\bm{q}_{2}\\\ \bm{p}_{3}\cdot\bm{q}_{3}\\\ \vdots\\\
\bm{p}_{c}\cdot\bm{q}_{c}\end{Bmatrix}.\qquad\qquad\qquad\end{split}$ (13)

The system of equations (12) is constructed at every time step and inverted
numerically to obtain the contact forces. The swimmer and colloid positions
are subsequently updated using eqn (10) and eqn (11), respectively. Note that
the present formulation can still result in occasional swimmer-colloid
overlaps $(|\bm{r}_{i}-\bm{R}|<1)$ due to the time discretization. As a
result, we check for overlaps at the end of each time step and, if necessary,
implement the additional correction for the swimmer position based on the
updated colloid location:

$\bm{r}_{i}(t+\Delta t)=\bm{R}(t+\Delta t)-\bm{q}_{i}.$ (14)

##  III Numerical results and discussion

Figure 2: Simulated trajectories of the passive colloid for different
parameter values. The three distinct color gradients are for different: (a)
number densities of swimmers: $\rho=0.08$ (blue), $0.8$ (green) and $8$ (red)
(with $Pe=5.0$ and $\mu=0.5$); (b) Péclet numbers: $Pe=0.05$ (purple), $0.5$
(orange) and $5$ (grey) (with $\rho=0.8$ and $\mu=0.5$). The increasing
intensity of color captures increasing time. All the trajectories are for
$t=0$ to $5000$. The black dot shows the initial colloid location. See ESI for
a video showing a typical simulation. Figure 3: Variation of the mean squared
displacement with time, for different: (a) number densities of swimmers:
$\rho=0.08$ (blue), $0.8$ (green) and $8$ (red) (with $Pe=5.0$ and $\mu=0.5$);
(b) Péclet numbers: $Pe=0.05$ (purple), $0.5$ (orange) and $5$ (grey) (with
$\rho=0.8$ and $\mu=0.5$). The dashed black lines show the linear diffusive
scaling at long times.

In this section, we present results from numerical simulations for a wide
range of system parameters. Figure 2 shows several representative colloid
trajectories (for $t=0$ to $5000$) at different densities $\rho$ and different
Péclet numbers $Pe$ in panels a and b, respectively (see figure caption for
details, and also see ESI for a video of a typical simulation). The
trajectories, along which increasing color intensity is used to show the
passage of time, are correlated random walks, albeit with a short correlation
time as we discuss further below. Visual inspection suggests that dispersion
gets stronger with either increasing $\rho$ (compare blue and green curves in
a) or $Pe$ (compare purple and orange curves in b), trends that we quantify
and explain in the following discussion.

Figure 4: Variation of the diffusivity $\mathcal{D}$ of the colloid with: (a)
number density $\rho$ of swimmers for different values of $Pe$; (b) Péclet
number $Pe$, for the different values of $\rho$; (c) mobility ratio $\mu$ of
the colloid to the swimmer. The mobility ratio is kept constant at $\mu=0.5$
in panels a and b. The dashed lines in each panel highlight asymptotic
scalings at low values of $\rho$, $Pe$ and $\mu$.

Figure 3 shows the time evolution of the mean squared displacement of the
colloid, defined as
$\langle|\Delta\bm{R}(t)|^{2}\rangle=\langle|\bm{R}(t)-\bm{R}(0)|^{2}\rangle$,
where $\langle\cdot\rangle$ denotes the ensemble average over multiple
realizations of single colloid trajectories. The system parameters are the
same as in Fig. 2. At short time scales ($t\sim 10^{-2}-10^{-1}$), the mean
squared displacement grows as $t^{\gamma}$ with $\gamma\approx 2$. At later
times, a transition to diffusive behavior (with $\gamma\approx 1$) is
observed, consistent with previous experimental studies. The crossover time
from the ballistic to the diffusive regime varies for different system
parameters as is evident in Fig. 3a,b. We quantify the long-time dispersion of
the colloid in terms of its diffusivity $\mathcal{D}$ defined as:

$\mathcal{D}=\lim_{t\rightarrow\infty}\frac{1}{4}\frac{\mathrm{d}}{\mathrm{d}t}\langle|\Delta\bm{R}(t)|^{2}\rangle.$
(15)

The diffusivity was numerically determined for a wide range of number
densities $\rho$, Péclet numbers $Pe$, and mobility ratios $\mu$, and the
results are summarized in Fig. 4.

The variation of $\mathcal{D}$ with number density $\rho$ is illustrated in
Fig. 4a for different values of $Pe$ at $\mu=0.5$. At low to moderate swimmer
densities, increasing $\rho$ leads to a linear increase of $\mathcal{D}$,
which can be rationalized by a concomitant increase in the number of swimmer-
colloid collisions as we analyze further below. This linear dependence on
$\rho$ is in agreement with previous experimental studies [3, 15, 18, 4]. Upon
further increasing the density of the active bath, the diffusivity
$\mathcal{D}$ is found to reach a peak and then slightly decrease with $\rho$
at high densities. A similar departure of the diffusivity from linear growth
with increasing swimmer density was previously reported by Lagarde  _et al._
[18], in both experiments and numerical simulations. The explanation provided
by Lagarde  _et al._ [18] is based on the hypothesis that multiple bacteria
simultaneously interact with the colloid at high densities, resulting in a
trend that breaks the linear growth of $\mathcal{D}$. This explanation is
consistent with our simulations: the departure from the linear increase in
Fig. 4a occurs for $\rho\gtrsim 1$, which coincides approximately with the
density at which the mean number of swimmers on the surface of the colloid at
any given time exceeds 1 (see Fig. 6 below). The slight decrease with $\rho$
observed at high densities in Fig. 4a can be understood by the same effect:
when many swimmers interact with the colloid at the same time, partial
cancellation of the forces they exert on the colloid occurs, resulting in a
lower diffusivity; this will be confirmed in Fig. 8 when we discuss the
dependence of the net force variance on $\rho$. To the best of our knowledge,
this decrease in $\mathcal{D}$ with respect to density at large $\rho$ has not
been experimentally observed in bacterial suspensions. Indeed, the regime
where it occurs in our model may be challenging to realize experimentally, as
it requires moderate to high mobility ratios (i.e., small colloids) as well as
many swimmers interacting with the colloid at any given time. In a system with
finite-sized swimmers, steric hindrance would likely impose an upper bound on
the number of active particles capable of simultaneously colliding with the
colloid surface. Furthermore, bacterial suspensions at such high densities may
also exhibit collective motion, which would further act to enhance
$\mathcal{D}$.

The dependence of diffusivity on Péclet number is shown in Fig. 4b for
different swimmer densities. At lower values of $Pe$ ($Pe\lesssim 1)$,
$\mathcal{D}$ increases linearly with $Pe$, since persistent swimmers push on
the colloid surface for a longer time before they escape or tumble into the
bulk as the Péclet number is increased. For sufficiently large Péclet number
($Pe\gtrsim 1$), the diffusivity saturates. This can be attributed to the fact
that at large values of $Pe$ most collisions result in an escape, and
therefore the amount of time that a swimmer interacts with the colloid becomes
independent of $Pe$; this trend will be confirmed in Fig. 5(c) when we discuss
the autocorrelation time for the net force on the colloid. Finally, panel c of
Fig. 4 shows the dependence of $\mathcal{D}$ on the mobility ratio $\mu$. In
the limit of large colloids ($\mu\lesssim 1$), the dependence is quadratic as
$\mathcal{D}\sim\mu^{2}$, but $\mathcal{D}$ is found to eventually saturate in
the limit of small colloids ($\mu\gtrsim 1$). Next, we rationalize some of
these trends by deriving a semi-analytical model for the diffusivity in the
limits of low swimmer density, high Péclet number and low mobility ratio.

##  IV Semi-analytical description

Figure 5: (a) Variations of $x$ and $y$ components of the net active force on
the colloid with time for Péclet numbers $Pe=0.1$ ($i$), $1$ ($ii$) and $10$
($iii$), for constant density of swimmers $\rho=8.04$ and mobility ratio
$\mu=0.5$. (b) Time autocorrelation $C_{F}(\Delta t)$ of the net active force,
defined in eqn (20), for the same conditions as in (a). (c) Time constant
$\xi$ of the force autocorrelation function, obtained by exponential fit, as a
function of ($i$) Péclet number (for $\rho=1$ and $\mu=0.01$), and $(ii)$
swimmer density (for $Pe=8$ and different mobility ratios). In c($i$), the
solid black curve shows the lower-bound estimate derived in eqn (26).

In this section, we present an analytical model for the diffusivity of a
passive colloid suspended in the active bath. Following Hinch [59], we start
from the equations of motion for the suspended colloid and show that the
effective dispersion coefficient can be related to the time autocorrelation
function of the net active force it experiences due to collisions with the
active bath. We then show how that autocorrelation function can be estimated
analytically in the asymptotic limit of low mobility ratios
$(\mu\longrightarrow 0)$, in which we obtain a closed-form expression for the
diffusivity. We then compare our theory to results from simulations and find
excellent agreement in the limit of validity of the model.

###  IV.1 Diffusivity calculation

The run-and-tumble random walk of the active microswimmers translates to
fluctuations in the motion of the colloid. The starting point for our
diffusivity calculation in the absence of Brownian fluctuations is the
equation of motion (3) of the passive colloid, written as

$\dot{\bm{R}}=\mu\bm{F}^{a}(t),$ (16)

where $\bm{F}^{a}(t)$ is the net instantaneous fluctuating active force
resulting from collisions with the swimmers:

$\bm{F}^{a}(t)=-\sum_{i=1}^{c(t)}{F}_{i}^{c}(t)\bm{q}_{i}(t).$ (17)

As explained in Sec. II.1, the net force on the colloid results from contact
forces with the swimmers on its surface, and thus depends on the number $c(t)$
of swimmers in contact, as well as their positions and orientations on the
colloid surface, which determine the magnitude of the contact forces
$F_{i}^{c}(t)$ and their directions $\bm{q}_{i}(t)$. Integrating eqn (16) and
choosing the origin such that $\bm{R}(0)=\mathbf{0}$, the displacement of the
passive colloid is found to be,

$\Delta\bm{R}(t)=\bm{R}(t)-\bm{R}(0)=\int_{0}^{t}\mu\bm{F}^{a}(t^{\prime})\,\mathrm{d}t^{\prime}.$
(18)

We use this expression to calculate the mean square displacement as

$\langle|\Delta\bm{R}(t)|^{2}\rangle=\mu^{2}\int_{0}^{t}\int_{0}^{t}\langle\bm{F}^{a}(t^{\prime})\cdot\bm{F}^{a}(t^{\prime\prime})\rangle\,\mathrm{d}t^{\prime}\mathrm{d}t^{\prime\prime},$
(19)

where the angle brackets $\langle\cdot\rangle$ denote the ensemble average.
Eqn (19) is the well-known Green-Kubo relation [60].

We gain intuition on the force autocorrelation function by plotting data from
simulations in Fig. 5a, showing the $x$ and $y$ components of the active force
as functions of time for different Péclet numbers. As expected, the net active
force fluctuates around zero, $\langle\bm{F}^{a}(t)\rangle=\bm{0}$. As the
Péclet number increases, time fluctuations become slower, indicating an
increase in the autocorrelation time scale. This is quantified in Fig. 5b,
where we plot the autocorrelation function $C_{F}(\Delta t)$ defined as

$C_{F}(\Delta t)=\frac{\langle\bm{F}^{a}(t)\cdot\bm{F}^{a}(t+\Delta
t)\rangle}{\langle|\bm{F}^{a}|^{2}\rangle},$ (20)

where the denominator $\langle|\bm{F}^{a}|^{2}\rangle$ is the active force
variance. Initially, the autocorrelation function decays exponentially with
time as $C_{F}(\Delta t)=\exp(-\Delta t/\xi)$, followed by a sharp drop at a
given value of $\Delta t$ that depends on $Pe$. Leveraging this exponential
decay, we can rewrite the mean-squared displacement of eqn (19) as

$\displaystyle\begin{split}\langle|\Delta\bm{R}(t)|^{2}\rangle&=2\mu^{2}\int_{0}^{t}(t-t^{\prime})\langle|\bm{F}^{a}|^{2}\rangle\exp(-t^{\prime}/\xi)\,\mathrm{d}t^{\prime},\\\
&=2(\xi\mu)^{2}\langle|\bm{F}^{a}|^{2}\rangle\left[\exp(-t/\xi)+(t/\xi)-1\right].\end{split}$
(21)

Eqn (15) then provides the long-time diffusivity $\mathcal{D}$ as

$\begin{split}\mathcal{D}&=\lim_{t\rightarrow\infty}\frac{(\xi\mu)^{2}\langle|\bm{F}^{a}|^{2}\rangle}{2}\left[\frac{-1}{\xi}\exp(-t/\xi)+\frac{1}{\xi}\right]\\\
&=\frac{\xi\mu^{2}\langle|\bm{F}^{a}|^{2}\rangle}{2}.\end{split}$ (22)

Calculation of $\mathcal{D}$ using this formula requires the determination of
the net active force autocorrelation time scale $\xi$ and variance
$\langle|\bm{F}^{a}|^{2}\rangle$. Next, we explain how these two quantities
can be estimated in some limits.

###  IV.2 Time-autocorrelation constant

The autocorrelation time scale $\xi$ is a function of $Pe$, $\rho$ and $\mu$,
and can be extracted numerically from the data of Fig. 5b. Its dependence on
Péclet number is illustrated in Fig. 5c($i$). Consistent with the observations
of panels a and b, we find that the correlation time increases with $Pe$. At
low Péclet numbers, that increase is nearly linear; indeed, at low $Pe$, the
typical duration of a collision scales with $Pe$ as we will explain in more
detail below. At high Péclet numbers, the correlation time plateaus and
asymptotes to a value slightly above 1; indeed, in that regime, most swimmers
slide off the colloid before the end of their run, so that the typical
duration of a collision is of order 1. The dependence of $\xi$ on density
$\rho$ and mobility ratio $\mu$ is shown in Fig. 5c($ii$): the correlation
time is found to be roughly independent of $\mu$ and shows a slight decrease
with $\rho$, albeit very weak.

A lower-bound estimate for the time constant $\xi$ can be derived analytically
as the average time $\langle\tau_{int}\rangle$ that a swimmer interacts with
the colloid over the course of one collision. This average time can be
calculated exactly in the limit of low mobility ratios ($\mu\to 0$) based on
the model of Saintillan [38]. When a collision occurs with incidence angle
$\alpha_{0}$, there are two possible outcomes: either the swimmer will finish
its current run on the colloid surface, or it will escape the surface by
sliding off tangentially. This outcome depends on the remaining time
$\tau_{r}$ in the run after collision. In dimensionless variables, the time to
escape tangentially, in the case of a fixed colloid ($\mu=0$), can be
calculated as

$t_{e}(\alpha_{0})=-\log\tan(\alpha_{0}/2).$ (23)

Escape will only occur if the remaining time $\tau_{r}$ exceeds the escape
time $t_{e}$, which defines a critical incidence angle for escape,
$\alpha_{c}(\tau)=2\tan^{-1}[\exp(-\tau_{r})]$ (see Saintillan [38] for more
details). Note that the collision time $\tau_{r}$ is a random variable over
the interval $[0,\tau]$; for the purpose of estimating the longest possible
interaction time during a run, we set $\tau_{r}=\tau$. The interaction time in
that case is

$\tau_{int}(\alpha_{0},\tau)=\begin{cases}\tau&\text{if
$0\leq\alpha_{0}<\alpha_{c}(\tau)$},\\\ t_{{e}}&\text{if
$\alpha_{c}(\tau)\leq\alpha_{0}\leq\pi/2$}.\\\ \end{cases}$ (24)

In this expression, both $\alpha_{0}$ and $\tau$ are random variables. We
first take an average over the incidence angle, which is uniformly distributed
over $[0,\pi/2]$ [38], yielding

$\langle\tau_{int}\rangle_{\alpha_{0}}(\tau)=\frac{2}{\pi}\tau\alpha_{c}(\tau)-\frac{2}{\pi}\int_{\alpha_{c}(\tau)}^{\pi/2}\log\tan(\alpha_{0}/2)\,\mathrm{d}\alpha_{0}.$
(25)

Next, we integrate over exponentially distributed run times, providing the
estimate for the time autocorrelation constant as:

$\xi=\langle\tau_{int}\rangle_{\alpha_{0},\tau}=\int_{0}^{\infty}\langle\tau_{int}\rangle_{\alpha_{0}}(\tau)\frac{\exp(-\tau/Pe)}{Pe}\,\mathrm{d}\tau.$
(26)

This estimate is plotted in Fig. 5c($i$), where it is found to underpredict
the correlation time at low $Pe$, but captures the correct asymptote at high
$Pe$. The departure at low $Pe$ can be rationalized as follows. When the
Péclet number is small, most runs involving a collision end on the surface of
the colloid; half of the subsequent runs then start with a collision, leading
to a longer interaction time than estimated here. On the other hand, at high
$Pe$, nearly all collisions result in an escape, which explains why the
estimate of eqn (26) matches the data well in that limit.

###  IV.3 Fluctuating active force variance

We now seek to estimate the force variance $\langle|\bm{F}^{a}|^{2}\rangle$,
where the net active force was defined in eqn (17). Here again, we focus on
the limit of low-mobility ratio, $\mu\to 0$. In that limit, the contact force
magnitudes in eqn (12) become decoupled, and are simply given by

$F_{i}^{c}(t)=\bm{p}_{i}(t)\cdot\bm{q}_{i}(t)=-\cos\alpha_{i}(t).$ (27)

The net instantaneous active force on the colloid resulting from swimmer
collisions is therefore

$\bm{F}^{a}(t)=-\sum_{i=1}^{c(t)}\cos\alpha_{i}(t)\bigg{(}\begin{matrix}\cos\beta_{i}(t)\\\
\sin\beta_{i}(t)\end{matrix}\bigg{)},$ (28)

and its variance can be obtained as

$\langle|\bm{F}^{a}|^{2}\rangle=\bigg{\langle}\sum_{i=1}^{c(t)}\sum_{j=1}^{c(t)}\Theta(\alpha_{i},\alpha_{j})\cdot\Xi(\beta_{i},\beta_{j})\bigg{\rangle},$
(29)

where $\Theta(\alpha_{i},\alpha_{j})=\cos\alpha_{i}(t)\cdot\cos\alpha_{j}(t)$
and
$\Xi(\beta_{i},\beta_{j})=\cos\beta_{i}(t)\cdot\cos\beta_{j}(t)+\sin\beta_{i}(t)\cdot\sin\beta_{j}(t)$.
The ensemble average is over all possible values of $c(t)$, $\alpha_{i}$ and
$\beta_{i}$. The angular positions $\beta_{i}$ of the swimmers on the colloid
surface are uniformly distributed between $0$ and $2\pi$, and are
uncorrelated. Averaging over $\beta_{i}$ and $\beta_{j}$ therefore gives

$\int_{0}^{2\pi}\int_{0}^{2\pi}\Xi(\beta_{i},\beta_{j})p(\beta_{i})p(\beta_{j})\,\mathrm{d}\beta_{i}\,\mathrm{d}\beta_{j}=\delta_{ij},$
(30)

so that the variance becomes

$\langle|\bm{F}^{a}|^{2}\rangle=\bigg{\langle}\sum_{i=1}^{c(t)}\sum_{j=1}^{c(t)}\Theta(\alpha_{i},\alpha_{j})\delta_{ij}\bigg{\rangle}=\bigg{\langle}\sum_{i=1}^{c(t)}\cos^{2}\alpha_{i}(t)\bigg{\rangle}.$
(31)

In the low-mobility-ratio limit, the dynamics of the swimmers on the surface
are uncorrelated, therefore the variance further simplifies to

$\langle|\bm{F}^{a}|^{2}\rangle=\langle c\rangle\langle\cos^{2}\alpha\rangle,$
(32)

where $\langle c\rangle$ is the average number of swimmers interacting with
the colloid at any given time, and $\langle\cos^{2}\alpha\rangle$ is the
average squared cosine of the contact angle. Next, we turn to the
determination of these two quantities.

####  IV.3.1 Surface swimmer density

The mean number of swimmers on the surface of the colloid can also be
estimated based on the prior work of Saintillan [38] for a fixed colloid. As
explained in that model, swimmer-colloid collisions are of two types:
collisions of type A involve a swimmer starting its run in the bulk and
encountering the colloid, while collisions of type B involve a swimmer
starting its run on the colloid surface after a tumble leading to a new
orientation pointing towards the colloid. For a collision of type B to occur,
the swimmer should had ended its previous run on the colloid surface. As shown
by Saintillan [38], the probabilities for any swimmer to undergo a collision
of type A or B during any given run can be calculated as

$P^{c}_{\\!\mathrm{A}}=1-\exp\bigg(-\frac{2Pe}{\pi}\phi\bigg{missing}),\quad
P^{c}_{\\!\mathrm{B}}=\bigg{(}\frac{1-P^{esc}_{\\!\mathrm{A}}}{1+P^{esc}_{\\!\mathrm{B}}}\bigg{)}P^{c}_{\\!\mathrm{A}},$
(33)

where $\phi=\pi/L^{2}$ is the area fraction of the colloid in the square
domain of size $L$, and $P^{esc}_{\\!\mathrm{A}}$ and
$P^{esc}_{\\!\mathrm{B}}$ are the probabilities for a collision of type A or B
to lead to an escape from the pillar surface. These escape probabilities can
be obtained by analysis of collision dynamics and were found to be

$P^{esc}_{\\!\mathrm{A}}=1+\frac{1}{Pe}\left(\bar{\alpha}-\frac{\pi}{2}\right),\quad
P^{esc}_{\\!\mathrm{B}}=1-\frac{2}{\pi}\bar{\alpha},$ (34)

with $\bar{\alpha}=2\tan^{-1}[\exp(-Pe)]$. Note that eqn (33) and (34) were
technically derived for constant run time [38], here taken equal to $Pe$; we
will show below that they yield a quantitative estimate for $\langle c\rangle$
despite this approximation.

Figure 6: Dependence of the mean number $\langle c\rangle$ of swimmers in
contact with the colloid with (a) Péclet number $Pe$, for $\rho=8.04$ and
$\mu=0.5$; and (b) density of the swimmers $\rho$ for $Pe=8.0$ and different
mobility ratios. In (a), the error bars show half a standard deviation around
the mean, and the inset illustrates temporal variations of $c(t)$ at $Pe=10$.
In both panels, the solid black curves show the theoretical prediction of eqn
(36) .

We can now estimate the probability $P^{s}$ that a swimmer will end its
current run on the surface of the colloid. For this to happen, the swimmer
must undergo a collision of either type A or B and finish its run on the
surface. Therefore,

$\displaystyle\begin{split}P^{s}&=P^{c}_{\\!\mathrm{A}}(1-P^{esc}_{\\!\mathrm{A}})+P^{c}_{\\!\mathrm{B}}(1-P^{esc}_{\\!\mathrm{B}})\\\
&=2\bigg{(}\frac{1-P^{esc}_{\\!\mathrm{A}}}{1+P^{esc}_{\\!\mathrm{B}}}\bigg{)}\left[1-\exp\bigg(-\frac{2Pe}{L^{2}}\bigg{missing})\right].\end{split}$
(35)

The mean number of swimmers in contact with the colloid can then be estimated
as $\langle c\rangle=\nu P^{s}$, where $\nu$ is the total number of swimmers
in the domain. In the limit of $L\gg 1$ (large domain size), we can expand the
exponential in a Taylor series, yielding the simple form

$\langle
c\rangle\approx\frac{\pi^{2}-4\pi\tan^{-1}[\exp(-Pe)]}{\pi-2\tan^{-1}[\exp(-Pe)]}\rho,$
(36)

where $\rho$ is the mean swimmer number density introduced in eqn (9). The
linear dependence on $\rho$ is unsurprising, since swimmers are non-
interacting in our model. At low Péclet number, the number of swimmers on the
surface grows linearly with $Pe$ as $\langle c\rangle\approx 4Pe\rho$: indeed,
in that regime, most swimmers incurring a collision will finish their run on
the colloid surface and therefore spend a time on the surface that scales
linearly with $Pe$. On the other hand, at high Péclet number, $\langle
c\rangle$ becomes independent of $Pe$ and asymptotes to $\pi\rho$: in this
regime, most swimmers colliding with the colloid will escape the colloid
surface before the end of their run, and therefore spend a time on the colloid
surface that is independent of $Pe$.

We test the prediction of eqn (36) against numerical data in Fig. 6. The
dependence on $Pe$ is shown in panel a for a density of $\rho=8.04$ and a
mobility ratio of $\mu=0.5$, and excellent agreement is found between the
theory and the numerical data, despite the approximations made (constant run
length, fixed colloid). In particular, the low- and high-Péclet-number
asymptotes match the data perfectly; the largest deviations are observed for
$Pe=O(1)$, where the agreement remains very good. The inset displays the
temporal variation of the number of swimmers in contact $c(t)$ with the
colloid for $Pe=10$ and highlights significant variations around the mean. The
variance of $c(t)$ is in fact found to be equal to the mean,
$\mathrm{Var}(c)=\langle c\rangle$, consistent with Poisson statistics. Panel
b shows the dependence of $\langle c\rangle$ with $\rho$ for $Pe=8.0$ and for
different mobility ratios. Here again, excellent match is obtained between
theory and data, independent of the value of $\mu$.

Figure 7: (a) Probability density functions of the squared cosine of the
contact angle $\alpha$ for different Péclet numbers $Pe$, obtained from
numerical simulations with $\rho=8.04$ and $\mu=0.01$. (b) Deviation of
$\langle\cos^{2}\alpha\rangle$ about 1/2 as a function of Péclet number for
different swimmer densities and $\mu=0.01$.

####  IV.3.2 Contact angle distribution

To complete the calculation of the force variance, we need an estimate for the
average squared cosine of the contact angle $\alpha$. The probability density
function for $\alpha$ is not easily obtained theoretically: indeed, it depends
on the distribution of the incidence angle $\alpha_{0}$ at the start of a
collision, as well as on the dynamics of the swimmer on the colloid surface
during contact and on any possible tumbles that may occur during the
collision. In particular, we expect a dependence on $Pe$. To estimate
$\langle\cos^{2}\alpha\rangle$, we instead turn to numerical data, and plot in
Fig. 7a the probability density function of $\cos^{2}\alpha$ for different
Péclet numbers. The probability density functions peak near $\cos^{2}\alpha=0$
and $1$, and are nearly symmetric about $1/2$, with a slight bias towards $1$
that is very weak and most visible at low values of $Pe$. As a result, we
expect $\langle\cos^{2}\alpha\rangle\approx 1/2$ over a wide range of $Pe$.
Departures from this value are quantified more precisely in Fig. 7b, where
they are most significant at low $Pe$ but never exceed 10% over the whole
range of Péclet numbers considered in this study.

Figure 8: Variance $\langle|\bm{F}^{a}|^{2}\rangle$ of the net active force on
the passive colloid as a function of swimmer number density $\rho$ for
different mobility ratios $\mu$ for $Pe=8$. The solid black line shows the
analytical expression of eqn (32).

####  IV.3.3 Force variance

Figure 8 shows the active force variance $\langle|\bm{F}^{a}|^{2}\rangle$ as a
function of swimmer number density $\rho$, for a Péclet number of $Pe=8$. The
plot compares numerical data for different values of the mobility ratio with
the derived theoretical estimate of $\langle
c\rangle\langle\cos^{2}\alpha\rangle$, which is technically valid in the limit
of $\mu\to 0$. At zero mobility ratio, there is a quantitative agreement
between data and theory. As the mobility ratio increases, the force variance
is found to decrease; the dependence on $\rho$ remains linear at low
densities, but significant departures are observed for moderate values of
$\mu$ at high densities. Since our estimates for both $\langle c\rangle$ and
$\langle\cos^{2}\alpha\rangle$ are valid independent of $\rho$ and $\mu$ (see
Sec. IV.3.1 and IV.3.2), this suggests that it is the derivation of eqn (32)
itself that breaks down. Indeed, when deriving eqn (32), we assumed $\mu=0$,
which entirely decouples the dynamics of the swimmers on the colloid surface
as can be seen from eqn (12). At moderate mobility ratios, correlations
between swimmers will affect the contact forces, and these correlations should
become all the more significant at high densities when many swimmers interact
with the colloid at any given time. In particular, the decrease in
$\langle|\bm{F}^{a}|^{2}\rangle$ seen in Fig. 8 suggests that correlations may
result in a partial cancelation of forces, which is not captured by our simple
theoretical model.

###  IV.4 Effective dispersion

Combining the results of eqn (22), (32) and (36), we obtain the following
theoretical prediction for the long-time diffusivity:

$\mathcal{D}\approx\bigg{(}\frac{\pi^{2}-4\pi\tan^{-1}[\exp(-Pe)]}{\pi-2\tan^{-1}[\exp(-Pe)]}\bigg{)}\frac{\xi(Pe)\mu^{2}\rho}{4}.$
(37)

The correlation time $\xi(Pe)$ can either be obtained numerically from
simulation data as in Fig. 5, or estimated theoretically using eqn (26), with
the caveat that this estimate is quantitative only at high Péclet number. In
that limit ($Pe\to\infty$), eqn (37) can be expanded to obtain the asymptotic
result,

$\mathcal{D}\approx 0.91\mu^{2}\rho,$ (38)

or, in dimensional variables,

$\mathcal{D}\approx 0.91\mu^{2}\rho uA^{3}.$ (39)

We compare the prediction of eqn (37) to simulation data in Fig. 9, showing
the variation of the effective diffusivity $\mathcal{D}$, scaled by $\mu^{2}$,
with Péclet number and swimmer density. The theoretical prediction in Fig. 9
uses the analytical estimate of eqn (26) for $\xi(Pe)$. In both plots,
rescaling $\mathcal{D}$ by $\mu^{2}$ collapses the data at low mobility ratios
in agreement with the model, with some departures observed for $\mu\gtrsim
0.25$, especially at large densities, consistent with the observations of
Figs. 4c and 8. Figure 9a shows the variation of $\mathcal{D}/\mu^{2}$ with
Péclet number. The theoretical model captures the data very well in the limit
of $Pe\geq 1$ and $\mu\longrightarrow 0$. At lower Péclet numbers, the theory
departs from the simulation results as expected, and could be improved by
deriving more accurate estimates for $\xi(Pe)$ and
$\langle\cos^{2}\alpha\rangle$. The dependence on swimmer density $\rho$ is
investigated in Fig. 9b, where the linear prediction of eqn (9) captures the
data quantitatively at low mobility ratios and low to moderate densities. As
expected based on the model assumptions, departures are observed at high $\mu$
and $\rho$ and can be traced back to departures in the force variance in Fig.
8.

Figure 9: Variation of the diffusivity $\mathcal{D}$, scaled by the square of
the mobility ratio $\mu^{2}$, with: (a) Péclet number $Pe$, for a fixed
swimmer density of $\rho=0.2$; and (b) swimmer density $\rho$ for $Pe=8$.
Symbols show results from stochastic simulations for different values of
$\mu$, while the solid black curves show the semi-analytical model of eqn
(37).

In summary, our theory agrees very well with particle simulations in the
regimes of low swimmer density, low mobility ratio, and high Péclet numbers,
and provides a simple analytical framework for the description of long-time
colloidal dispersion in baths of run-and-tumble microswimmers. It should be
noted that the parameter regime in which our model works best is
experimentally relevant. Using an experimental estimate of $\zeta\sim
10^{-8}-10^{-7}\,$Ns/m for the drag on a bacterium [61, 18], along with Stokes
drag $Z=1/M=6\pi\eta A$ for a colloid in water with radius $A=0.5-10\,\mu$m
[3, 62, 18, 4], we estimate the mobility ratio to be in the range of
$\mu\in[0.1,10]$ in typical experiments. Yet lower mobility ratios, reaching
down to $0.01$, occur in systems involving colloidal clusters [4]. Based on
experimental estimates of typical bacterial swim speeds ($u=5-25\,\mu$m/sec)
[61, 3, 18, 4] and mean run time ($\bar{\tau}\sim 1$ sec) [4], as well as
colloid sizes, we estimate the Péclet number to vary in the range of
$Pe\in[1,50]$. Finally, the typical 2D density of swimmer suspensions, based
on recent experiments near flat substrates [18, 4], is in the range of
$\rho\in[0.01,1]$.

We also note that our model, and in particular the predicted linear dependence
on $\rho$ and $u$ at high Péclet numbers, is consistent with various past
experiments and models. In early experiments using C. reinhardtii, Leptos  _et
al._ [21] experimentally measured the diffusivity of suspended tracers and
observed a clear linear dependence with number density. Subsequent experiments
performed by Miño  _et al._ [3] on the diffusion of passive particles near a
solid substrate in E. coli suspensions found a linear dependence on the active
flux $J=\rho_{v}u$, where $\rho_{v}$ is the volume density of bacteria. This
result again agrees with eqn (39), although they found only a weak dependence
on the size of the colloid, which they attributed to a hydrodynamic mechanism
for the diffusivity enhancement. A similar result was obtained Jepson  _et
al._ [12] for the diffusion of non-motile bacteria in a motile bath.
Hydrodynamic theories such as those of Lin  _et al._ [32] or Kasyap  _et al._
[25], which explain the diffusivity enhancement based on dipolar flow
disturbances induced by microswimmers, indeed predict a linear dependence on
the active flux $J$, but with a prefactor that is independent of $\mu$ and
depends on the fourth power of microswimmer size rather than colloid size.
Lagarde  _et al._ [18] performed both experiments and numerical simulations
based on steric interactions; they found a linear dependence of $\mathcal{D}$
on $u$, but did not characterize the dependence on swimmer density and
mobility ratio. More recently, Burkholder and Brady [27] derived a model for
the diffusivity of a passive colloid resulting from steric interactions with a
bath of active Brownian particles, and predicted a quadratic dependence on
$\mu$ as well as a linear dependence on $\rho u$ in the limit of persistent
swimmers. They also predicted a quadratic dependence on $u$ at low Péclet
number, which is also consistent with our data in Fig. 4b. To the best of our
knowledge, our model is the first to specifically address the case of athermal
run-and-tumble swimmers, and to provide a semi-analytical prediction for the
dependence of $\mathcal{D}$ on all system parameters for that particular
system.

##  V Conclusions

Using stochastic simulations and an asymptotic theory, we have analyzed the
dispersion of a passive colloid immersed in an unbounded suspension of non-
interacting active run-and-tumble particles in two dimensions. We found a
linear increase of the colloid diffusivity with swimmer density at low to
moderate densities, consistent with past experiments and models [21, 3, 12,
26, 27, 4], followed by a peak and slight decrease at very high densities, in
a regime that is not experimentally relevant. The variation of $\mathcal{D}$
with the persistence of the swimmer trajectories, as captured by the Péclet
number, shows a linear increase with $Pe$ that slows down and plateaus for
$Pe\gtrsim 1$; in dimensional variables, that corresponds to a diffusivity
that scales quadratically then linearly with the microswimmer speed $u$, once
again in agreement with past models [27]. The dependence of $\mathcal{D}$ on
the mobility ratio of the colloid to the swimmer was found to be quadratic at
low mobility ratios but to saturate for $\mu\gtrsim 1$ (small colloids).

Using a theoretical model based on the Green-Kubo formula, we related the
long-time colloid diffusivity to the variance of the net fluctuating force on
the colloid resulting from collisions with the swimmers. A semi-analytical
expression for the force variance was obtained under the assumptions of low
mobility ratio ($\mu\lesssim 0.25$), low swimmer density ($\rho\lesssim 1$),
and high Péclet number ($Pe\gtrsim 1$), a regime relevant to many experimental
systems. In that regime, the dependence of $\mathcal{D}$ on system parameters
is simple and given by eqn (39), and quantitative agreement is found between
theory and stochastic simulations.

The model introduced herein remains minimal and neglects many complexities
inherent to biological and synthetic active systems [63, 64, 53]. For
instance, several experiments demonstrate that biochemical signaling [65, 66,
67] and the large variability of run times of the E. Coli cells [68] can lead
to a non-Poissonian distribution for the run-and-tumble statistics. Such
distributions could be added to our current model with relative ease. Many
experiments involve a monolayer of colloids sedimented on a flat substrate in
a bacterial suspension. In these systems, hydrodynamic interactions with the
wall can have various effects including enhanced drag, bacterial accumulation
at the wall [50, 69], swimming in circles [47, 51, 70]. In some cases,
bacteria navigate either above or below the colloid [18], an effect that could
be captured by extending the model to three dimensions. Such extension to 3D
was recently discussed by Saintillan [38], and may allow further studies of
colloid sedimentation in active suspensions [71, 72], or of melting of
colloidal crystals by active baths [73]. Other effects that are of interest
are hydrodynamic interactions between the swimmers and colloid [32, 25], which
can significantly alter scattering dynamics [52], as well as the effect of
swimmer [55, 74] and colloid [6, 7, 75, 76] shape. Some of these effects are
straightforward extensions of the present model and may be addressed in future
work.

## Acknowledgements

The authors gratefully acknowledge discussions with Jérémie Palacci and Daniel
Grober, as well as funding from National Science Foundation Grant No.
CBET-1934199 (D.S.).

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