# Interplay between Brownian and hydrodynamic tracer diffusion in suspensions
of swimming microorganisms

Henrik Nordanger1  Alexander Morozov2  Joakim Stenhammar1
<EMAIL_ADDRESS>1Division of Physical Chemistry, Lund
University, Box 124, S-221 00 Lund, Sweden 2SUPA, School of Physics and
Astronomy, The University of Edinburgh, James Clerk Maxwell Building, Peter
Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom

###### Abstract

The general problem of tracer diffusion in non-equilibrium baths is important
in a wide range of systems, from the cellular level to geographical
lengthscales. In this paper, we revisit the archetypical example of such a
system: a collection of small passive particles immersed in a dilute
suspension of non-interacting dipolar microswimmers, representing bacteria or
algae. In particular, we consider the interplay between thermal (Brownian)
diffusion and hydrodynamic (active) diffusion due to the persistent advection
of tracers by microswimmer flow fields. Previously, it has been argued that
even a moderate amount of Brownian diffusion is sufficient to significantly
reduce the persistence time of tracer advection, leading to a significantly
reduced value of the effective active diffusion coefficient $D_{A}$ compared
to the non-Brownian case. Here, we show by large-scale simulations and kinetic
theory that this effect is in fact only practically relevant for microswimmers
that effectively remain stationary while still stirring up the surrounding
fluid, so-called _shakers_. In contrast, for moderate and high values of the
swimming speed $v_{s}$, relevant for biological microswimmer suspensions, the
effect of Brownian motion on $D_{A}$ is negligible, leading to the effects of
advection by microswimmers and Brownian motion being additive. This conclusion
contrasts with previous results from the literature, and encourages a
reinterpretation of recent experimental measurements of $D_{A}$ for tracer
particles of varying size in bacterial suspensions.

##  1 Introduction

Understanding the mass transport of colloidal and molecular species in non-
equilibrium environments is crucial for various processes, ranging from active
intracellular transport (Mogre et al., 2020) to the dispersion of nutrients in
world oceans (Katija, 2012). Apart from its practical importance, the
transport properties of tracer particles in generic “active baths” has
attracted much interest from a statistical physics perspective, where they can
be viewed as a minimal example of particles driven by external, non-
equilibrium noise (Argun et al., 2016; Park et al., 2020). Beyond the level of
tracer particles driven by generic non-equilibrium noise, the archetypical
example of a tracer particle in an active bath is a collection of point-like
tracers being advected by a set of microswimmers such as bacteria or algae
(Lauga & Powers, 2009). When swimming through a viscous fluid, these swimmers
create long-ranged flow fields that advect the tracers, leading to tracer
dynamics that is ballistic at short times and diffusive over timescales longer
than the autocorrelation time of the local flow field (Lin et al., 2011).
Realisations of this system have been extensively studied both experimentally,
typically in suspensions of _E. coli_ bacteria (Wu & Libchaber, 2000; Drescher
et al., 2011; Jepson et al., 2013; Kim & Breuer, 2004; Koumakis et al., 2013;
Miño et al., 2013, 2011; Patteson et al., 2016; Peng et al., 2016; Semeraro et
al., 2018) or _Chlamydomonas_ algae (Leptos et al., 2009; Ortlieb et al.,
2019; Yang et al., 2016; von Rüling et al., 2021), and theoretically, with
microswimmers typically being modelled either as force dipoles acting on the
surrounding fluid (Morozov & Marenduzzo, 2014; Pushkin et al., 2013; Pushkin &
Yeomans, 2013; Nordanger et al., 2022), as spherical “squirmers” with an
imposed slip velocity along their body (Lin et al., 2011; Thiffeault &
Childress, 2010; Thiffeault, 2015), or as needle-shaped “slender swimmers”
with imposed stresses along their body lengths (Krishnamurthy & Subramanian,
2015; Saintillan & Shelley, 2012). While the details of these three
microswimmer models differ, the results regarding enhanced tracer diffusion
are largely generic and consistent with experimental results, which have shown
the swimmer-induced, hydrodynamic diffusivity $D_{A}$ to scale linearly with
microswimmer density $n$ in the dilute limit where swimmer-swimmer
correlations can be neglected (Lin et al., 2011; Thiffeault & Childress, 2010;
Miño et al., 2013). In this limit, a fruitful way of calculating $D_{A}$ is to
consider the net displacement due to binary swimmer-tracer scattering events
(Pushkin & Yeomans, 2013; Morozov & Marenduzzo, 2014); two examples of
resulting (deterministic) tracer trajectories for scattering events are shown
in Fig. 1. For a non-tumbling swimmer, starting and ending at $x=\pm\infty$,
the resulting loop is closed, leading to a vanishing tracer net displacement
$\Delta$ (Fig. 1a). For tumbling swimmers with a finite persistence length,
the trajectory is however punctuated mid-way through the tracer loop, leading
to significantly larger values of $\Delta$ (Fig. 1b). The resulting value of
$D_{A}$ due to a large set of such scattering events can then be obtained by
explicitly summing over all possible sets of scattering parameters.

Figure 1: Tracer trajectories for infinite and finite swimmer paths. Panel (a)
shows a typical trajectory for a non-diffusing tracer advected by a non-
tumbling, point-dipole swimmer following an effectively infinite, straight
path, while (b) shows the corresponding trajectory terminated due to a
tumbling event. Note that, per Eq. (1), the effective tracer diffusion $D_{A}$
is independent of tumbling rate $\lambda$ for sufficiently high swimming
speeds when averaged over all possible swimmer-tracer configurations even
though the net displacement $\Delta$ is much larger for the tumbling swimmer.
Tracer trajectories were obtained through direct numerical integration using a
non-regularised dipolar flow field, as described by Morozov & Marenduzzo
(2014).

In spite of the dependence of $\Delta$ on the microswimmer tumbling rate
$\lambda$ for single scattering events such as that in Fig. 1, Pushkin &
Yeomans (2013) showed that, in the limit of large swimming speeds $v_{s}$,
$D_{A}$ is in fact _independent_ of $\lambda$ when summed over all possible
swimmer trajectories. This result was later generalised by Škultéty et al.
(2020) to arbitrary swimming speeds, leading to the following approximate
expression for $D_{A}$:

$D_{A}\approx\frac{7\kappa^{2}n}{2048\lambda\varepsilon+336\pi v_{s}}.$ (1)

Here, $\kappa$ is the magnitude of the microswimmer dipole (in units of volume
over time) and $\varepsilon$ is a characteristic linear size of the
microswimmer, which we take to be equal to the short-range regularisation
length of the dipolar flow field; unless stated otherwise, we will in the
following use $\kappa$ and $\varepsilon$ to non-dimensionalise the numerical
data.

Rather than using the scattering approach outlined above, Eq. (1) was derived
by formulating a kinetic theory for the spatiotemporal correlations of the
disturbance velocity field ${\bm{U}}$ created by the swimmers. Due to the
linearity of Stokes flow, we can write ${\bm{U}}$ as the superposition of the
individual swimmer flow fields ${\bm{u}}_{s}$:

${\bm{U}}({\bm{r}},t)=\sum_{i=1}^{N}{\bm{u}}_{s}({\bm{r}};{\bm{r}}_{i},{\bm{p}}_{i}),$
(2)

where ${\bm{r}}_{i}$ and ${\bm{p}}_{i}$ is, respectively, the position and
orientation of swimmer $i$. Knowing the statistical properties of ${\bm{U}}$,
$D_{A}$ can readily be calculated _via_ the Green-Kubo relation

$\displaystyle
D_{A}=\frac{1}{3}\int_{0}^{\infty}\langle\dot{{\bm{r}}}_{T}(t)\cdot\dot{{\bm{r}}}_{T}(0)\rangle
dt=\frac{1}{3}\int_{0}^{\infty}\langle{\bm{U}}({\bm{r}}_{T},t)\cdot{\bm{U}}({\bm{r}}_{T},0)\rangle
dt\equiv\frac{1}{3}\int_{0}^{\infty}C_{T}(t)dt,$ (3)

where, in the second equality, we have assumed point-like tracers advected by
the disturbance flow, so that $\dot{{\bm{r}}}_{T}={\bm{U}}({\bm{r}}_{T})$, and
the third equality defines the velocity autocorrelation function $C_{T}(t)$ in
the co-moving tracer frame. While yielding identical results for dilute
suspensions as the scattering approach discussed above, kinetic theories are
however more readily extended to accommodate the effect of swimmer-swimmer
correlations due to the mutual advection and reorientation of swimmers
(Škultéty et al., 2020). Importantly, these interactions break the symmetry
between rear-actuated (pusher) swimmers, such as most bacteria, and front-
actuated (puller) ones, such as _Chlamydomonas_ , leading to a super-linear
scaling of $D_{A}$ with $n$ for pushers and a corresponding sub-linear scaling
for pullers (Stenhammar et al., 2017).

Equation (1) shows two qualitatively different regimes for high and low
$v_{s}$: For $v_{s}\rightarrow 0$ – the so-called _shaker_ limit – the
dominant mechanism controlling the decorrelation of $C_{T}(t)$ is tumbling of
the swimmer. For fast swimmers, with $v_{s}\gg\lambda\varepsilon$, the
decorrelation of the fluid velocity is instead dominated by the swimmer self-
propulsion and thus independent of $\lambda$, and Eq. (1) reduces to the
expression derived by Pushkin & Yeomans (2013). A third, somewhat less
explored, mechanism affecting $C_{T}(t)$ is Brownian translational diffusion
of the tracer: even though Brownian diffusion does not affect the statistics
of the flow field ${\bm{U}}({\bm{r}},t)$ as measured in the lab frame, the
positional noise of the tracer particle will cause it to cross the streamlines
of the disturbance flow, thus perturbing its trajectory compared to the
athermal case shown in Fig. 1 and leading to a lower $D_{A}$. Since the
Brownian diffusion constant $D_{0}$ depends inversely on the tracer radius per
the Stokes-Einstein relation, the magnitude of this effect is expected to be
significant primarily for small tracer particles, and it has been hypothesised
to explain the non-monotonic size dependence of enhanced tracer diffusion seen
in experiments with colloids in _E. coli_ suspensions, where Patteson et al.
(2016) observed a maximum in $D_{A}$ for a tracer radius of approximately 5
$\mu$m. The effect of Brownian motion on enhanced tracer diffusion was
furthermore theoretically analysed by Kasyap et al. (2014) for a model of
slender-body swimmers, showing that $D_{A}$ is a non-monotonic function of
$D_{0}$, with $D_{A}$ first showing a small increase for intermediate $D_{0}$,
before falling below the athermal value as $D_{0}$ grows larger. In a more
recent study of finite-size spherical tracers in microswimmer suspensions,
Dyer & Ball (2021) numerically analysed the combined effect of thermal
fluctuations and near-field flows on the size-dependent tracer dynamics,
finding a similar non-monotonic behaviour as observed experimentally. In this
Paper, we will revisit the problem of the interplay between Brownian and
hydrodynamic diffusion for the simple case of point-like tracers immersed in a
dilute suspension of microswimmers described via a regularised dipolar flow
field. Using kinetic theory and large-scale lattice Boltzmann simulations of
_E. coli_ -like suspensions, we show that the effect of Brownian diffusion on
active diffusion is only practically relevant whenever
$v_{s}<\lambda\varepsilon$, which corresponds to extremely slow (or frequently
tumbling) swimmers. For biologically relevant values of $v_{s}$ and $\lambda$,
swimming is instead the dominant decorrelation mechanism, so that $D_{A}$
becomes independent of both $D_{0}$ and $\lambda$. In contrast to previous
studies, our results thus indicate that the effect of Brownian motion on the
enhanced diffusion is in fact negligible for most microswimmer realisations,
and thus that the experimentally observed non-monotonic size dependence of
$D_{A}$ on tracer size reported by Patteson et al. (2016) has other
explanations.

##  2 Model and Method

We consider a collection of $N$ non-interacting microswimmers at number
density $n=N/V$ moving through a three-dimensional viscous fluid of viscosity
$\mu$. Each microswimmer is composed of two equal and opposite point forces of
magnitude $F$ separated by a length $\ell$ and swims with a constant speed
$v_{s}$. The resulting reduced hydrodynamic dipole strength is
$\kappa=F\ell/\mu$. The swimming direction ${\bm{p}}_{i}$ furthermore relaxes
through Poisson-distributed random tumbles with uncorrelated directions
occurring with average frequency $\lambda$.

The position ${\bm{r}}_{T}$ of a point-like tracer obeys the equation of
motion

$\dot{{\bm{r}}}_{T}={\bm{U}}({\bm{r}}_{T})+\sqrt{2D_{0}}\bm{\eta},$ (4)

where $\bm{\eta}$ is a unit-variance white noise, $\delta$-correlated in time,
and $D_{0}$ is the Brownian diffusion constant. Thus, the effect of Brownian
motion is fully contained in the tracer dynamics, while we assume the effects
of thermal fluctuations on the fluid and on the pairwise swimmer-tracer
dynamics to be subdominant. The fluid disturbance velocity
${\bm{U}}({\bm{r}}_{T})$ due to the presence of all swimmers is given by Eq.
(2) and can, in principle, be explicitly summed up on each timestep. However,
to avoid the (prohibitively costly) pairwise summation over all swimmers and
tracers, we instead numerically solve for the flow field using an efficient
point-force implementation of the lattice Boltzmann (LB) method described
previously (Bárdfalvy et al., 2020; Nash et al., 2008). Our simulations
comprise a system with periodic boundaries and a size of $100^{3}$ lattice
units. In LB units, set by the LB lattice spacing $\Delta l$ and time step
$\Delta t$, the microswimmer density was kept constant at $n=0.01$ unless
otherwise stated, corresponding to $N=10^{4}$ microswimmers. Furthermore,
$N_{T}=5\times 10^{4}$ point tracers were included for statistical averaging.
In addition to the LB simulations, in Section 3 we will furthermore extend the
kinetic theory developed previously by Škultéty et al. (2020) to the case of
microswimmers undergoing Brownian diffusion and show that, in the limit of
non-interacting swimmers, the derived expression for $D_{A}$ is equivalent to
that for Brownian tracers in a suspension of non-Brownian microswimmers.

Inserting Eq. (4) into the Green-Kubo relation (3) yields

$D_{\mathrm{Tot}}=D_{0}+\frac{1}{3}\int_{0}^{\infty}C_{T}(t)dt=D_{0}+D_{A},$
(5)

where $C_{T}$ was defined in Eq. (3). Thus, to obtain $D_{A}$, we numerically
evaluate the time correlation of the disturbance velocity measured in the co-
moving tracer frame. Since the tracer position ${\bm{r}}_{T}$ depends on
$D_{0}$, $C_{T}(t)$, and thus $D_{A}$, will depend implicitly on $D_{0}$.

To characterise the system, we will use three dimensionless quantities.
Firstly, we define the Péclet number, which measures the relative importance
of active and thermal forces, as

$\mathrm{Pe}\equiv\frac{D_{A}(D_{0}=0)}{D_{0}},$ (6)

where $D_{A}(D_{0}=0)$ is the active diffusivity of a tracer immersed in an
equivalent microswimmer suspension but in the absence of Brownian tracer
motion. It should be noted that our definition of Pe is qualitatively
different from that of Kasyap et al. (2014), who, instead of $D_{A}(D_{0}=0)$,
use the swimming speed $v_{s}$ to characterise the active forces. We however
argue that the activity experienced by the tracers depend on the magnitude of
the velocity fields generated by the swimmers, and is thus dependent on
$\kappa$ and encoded in $D_{A}(D_{0}=0)$, while $v_{s}$ is instead a measure
of the swimmers’ self-propulsion. For experimental realisations of
microswimmers, $\kappa$ and $v_{s}$ are directly proportional to each other;
however, the specific relation between between them will nevertheless be
specific to each type (or species) of swimmer, and decoupling them
conveniently enables us to study separately the effects of self-propulsion and
fluid forcing, as we demonstrate further below.

Secondly, we measure the change in active diffusion due to Brownian motion
through the quantity

$\xi\equiv\frac{D_{A}(D_{0})}{D_{A}(D_{0}=0)}.$ (7)

In the limit $\mathrm{Pe}\rightarrow\infty$, where Brownian motion becomes
negligible, we thus expect that $\xi\rightarrow 1$ as $D_{A}$ approaches its
non-Brownian value. Finally, in accordance with Škultéty et al. (2020), we
account for the effect of microswimmer self-propulsion using the reduced
swimmer persistence length $L$, defined by

$L\equiv\frac{v_{s}}{\varepsilon\lambda}.$ (8)

##  3 Kinetic theory

In this Section, we will outline the main steps in the derivation of $D_{A}$
for a suspension of Brownian tracer particles immersed in a dilute
microswimmer suspension, whose dynamics are governed by Eq. (4). Just as in
our previous works (Škultéty et al., 2020; Stenhammar et al., 2017) , we
describe the flow field measured at ${\bm{r}}$ due to a swimmer with position
${\bm{r}}_{i}$ and orientation ${\bm{p}}_{i}$ by a regularised dipolar flow
field ${\bm{u}}_{s}({\bm{r}})$

${\bm{u}}_{s}({\bm{r}};{\bm{r}}_{i},{\bm{p}}_{i})=\frac{\kappa}{8\pi}\left[3\frac{({\bm{p}}_{i}\cdot{\bm{r}}^{\prime})^{2}{\bm{r}}^{\prime}+\varepsilon^{2}({\bm{p}}_{i}\cdot{\bm{r}}^{\prime}){\bm{p}}_{i}}{(r^{\prime
2}+\varepsilon^{2})^{5/2}}-\frac{{\bm{r}}^{\prime}}{(r^{\prime
2}+\varepsilon^{2})^{3/2}}\right],$ (9)

where ${\bm{r}}^{\prime}={\bm{r}}-{\bm{r}}_{i}$,
$r^{\prime}=|{\bm{r}}^{\prime}|$, and $\varepsilon$ is the regularisation
length. Our starting point is the derivation of Škultéty et al. (2020), where
we formulated and solved a kinetic theory describing the fluctuations of the
velocity field ${\bm{U}}({\bm{r}},t)$ due to a superposition of single-swimmer
flow fields. In the limit of non-interacting swimmers, which is the case that
we consider here, the temporal correlations of the steady-state velocity field
${\bm{U}}$ measured in the lab frame,
$C_{U}(t)\equiv\langle{\bm{U}}({\bm{r}},t)\cdot{\bm{U}}({\bm{r}},0)\rangle$,
is given by

$C_{U}(t)=\frac{\kappa^{2}n}{15\pi^{2}\varepsilon}\int_{0}^{\infty}A^{2}(\zeta)e^{-\tau}\mathcal{F}(L\zeta\tau)d\zeta,$
(10)

where

$A(x)=\frac{1}{2}x^{2}K_{2}(x),$ (11)

with $K_{2}$ being the modified Bessel function of the second kind, and

$\mathcal{F}(x)=15\frac{(5x^{2}-12)\sin x-x(x^{2}-12)\cos x}{x^{5}},$ (12)

defined such that $\mathcal{F}(0)=1$. We furthermore used the dimensionless
variables $L=v_{s}/(\lambda\varepsilon)$, $\tau=t\lambda$, and
$\zeta=k\varepsilon$, where $k=|{\bm{k}}|$ is the wavevector magnitude.
Equation (10) can equivalently be expressed in closed form in terms of
elliptic integrals, as given by Eq. (72) of Škultéty et al. (2020). Equation
(10) contains decorrelation of the flow field due to two separate mechanisms:
exponential decay of $C_{U}(t)$ due to tumbling, and a more complex,
oscillatory behaviour due to swimming, encoded in the function $\mathcal{F}$.

To obtain the hydrodynamic diffusivity of a passive tracer, Škultéty et al.
(2020) used a _stationary tracer_ approximation, implying that the tracer
advection by the swimmer flow field is negligible compared to the self-
propulsion of the swimmer. This implies that ${\bm{r}}_{T}$ remains
effectively constant over the time it takes for ${\bm{U}}$ to relax, such that
$C_{T}(t)=\langle{\bm{U}}({\bm{r}}_{T}[t],t)\cdot{\bm{U}}({\bm{r}}_{T}[t=0],0)\rangle\approx\langle{\bm{U}}({\bm{r}}_{T}[t=0],t)\cdot{\bm{U}}({\bm{r}}_{T}[t=0],0)\rangle=C_{U}(t)$,
where, in the last equality, we have made the additional assumption that the
tracers are homogeneously distributed in space so that they sample an
unweighted spatial average of the flow field. Thus, if we can replace the
correlation function $C_{T}$ in the co-moving tracer frame with that in the
stationary lab frame, $C_{U}$, we can insert Eq. (10) into the Green-Kubo
relation (3) and integrate over time to yield the following expression for
$D_{A}$, identical to Eq. (85) of Škultéty et al. (2020):

$D_{A}=\frac{\kappa^{2}n}{45\pi^{2}\lambda\varepsilon}\int_{0}^{\infty}A^{2}(\zeta)\mathcal{G}(L\zeta)d\zeta,$
(13)

where

$\mathcal{G}(x)=\frac{5}{2}\frac{3x+2x^{3}-3(1+x^{2})\arctan x}{x^{5}},$ (14)

defined such that $\mathcal{G}(0)=1$. By matching the asymptotic behaviours
for $L\rightarrow 0$ and $L\rightarrow\infty$, Equation (13) can furthermore
be approximated by the simple expression given in Eq. (1)

The generalisation of Eq. (13) to the case of Brownian tracers might seem
straightforward, but unfortunately is not: Since the approximation
$C_{T}(t)=C_{U}(t)$ amounts to the tracer remaining effectively stationary
during a swimmer-tracer scattering event, this approximation will, by
construction, not capture any effects on $D_{A}$ coming from Brownian
diffusion across streamlines. This can easily be realised by noticing that
$C_{U}(t)$ is solely a property of the swimmer suspension, and will be
strictly unaffected by the tracer dynamics; thus, $D_{A}$ in Eq. (13) remains
unaffected by the inclusion of tracer diffusion. Instead of going beyond the
stationary tracer approximation, we circumvent this problem by noticing that,
in a suspension of non-interacting microswimmers, the dynamics of a Brownian
tracer will be _statistically identical_ to that of a non-Brownian tracer in a
suspension of Brownian _swimmers_ with the same translational diffusivity
$D_{0}$. This is because the single-swimmer flow field in Eq. (9) solely
depends on the separation vector ${\bm{r}}_{T}-{\bm{r}}_{i}$, implying that
the flow field experienced by a diffusing tracer (_i.e._ , noise acting on
${\bm{r}}_{T}$) is identical to that experienced by a non-Brownian tracer
sampling the flow field from a swimmer with the same noise instead applied to
${\bm{r}}_{i}$. In a non-interacting microswimmer suspension this equivalence
is exact as long as the noise has zero mean and identical spectral properties,
and we verify it numerically in Fig. 3. It however breaks down as soon as
swimmer-swimmer correlations become significant since swimmer diffusion will
then affect the magnitude of such correlations, which tracer diffusion will
not.

Thus, as outlined in Appendix A, we instead calculate $C_{U}(t)$ for the case
of a suspension of diffusing swimmers, described by the dynamics

$\dot{{\bm{r}}}_{i}=v_{s}{\bm{p}}_{i}+\sqrt{2D_{0}}\bm{\eta},$ (15)

including the same tumbling dynamics as before. This yields the following
generalisation of Eq. (10):

$C_{U}(t;D_{0})=\frac{\kappa^{2}n}{15\pi^{2}\varepsilon}\int_{0}^{\infty}A^{2}(\zeta)e^{-(1+\tilde{D}\zeta^{2})\tau}\mathcal{F}(L\zeta\tau)d\zeta,$
(16)

where we have additionally defined the non-dimensional diffusivity
$\tilde{D}=D_{0}/(\lambda\varepsilon^{2})$. Since the effect of Brownian
diffusion is now fully incorporated into the properties of $C_{U}$, we again
use the stationary-tracer approximation and insert this expression into the
Green-Kubo relation (3), leading to the following expression for $D_{A}$ in
the presence of Brownian diffusion:

$D_{A}(D_{0})=\frac{\kappa^{2}n}{45\pi^{2}\lambda\varepsilon}\int_{0}^{\infty}\frac{A^{2}(\zeta)}{1+\tilde{D}\zeta^{2}}\mathcal{G}\left(\frac{L\zeta}{1+\tilde{D}\zeta^{2}}\right)d\zeta.$
(17)

In Section 4, we numerically evaluate Eqs. (16) and (17) and compare the
results with direct numerical simulations of microswimmer suspensions.

##  4 Results and Discussion

Figure 2: Effective tracer diffusion in the absence of Brownian motion. Panel
(a) shows the $\lambda^{-1}$-dependence of $D_{A}(D_{0}=0)$ for shakers with
$v_{s}=0$, and panel (b) its $v_{s}$-dependence at constant $\lambda=10^{-4}$.
Symbols denote simulation results and solid lines show results from Eq. (13)
using $\varepsilon$ Newas a fitting parameter. Error bars represent one
standard deviation as obtained from averaging over four separate runs with
different initial conditions. The results are nondimensionalised in terms of
$\kappa$ and $\varepsilon$.

In Fig. 2 we begin by verifying Eq. (13) for the hydrodynamic diffusion
coefficient $D_{A}$ in the limit $D_{0}=0$. In Fig. 2a, we demonstrate the
$\lambda^{-1}$ dependence of $D_{A}$ in the shaker limit $v_{s}=0$, while Fig.
2b shows its more complex dependence on $v_{s}$ for constant $\lambda$. Apart
from numerically verifying the kinetic theory expression (13), these results
illustrate how $D_{A}$ decreases abruptly due to the temporal decorrelation of
the flow field ${\bm{U}}({\bm{r}}_{T},t)$ induced respectively by tumbling and
swimmer self-propulsion. The slight deviation between the theoretical curve
and simulation results at small $v_{s}$ in Fig. 2b is likely due to the
specific form for the short-range regularisation, which becomes important as
$v_{s}\rightarrow 0$. In our derivation of Eq. (13) we use the regularised
flow field (9), based on the regularisation first introduced by Cortez et al.
(2005). In the LB simulations we instead use a numerical interpolation scheme
based on a regularisation of the $\delta$ function (Peskin, 2002) acting
separately on the two point forces that make up each microswimmer. Unlike the
expression in (9), this numerical regularisation does not allow a direct
mapping (or adjustment) of the regularisation length $\varepsilon$. We thus do
not expect perfect agreement between kinetic theory and simulation in the
low-$v_{s}$ regime where the short-range regularisation becomes important, and
therefore treat $\varepsilon$ as a fitting parameter when comparing data from
LB simulations with kinetic theory predictions. However, we find that the
fitted value of $\varepsilon$ only varies slightly ($\varepsilon\in[1.9\Delta
l,2.5\Delta l]$) for the values of $v_{s}$ used throughout this work, in good
accordance with the fact that the regularised $\delta$ function is
interpolated over a support of $2\Delta l$ in each Cartesian direction; for a
more in-depth discussion of the effect of the interpolation scheme on the
tracer dynamics, see de Graaf & Stenhammar (2017).

Figure 3: Brownian motion suppresses active diffusion for slow swimming
speeds. Panel (a) shows values of $D_{A}$ measured from LB simulations
(diamonds) and calculated from Eq. (17) (solid lines), both expressed in LB
units. Panel (b) shows the same data but expressed in the dimensionless
quantities $\xi$ and Pe. For very slow swimmers with $L\lesssim 1$, $D_{A}$ is
reduced compared to the non-Brownian value ($\xi=1$) whenever $\mathrm{Pe}<1$,
while for faster swimmers, significantly lower values of Pe are necessary to
affect $D_{A}$. The circles for $L=0$ and 2.4 correspond to the hydrodynamic
diffusion of non-Brownian tracers measured in a suspension of Brownian
_swimmers_ of the same $D_{0}$, verifying the statistical equivalence between
tracer and swimmer diffusion in the non-interacting limit. Error bars
represent one standard deviation as obtained from averaging over four separate
runs with different initial conditions. The results in panel (a) are
nondimensionalised using $\kappa$ and $\varepsilon$.

In Fig. 3, we study the additional effect of varying the Brownian diffusion
coefficient $D_{0}$, as encoded in Eq. (17). From the data in Fig. 3a, it is
clear that, for large enough $D_{0}$, the active diffusivity $D_{A}$ decreases
compared to its non-Brownian value. To enable an easier analysis of the effect
of varying swimming speed, in panel (b) we present the same data instead
plotted as a function of the reduced variables $\xi$ and Pe. For shakers with
$L=0$ (blue curve in Fig. 3b), $D_{A}$ is reduced compared to its non-Brownian
value ($\xi<1$) as soon as $\mathrm{Pe}<1$, reaching a value as low as
$\xi=0.2$ for $\mathrm{Pe}\approx 10^{-3}$. For finite values of $v_{s}$, this
effect on $D_{A}$ however occurs for gradually lower values of Pe; for the
fastest swimmers considered here, with $L=4.0$, no significant reduction of
$D_{A}$ is observed even for Pe as low as $10^{-3}$. Instead, we observe a
small but significant _increase_ in the active diffusion compared to its non-
Brownian value, in accordance with what was previously observed for slender
swimmers by Kasyap et al. (2014); we discuss this effect further below.
Crucially, a reduced persistence length $L=4$ nevertheless corresponds to
relatively slow swimming from a biological perspective: According to the
approximate calculation in Škultéty et al. (2020), $L$ for _E. coli_ bacteria
lies somewhere in the range between 5 and 20, indicating that the effect of
Brownian motion on active diffusion is likely negligible in suspensions of
swimming bacteria due to their fast self-propulsion. In our LB simulations,
studying values higher than $L\approx 4$ is challenging, as these large
swimming speeds both require very large systems to avoid significant finite-
size effects and yields artifacts due to the effect of finite Reynolds number
in the swimmer-tracer scattering dynamics (de Graaf & Stenhammar, 2017). We
nevertheless numerically studied $\xi(\mathrm{Pe})$ using Eq. (17) for larger
values of $L$, verifying that both the peak and the subsequent decrease in
$\xi$ continues to move to even lower values of Pe as $L$ is increased.

To further understand the mechanism behind the reduction in $D_{A}$ with
$D_{0}$, we consider the two autocorrelation functions $C_{T}(t)$ and
$C_{U}(t)$, which respectively measure the fluid autocorrelation in the co-
moving tracer frame and in the lab frame. Figure 4 shows these correlation
functions for $L=0$ and $L=2.4$, with the top row corresponding to LB results
for $C_{T}$ and the bottom row to kinetic theory results from Eq. (16) for
$C_{U}$. First, we notice that the two sets of curves are very similar,
implying that the stationary tracer approximation $C_{T}\approx C_{U}$ is
indeed accurate. Secondly, we notice that the decay of the correlation
function is significantly faster for swimmers than for shakers, again
illustrating that self-propulsion acts an efficient decorrelation mechanism
for ${\bm{U}}$. The effect of finite $D_{0}$ for shakers (left column) is
simply to decrease the relaxation time of the exponential decay, in accordance
with the $L=0$ limit of Eq. (16). For swimmers, the situation is more complex:
For short times, the flow field decays faster with decreasing Pe, while the
long-time tail of $C_{T}$ and $C_{U}$ instead becomes somewhat more extended
with decreasing Pe. For fast enough swimmers, the latter effect leads to the
local maximum at $\xi>1$ for intermediate Pe observed in Fig. 3b for the two
highest $L$. Finally, we note that the equal-time fluid velocity variance
$\langle U^{2}({\bm{r}}_{T})\rangle$, corresponding to the $t=0$ values of
$C_{T}$ and $C_{U}$, is independent of Pe. This means that, regardless of the
ratio between diffusive and active motion, the tracer particles sample the
overall flow field homogeneously. This fact is non-trivial, since dry active
particles that move autonomously on a solid substrate are known to
preferentially sample regions where they move slowly (Stenhammar et al.,
2016). Our results thus highlight that this generic mechanism is absent for
inertialess point tracers advected by an incompressible fluid. However, for
microswimmer systems where entrainment due to tracers being captured by the
near-field flows of passing swimmers (Jeanneret et al., 2016), we would expect
the fluid flow sampled by tracers to be significantly different from the
average flow field in the system.

Figure 4: Brownian motion decorrelates tracer trajectories. Panels (a) and (b)
show the fluid velocity autocorrelation $C_{T}(t)$ in the co-moving tracer
frame measured from LB simulations for (a) shakers ($L=0$) and (b) swimmers
with $L=2.4$ at indicated values of Pe. The dotted line shows the correlation
function $C_{U}(t)$ of the fluid velocity in the lab frame, demonstrating that
the stationary-tracer approximation $C_{T}(t)\approx C_{U}(t)$ is excellent in
the absence of Brownian tracer diffusion (Pe $\rightarrow\infty$). Panels (c)
and (d) show the corresponding lab-frame correlation function $C_{U}(t)$,
obtained from kinetic theory (Eq. (17)) for a suspension of diffusing
swimmers, as described in Section 3. $C_{T}$ and $C_{U}$ are
nondimensionalised using $\kappa$ and $\varepsilon$.

In Fig. 5, we study the dependence of the suppression of active diffusion on
the microswimmer density $n$ in the shaker limit $L=0$. At first sight, this
dependence might appear trivial, since $D_{A}$ is well-known to be linearly
dependent on $n$ (Jepson et al., 2013; Miño et al., 2013) in the limit of non-
interacting swimmers, a fact which is unaffected by Brownian motion as shown
by Eq. (17). Since $\xi$ measures the ratio between the Brownian and non-
Brownian values of $D_{A}$, one would naïvely expect $\xi$ to be independent
of $n$. However, since the Péclet number itself, as defined in Eq. (6),
increases with $n$ for constant $D_{0}$, the relative effect of Brownian
motion on $D_{A}$ is in fact a complex function of $n$ even for noninteracting
swimmers, as shown in Fig. 5. More specifically, according to Fig. 5b the
suppression of active diffusion becomes more significant with increasing
microswimmer density. In physical units, the highest concentration considered
($n\varepsilon^{3}\approx 0.08$) approximately corresponds to a bacterial
concentration of $10^{9}$ mL-1, which is somewhat higher than the highest _E.
coli_ concentration considered by Jepson et al. (2013) but still within the
range of concentrations where swimmer-swimmer correlations are reasonably
small (Stenhammar et al., 2017). In summary, our results thus show that, for
Brownian diffusion to have any measurable influence on the hydrodynamic
diffusion, it is necessary to create a system with a relatively high density
of very slow microswimmers; as we discuss in Section 5, this set of parameters
is likely not achievable for suspensions of biological microswimmers.

##  5 Summary and conclusions

In this study we have demonstrated a number of theoretical and computational
results regarding the effect of Brownian diffusion on the swimmer-induced
hydrodynamic diffusion of tracer particles in a suspension of dipolar
microswimmers. Our key finding is that the effect of Brownian diffusivity
$D_{0}$ on the activity-induced, hydrodynamic diffusivity $D_{A}$ is only
significant when the Péclet number as defined by Eq. (6) is below unity,
meaning that Brownian diffusivity needs to dominate over the hydrodynamic one.
However, the necessary requirement $\mathrm{Pe}<1$ is only sufficient in the
shaker limit $v_{s}\rightarrow 0$: for swimmers with persistence lenghts
larger than the organism size, significantly lower values of Pe are required
to perturb the tracer trajectories sufficiently to affect $D_{A}$. This
conclusion is analogous to the independence of $D_{A}$ on the tumbling rate
$\lambda$ for large $v_{s}$ illustrated in Eq. (1): Whenever $v_{s}$ is large,
the decorrelation of ${\bm{U}}$ by swimming will dominate over the
decorrelation due to tumbling and translational diffusion, and the dependence
on $\lambda$ and $D_{0}$ will thus vanish in the limit
$v_{s}\rightarrow\infty$. While this effect is expected, what is perhaps
surprising is the rather moderate values of $L=v_{s}/(\varepsilon\lambda)$
necessary to render the coupling between $D_{0}$ and $D_{A}$ negligible, as
illustrated in Fig. 3b. To put these values into perspective, we use the
conservative estimate $L=5$ for _E. coli_. By virtue of Fig. 3b, we require
that $\mathrm{Pe}\leq 10^{-3}$ for Brownian motion to have an effect of $\sim
5$ percent on $D_{A}$. Using as an example the minimum value
$D_{A}\approx${10}^{-2}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{2}\text{\,}{\mathrm{s}}^{-1}$$ measured by Jepson et al. (2013) in a
3-dimensional _E. coli_ suspension, this requirement thus implies that
$D_{0}\geq$10\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{2}\text{\,}{\mathrm{s}}^{-1}$$, which by virtue of the Stokes-Einstein
relation corresponds to a tracer radius of $R_{0}\sim 20$ nm. While this is
significantly smaller than used in typical measurements on colloidal tracers
(Miño et al., 2013; Patteson et al., 2016; Leptos et al., 2009), this
diffusion coefficient is close to the value of $D_{0}$ measured for dextran in
_E. coli_ suspensions by Kim & Breuer (2004). It is also fully feasible to
realise such low Péclet numbers for micron-sized spheres by instead decreasing
the bacterial density to very low values; however, measuring the correction to
$D_{A}$ for $\mathrm{Pe}\sim 10^{-3}$ represents a major difficulty, since it
amounts to measuring a $\sim$ 5 percent deviation of an effective diffusivity
which is itself a thousand times smaller than the Brownian diffusion. It would
thus require an extremely accurate determination of $D_{0}$, which then needs
to be subtracted from the total measured diffusion constant to determine
$D_{A}$. Obtaining this accuracy in a colloidal suspension would be very
challenging due to particle polydispersity, interactions with boundaries,
temperature gradients, and other system-specific complications. We thus
conclude that, for typical $L$ values relevant for biological microswimmers,
our results imply that the effect of Brownian motion on $D_{A}$ is likely
negligible for all practical purposes. To experimentally observe the reduction
in $D_{A}$, one would instead need to study a system of dipolar shakers, which
stir up the surrounding fluid without self-propelling. While this is a
somewhat exotic type of system, it could potentially be realised by anchoring
molecular motors or biological microswimmers to a surface. In a biological
setting, the shaker limit furthermore resembles previously developed models of
enzymes anchored to lipid bilayers that induce dipolar flows through cyclical
conformation changes (Hosaka et al., 2020).

Figure 5: Reduction of $D_{A}$ varies with microswimmer density. Panel (a)
shows $\xi$ as a function of Pe for shakers ($L=0$) at various densities $n$,
as indicated, while panel (b) shows $\xi$ as a function of $n$, at a fixed
value of $\mathrm{Pe}=0.013$. Simulation data are given by the symbols, with
error bars obtained from averaging over four separate runs with different
initial conditions, while solid lines are computed from Eq. (17).

Our results are qualitatively consistent with the previous theoretical results
obtained by Kasyap et al. (2014), including the non-monotonic behaviour of
$D_{A}$ with respect to $D_{0}$ at high swimming speed. Their results however
differ in three important ways. First of all, their results consider a
“slender swimmer” model, where the fluid is forced via a stress applied along
a line representing the swimmer body, rather than by two point forces. While
this model should lead to a dipolar flow in the far field, the near-field
differences are significant, as illustrated in Fig. 8 of Kasyap et al. (2014).
Furthermore, they consider only the fast-swimming limit where, according to
Eq. (1), $D_{A}$ for dipolar microswimmers is independent of $\lambda$.
Finally, they parametrise their model in a qualitatively different way than
us: as discussed above, they adopt a definition of $\mathrm{Pe}$ based on
$v_{s}$ rather than on the tracer diffusivity as in Eq. (6). In this
description, $v_{s}$ is furthermore directly coupled to $\kappa$, so that
changing Pe simultaneously changes the activity of the bath (via $\kappa$) and
the swimming speed $v_{s}$. These differences makes it difficult to compare
directly with our results, as we consider the effects of fluid advection and
self-propulsion separately via Pe and $L$.

Thus, even though Brownian motion is unlikely to provide a significant
dependence of $D_{A}$ on $R_{0}$ for tracers immersed in suspensions of
biological microswimmers, there are several other mechanisms that need to be
studied to explain the non-monotonic dependence observed experimentally
(Patteson et al., 2016) and computationally (Dyer & Ball, 2021). First of all,
the effect of tracer entrainment by the near field flows of the swimmer is
strongly dependent on the size ratio between the swimmer and the tracer
(Jeanneret et al., 2016), although we expect this term to be small for micron-
sized tracer particles in _E. coli_ suspensions. Secondly, the finite size of
the tracer will change the equation of motion (4) into the Faxén equation that
takes into account the non-linearity of the flow field (Kim & Karrila, 1991),
an effect which was implicitly included in the wavelet Monte Carlo simulations
by Dyer & Ball (2021) and, together with tracer entrainment, is a significant
explanation of their observed $R_{0}$ dependence of $D_{A}$. Finally, non-
hydrodynamic interactions such as direct collisions, electrostatic
interactions, and artifacts due to container walls are likely to depend in a
non-trivial manner on the tracer size for each system in question. We thus
conclude by noting that significant further experimental and theoretical work
is necessary to disentangle the system-specific properties from the generic
properties of tracer dynamics in microswimmer suspensions.

## Acknowledgement

Discussions with Cesare Nardini are kindly acknowledged. The computations were
enabled by resources provided by LUNARC. For the purpose of open access, the
authors have applied a Creative Commons Attribution (CC BY) licence to any
Author Accepted Manuscript version arising from this submission.

## Funding

This work was financed through the Knut and Alice Wallenberg Foundation
(project grant KAW 2014.0052). JS acknowledges financial support from the
Swedish Research Council (Project No. 2019-03718).

## Competing Interests

The authors report no competing interests.

##  Appendix A Kinetic theory for $D_{A}(D_{0})$.

In this Appendix, we demonstrate the main steps in the derivation of Eq. (16).
The analysis follows closely a similar derivation presented in Škultéty et al.
(2020), albeit with two major differences. First, in addition to that work, we
include the effect of microswimmer Brownian diffusivity, as discussed in
Section 3. Second, we consider the case of non-interacting microswimmers,
which significantly simplifies the analysis. Due to the similarity with the
derivation in Škultéty et al. (2020), we here present the key steps of the
derivation and refer the interested reader to that paper for technical
details.

The quantity of interest is the fluid velocity autocorrelation function
$C_{U}(t)$, formally defined as

$\displaystyle C_{U}(t)=\lim_{t^{\prime}\rightarrow\infty}\frac{1}{V}\int
d{\bm{r}}\,\overline{U^{\alpha}\left({\bm{r}},t^{\prime}\right)U^{\alpha}\left({\bm{r}},t+t^{\prime}\right)},$
(18)

where $V$ is the volume of the system, and the overbar denotes an average over
the stochastic history, _i.e._ , the history of tumble events, and the long-
time limit ensures independence of the initial state of the system. Here and
in the following, the superscript indices denote Cartesian components of
vectors. For a given state of the system, the instantaneous fluid velocity
$U^{\alpha}$ at a position ${\bm{r}}$ and time $t$ is readily obtained as a
superposition of individual velocity fields generated by the microswimmers

$\displaystyle
U^{\alpha}\left({\bm{r}},t\right)=\sum_{i=1}^{N}u_{s}^{\alpha}({{\bm{r}}};{{\bm{r}}}_{i}(t),{{\bm{p}}}_{i}(t)).$
(19)

Here, ${\bm{r}}_{i}(t)$ gives the instantaneous position of particle $i$,
while the unit vector ${\bm{p}}_{i}(t)$ gives its instantaneous orientation;
the index $i=1\dots N$ enumerates the particles, where $N$ is the total number
of microswimmers. In the following, we assume ${\bm{u}}_{s}$ to be given by
the regularised hydrodynamic dipole (9).

Time evolution of the suspension comprises spatial motion of the microswimmers
according to Eq. (15) and their random re-orientation with rate $\lambda$, as
discussed in Section 1. These dynamics can be alternatively described by the
master equation

$\displaystyle\partial_{t}F_{N}+\sum_{i=1}^{N}\Big{(}v_{s}p_{i}^{\alpha}\nabla_{i}^{\alpha}-D_{0}\nabla_{i}^{2}\Big{)}F_{N}=-N\lambda
F_{N}+\frac{\lambda}{4\pi}\sum_{i=1}^{N}\int\mathrm{d}{\bm{p}}_{i}F_{N},$ (20)

where
$F_{N}=F_{N}({{\bm{r}}}_{1},\dots,{{\bm{r}}}_{N},{{\bm{p}}}_{1},\dots,{{\bm{p}}}_{N},t)$
is the $N$-particle probability distribution function. Here,
${\bm{\nabla}}_{i}$ denotes spatial derivatives with respect to the
coordinates of particle $i$. As shown in Škultéty et al. (2020), the same
dynamics can be conveniently encoded by the following equation:

$\displaystyle\partial_{t}h+v_{s}p^{\alpha}\nabla^{\alpha}h-D_{0}\nabla^{2}h+\lambda
h-\frac{\lambda}{4\pi}\int d{\bm{p}}\,h=\chi({\bm{r}},{\bm{p}},t),$ (21)

where $h=h({\bm{r}},{\bm{p}},t)$ is an auxiliary field related to the
Klimontovich correlation function (Klimontovich & Silin, 1962), and $\chi$ is
a noise term with the following properties:

$\displaystyle\langle\chi({\bm{r}},{\bm{p}},t)\rangle=0,$ (22)
$\displaystyle\langle\chi({\bm{r}},{\bm{p}},t)\chi({\bm{r}}^{\prime},{\bm{p}}^{\prime},t^{\prime})\rangle=\frac{n}{4\pi}\delta(t-t^{\prime})\Bigg{[}2\lambda\delta({\bm{r}}-{\bm{r}}^{\prime})\left(\delta({\bm{p}}-{\bm{p}}^{\prime})-\frac{1}{4\pi}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-
D_{0}\delta({\bm{p}}-{\bm{p}}^{\prime})\Big{(}\nabla^{2}+\nabla^{\prime
2}\Big{)}\delta({\bm{r}}-{\bm{r}}^{\prime})\Bigg{]}.$ (23)

The advantage of this representation lies in its direct relation to the phase-
space density correlation function (Klimontovich & Silin, 1962), which allows
us to express the fluid velocity autocorrelation function as

$\displaystyle C_{U}(t)=\lim_{t^{\prime}\rightarrow\infty}\frac{1}{V}\int
d{\bm{r}}\int
d{\bm{r}}^{\prime}d{\bm{r}}^{\prime\prime}d{\bm{p}}^{\prime}d{\bm{p}}^{\prime\prime}u_{s}^{\alpha}({\bm{r}};{\bm{r}}^{\prime},{\bm{p}}^{\prime})u_{s}^{\alpha}({\bm{r}};{\bm{r}}^{\prime\prime},{\bm{p}}^{\prime\prime})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\langle
h({\bm{r}}^{\prime},{\bm{p}}^{\prime},t)h({\bm{r}}^{\prime\prime},{\bm{p}}^{\prime\prime},t+t^{\prime})\rangle_{\chi},$
(24)

where the angular brackets denote the average of the (fictitious) noise $\chi$
(see Škultéty et al. (2020) for details). The linear equation (21) is readily
solved by introducing the Fourier–Laplace transform $\hat{h}$ of the auxiliary
field $h$:

$\displaystyle\hat{h}({\bm{k}},{\bm{p}},s)=\int_{0}^{\infty}dte^{-st}\int
d{\bm{r}}e^{i{\bm{k}}\cdot{\bm{r}}}h({\bm{r}},{\bm{p}},t),$ (25)

which yields

$\displaystyle\hat{h}({\bm{k}},{\bm{p}},s)=\frac{\hat{\chi}({\bm{k}},{\bm{p}},s)}{\sigma({\bm{k}},{\bm{p}},s)}+\frac{\lambda}{4\pi\sigma({\bm{k}},{\bm{p}},s)}\frac{\int
d{\bm{p}}^{\prime}\frac{\hat{\chi}({\bm{k}},{\bm{p}}^{\prime},s)}{\sigma({\bm{k}},{\bm{p}}^{\prime},s)}}{1-\frac{\lambda}{4\pi}\int\frac{d{\bm{p}}^{\prime}}{\sigma({\bm{k}},{\bm{p}}^{\prime},s)}}.$
(26)

Here,
$\sigma({\bm{k}},{\bm{p}},s)=s+\lambda+D_{0}k^{2}+iv_{s}({\bm{k}}\cdot{\bm{p}})$,
$\hat{\chi}({\bm{k}},{\bm{p}},s)$ is the Fourier–Laplace transform of the
noise, and we have dropped the initial condition
$\hat{h}({\bm{k}},{\bm{p}},t=0)$ which does not contribute in the large-$t$
limit.

As demonstrated by Škultéty et al. (2020), only the first term in Eq. (26)
contributes to $C_{U}(t)$. Performing the Fourier–Laplace transform in Eq.
(24) and combining it with Eq. (26) yields

$\displaystyle
C_{U}(t)=\frac{n\kappa^{2}}{16\pi^{4}}\lim_{t^{\prime}\rightarrow\infty}{\mathcal{L}}^{-1}_{s_{1},t^{\prime}}{\mathcal{L}}^{-1}_{s_{2},t^{\prime}+t}\int
d{\bm{k}}\frac{A^{2}(k\epsilon)}{k^{4}}\left(\lambda+D_{0}k^{2}\right)$
$\displaystyle\quad\times\int
d{\bm{p}}\,(\bm{k}\cdot{\bm{p}})^{2}\left[1-\frac{(\bm{k}\cdot{\bm{p}})^{2}}{k^{2}}\right]\frac{1}{s_{1}+s_{2}}$
$\displaystyle\qquad\qquad\times\frac{1}{\lambda+s_{1}+D_{0}k^{2}+iv_{s}(\bm{k}\cdot{\bm{p}})}\frac{1}{\lambda+s_{2}+D_{0}k^{2}-iv_{s}(\bm{k}\cdot{\bm{p}})}.$
(27)

Performing the inverse Laplace transforms, denoted symbolically by
${\mathcal{L}}^{-1}$, and integrating over $\bm{p}$ and the orientation of
$\bm{k}$, we finally arrive at Eq. (16) in the main text.

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