# Odd mobility of a passive tracer in a chiral active fluid

Anthony R. Poggioli<EMAIL_ADDRESS>Department of Chemistry,
University of California, Berkeley  Kavli Energy NanoScience Institute,
Berkeley, California  David T. Limmer<EMAIL_ADDRESS>Department of
Chemistry, University of California, Berkeley  Kavli Energy NanoScience
Institute, Berkeley, California  Materials Science Division, Lawrence Berkeley
National Laboratory  Chemical Science Division, Lawrence Berkeley National
Laboratory

###### Abstract

Chiral active fluids break both time-reversal and parity symmetry, leading to
exotic transport phenomena unobservable in ordinary passive fluids. We develop
a generalized Green-Kubo relation for the anomalous lift experienced by a
passive tracer suspended in a two-dimensional chiral active fluid subjected to
an applied force. This anomalous lift is characterized by a transport
coefficient termed the odd mobility. We validate our generalized response
theory using molecular dynamics simulations, and we show that the asymmetric
tracer mobility may be understood mechanically in terms of asymmetric
deformations of the tracer-fluid density distribution function. We show that
the even and odd components of the mobility decay at different rates with
tracer size, suggesting the possibility of size-based particle separation
using a chiral active working fluid.

Chiral active fluids are composed of constituents that convert energy into
directed rotational motion Banerjee  _et al._ (2017); Hargus  _et al._ (2020);
Han  _et al._ (2021). The quiescent state of such a fluid breaks both parity
and time-reversal symmetry and accordingly is not a state of thermodynamic
equilibrium. These broken symmetries lead to a variety of exotic transport
phenomena Avron (1998); Banerjee  _et al._ (2017); Dasbiswas  _et al._ (2019);
Ganeshan and Abanov (2017); Han  _et al._ (2021); Hargus  _et al._ (2020,
2021); Hosaka  _et al._ (2021); Lou  _et al._ (2022a); Markovich and Lubensky
(2021); Souslov  _et al._ (2019); van Zuiden  _et al._ (2016); Yang  _et al._
(2021); Lou  _et al._ (2022b); Soni  _et al._ (2019). For example, an odd
viscosity that links dilational flow to shear stress and sheared flow to
normal stress can translate forcing into fluid motion in an orthogonal
direction Avron (1998); Lou  _et al._ (2022a); Markovich and Lubensky (2021).
Describing transport phenomena in such nonequilibrium fluids is challenging
because classical linear response theories are valid perturbatively only about
an equilibrium steady state DeGroot and Mazur (2011). Here, we consider a
similar transport process, the motion of a passive tracer suspended in a two-
dimensional, odd viscous chiral active fluid and subjected to a constant
force, and we develop a response theory based on ensembles of trajectories Gao
and Limmer (2017, 2019). Recent work has examined the advective (Yang  _et
al._ , 2021) and driven (Hosaka  _et al._ , 2021) transport of passive tracers
in odd viscous fluids. These theoretical treatments are phenomenological,
however, illustrating the need for a microscopic description of tracer
mobility in chiral active fluids. The framework employed here can be extended
to arbitrary order in the applied forcing and is valid for a reference state
arbitrarily far from equilibrium Limmer  _et al._ (2021).

In general, the velocity response of a passive tracer is characterized by a
second rank tensor called the mobility tensor by the relation
$\left\langle{\bm{V}}\right\rangle_{\bm{F}}=\bm{\mu}\cdot{\bm{F}}$, where
${\bm{V}}$ is the tracer velocity, $\bm{\mu}$ is the mobility tensor,
${\bm{F}}$ is an external force applied to the tracer, and angled brackets
denote a nonequilibrium ensemble average Hargus  _et al._ (2021); Hosaka  _et
al._ (2021). In passive fluids, isotropy requires that the mobility tensor be
proportional to the unit tensor, introducing a single transport coefficient,
$\mu_{\mathrm{e}}$, termed the even mobility. However, in two dimensional
fluids breaking parity and time-reversal symmetry, the isotropy of the second
rank two-dimensional Levi-Civita tensor permits the emergence of an additional
transport coefficient, $\mu_{\mathrm{o}}$, termed the odd mobility, analogous
to the emergence of the odd viscosity Avron (1998). The general expression for
the mobility tensor is given in Cartesian coordinates by

$\bm{\mu}=\mu_{\mathrm{e}}{\bm{1}}+\mu_{\mathrm{o}}\bm{\epsilon}=\begin{pmatrix}\mu_{\mathrm{e}}&\mu_{\mathrm{o}}\\\
-\mu_{\mathrm{o}}&\mu_{\mathrm{e}},\end{pmatrix}$ (1)

where ${\bm{1}}$ is the unit tensor, and $\bm{\epsilon}$ is the Levi-Civita
tensor. Under a force ${\bm{F}}=F\hat{\bm{x}}$ applied in the positive
$x$-direction, the resulting tracer velocity is then
$\left\langle{\bm{V}}\right\rangle_{\bm{F}}=\mu_{\mathrm{e}}F\hat{\bm{x}}-\mu_{\mathrm{o}}F\hat{\bm{y}}$,
indicating that the particle experiences an anomalous lift proportional to the
odd mobility in the direction orthogonal to the applied force. This lift
corresponds to the conversion of directed rotational motion of the underlying
fluid into a uniform deflection of the tracer particle. Such a behavior could
be observed experimentally by subjecting tracers in an odd viscous fluid Soni
_et al._ (2019) to an external force.

Here, we consider odd mobility numerically with a simple model containing a
passive Weeks-Chandler-Andersen (WCA) particle Weeks  _et al._ (1971)
suspended in a fluid of active dimers Hargus  _et al._ (2020). The dimers are
composed of monomers linked by stiff harmonic bonds that interact with each
other and the passive tracer via a purely repulsive WCA potential
characterized by an energy parameter $\beta\epsilon=1$, where $\beta\equiv
1/k_{B}T$ is the inverse of the temperature times Boltzmann’s constant. Each
monomer is also subjected to an active force of constant magnitude $F_{a}$,
directed perpendicular to the instantaneous dimer bond. This active force
results in a directed active torque experienced by each dimer of average
magnitude $F_{a}d_{0}$, where $d_{0}=\sigma$ is the equilibrium bond length
set equal to the monomer diameter $\sigma$. The active forces are oriented
such that $F_{a}>0$ corresponds to a counterclockwise active torque. The
monomers undergo underdamped Langevin dynamics with the equation of motion

$m\dot{\bm{v}}_{i}^{\alpha}=-\gamma{\bm{v}}_{i}^{\alpha}+{\bm{F}}_{{\rm
c},i}^{\alpha}+{\bm{F}}_{a,i}^{\alpha}+\sqrt{2\gamma/\beta}{\bm{N}}_{i}^{\alpha}(t),$
(2)

where $i$ is the dimer index and $\alpha\in\left\\{1,2\right\\}$ indexes the
monomers on dimer $i$, $m$ is monomer mass, ${\bm{F}}_{{\rm c},i}^{\alpha}$
are the conservative forces acting on monomer $i$-$\alpha$,
${\bm{F}}_{a,i}^{\alpha}$ is the active force, and $\gamma$ is a linear
friction coefficient describing dissipation from the particles to an
underlying quiescent fluid bath. (See Supplemental Materials, including Refs.
Marchetti  _et al._ (2013); Weisenborn and Mazur (1984).) ${\bm{N}}_{i}(t)$ is
a delta-correlated Gaussian fluctuating force with mean zero and unit
variance. A simulation snapshot is shown in the inset in Fig. 1A, and details
of the dimer model can be found in Refs. Hargus  _et al._ (2020); Hargus and
Mandadapu (2022). This model represents a minimal model of a chiral active
fluid exhibiting odd viscosity Hargus  _et al._ (2020). The Supplemental
Materials contain further simulation details.

The tracer equation of motion is

$m\dot{\bm{V}}=-\gamma{\bm{V}}+{\bm{F}}_{\rm
WCA}+{\bm{F}}+\sqrt{2\gamma/\beta}{\bm{N}}(t),$ (3)

where ${\bm{V}}\equiv\dot{\bm{X}}$ is the tracer velocity, ${\bm{X}}$ is the
tracer position, ${\bm{F}}_{\rm WCA}$ is the total WCA force on the tracer
from the surrounding monomers, and ${\bm{F}}$ is a constant external force. We
take the tracer mass and friction coefficient equal to that for the monomers
in all simulations. Except in Fig. 2C, we also take the tracer diameter
$\sigma_{\rm tracer}$ to be equal to $\sigma$. The dimer activity has a
profound influence on the tracer dynamics. This effect is contained in the
average structure of ${\bm{F}}_{\rm WCA}$, as we will explore below.

The Gaussian nature of the noise allows us to write the path probability
density as Cugliandolo  _et al._ (2019)

$\mathbb{P}_{\bm{F}}\left[{\rm X}\right]\propto e^{-\beta S_{\bm{F}}\left[{\rm
X}\right]},$ (4)

where ${\rm X}\equiv\left\\{{\bm{X}}(t)\right\\}_{t=0}^{t_{N}}$ is a tracer
trajectory of duration $t_{N}\rightarrow\infty$, and the path action under an
applied force ${\bm{F}}$ is given by the Onsager-Machlup construction Onsager
and Machlup (1953); Machlup and Onsager (1953) as

$\begin{split}S_{\bm{F}}\left[{\rm
X}\right]&=\frac{1}{4\gamma}\int_{0}^{t_{N}}{\rm
d}t\,\left(m\dot{\bm{V}}+\gamma{\bm{V}}-{\bm{F}}_{\rm
WCA}-{\bm{F}}\right)^{2}\\\ &\equiv S_{0}\left[{\rm X}\right]+\Delta
S_{\bm{F}}\left[{\rm X}\right].\end{split}$ (5)

In the second equality we have extracted the unbiased path action
$S_{0}\left[{\rm X}\right]$ obtained when ${\bm{F}}=0$. The relative path
action $\Delta S_{\bm{F}}\left[{\rm X}\right]$ is defined by

$\frac{\mathbb{P}_{\bm{F}}\left[{\rm X}\right]}{\mathbb{P}_{0}\left[{\rm
X}\right]}=e^{+\beta\Delta S_{\bm{F}}\left[{\rm X}\right]},$ (6)

with

$\begin{split}\Delta S_{\bm{F}}\left[{\rm
X}\right]&\approx\frac{1}{2}{\bm{J}}\cdot{\bm{F}}+\frac{1}{2}{\bm{Q}}\cdot{\bm{F}},\end{split}$
(7)

where we have defined the time-extensive current ${\bm{J}}\equiv\int{\rm
d}t\,{\bm{V}}$ and frenesy ${\bm{Q}}\equiv-\int{\rm d}t\,\left({\bm{F}}_{\rm
WCA}/\gamma\right)$ Basu and Maes (2015). We have retained terms only to first
order in ${\bm{F}}$ as we are interested only in the linear response, and we
have dropped the temporal boundary term proportional to $\int{\rm
d}t\,m\dot{\bm{V}}$ because its contribution to the mobility exactly vanishes
in the limit $t_{N}\rightarrow\infty$. (See Supplemental Materials.)

Eq. 6 allows us to reweight path ensemble averages taken in a forced
(${\bm{F}}\neq 0$) ensemble to those in an unforced one, which we may use to
develop a linear response expression for the mobility. Rewriting the
trajectory average of ${\bm{J}}$ in the presence of ${\bm{F}}$ as one in its
absence using Eq. 6 and expanding to first order in ${\bm{F}}$, we obtain

$\left\langle\delta{\bm{J}}\right\rangle_{\bm{F}}=t_{N}\left\langle{\bm{V}}\right\rangle_{\bm{F}}=\frac{\beta}{2}\left(\left\langle\delta{\bm{J}}\otimes\delta{\bm{J}}\right\rangle_{0}+\left\langle\delta{\bm{J}}\otimes\delta{\bm{Q}}\right\rangle_{0}\right)\cdot{\bm{F}},$
(8)

where
$\delta\mathcal{O}\equiv\mathcal{O}-\left\langle\mathcal{O}\right\rangle_{0}$.
We insert our definition for the mobility tensor
$\left\langle{\bm{V}}\right\rangle={\bm{\mu}}\cdot{\bm{F}}$ and differentiate
with respect to ${\bm{F}}$ to obtain a linear response relationship for the
mobility tensor directly:

$\bm{\mu}=\frac{\beta}{2t_{N}}\left(\left\langle\delta{\bm{J}}\otimes\delta{\bm{J}}\right\rangle_{0}+\left\langle\delta{\bm{J}}\otimes\delta{\bm{Q}}\right\rangle_{0}\right).$
(9)

This equation represents the key theoretical result of this work. The first
term on the right hand side of Eq. 9 contains only the steady state
fluctuations in the current and corresponds to the ordinary ‘thermodynamic’
response obtained about equilibrium steady states. The second term contains
the current-frenesy correlation and is nonvanishing only in nonequilibrium
reference states. Physically, the frenesy represents microscopic motion
unassociated with the current. Though the particular form of the frenesy
depends on the nature of the underlying microscopic dynamics Baiesi  _et al._
(2009); Lesnicki  _et al._ (2020, 2021), for tracer motion well described by
the underdamped Langevin equation given in Eq. 3, the frenesy requires
knowledge only of the forces and their time correlations. It therefore in
principle may be measured experimentally for such a system.

We can make explicit the time-integrals appearing in the definitions of
${\bm{J}}$ and ${\bm{Q}}$ in Eq. 9. Using time-translation invariance and
averaging the diagonal and off-diagonal components of the mobility tensor, we
find

$\mu_{\mathrm{e}}=\beta\int_{0}^{\infty}{\rm
d}t\,\left[C_{jj}^{\mathrm{e}}(t)+C_{jq}^{\mathrm{e}}(t)\right],$ (10)

and

$\mu_{\mathrm{o}}=\beta\int_{0}^{\infty}{\rm d}t\,C_{jq}^{\mathrm{o}}(t),$
(11)

with

$C_{jj}^{\mathrm{e}}(t)\equiv\frac{1}{2}\sum_{\alpha=x,y}\left\langle\delta
j_{\alpha}(t)\delta j_{\alpha}(0)\right\rangle_{0},$ (12)
$C_{jq}^{\mathrm{e}}(t)\equiv\frac{1}{4}\sum_{\alpha=x,y}\left[\left\langle\delta
j_{\alpha}(t)\delta q_{\alpha}(0)\right\rangle_{0}+\left\langle\delta
q_{\alpha}(t)\delta j_{\alpha}(0)\right\rangle_{0}\right]\,,$ (13)

and

$\displaystyle C_{jq}^{\mathrm{o}}(t)\equiv\frac{1}{4}$
$\displaystyle\left[\left\langle\delta j_{x}(t)\delta
q_{y}(0)\right\rangle_{0}+\left\langle\delta q_{y}(t)\delta
j_{x}(0)\right\rangle_{0}\right]$
$\displaystyle-\frac{1}{4}\left[\left\langle\delta j_{y}(t)\delta
q_{x}(0)\right\rangle_{0}+\left\langle\delta q_{x}(t)\delta
j_{y}(0)\right\rangle_{0}\right].$

In the above, ${\bm{j}}\equiv{\bm{V}}$, and ${\bm{q}}\equiv-{\bm{F}}_{\rm
WCA}/\gamma$. The contribution of the current autocorrelation to the odd
mobility exactly vanishes. This can be seen clearly from Eq. 9 where the
thermodynamic contribution to the mobility is proportional to the explicitly
symmetric variance of the time extensive current.

Fig. 1 shows the contributions of each correlation function to the mobilities.
We plot the results for several different activities, quantified by a Péclet
number ${\rm Pe}\equiv 2\beta F_{a}d_{0}$. We include the results for
equilibrium, ${\rm Pe}=0$, for which the odd mobility must vanish. Indeed, we
observe that $C_{jq}^{\mathrm{o}}$ is zero. For finite ${\rm Pe}$, both the
current (thermodynamic) and current-frenesy (frenetic) contributions to
$\mu_{e}$ are invariant under change of sign of ${\rm Pe}$. This is a
reflection of the fact that $\mu_{\mathrm{e}}$ enters into the dissipation,
$\left\langle{\bm{V}}\right\rangle_{\bm{F}}\cdot{\bm{F}}=\mu_{\mathrm{e}}F^{2}$,
and hence is constrained to remain positive under time-reversal. It therefore
must be invariant under parity inversion, $F_{a}\rightarrow-F_{a}$, in order
to preserve overall parity-time symmetry.

Figure 1: Correlation functions contributing to the even and odd mobility
response of a passive tracer for several different Péclet numbers and density
$\bar{n}=0.2\,\sigma^{-2}$. A) The current autocorrelation for the even
mobility. B) The current-frenesy cross-correlation for the even mobility. C)
and D) The current autocorrelation (C) and current-frenesy cross-correlation
for the odd mobility. The inset in panel B shows a simulation snapshot and the
decomposition of the velocity response of a passive tracer subjected to an
applied force in terms of the even and odd mobility coefficients. The blue
particle is the passive tracer, and the orange particles are the actively
rotating dimers. Errors are smaller than the line thickness.

The initial values of the thermodynamic and frenetic correlation functions for
the even mobility both grow with increasing magnitude ${\rm Pe}$, though the
current variance grows more rapidly. Adding these curves shows that the growth
of the frenetic contribution acts to maintain a constant initial value of the
overall correlation function, while the relatively larger frenetic integral
relaxation time results in a decreasing overall relaxation time with
increasing activity, and hence a decreasing even mobility. (See Supplemental
Materials.) This is consistent with the decrease in overall even mobility with
increasing activity shown in Fig. 2A.

The thermodynamic contribution to the odd mobility, $C_{jj}^{\mathrm{o}}$,
exactly vanishes for all activities. As noted above this is predicted
analytically and reflects the fact that the odd mobility is a purely
nonequilibrium phenomenon. The frenetic correlation functions show a rich
structure, possessing an initial shift in their decay rate at short times and
a change in sign at longer times. The curves are inverted under a change in
sign of $F_{a}$, reflecting the parity – and hence time-inversion –
antisymmetry of the odd mobility. This is permitted because the odd mobility
does not contribute directly to the tracer dissipation, analogous to the odd
viscosity of the underlying fluid Avron (1998).

In Fig. 2, we show the even and odd mobility responses as measured by Eq. 9 as
a function of activity for several different densities. As anticipated, both
the even and odd mobility responses are suppressed as density is increased.
The maximum ratio of odd to even mobilities is obtained for the lowest density
examined, $\bar{n}=0.2\,\sigma^{-2}$, and is roughly $20\%$ for $\left|{\rm
Pe}\right|=20$. Surprisingly, we find that the even mobility is maximized for
${\rm Pe}=0$ and decreases with increasing activity, demonstrating that a
simple understanding of the activity in terms of an effective temperature does
not hold in this system, as previous results have reported an increase in the
mean squared diffusivity with ${\rm Pe}$ Hargus  _et al._ (2021).

Figure 2: Mobility as a function of activity, density, and tracer diameter. A)
and B) show even and odd mobility as a function of Péclet number for three
different densities. C) Ratio of odd to even mobility as a function of tracer
diameter for $\bar{n}=0.2\,\sigma^{-2}$. Shaded regions indicate standard
errors. The black squares in panels A and B show the results of forced
mobility calculations. The dashed black lines in panel B are guides to the
eye.

At the highest density, $\bar{n}=0.8\,\sigma^{-2}$, the odd mobility is
observed to vary linearly over the range of Péclet numbers examined, with
positive values, corresponding to downward deflections under a force applied
in the postive $x$-direction, obtained for counterclockwise dimer rotations
and vice verse. However, we find that at lower densities the odd mobility
appears to saturate to a constant value as $\left|{\rm Pe}\right|$ is
increased. Combined with the observed monotonically decreasing nature of
$\mu_{\mathrm{e}}$ with increasing $\left|{\rm Pe}\right|$, this indicates
that the ratio of odd to even mobility may be made arbitrarily large for high
activity, suggesting the possibility of a regime in which the particle is
deflected by essentially $90^{\circ}$ from the applied force. We note that in
Figs. 2A and B we have compared our simulation results in the reference steady
state to runs in which a force is applied directly to the tracer particle and
its velocity is directly measured, and the results are in excellent agreement.

Finally, in Fig. 2C, we plot the ratio of even to odd mobility as a function
of tracer diameter, $\sigma_{\rm tracer}$. We plot only for a single ${\rm
Pe}$ as we observe that this curve is essentially independent of ${\rm Pe}$
for finite activity. We find that the ratio of mobilities decreases with
increasing $\sigma_{\rm tracer}$, indicating that $\mu_{\mathrm{o}}$ decreases
more rapidly than $\mu_{\mathrm{e}}$ as tracer size is increased. The angle
$\theta$ between applied force and resultant tracer velocity is given by ${\rm
tan}\,\theta=-\mu_{\mathrm{o}}/\mu_{\mathrm{e}}$ Reichhardt and Reichhardt
(2019, 2022), indicating that the particle deflection angle is therefore size-
dependent. This suggests the possibility of size-based particle separation
using an odd viscous fluid as a working fluid.

We can understand the origin of the trends in mobility mechanically by
deriving a relationship between the tracer-monomer pair distribution function
(PDF) and the mobility response Lesnicki  _et al._ (2020, 2021); Liao  _et
al._ (2019). Averaging the tracer equation of motion, Eq. 3, over the noise,
we obtain

$\left\langle{\bm{V}}\right\rangle_{\bm{F}}=\frac{1}{\gamma}\left({\bm{F}}+\left\langle{\bm{F}}_{\rm
WCA}\right\rangle_{\bm{F}}\right).$ (15)

Given the pairwise nature of the repulsive interaction, the average of the
components of the velocity parallel and orthogonal to the applied force can be
written as

$V_{||}=\frac{F}{\gamma}+\frac{\bar{n}}{\gamma}\int{\rm
d}{\bm{x}}\,g\left({\bm{x}}\left|{\bm{F}}\right.\right)F_{{\rm
WCA},||}\left({\bm{x}}\right),$ (16)

and

$V_{\bot}=\frac{\bar{n}}{\gamma}\int{\rm
d}{\bm{x}}\,g\left({\bm{x}}\left|{\bm{F}}\right.\right)F_{{\rm
WCA},\bot}\left({\bm{x}}\right),$ (17)

respectively, where $g\left({\bm{x}}\left|{\bm{F}}\right.\right)$ is the
tracer-monomer PDF in the presence of an applied force ${\bm{F}}$ Hansen and
McDonald (2013); Onsager (1927), and ${\bm{F}}_{\rm WCA}\left({\bm{x}}\right)$
is the force field characterizing the WCA interaction between monomers and the
tracer particle held fixed at ${\bm{X}}=0$.

We take ${\bm{F}}$ to be directed along the positive $x$-direction.
Intuitively, we anticipate that a relative decrease in $V_{||}$ – and hence
$\mu_{\mathrm{e}}$ – with increasing $\left|{\rm Pe}\right|$ will be
associated with an increase in monomer density accumulated on the front of the
particle. However, we observe the opposite in Figs. 3A and B, where we plot
the PDF for ${\rm Pe}=0$ and $20$ under an applied force. Instead, we observe
that the ridge of monomer density accumulated on the front of the tracer
decreases substantially in magnitude with increasing activity, ostensibly
suggesting that $\mu_{\mathrm{e}}$ should increase with increasing magnitude
${\rm Pe}$ – the opposite of what is observed in Fig. 2. This apparent
inconsistency is resolved by considering the relative density distribution.

Though the accumulation of monomer density in front of the tracer decreases
with increasing activity, a high density ridge is pushed further into the
interaction region with the tracer, resulting in a much stronger resistive
interaction. This is clearly illustrated in the inset in Fig. 3B, which shows
the PDF obtained for ${\rm Pe}=20$ less that obtained for ${\rm Pe}=0$. Direct
integration confirms that this shift in particle distribution results in the
suppression of the even mobility at finite activity. Interestingly, this shift
in the high density ridge is not observed in the PDF calculated using the
dimer center of mass coordinates, indicating that the dimers accumulated at
the front of the tracer are preferentially oriented with a component of their
bond vector orthogonal to the face of the tracer, and that this ‘orientational
locking’ increases with activity. This mechanism is fundamentally linked to
the biased nature of the rotational motion of the dimers and is enhanced with
greater activity. It therefore cannot be renormalized through an effective
temperature that implicitly treats the effect of the activity as enhanced
unbiased random noise of the dimer particles.

Figure 3: Tracer-monomer pair distribution function (PDF) for a density of
$\bar{n}=0.2\,\sigma^{-2}$ and different Péclet numbers under a force applied
to the tracer in the positive $x$-direction of magnitude
$F=10\,\epsilon\,\sigma^{-1}$. The left and right columns correspond to Péclet
numbers of $0$ and $20$. The upper row shows the PDF in the upper half plane,
while the lower row shows the PDF in the upper half plane minus the PDF in the
lower half plane, illustrating the reflection asymmetry in $y$. The solid blue
half circle indicates the location of the tracer particle. The inset in panel
B shows the difference in PDFs obtained at ${\rm Pe}=20$ and $0$, and the
dashed red line in panels A and B and the inset indicates the cutoff of the
WCA interaction.

Given the central nature of the WCA interaction, $F_{{\rm WCA},\bot}$ is
necessarily antisymmetric in $y$. Therefore, the deflection associated with
the odd mobility must be the result of an asymmetry in the PDF under an
applied force for finite ${\rm Pe}$ such that the integral in Eq. 17 is
nonvanishing. Indeed, this is observed in Figs. 3C and D, where we plot the
PDF in the upper half plane less that in the lower half plane,
$g(x,y)-g(x,-y)$. This is a direct measure of the asymmetry under the
transformation $y\rightarrow-y$ and hence of a nonvanishing odd mobility. We
observe that there is measurable asymmetry only in the case of finite Péclet
number, and the odd mobility is therefore due to asymmetric deformations of
the particle distribution about the tracer particle for finite activities.

In this letter, we have derived a generalized Green-Kubo relation relating
current and frenesy fluctuations in a steady state far from equilibrium to the
mobility of a passive tracer suspended in a two-dimensional chiral active
fluid. We have validated our results using molecular dynamics simulations and
shown that, whereas the even mobility counterintuitively decreases with
increasing activity, the odd mobility increases until it saturates to a finite
value. The former result indicates that an effective temperature description
of the mobility response is invalid in this system. We have further shown that
the odd mobility decays more rapidly than the even mobility with increasing
tracer particle size, independent of activity, indicating the possibility of
size-based particle separation using an odd viscous working fluid. Our results
provide strong evidence for the generality of the path integral framework for
nonequilibrium response and a microscopic picture of the mobility response.
Future work will focus on the development of effective hydrodynamic
descriptions of the underlying fluid and tracer mobility.

Acknowledgments A. R. P. and D. T. L. acknowledge helpful discussions with
Kranthi Mandadapu and Cory Hargus. This work has been supported by NSF Grant
CHE-1954580. A. R. P was also supported by the Heising-Simons Fellowship from
the Kavli Energy Nanoscience Institute at UC Berkeley and D. T. L acknowledges
support from thr Alfred P. Sloan Foundation.

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