††thanks: Present address: Laboratoire Physico Chimie Curie, Institut Curie,
PSL Research University, CNRS UMR168, 75005 Paris, France

# Breakdown of effective temperature, power law interactions and self-
propulsion in a momentum conserving active fluid

Amit Singh Vishen  Simons Centre for the Study of Living Machines, National
Centre for Biological Sciences (TIFR), Bangalore 560065, India  Jacques Prost
jacques<EMAIL_ADDRESS>Mechanobiology Institute and Department of Biological
Sciences, National University of Singapore, 117411 Singapore  Laboratoire
Physico Chimie Curie, Institut Curie, PSL Research University, CNRS UMR168,
75005 Paris, France  Madan Rao<EMAIL_ADDRESS>Simons Centre for the Study
of Living Machines, National Centre for Biological Sciences (TIFR), Bangalore
560065, India

###### Abstract

The simplest extensions of single particle dynamics in a momentum conserving
active fluid - an active suspension of two colloidal particles or a single
particle confined by a wall - exhibit strong departures from Boltzmann
behavior, resulting in either a breakdown of an effective temperature
description or a steady state with nonzero entropy production rate. This is a
consequence of hydrodynamic interactions that introduce multiplicative noise
in the stochastic description of particle positions. This results in
fluctuation-induced interactions that depend on distance as a power law. We
find that the dynamics of activated colloids in a passive fluid, with
stochastic forcing localized on the particle, is different from that of
passive colloids in an active fluctuating fluid.

##  I Introduction

Fluctuations of a dilute active suspension (e.g., bacterial bath, Wu and
Libchaber (2000); Lau  _et al._ (2003)) have often been described in terms of
an equilibrium system with a (large) effective temperature Palacci  _et al._
(2010); Takatori and Brady (2015), with theoretical rationalisations provided
by studies of the dynamics of a single particle in an active fluctuating fluid
or a single active particle embedded in a passive fluid Takatori and Brady
(2015); Burkholder and Brady (2017).

However, as we find here, even the simplest extensions - the stochastic
dynamics of two particles embedded in an unbounded isotropic active
fluctuating gel or a particle in an isotropic active fluctuating gel bound
within confining walls - do not allow for an effective temperature
description, since the corresponding steady state probability distribution
shows strong departures from the Boltzmann form. This is a consequence of a
drift that arises from hydrodynamic interactions, that introduces a
multiplicative noise in the stochastic description of particle positions. The
form of the drift can only be decided after solving the full hydrodynamics
problem. This has important implications for current discussions on active
contributions to pressure, osmotic pressure and surface tension in momentum
conserving active fluids Takatori and Brady (2015); Marini  _et al._ (2015);
Rodenburg  _et al._ (2017).

Indeed, deviations from Boltzmann behaviour and consequent breakdown of an
effective temperature description have been systematically analysed in a dry
system of active Ornstein-Uhlenbeck particles (AOUP) Fodor  _et al._ (2016),
where momentum is not conserved. Within a systematic perturbation expansion in
the active noise correlation time $\tau_{n}$, the nonequilibrium nature of the
steady state distribution first shows up at order $\tau_{n}$ (characterised by
non-Boltzmann probability distribution but zero entropy dissipation), while
the full nonequilibrium aspect with nonzero entropy production shows up at
order $\tau_{n}^{3/2}$ Fodor  _et al._ (2016). Moreover, there have been
several studies on the form of the effective interaction between passive
particles (both fixed and mobile) embedded in a bath of dry active particles.
For instance, passive colloids in a suspension of self-propelled particles
interact through a non-equilibrium analog of depletion forces Angelani  _et
al._ (2011); Ray  _et al._ (2014); Ni  _et al._ (2015). The range of these
interactions depends on the shape of tracers - being short-range between
passive spheres Angelani  _et al._ (2011), and long-range between two parallel
walls Ni  _et al._ (2015). In addition, passive spheres can have large
interactions due to density fluctuations of dry active particles Baek  _et
al._ (2018).

In this paper, we ask, what is the nature of effective interactions and
departures from Boltzmann distribution, in momentum conserving active systems?
We find that the simplest extensions of single-particle dynamics, viz., that
of colloid particle-wall and colloid particle-particle interactions embedded
in a three dimensional active fluctuating gel, exhibits a clear non-Boltzmann
steady state distribution, characterized by an effective attractive potential
($\propto 1/r$, for particle-wall separated by $r$ and $\propto 1/r^{4}$, for
particle-particle). This is a consequence of the active (fluctuation-
dissipation relation violating) fluctuations and hydrodynamic interactions
that introduce a multiplicative noise. A dimer of unequal sized spherical
particles embedded in an active fluctuating gel self-propels with a velocity
proportional to the fluctuation amplitude. Interestingly, for this momentum
conserving active system, both the breakdown of the effective temperature
description and finite entropy production rate, appear even in the limit
$\tau_{n}\to 0$. We next study the statistics of fluctuations of activated
particles in a passive fluid, i.e., particles directly subject to a stochastic
driving force. We find that the dynamics of activated particles in a passive
medium is not the same as the dynamics of passive particles in an active
medium - for instance, the effective particle-particle interaction is
repulsive and long ranged ($\propto 1/r^{2}$). We proceed to demonstrate these
results below.

Consider an incompressible, isotropic, actively fluctuating viscoelastic gel,
described by a local stress,

$\left(1+\tau_{v}\partial_{t}\right)\sigma_{ij}=-p+\eta\left(\partial_{i}v_{j}+\partial_{j}v_{i}\right)+\sigma^{n}_{ij},$
(1)

where $i,j\in(x,y,z)$, $\tau_{v}$ is a Maxwell time, $\eta$ is the viscosity,
$p$ is the pressure which includes, a priori, the isotropic component of the
mean and fluctuating active stress, and $\sigma^{n}_{ij}$ is the fluctuating
component of the active deviatoric stress, with zero mean and correlation Lau
and Lubensky (2009); Basu  _et al._ (2008)

$\displaystyle\langle\sigma^{n}_{ij}({\bf r},t)$
$\displaystyle\sigma^{n}_{kl}({\bf r^{\prime}},t^{\prime})\rangle=$ (2)
$\displaystyle 2\pi\delta({\bf
r-r^{\prime}})\Delta(t-t^{\prime})\left[\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}-\frac{2}{3}\delta_{ij}\delta_{kl}\right],\quad$

with $\Delta(t-t^{\prime})=\Lambda\tau_{n}^{-1}e^{-|t-t^{\prime}|/\tau_{n}}$.
For simplicity, we have taken the variance of the anisotropic stress
fluctuation to be a scalar $\Delta(t-t^{\prime})$. In general,
$\Delta_{ijkl}(t-t^{\prime})$ is a fourth rank tensor, which can arise from
fluctuations of the nematic order parameter Lau and Lubensky (2009). Since the
temporal correlations of the noise are unrelated to the drag
($\tau_{v}\neq\tau_{n}$), this system does not satisfy the generalized Stokes-
Einstein relation at a microscopic scale. This is a minimal active system, the
non-equilibrium component is included as active fluctuations that violate
fluctuation dissipation relation. Note that the stress in Eq. 1 should also
have a stochastic contribution due to the thermal fluctuations, which we take
to be much smaller than the active fluctuations.

Throughout this paper, we work at timescales larger than $\tau_{n}$ and
$\tau_{v}$, thus Eq. 1 becomes

$\sigma_{ij}=-p+\eta\left(\partial_{i}v_{j}+\partial_{j}v_{i}\right)+\sigma^{n}_{ij}.$
(3)

with $\Delta(t-t^{\prime})=2\Lambda\delta(t-t^{\prime})$, obtained by taking
the limit $\tau_{n}\to 0$. The dynamics in the Stokes limit is
$\nabla\cdot{\bm{\sigma}}=0$, along with the incompressibility condition
$\nabla\cdot{\bm{v}}=0$. We emphasize that at long times, the stress as given
by Eq. 3 is identical to that of a passive viscous fluid, however, as we show
in the following, the stochastic dynamics of embedded colloids is, in general,
not the same as that of colloids in a passive fluid.

The generalized Langevin dynamics of a spherical colloidal particle of radius
$a$ embedded in an unbounded fluid defined by Eq. 1 is obtained by integrating
out the stress Eq. 1 and using no-slip boundary condition at its surface (see,
appendix of ref. Lau and Lubensky (2009))

$\tau_{m}\ddot{\bf
R}+\int_{-\infty}^{t}dt^{\prime}\gamma(t-t^{\prime})\dot{\bf
R}(t^{\prime})=-\mu\nabla\cdot{\sf U}+\sqrt{2\lambda\mu}\,{\bm{\theta}(t)},$
(4)

where $\lambda=\Lambda/\eta$, $\mu=1/6\pi\eta a$, $\tau_{m}=m\,\mu$,
$\gamma(t)=\tau_{v}\exp(-t/\tau_{v})$ is the friction coefficient,
$\mathrm{\sf U}$ is an externally applied potential, and ${\bm{\theta}}(t)$ is
a Gaussian noise with correlation

$\langle\theta_{i}(t)\theta_{j}(t^{\prime})\rangle=\delta_{ij}\tau_{n}\exp(-(t-t^{\prime})/\tau_{n}).$
(5)

In the limit $\tau_{m},\tau_{v},\tau_{n}\to 0$, Eq. 4 reduces to

$\partial_{t}{\bf R}=-\mu\nabla\cdot{\sf
U}+\sqrt{2\lambda\mu}\,{\bm{\vartheta}},$ (6)

where ${\bm{\vartheta}}(t)$ is a unit variance Gaussian white noise. As
expected, the overdamped Langevin dynamics obtained by integrating out the
fluid stress Eq. 3 and using no-slip boundary condition at the surface Fox and
Uhlenbeck (1970); Noetinger (1990); M. Doi. and S. F. Edwards (1988) is
identical to Eq. 6. Henceforth, wherever required, we use the standard form of
Langevin equations corresponding to Eq. 3.

The steady state probability distribution of the position of the colloidal
sphere obtained from the Fokker-Planck equation corresponding to Eq. 6 has a
Boltzmann form $P(z)\propto e^{-\mathrm{\sf U}/\lambda}$, with an effective
temperature $k_{B}\mathrm{T_{eff}}\equiv\lambda$.

Figure 1:  Schematic of (A) a sphere of radius $a$ at distance $z$ from a
fixed wall at $z=0$, and (B) two spheres of radius $a$ and $b$ centred at
${\bf R_{1}}$ and ${\bf R_{2}}$ respectively, in an unbound fluctuating
viscous fluid.

##  II Particle-Wall Interaction

The Langevin dynamics of a spherical colloid of radius $a$ at a distance $z$
from a fixed wall at $z=0$ (Fig. 1(a)), is obtained by integrating out Eq. 3
and using the no-slip boundary condition at the surface of the colloid M. Doi.
and S. F. Edwards (1988),

$\displaystyle\partial_{t}z$ $\displaystyle=$
$\displaystyle-\mathrm{H}_{z}\partial_{z}\mathrm{U}+\sqrt{2\lambda\mathrm{H}_{z}}\vartheta_{z},$
(7) $\displaystyle\partial_{t}x_{\perp}$ $\displaystyle=$
$\displaystyle-\mathrm{H}_{x_{\perp}}\partial_{x_{\perp}}\mathrm{U}+\sqrt{2\lambda\mathrm{H}_{x_{\perp}}}\vartheta_{x_{\perp}},$
(8)

where $x_{\perp}\equiv(x,y)$, $\mathrm{H}_{z}(\mathrm{H}_{x_{\perp}})$ is the
mobility in the longitudinal (transverse) direction to the wall,
$\mathrm{U}(z,x_{\perp})$ is the particle-wall potential, and $\vartheta_{z}$
and $\vartheta_{x_{\perp}}$ are zero mean Gaussian white noise with
correlation

$\langle\vartheta_{i}(t)\vartheta_{j}(t^{\prime})\rangle=2\lambda\mathrm{H}^{-1}_{i}\delta_{ij}\delta(t-t^{\prime}),$
(9)

where $i,j\in(x,y,z)$. The mobilities $\mathrm{H}_{i}(z)$ are functions of the
separation $z$ from the wall, which can be evaluated as power series in $a/z$
(Appendix B and Kim and Karilla (1991)). The appearance of a multiplicative
and correlated noise is typical of a stochastic dynamics with hydrodynamic
interactions. As is well known van Kampen (1981); Gardiner (2009), this
Langevin equation is meaningless unless supplemented with a stochastic
calculus convention for the multiplicative noise. The choice of convention
depends on the fast timescales that have been integrated out - viscoelastic
relaxation time ($\tau_{v}$), particle inertial relaxation time ($\tau_{m}\sim
m/\eta\,a$, where $m$ is the particle mass and $a$ is the particle size), and
noise correlation time ($\tau_{n}$). In ref. Rupprecht  _et al._ (2018); Singh
Vishen  _et al._ (2018) it was shown that for an active noise with
$\tau_{n}\gg\sqrt{\tau_{m}\tau_{v}}$, the appropriate convention is
Stratonovich Stratonovich (1966); Gardiner (2009). For the cell cortex, for
instance, the timescales are $\tau_{v}\sim 1$-$10$ s Saha  _et al._ (2016);
Rupprecht  _et al._ (2018), $\tau_{m}\sim 10^{-9}$ s, and $\tau_{n}\sim 10$ s
Rupprecht  _et al._ (2018), which makes $\sqrt{\tau_{m}\tau_{v}}\sim
0.1\,\mathrm{ms}\ll\tau_{n}$. In the following, we work in the limit
$\tau_{n}\gg\sqrt{\tau_{m}\tau_{v}}$, we believe this limit is reasonable for
many biological systems. For the appropriate choice of noise convention when
this limit does not hold, we refer to Rupprecht  _et al._ (2018); Singh Vishen
_et al._ (2018).

The Fokker-Planck equation corresponding to Eq. 7 and 8, interpreted in
Stratonovich convention is (Appendix A)

$\displaystyle\partial_{t}P$ $\displaystyle=$
$\displaystyle\partial_{z}\left(\mathrm{H}_{z}\partial_{z}\mathrm{U}+\frac{1}{2}\lambda\left(\partial_{z}\mathrm{H}_{z}\right)+\lambda\mathrm{H}_{z}\partial_{z}\right)P$
(10) $\displaystyle+$
$\displaystyle\partial_{x_{\perp}}\left(\mathrm{H}_{x_{\perp}}\partial_{x_{\perp}}\mathrm{U}+\lambda\mathrm{H}_{x_{\perp}}\partial_{x_{\perp}}\right)P.$

which leads to a steady state probability distribution $P(z,x_{\perp})\propto
e^{-\Phi/\lambda}$, with an effective potential,

$\Phi=\mathrm{U}+\frac{1}{2}\lambda\log\mathrm{H}_{z}.$ (11)

This effective interaction between the wall and the particle, as a result of
the active noise and hydrodynamics, is long-range (goes as $a/z$, for large
$z$), attractive and anisotropic. The additional fluctuation term along with
the applied potential cannot be captured by a simple effective temperature
definition. Nevertheless, the steady state has zero current and obeys time
reversal symmetry, making this a nonthermal equilibrium model Fodor  _et al._
(2016). We emphasize however that unlike in Fodor  _et al._ (2016), this
deviation from Boltzmann measure occurs even in the limit $\tau_{n}\to 0$.
Further, in contrast to thermal fluctuations where hydrodynamic interactions
only affect dynamics, active fluctuations in a fluid affect both the dynamics
and the steady state.

##  III Pressure

The contribution to the force on the wall due to the bare colloid-wall
potential $\mathrm{U}(z)$ is,

$F_{p}=\int_{0}^{\infty}dzP(z)\partial_{z}\mathrm{U}(z)\propto\int_{0}^{\infty}dz\frac{1}{\sqrt{\mathrm{H}_{z}}}e^{-\mathrm{U}/\lambda}\partial_{z}\mathrm{U}(z).$
(12)

The observation that the force on the wall depends on the form of wall-
particle interaction, is directly related to the fact that the probability
distribution is non-Boltzmann. This is analogous to the situation in dry
active particle systems Solon  _et al._ (2015), where, apart from the kinetic
contribution, this would have sufficed to give the pressure. However, in
momentum conserving Stokesian fluid systems, the net force due to particle-
wall interactions is balanced by the force due to the embedding fluid.

Consider a suspension of $N$ colloidal particles in a semi-infinite active
fluid confined by a wall. Assuming particles do not interact directly with
each other via forces such as van der Waals, the total force on the $I^{th}$
particle reads:

$f_{i}^{tot}=-\int_{S_{p}}\sigma_{ij}dS_{j}+\int_{V_{p}}dv\,g_{i}^{p},$ (13)

where $\sigma_{ij}$ is the fluid stress acting on the particle $I$, $S_{p}$ is
the surface with normal pointing out of the fluid, $g^{p}_{1}$ is the force
density on the colloid due to wall-particle interaction, and $V_{p}$ is the
volume of the particle $I$. Force balance on the particle $I$: $f_{i}^{tot}=0$
implies

$\int_{S_{p}}\sigma_{ij}dS_{j}=\int_{V_{p}}dv\,g_{i}^{p}.$ (14)

The dynamics of the Stokesian fluid is given by,

$g^{f}_{i}+\partial_{j}\sigma_{ij}=0,$ (15)

where $g^{f}_{i}$ is the force density on the fluid due to interaction between
fluid particles and the wall. Integrating this relation over a volume $V$
bounded by two parallel surfaces $S_{f}$ and $S_{f}^{\prime}$, the first at
the wall, the second in the fluid at a distance such that the body forces
$g_{i}^{p}$ and $g^{f}_{i}$ vanish:

$\int_{S_{f}}\sigma_{ij}\,dS_{j}+\int_{S_{f}^{\prime}}\sigma_{ij}\,dS_{j}+\int_{V-NV_{p}}dVg^{f}_{i}=-\sum_{I=1}^{N}\int_{S_{p}}\sigma_{ij}\,dS_{j},$
(16)

where $N$ are the number of colloidal particles within the volume $V$ and the
surface normals point outside the considered fluid volume. Substituting Eq. 14
in Eq. 16 gives

$\int_{S_{f}}\sigma_{ij}\,dS+\int_{S_{f}^{\prime}}\sigma_{ij}\,dS_{j}=-\int_{V-V_{p}}dVg^{f}_{i}-\sum_{I=1}^{N}\int_{V_{p}}dv\,g_{i}^{p}.$
(17)

Now, the force exerted by the suspension on the wall is,

$F_{i}=-\int_{S_{f}}\sigma_{ij}dS_{j}-\int_{V-V_{p}}dVg^{f}_{i}-\sum_{I=1}^{N}\int_{V_{p}}dv\,g_{i}^{p},$
(18)

where the first term is the force on the confining wall due to the fluid, the
second term is the force on the wall due to the fluid particles, which is
equal and opposite to the force on the fluid due to the wall by Newton third
law, similarly, the third term is due to the interaction between the colloidal
particles and the wall. Substituting Eq. 17 in Eq. 18 we get

$F_{i}=\int_{S_{f}^{\prime}}\sigma_{ij}\,dS_{j}.$ (19)

Since $S_{f}^{\prime}$ is an arbitrary surface chosen to be far from the wall,
we see that there is no net added force on the wall even though the particles
feel an effective attraction towards it. The total force exerted on the wall
by the system particles$+$fluid vanishes exactly. This point appears to have
been disregarded in recent theoretical studies of the active contribution to
osmotic pressure and surface tension in momentum conserving active fluids
Takatori and Brady (2015); Marini  _et al._ (2015); Rodenburg  _et al._
(2017).

##  IV  Two particles embedded in active fluid

We now consider the dynamics of two spherical colloids of radius $a$ and $b$,
centered at ${\bf R}_{1}$, and ${\bf R}_{2}$ respectively (Fig. 1(b)), in an
unbounded active fluctuating fluid. Since for spherical colloids, the
translational motion of the center of mass is decoupled from the rotational
motion, we will consider only the hydrodynamic coupling between the
translational degree of freedom.

The Langevin dynamics for the centers of the spherical colloids follows as
before and is given by Kim and Karilla (1991); Lau  _et al._ (2003),

$\frac{d{\bf R}_{\alpha}}{dt}=\sum_{\beta}{\sf H}_{\alpha\beta}\cdot({\bf
f}_{\beta}+{\bm{\vartheta}}_{\beta}),$ (20)

where $\alpha$, and $\beta$ are particle labels, ${\sf H}_{\alpha\beta}$, is
the $3\times 3$ hydrodynamic interaction tensor coupling the translation
motion of particle $\beta$ with that of particle $\alpha$ (see appendix B for
the form), ${\bf f_{\beta}}=-\nabla_{{\bf R_{\beta}}}\mathrm{U}$ is the
deterministic force from an externally applied potential $\mathrm{U}$. The
stochastic force ${\bm{\vartheta}}_{\beta}$ on the particle $\beta$, is a
three dimensional vector of Gaussian white noise with correlation

$\langle{\vartheta}_{i\,\alpha}(t){\vartheta}_{j\,\beta}(s)\rangle=2\lambda{\sf
H}_{\alpha\beta}^{-1}\delta_{ij}\delta(t-s).$ (21)

The Fokker-Planck equation corresponding to Eq. 20, interpreted in
Stratonovich convention is Méndez  _et al._ (2014) (see appendix A)

$\partial_{t}P=\nabla_{{\bf R_{\alpha}}}\cdot\left[-{\sf
H}_{\alpha\beta}\cdot{\bf f}_{\beta}+\frac{\lambda}{2}\left(\nabla_{{\bf
R_{\beta}}}\cdot{\sf H}_{\alpha\beta}\right)+\lambda{\sf
H}_{\alpha\beta}\cdot\nabla_{{\bf R_{\beta}}}\right]P.$ (22)

Due to incompressibility, $\nabla_{{\bf R_{\beta}}}\cdot{\sf H}_{\alpha\beta}$
is identically zero in Oseen approximation of ${\sf H}_{\alpha\beta}$ M. Doi.
and S. F. Edwards (1988), the first non-zero contribution is at order
$1/r^{4}$. In terms of variables, ${\bf R\equiv R_{1}+R_{2}}$ and ${\bf
r\equiv R_{2}-R_{1}}$, Eq. 22 is

$\displaystyle\partial_{t}P=\nabla_{{\bf R}}\cdot\left(-{\sf
M}_{11}\cdot{\bf\tilde{f}}_{1}-{\sf M}_{12}\cdot{\bf\tilde{f}}_{2}+\lambda{\sf
M}_{11}\cdot\nabla_{{\bf R}}\right)P$ (23) $\displaystyle+\lambda\nabla_{{\bf
R}}\cdot\left(\frac{1}{2}\nabla_{{\bf r}}\cdot{\sf M}_{12}+{\sf
M}_{12}\cdot\nabla_{{\bf r}}\right)P+\nabla_{{\bf r}}\cdot\left(\lambda{\sf
M}_{12}\cdot\nabla_{{\bf R}}\right)P$ $\displaystyle+\nabla_{{\bf
r}}\cdot\left(-{\sf M}_{12}\cdot{\bf\tilde{f}}_{1}-{\sf
M}_{22}\cdot{\bf\tilde{f}}_{2}+\frac{\lambda}{2}\nabla_{{\bf r}}\cdot{\sf
M}_{22}+\lambda{\sf M}_{22}\cdot\nabla_{{\bf r}}\right)P,$

where ${\bf\tilde{f}}_{1}=-\nabla_{\bf R}\mathrm{U}$,
${\bf\tilde{f}}_{2}=-\nabla_{{\bf r}}\mathrm{U}$, and the mobility matrix

$\displaystyle{\sf M}=\begin{bmatrix}{\sf H}_{11}+{\sf H}_{22}+2{\sf
H}_{12}&{\sf H}_{22}-{\sf H}_{11}\\\ {\sf H}_{22}-{\sf H}_{11}&{\sf
H}_{11}+{\sf H}_{22}-2{\sf H}_{12}\end{bmatrix}.$ (24)

Let us first look at equal sized colloids, $a=b$. In this case, the off-
diagonal block matrix ${\sf M_{12}}=0$, and Eq. 23 reduces to

$\displaystyle\partial_{t}P$ $\displaystyle=$ $\displaystyle\nabla_{{\bf
R}}\cdot\left(-{\sf M}_{11}\cdot{\bf\tilde{f}}_{1}+\lambda{\sf
M}_{11}\cdot\nabla_{{\bf R}}\right)P$ $\displaystyle+$
$\displaystyle\nabla_{{\bf r}}\cdot\left(-{\sf
M}_{22}\cdot{\bf\tilde{f}}_{2}+\frac{\lambda}{2}\nabla_{{\bf r}}\cdot{\sf
M}_{22}+\lambda{\sf M}_{22}\cdot\nabla_{{\bf r}}\right)P.$

Now, the necessary and sufficient condition for the steady state solution of
the Fokker-Plank equation of the from

$\partial_{t}P=\sum_{\alpha=1}^{N}\nabla_{\alpha}\cdot\left(-{\bf
F}_{\alpha}+{\sf D}_{\alpha\beta}\nabla_{\beta}\right)P,$ (26)

to have zero probability current is that it satisfies the potential condition
Gardiner (2009), defined as,

$\frac{\partial{Z}_{\beta j}}{\partial R_{\alpha i}}=\frac{\partial{Z}_{\alpha
i}}{\partial R_{\beta j}},$ (27)

where $i,j\in(x,y,z)$ and

${Z}_{\alpha i}={\sf D}^{-1}_{\alpha i,\beta j}{F}_{\beta j}.$ (28)

We see that at steady state, Eq. IV satisfies the potential condition Eq. 27.
Using this, we find that the steady state distribution has the form $P\propto
e^{-\Phi/\lambda}$, where

$\displaystyle\Phi({\bf
r,R})=\mathrm{U}-\frac{1}{2}\lambda\left[\frac{15b^{4}}{8r^{4}}+\mathcal{O}\left(\frac{1}{r^{5}}\right)\right].$
(29)

As in the colloid-wall interaction, the steady state distribution is non-
Boltzmann with a fluctuation-induced particle-particle interaction that is
attractive, though short-ranged.

For spheres of unequal size, $a\neq b$, the steady state distribution does not
obey the potential condition given by Eq. 27; thus the steady state has a
finite probability current and an associated entropy production rate,
resulting in a finite propulsion velocity at steady state. We emphasize that
unlike AOUP Fodor  _et al._ (2016), this nonequilibrium steady state with
finite entropy dissipation occurs even in the limit of $\tau_{n}\to 0$.

To see this, we describe the two particles as a dimer, characterised by the
separation $r$, the orientation ${\bf\hat{r}}$, and the center of mass
position ${\bf R}$ Illien  _et al._ (2017). In general, it is difficult to
obtain an analytic expression for the steady state distribution
$P(r,{\bf\hat{r}},{\bf R})$. However, in a well defined limit where there is a
time scale separation, we obtain analytic expressions for the steady state
distribution, fluctuation-induced potential and mean propulsion velocity.

Taking $\mathrm{U}$ to be a function of $r$ alone, ${\bf\tilde{f}}_{1}=0$ and
${\bf\tilde{f}}_{2}=-\mathrm{U}^{\prime}(r){\bf\hat{r}}$. With this, the
probability flux for r in Eq. IV is now independent of R, hence, we can
integrate out R to obtain the marginal dynamics of ${\bf r}$. This allows us
to solve for the steady state marginal distribution. The steady state for the
marginal distribution: $P(r)$, obtained by integrating Eq. $14$ in the main
text over ${\bf R}$, with no flux boundary condition gives

$\nabla_{{\bf r}}\log P({\bf
r})=\frac{1}{\lambda}{\bf\tilde{f}}_{2}-\frac{1}{2}{\sf
M}^{-1}_{22}\cdot\left(\nabla_{\bf r}\cdot{\sf M}_{22}\right).$ (30)

The mobility tensor ${\sf M}$ defined in Eq. $15$ decomposed as sum of
projection along $\hat{\bf r}$ (denoted by superscript $n$) and perpendicular
to $\hat{\bf r}$ (denoted by superscript $q$) is

${\sf M}_{\alpha\beta}=m_{\alpha\beta}^{q}(r)\left({\bf\mathrm{\sf
I}}-{\bf\mathrm{\bf\hat{r}\hat{r}}}\right)+m_{\alpha\beta}^{n}(r){\bf\mathrm{\bf\hat{r}\hat{r}}}.$
(31)

From this we see that the inverse is

${\sf M}_{\alpha\beta}^{-1}=\frac{1}{m^{q}_{\alpha\beta}}\left({\sf
I}-{\bf\hat{r}\hat{r}}\right)+\frac{1}{m^{n}_{\alpha\beta}}{\bf\hat{r}\hat{r}},$
(32)

and its divergence is

$\nabla_{\bf r}\cdot{\sf M}_{\alpha\beta}=\left(\frac{\partial}{\partial
r}m^{n}_{\alpha\beta}+2\frac{\left(m^{n}_{\alpha\beta}-m^{q}_{\alpha\beta}\right)}{r}\right){\bf\hat{r}}.$
(33)

Using Eq. 32 and Eq. 33 we get

${\sf M}^{-1}_{22}\cdot\left(\nabla_{\bf r}\cdot{\sf
M}_{22}\right)=\frac{1}{m_{22}^{n}}\left(\frac{\partial}{\partial
r}m^{n}_{22}+2\frac{\left(m^{n}_{22}-m^{q}_{22}\right)}{r}\right){\bf\hat{r}}.$
(34)

The form of ${\sf M}$ is given in Appendix B. We see that in Oseen and Rotne-
Pragar approximation of ${\sf M}$ the right hand side of Eq. 34 is zero. The
first nonzero contribution comes when ${\sf M}$ is of order $1/r^{4}$, at
which the self mobilities of the particles are also modified. Substituting
${\sf M}$ to the order $1/r^{4}$ (see Appendix B) in Eq. 34 gives

$\displaystyle{\sf M}^{-1}_{22}\cdot\left(\nabla_{\bf r}\cdot{\sf
M}_{22}\right)=\frac{15ab(a^{3}+b^{3})}{2(a+b)}\frac{1}{r^{5}}\hat{\bf r}.$
(35)

Substituting this expression in Eq. 30 and using
${\bf\tilde{f}}=-\mathrm{U}^{\prime}(r){\bf\hat{r}}$ gives

$\nabla_{{\bf r}}\log P({\bf
r})=-\frac{1}{\lambda}\mathrm{U}^{\prime}(r){\bf\hat{r}}-\frac{15ab(a^{3}+b^{3})}{4(a+b)}\frac{1}{r^{5}}\hat{\bf
r}.$ (36)

Integrating this gives

$\log P({\bf
r})\propto-\frac{1}{\lambda}\mathrm{U}(r)+\frac{15ab(a^{3}+b^{3})}{16(a+b)}\frac{1}{r^{4}},$
(37)

which gives $P({r})\propto e^{-\Phi({\bf r})/\lambda}$, where

$\Phi=\mathrm{U}(r)-\frac{1}{2}\lambda\left[\frac{15ab(a^{3}+b^{3})}{8(a+b)}\frac{1}{r^{4}}+\mathcal{O}\left(\frac{1}{r^{5}}\right)\right].$
(38)

Note that while we can define an effective potential for the marginal dynamics
of $r$ , there is no effective potential description in the full Fokker-Planck
description that includes $\hat{\bf r}$ and ${\bf R}$. If we now assume that
the dynamics of $r$ is fast, we can decompose the probability distribution as

$P({\bf R,\hat{r}},r)=P(r)\int drr^{2}P({\bf R,\hat{r}},r)=P(r)\tilde{P}({\bf
R,\hat{r}}),$ (39)

where we have defined $\tilde{P}({\bf R,\hat{r}})\equiv\int drr^{2}P({\bf
R,\hat{r}},r)$.

Decomposing the derivative as radial and rotational derivative

$\nabla_{{\bf r}}={\bf\hat{r}}\frac{\partial}{\partial
r}+\frac{1}{r}{\bf\hat{r}}\times\mathcal{R},$ (40)

where $\mathcal{R}$ is the rotation operator that rotates the ${\bf r}$
keeping $r$ fixed. In spherical co-ordinates it reads

${\bf\hat{r}}\times\mathcal{R}=\hat{\theta}\frac{\partial}{\partial\theta}+\hat{\phi}\frac{1}{\sin\theta}\frac{\partial}{\partial\phi}.$
(41)

Integrating out $r$ from Eq. IV we get Illien  _et al._ (2017)

$\displaystyle\partial_{t}\tilde{P}$ $\displaystyle=$
$\displaystyle\nabla_{{\bf
R}}\cdot\left[-v_{0}\,{\bf\hat{r}}+\lambda\left\langle\frac{{\sf
M}_{12}}{r}\right\rangle\cdot{\bf\hat{r}}\times\mathcal{R}+\lambda\left\langle{\sf
M}_{11}\right\rangle\cdot\nabla_{{\bf R}}\right]\tilde{P}$ (42)
$\displaystyle+$
$\displaystyle\lambda{\bf\hat{r}}\times\mathcal{R}\cdot\left[\left\langle\frac{{\sf
M}_{12}}{r}\right\rangle\cdot\nabla_{{\bf R}}+\left\langle\frac{{\sf
M}_{22}}{r^{2}}\right\rangle\cdot{\bf\hat{r}}\times\mathcal{R}\right]\tilde{P},$

where the averages are over the distributions $P(r)$,
$\langle\phi(r)\rangle=\int drr^{2}\phi(r)P(r)$, and the self-propelled
velocity

$v_{0}=\int r^{2}dr{\bf\hat{r}}\cdot\left({\sf
M}_{12}\cdot{\bf\tilde{f}}_{2}-\frac{\lambda}{2}\nabla_{{\bf r}}\cdot{\sf
M}_{12}-\lambda\sf{M}_{12}\cdot\nabla_{{\bf r}}\right)P(r).$ (43)

Note that if the fluctuations are thermal, the middle term in Eq. 43 is not
present, the probability distribution has the Boltzmann form $P(r)\propto
e^{-\mathrm{U}/\lambda}$, and the velocity vanishes as shown Appendix C. To
evaluate the average, we consider the bare inter-particle potential to be a
stiff spring, $\mathrm{U}=k(r-l)^{2}/2$, and $l\gg a,b$. In this limit, the
relaxation time scale of $r$ is set by $\sf{M_{22}}$, $k$, and $\lambda$;
$kl^{2}/2\lambda\gg 1$ ensures that the $r$-dynamics is fast Illien  _et al._
(2017). Using the Laplace approximation Bender and Orszag (1999), we obtain,
to leading order in $1/l$, the propulsion velocity

$v_{0}=\frac{5ab}{4\pi\eta}(b-a)\frac{\lambda}{l^{5}}+\mathcal{O}\left(\frac{1}{l^{6}}\right).$
(44)

directed along ${\bf\hat{r}}$, if $b>a$. Orientation decorrelation will lead
to diffusion over times longer than the orientation correlation time of
${\bf\hat{r}}$ ($\tau_{R}$). The enhancement of the diffusion constant due to
this self-propulsion in units of diffusion of a sphere of radius $l$ is
$v_{0}^{2}\tau_{R}/D_{t}=(1-\delta)^{2}\delta^{2}\epsilon^{6}$, where
$D_{t}=\lambda/6\pi\eta l$, $\tau_{R}\sim 3\eta l^{3}/\lambda$, $\delta=a/b$,
and $\epsilon=b/l$. Since $\delta<1$ and $\epsilon\ll 1$, this enhancement is
very small.

##  V  Activating the particle by a fluctuating force

Now consider two colloidal spheres of radii $a$ and $b$, embedded in an
unbounded passive fluid, each of which experiences a stochastic force,
localized on the colloids. To make the discussion simple, we set the stress
fluctuations of the embedding medium $\sigma^{n}_{ij}=0$; the dynamics of the
colloids is then given by Eq. 20, with ${\bm{\vartheta}_{\beta}}=0$, and the
applied force on the particles is a sum of deterministic and stochastic
components, ${\bf f_{\beta}}\equiv{\bf f_{\beta}}+{\bm{\xi}_{\beta}}$. The
fluctuating force on the two colloids is taken to be isotropic, zero mean
Gaussian white with correlations,

$\langle{\bf\xi}_{\alpha i}(t){\bf\xi}_{\beta
j}(s)\rangle=2\Lambda\delta_{\alpha\beta}\delta_{ij}\delta(t-s).$ (45)

Since the fluctuation is external and the dissipation is in the fluid bath,
the system is always active. Hence, for any choice of noise convention the
fluctuation dissipation relation is not satisfied. For consistency with the
approximation for passive fluid, we take the noise correlation time to be the
slowest timescale leading to Stratonovich noise convention. The corresponding
Fokker-Planck equation (in Stratonovich convention) is now (see Appendix A),

$\displaystyle\partial_{t}P=\nabla_{{\bf R}}\cdot\left(-{\sf
M}_{11}\cdot{\bf\tilde{f}}_{1}-{\sf M}_{12}\cdot{\bf\tilde{f}}_{2}+\Lambda{\sf
D}_{11}\cdot\nabla_{{\bf R}}\right)P$ (46) $\displaystyle+\Lambda\nabla_{{\bf
R}}\cdot\left(\frac{1}{2}\nabla_{{\bf r}}\cdot{\sf D}_{12}+{\sf
D}_{12}\cdot\nabla_{{\bf r}}\right)P+\nabla_{{\bf r}}\cdot\left(\Lambda{\sf
D}_{12}\cdot\nabla_{{\bf R}}\right)P$ $\displaystyle+\nabla_{{\bf
r}}\cdot\left(-{\sf M}_{12}\cdot{\bf\tilde{f}}_{1}-{\sf
M}_{22}\cdot{\bf\tilde{f}}_{2}+\frac{1}{2}\Lambda\nabla_{{\bf r}}\cdot{\sf
D}_{22}+\Lambda{\sf D}_{22}\cdot\nabla_{{\bf r}}\right)P,$

where, ${\sf M}$ is given by Eq. 24, and the diffusion matrix ${\sf
D}_{\alpha\beta}$ is $3\times 3$ matrix given in terms of ${\sf
H}_{\alpha\beta}$ by ${\sf D}_{11}=\left({\sf H}_{11}+{\sf
H}_{12}\right)^{2}+\left({\sf H}_{22}+{\sf H}_{12}\right)^{2}$, ${\sf
D}_{12}={\sf D}_{21}={\sf H}_{22}^{2}-{\sf H}_{11}^{2}$, and ${\sf
D}_{22}=\left({\sf H}_{11}-{\sf H}_{12}\right)^{2}+\left({\sf H}_{22}-{\sf
H}_{12}\right)^{2}$.

Once again, Eq. 46 does not satisfy the potential condition given by Eq. 27,
and hence does not have a zero probability current steady state, even when the
spheres are of the same size. This proves that the dynamics of particles in an
active medium is fundamentally different from the dynamics of activated
particles. In this context, we refer to recent experiments Berut  _et al._
(2014); Bérut  _et al._ (2016) in which two spheres embedded in a fluid are
held in two optical traps. A fluctuating force is applied on one sphere by
moving the position of its laser trap randomly. This has been modeled as a two
temperature system Hough and Ou-Yang (2002); Berut  _et al._ (2014); Bérut
_et al._ (2016), where the static particle feels the bath temperature and the
particle in the fluctuating trap, a higher temperature. Our study demonstrates
the inadequacy of such an effective temperature approach, and in principle (at
least numerically) provides a full solution to the steady state distribution.

We now consider the case when $\mathrm{U}$ is a function of $r$ alone implying
${\bf\tilde{f}}_{1}=0$ and ${\bf\tilde{f}}_{2}=-\mathrm{U}{\bf\hat{r}}$. For
this case Eq. 46 reduces to

$\displaystyle\partial_{t}P$ $\displaystyle=\nabla_{{\bf R}}\cdot\left(-{\sf
M}_{12}\cdot{\bf\tilde{f}}_{2}+\frac{1}{2}\nabla_{{\bf r}}\cdot{\sf
D}_{12}+\Lambda{\sf D}_{11}\cdot\nabla_{{\bf R}}\right)P$ (47)
$\displaystyle+\Lambda\nabla_{{\bf R}}\cdot\left({\sf D}_{12}\cdot\nabla_{{\bf
r}}\right)P+\nabla_{{\bf r}}\cdot\left(\Lambda{\sf D}_{12}\cdot\nabla_{{\bf
R}}\right)P$ $\displaystyle+\nabla_{{\bf r}}\cdot\left(-{\sf
M}_{22}\cdot{\bf\tilde{f}}_{2}+\frac{1}{2}\Lambda\nabla_{{\bf r}}\cdot{\sf
D}_{22}+\Lambda{\sf D}_{22}\cdot\nabla_{{\bf r}}\right)P.\qquad$

The steady state for the marginal distribution $P(r)$ obtained by integrating
Eq. 47 over ${\bf R}$, with no flux boundary condition gives

$\nabla_{{\bf r}}\log P({\bf r})=\frac{1}{\Lambda}{\sf D}_{22}^{-1}\cdot{\sf
M}_{22}\cdot{\bf\tilde{f}}_{2}-\frac{1}{2}{\sf D}_{22}^{-1}\cdot\nabla_{{\bf
r}}\cdot{\sf D}_{22}.$ (48)

The tensor ${\sf D}$ decomposed as sum of projection along $\hat{\bf r}$
(denoted by superscript $n$) and perpendicular to $\hat{\bf r}$ (denoted by
superscript $q$) is

${\sf D}_{\alpha\beta}=d_{\alpha\beta}^{q}(r)\left({\bf\mathrm{\sf
I}}-{\bf\mathrm{\bf\hat{r}\hat{r}}}\right)+d_{\alpha\beta}^{n}(r){\bf\mathrm{\bf\hat{r}\hat{r}}}.$
(49)

The inverse is

${\sf D}_{\alpha\beta}^{-1}=\frac{1}{d^{q}_{\alpha\beta}}\left({\sf
I}-{\bf\hat{r}\hat{r}}\right)+\frac{1}{d^{n}_{\alpha\beta}}{\bf\hat{r}\hat{r}},$
(50)

and the divergence is

$\nabla_{\bf r}\cdot{\sf D}_{\alpha\beta}=\left(\frac{\partial}{\partial
r}d^{n}_{\alpha\beta}+2\frac{\left(d^{n}_{\alpha\beta}-d^{q}_{\alpha\beta}\right)}{r}\right){\bf\hat{r}}.$
(51)

Using Eq. 50, and Eq. 51 we get

${\sf D}^{-1}_{22}\cdot\left(\nabla_{\bf r}\cdot{\sf
D}_{22}\right)=\frac{1}{d_{22}^{n}}\left(\frac{\partial}{\partial
r}d^{n}_{22}+2\frac{\left(d^{n}_{22}-d^{q}_{22}\right)}{r}\right){\bf\hat{r}}.$
(52)

Taking the diffusion tensor $\sf{D}$ (see Appendix D) to the order $1/r$ and
substituting it in Eq. 52 gives

${\sf D}^{-1}_{22}\cdot\left(\nabla_{\bf r}\cdot{\sf
D}_{22}\right)=-\frac{9(a^{2}b^{2})}{4(a^{2}+b^{2})}\frac{1}{r^{3}}\hat{\bf
r},$ (53)

and substituting the diffusion tensor in the first term on the right of Eq. 48
we obtain

${\sf D}_{22}^{-1}\cdot{\sf
M}_{22}\cdot{\bf\tilde{f}}_{2}=-\frac{m_{22}^{n}}{d_{22}^{n}}{\sf
U^{\prime}}(r)\hat{\bf r}$ (54)

Substituting Eq. 53 and 54 in Eq. 48 gives

$\nabla_{{\bf r}}\log P({\bf
r})=-\frac{1}{\Lambda}\frac{m_{22}^{n}}{d_{22}^{n}}{\sf
U^{\prime}}(r){\bf\hat{r}}+\frac{9(a^{2}b^{2})}{8(a^{2}+b^{2})}\frac{1}{r^{3}}\hat{\bf
r}.$ (55)

From this the effective potential $\Phi\equiv-\Lambda\log P$ upon integration
of Eq. 55 is

$\Phi=\int
drr^{2}\frac{m_{22}^{n}}{d_{22}^{n}}{\mathrm{U}^{\prime}}+\Lambda\frac{9(a^{2}b^{2})}{16(a^{2}+b^{2})}\frac{1}{r^{2}}+\mathcal{O}\left(\frac{1}{r^{3}}\right).$
(56)

Note that in this case $m_{22}/d_{22}$ depends on $r$, it is not possible to
define an effective free energy, keeping the energy $\mathrm{U}$ and a
constant effective temperature. As in the colloid-colloid interaction in an
active fluid, the steady state distribution is non-Boltzmann with a
fluctuation-induced particle-particle interaction. However, in contrast, the
interaction is repulsive, long-ranged, and depends on the form of the
interaction potential $\mathrm{U}$. This effect is similar to that of
effective colloid-wall interactions due interplay between hydrodynamic and
electrostatic interactions Saha and Ramaswamy (2014).

Integrating out $r$ from Eq. 47 leads to the similar form of self-propulsion
velocity as Eq. 43. To order $1/r^{3}$ ${\sf M}_{12}$ is a constant and self-
propulsion velocity is

$v_{0}=-N\int
drr^{2}\left(m^{n}_{12}\mathrm{U}^{\prime}-d^{n}_{12}\frac{\partial}{\partial
r}\Phi\right)e^{-\Phi/\Lambda}.$ (57)

Using Eq. 56 we get

$\frac{\partial}{\partial
r}\Phi=\frac{m_{22}^{n}}{d_{22}^{n}}{\mathrm{U}^{\prime}}-\Lambda\frac{9(a^{2}b^{2})}{8(a^{2}+b^{2})}\frac{1}{r^{3}}+\mathcal{O}\left(\frac{1}{r^{4}}\right).$
(58)

and

$\frac{m_{22}^{n}}{d_{22}^{n}}=\frac{(6\pi\eta
ba)(a+b)}{a^{2}+b^{2}}\left[1+\frac{6b^{2}a^{2}}{(a^{2}+b^{2})(a+b)}\frac{1}{r}+\mathcal{O}\left(\frac{1}{r^{2}}\right)\right].$
(59)

Thus to leading order in $1/r$ we obtain

$v_{0}=-N\int
drr^{2}m^{n}_{12}\left(1-\frac{d^{n}_{12}m^{n}_{22}}{d^{n}_{22}m^{n}_{12}}\right){\mathrm{U}^{\prime}}e^{-{\mathrm{U}}/\Lambda^{\prime}},$
(60)

where $\Lambda^{\prime}=\Lambda(a^{2}+b^{2})/6\pi\eta ab(a+b)$. As before,
taking $\mathrm{U}=k(r-l)^{2}/2$, with $k$ large and $l\gg a,b$, such that
$kl^{2}/2\Lambda\gg 1$, gives the mean self-propulsion velocity of the dimer
to be

$v_{0}\sim\frac{a-b}{18\pi^{2}\eta^{2}ab(a+b)}\frac{\Lambda}{l},$

directed along $\hat{\bf r}$. Note that this leading order contribution is due
to the interaction potential between the dimer. In contrast, the leading order
contribution in the fluctuation fluid case was from the fluctuation induced
interaction. The long time dynamics of the dimer is diffusive, thus resembling
an active Brownian particle Romanczuk  _et al._ (2012). The enhancement in
diffusion constant over the bare diffusion $D_{t}\sim\mathrm{k_{B}T}/6\pi\eta
l$ is

$\frac{v_{0}^{2}\tau_{R}}{D_{t}}\sim\frac{(1-\delta)^{2}}{(1+\delta^{2})^{2}\epsilon^{2}}\frac{\Lambda^{\prime
2}}{\mathrm{k_{B}T}^{2}}$ (61)

where the rotational correlation time is $\tau_{R}\sim 3\eta l^{3}/\lambda$,
$\delta=a/b$, $\epsilon=b/l$, and
$\Lambda^{\prime}=\Lambda(a^{2}+b^{2})/6\pi\eta ab(a+b)$. Since both
$\mathrm{k_{B}T}/\Lambda^{\prime},\epsilon\ll 1$, this enhancement can now be
large compared to the bare diffusion. For
$\Lambda^{\prime}\sim\mathrm{k_{B}T}$, $a=2\,\mathrm{nm}$, $b=3\,\mathrm{nm}$,
$l=10\,\mathrm{nm}$, and $\eta=10^{-3}\,\mathrm{Pa\,s}$ we get
$v_{0}=5\,\mathrm{mm/s}$ and $v_{0}^{2}\tau_{R}\sim 25\,\mu\mathrm{m^{2}/s}$,
a value which is comparable to thermal diffusivity $D_{t}$.

###  V.1 One-Dimensional example of self-propulsion

To obtain an intuitive understanding of the self-propulsion described above,
consider a simple example in which two spherical colloids interacting via a
harmonic potential are confined along the $x$-axis. The center of the spheres
are positioned at $x_{1}(t)$ and $x_{2}(t)$ with $x_{1}<x_{2}$. This dimer is
activated by an external force $f$ acting on the colloid at $x_{1}$. The one-
dimensional Langevin dynamics for this dimer, with the hydrodynamic
interaction tensor Kim and Karilla (1991) (Appendix B) can be obtained from
Eq. 20,

$\displaystyle\dot{x}_{1}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{6\pi\eta a}-\frac{1}{4\pi\eta
r}\right)k(r-l)+\frac{f}{6\pi\eta a},$ (62) $\displaystyle\dot{x}_{2}$
$\displaystyle=$ $\displaystyle-\left(\frac{1}{6\pi\eta a}-\frac{1}{4\pi\eta
r}\right)k(r-l)+\frac{f}{4\pi\eta a},$ (63)

where $r=x_{2}-x_{1}$. In terms of the separation $r$ and center of mass
$R=x_{1}+x_{2}$ we get

$\displaystyle\dot{r}$ $\displaystyle=$ $\displaystyle-\left(\frac{1}{6\pi\eta
a}-\frac{1}{4\pi\eta r}\right)(2k(r-l)+f),$ (64) $\displaystyle\dot{R}$
$\displaystyle=$ $\displaystyle\left(\frac{1}{6\pi\eta a}+\frac{1}{4\pi\eta
r}\right)f.$ (65)

At steady state $r=l-f/2k$ and the center of mass velocity is

$\dot{R}=\frac{f}{6\pi\eta a}+\frac{f}{4\pi\eta\left(l-f/2k\right)}.$ (66)

Thus we see that the magnitude of $\dot{R}$ are not equal for $f\to-f$. For
positive $f$ the harmonic spring is compressed leading to a larger speed in
comparison to the case when $f$ is negative and the spring is stretched
leading to a lower speed. Now consider a periodic symmetric driving where $f$
switches direction after fixed time intervals of duration $\Delta t$. The
average force applied is zero, however, the dimer will have a net positive
velocity given by

$\langle\dot{R}\rangle=\frac{f^{2}/k}{8\pi\eta\left(l^{2}-(f/2k)^{2}\right)}.$
(67)

This mechanism is similar to that proposed in ref. Kumar  _et al._ (2008);
Baule  _et al._ (2008) for the self-propulsion of an elastic dimer on a
frictional substrate. This setup can be experimentally realized by connecting
a paramagnetic and a diamagnetic bead by a polymer and applying a magnetic
field gradient which periodically switches direction.

##  VI Discussion

We have seen that the interplay between nonequilibrium fluctuations and
hydrodynamics, even in the simplest extension of a single particle embedded in
a momentum conserving fluid, viz., two particles embedded in an unbounded
active gel or a particle in an active gel bound within confining walls, brings
out the inadequacy of the effective temperature description, since the
corresponding steady state probability distribution shows strong departures
from the equilibrium Boltzmann form. This leads to a non-equilibrium effective
“Casimir-like” power law interaction Kirkpatrick  _et al._ (2013); Aminov  _et
al._ (2015); Cattuto  _et al._ (2006); Bartolo  _et al._ (2003); Parra-Rojas
and Soto (2014); Brito  _et al._ (2007). Furthermore, both in an active fluid
or in a passive fluid activated by stochastic forcing, unequal size particles
exhibit short time ballistic motion. This shows that the effective equilibrium
limits, that have been obtained for “dry” active systems, do not hold for
“wet” active systems. The effective interaction between activated particles
can be experimentally verified by more precise measurements of the particle
position in the optical trap setups used in the experiments reported in Ref.
Bérut  _et al._ (2016).

We expect the interplay between hydrodynamic interactions and active
fluctuations will also lead to effective interaction between self-propelled
particles in a passive fluid. Since the effective interactions are sensitive
to the precise origin of the fluctuation, the form of effective interaction
may be different for different models of self-propelled particles. There are
various extensions of this work that will be useful to explore. So far in this
work we only look at incompressible fluid, it would be of interest to analyze
the behavior for compressible fluids. The form of the effective interaction
and self-propulsion velocity was computed in the far-field approximation of
the hydrodynamic interaction. We expect the effective interaction and self-
propulsion to exist in lubrication limit as well.

##  VII Acknowledgements

We thank M. Wyart, J.-F. Rupprecht, R. Morris, K.B. Husain and S.A. Rautu for
useful discussions.

##  Appendix A Fokker-Planck from Langevin

Following Méndez  _et al._ (2014) we derive the Fokker-Planck equation from a
multivariate overdamped Langevin equation with multiplicative noise, for
general choice of stochastic calculus. The Langevin equations are

$\displaystyle\dot{x}_{i}=F_{i}({\bf x})+G_{ij}({\bf x})\,\vartheta_{j},$ (68)

where $i\in(1,...,N)$ and $j\in(1,...,M)$, $\vartheta_{j}$ is a zero mean
Gaussian white noise with correlation

$\langle\vartheta_{k}(t)\vartheta_{j}(t^{\prime})\rangle=C_{kj}({\bf
x})\delta(t-t^{\prime}),$ (69)

where $(k,j)\in(1,...,M)$. Integrating Eq. 68 over a small time interval
$\Delta t$ gives

$\Delta x_{i}=\int_{t}^{t+\Delta t}dt^{\prime}F_{i}({\bf
x})+\int_{t}^{t+\Delta t}dt^{\prime}\,G_{ij}({\bf x})\vartheta_{j}.$ (70)

The first term on the right is unambiguously approximated for small $\Delta t$
using a straightforward Taylor expansion of $F_{i}$, in contrast, the limit of
the second term is not well defined Gardiner (2009); Lau and Lubensky (2007).
Unlike deterministic calculus, there are multiple choices for stochastic
calculus. This amounts to choosing the time between $t$ and $t+\Delta t$ at
which ${\bf x}$ in $G({\bf x})$ is evaluated. If $G({\bf x})$ is evaluated at
$t$ it is Ito calculus Gardiner (2009), at $t+\Delta t/2$ it is Stratonovich
calculus Stratonovich (1966); Gardiner (2009), and at $t+\Delta t$ it is
Hanggi-Klimontovich calculus Hänggi and Thomas (1982); Klimontovich (1990).
Using the definition introduced in Lau and Lubensky (2007), ${\bf x}(t)$
evaluated at any generic point between $t$ and $t+\Delta t$, parameterized by
$\epsilon\in(0,1)$ gives

$\Delta x_{i}=F_{i}({\bf x}_{t})\Delta t+G_{ij}({\bf x}_{t}+\epsilon\Delta{\bf
x}))\int_{t}^{t+\Delta t}dt^{\prime}\vartheta_{j}(t^{\prime}).$ (71)

Taylor expanding $G({\bf x})$ around ${\bf x}(t)$

$\displaystyle\Delta x_{i}$ $\displaystyle=$ $\displaystyle F_{i}({\bf
x}_{t})\Delta t+G_{ij}({\bf x}_{t})\int_{t}^{t+\Delta
t}dt^{\prime}\vartheta_{j}(t^{\prime})$ (72) $\displaystyle+$
$\displaystyle\epsilon\frac{\partial G_{ij}({\bf x}_{t})}{\partial
x_{k}}\Delta x_{k}\int_{t}^{t+\Delta t}dt^{\prime}\vartheta_{j}(t^{\prime}).$

$\Delta x_{k}$ has a term of order $\sqrt{\Delta t}$ hence the $\epsilon$ term
has a contribution of order $\Delta t$, substituting $\Delta x_{k}$ back in
the equation and keeping terms to order $\Delta t$ we get

$\displaystyle\Delta x_{i}$ $\displaystyle=$ $\displaystyle F_{i}({\bf
x}_{t})\Delta t+G_{ij}({\bf x}_{t})\int_{t}^{t+\Delta
t}dt^{\prime}\vartheta_{j}(t^{\prime})$ $\displaystyle+$
$\displaystyle\epsilon\frac{\partial G_{ij}({\bf x}_{t})}{\partial
x_{k}}G_{kl}({\bf x}_{t})\int_{t}^{t+\Delta
t}dt^{\prime}\vartheta_{l}(t^{\prime})\int_{t}^{t+\Delta
t}dt^{\prime\prime}\vartheta_{j}(t^{\prime\prime}).$

The first and the second moment of $\Delta x$ are

$\displaystyle\langle\Delta x_{i}\rangle=F_{i}({\bf x}_{t})\Delta
t+\epsilon\frac{\partial G_{ij}({\bf x}_{t})}{\partial x_{k}}G_{kl}({\bf
x}_{t})C_{lj}({\bf x}_{t})\Delta t,\quad$ (74) $\displaystyle\langle\Delta
x_{i}\Delta x_{l}\rangle=G_{ij}G_{lk}C_{jk}\Delta t.$ (75)

The corresponding Fokker-Plank equation is Gardiner (2009)

$\partial_{t}P=\frac{\partial}{\partial
x_{i}}\left(-F_{i}-\epsilon\frac{\partial G_{ik}}{\partial
x_{j}}S_{jk}+\frac{1}{2}\frac{\partial}{\partial x_{j}}G_{ik}S_{jk}\right)P,$
(76)

where $S_{ij}=C_{ik}G_{jk}$. Thus we see that different choices of stochastic
calculus ($\epsilon$) leads to different Fokker-Planck equations and hence
different physics.

For a given problem, the relevant value of $\epsilon$ depends on the fast
timescales which have been integrated out. Furthermore the existence of a
simple convention choice is not always guaranteed Rupprecht  _et al._ (2018).
In this paper, the effective description of aviscoelastic gel is obtained by
integrating out: the viscoelastic relaxation time ($\tau_{v}$), the inertial
relaxation time ($\tau_{m}=m/\eta$), and the noise correlation time
($\tau_{n}$). In Rupprecht  _et al._ (2018); Singh Vishen  _et al._ (2018) it
was shown that for an exponentially correlated noise with
$\tau_{n}\gg\sqrt{\tau_{m}\tau_{v}}$ $\epsilon=1/2$ (Stratonovich convention)
is the right value. For thermal noise, in general, no simple convention works.
But if $\frac{G_{ik}}{\partial x_{j}}S_{jk}=G_{ik}\frac{S_{jk}}{\partial
x_{j}}$, $\epsilon=1$ (Hanggi-Klimontovich convention) is the right value (see
Lau and Lubensky (2007) for a detailed discussion on noise convention for
thermal fluctuations). These are the convention choices used throughout this
paper.

The Langevin equations in the main text are of the form given by Eq. 68 and
the corresponding Fokker-Planck will be given by Eq. 76 by making the
following identifications in different cases

  * •

Wall-particle: Comparing Eq. 68 with Eq. 7,8 and noise correlation given by
Eq. 9 we get: ${\bf F}=-\nabla\mathrm{U}\,\hat{z}$, ${\sf G}={\sf H}$, and
${\sf C}=2\lambda{\sf H}^{-1}$, which gives ${\sf S}=2\lambda{\sf
H}^{-1}\cdot{\sf H}=2\lambda\,{\sf I}$,where ${\sf I}$ is $3\times 3$ identity
matrix. Using these values in Eq. 76 leads to Eq. $8$.

  * •

Fluctuating fluid: Comparing Eq. 68 with Eq. 20 and noise correlation given by
Eq. 21 we get: ${\sf G}={\sf H}$, and ${\sf C}=2\lambda{\sf H}^{-1}$, and
${\sf S}={\sf C\cdot H}=2\lambda{\sf I}$, where ${\sf I}$ is $6\times 6$
identity matrix. Using the above values in Eq. 76 we get Eq. $13$.

  * •

Fluctuating force: Comparing Eq. 68 and Eq. $11$ and the noise correlation
given by Eq. 45 we get: ${\sf G}={\sf H}$, and ${\sf C}=2\Lambda{\sf I}$, and
${\sf S}={\sf C\cdot H}=2\Lambda{\sf H}$, where ${\sf I}$ is the $6\times 6$
identity matrix. Using the above values in Eq. 76 and changing of variables to
${\bf R}$ and ${\bf r}$ we get Eq. 46.

##  Appendix B Hydrodynamic interaction tensor

###  B.1 Wall-Particle

For distances larger than the particle size ($z\gg a$), the mobilities can be
calculated as a power series in the inverse of separation from the wall
($1/z$). To second order (Rotne-Prager approximation) the mobility
longitudinal to the wall is Kim and Karilla (1991); Rotne and Prager (1969)

$\mathrm{H}_{z}=\frac{1}{6\pi\eta
a}\left(1-\frac{9}{8}\frac{a}{z}+\frac{1}{2}\frac{a^{3}}{z^{3}}\right),$ (77)

and the mobility transverse to the wall is

${\mathrm{H}}_{x_{\perp}}=\frac{1}{6\pi\eta
a}\left(1-\frac{9}{16}\frac{a}{z}+\frac{1}{8}\frac{a^{3}}{z^{3}}\right).$ (78)

###  B.2 Two spheres

The hydrodynamic interaction tensor ${\sf H_{\alpha\beta}}$ coupling the
translational degree of freedom is a $3\times 3$ tensor. This tensor can be
calculated as a power series in the inverse of separation ($1/r$) between the
center of the two spheres. To the fourth power it is given by Kim and Karilla
(1991)

$\displaystyle{\sf H_{11}}$ $\displaystyle=$ $\displaystyle\frac{1}{6\pi\eta
a}({\sf I}-{\bf\hat{r}\hat{r}})+\left(\frac{1}{6\pi\eta
a}-\frac{5}{8}\frac{b^{3}}{\pi\eta r^{4}}\right){\bf\hat{r}\hat{r}},$
$\displaystyle{\sf H_{22}}$ $\displaystyle=$ $\displaystyle\frac{1}{6\pi\eta
b}({\sf I}-{\bf\hat{r}\hat{r}})+\left(\frac{1}{6\pi\eta
b}-\frac{5}{8}\frac{a^{3}}{\pi\eta r^{4}}\right){\bf\hat{r}\hat{r}},$
$\displaystyle{\sf H_{12}}={\sf H_{21}}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{8\pi\eta
r}+\frac{1}{24}\frac{(a^{2}+b^{2})}{\pi\eta r^{3}}\right)({\sf
I}-{\bf\hat{r}\hat{r}})$ (79) $\displaystyle+$
$\displaystyle\left(\frac{1}{4\pi\eta
r}+\frac{1}{12}\frac{(a^{2}+b^{2})}{\pi\eta r^{3}}\right){\bf\hat{r}\hat{r}}.$

The mobility tensor ${\sf M}$ defined in Eq. $15$ of the main text decomposed
as sum of projection along $\hat{\bf r}$ (denoted by superscript $n$) and
perpendicular to $\hat{\bf r}$ (denoted by superscript $q$) is

${\sf M}_{\alpha\beta}=m_{\alpha\beta}^{q}(r)\left({\bf\mathrm{\sf
I}}-{\bf\mathrm{\bf\hat{r}\hat{r}}}\right)+m_{\alpha\beta}^{n}(r){\bf\mathrm{\bf\hat{r}\hat{r}}}.$
(80)

By substituting Eq. B.2 into Eq. 80 we get

$\displaystyle m_{11}^{q}$ $\displaystyle=$
$\displaystyle\frac{(a+b)}{6\pi\eta ab}+\frac{1}{4\pi\eta
r}+\frac{1}{12}\frac{(a^{2}+b^{2})}{\pi\eta r^{3}},$ (81) $\displaystyle
m_{11}^{n}$ $\displaystyle=$ $\displaystyle\frac{(a+b)}{6\pi\eta
ab}+\frac{1}{2\pi\eta r}-\frac{1}{6}\frac{(a^{2}+b^{2})}{\pi\eta
r^{3}}-\frac{5}{8}\frac{a^{3}+b^{3}}{\pi\eta r^{4}},\qquad$ (82)
$\displaystyle m_{12}^{n}$ $\displaystyle=$ $\displaystyle\frac{a-b}{6\pi\eta
ab}+\frac{5}{8}\frac{(b^{3}-a^{3})}{\pi\eta r^{4}},$ (83) $\displaystyle
m_{12}^{q}$ $\displaystyle=$ $\displaystyle\frac{a-b}{6\pi\eta ab},$ (84)
$\displaystyle m_{22}^{q}$ $\displaystyle=$
$\displaystyle\frac{(a+b)}{6\pi\eta ab}-\frac{1}{4\pi\eta
r}-\frac{1}{12}\frac{(a^{2}+b^{2})}{\pi\eta r^{3}},$ (85) $\displaystyle
m_{22}^{n}$ $\displaystyle=$ $\displaystyle\frac{(a+b)}{6\pi\eta
ab}-\frac{1}{2\pi\eta r}+\frac{1}{6}\frac{(a^{2}+b^{2})}{\pi\eta
r^{3}}-\frac{5}{8}\frac{a^{3}+b^{3}}{\pi\eta r^{4}}.\qquad$ (86)

##  Appendix C Passive colloids in active fluid - Thermal Fluctuations

For thermal fluctuations, $\lambda=k_{B}T$, and the Fokker-Planck equation
corresponding to Eq. 20 with noise correlation given by Eq. 21 in Hanggi-
Klimontovich convention is

$\displaystyle\partial_{t}P$ $\displaystyle=\nabla_{{\bf R}}\cdot\left(-{\sf
M}_{12}\cdot{\bf\tilde{f}}_{2}+\lambda{\sf M}_{12}\cdot\nabla_{{\bf
r}}+\lambda{\sf M}_{11}\cdot\nabla_{{\bf R}}\right)P$ (87)
$\displaystyle+\nabla_{{\bf r}}\cdot\left(-{\sf
M}_{22}\cdot{\bf\tilde{f}}_{2}+\lambda{\sf M}_{12}\cdot\nabla_{{\bf
R}}+\lambda{\sf M}_{22}\cdot\nabla_{{\bf r}}\right)P.\qquad$

The marginal of steady state distribution of $r$ is $P(r)\propto
e^{-\mathrm{U}/\lambda}$. Integrating out separation $r$ as for active
fluctuation we get the self-propulsion velocity of the form

$v_{0}=\int r^{2}dr{\bf\hat{r}}\cdot\left({\sf
M}_{12}\cdot{\bf\tilde{f}}_{2}-\lambda\sf{M}_{12}\cdot\nabla_{{\bf
r}}\right)P(r).$ (88)

Expanding this we get

$v_{0}=-\int
drr^{2}\left(m^{n}_{12}\mathrm{U}^{\prime}e^{-\mathrm{U}/\lambda}+\lambda
m_{12}^{n}\frac{\partial}{\partial r}e^{-\mathrm{U}/\lambda}\right)=0.$ (89)

As expected for thermal fluctuations the self-propulsion velocity is
identically zero.

##  Appendix D Activated particles in a passive fluid

The effective diffusion tensor for an activated particle in a passive fluid as
defined in Eq. 46 is

${\sf D}=\begin{bmatrix}{\sf L}_{11}+{\sf L}_{22}+{\sf L}_{12}+{\sf
L}_{21}&{\sf L}_{22}-{\sf L}_{11}\\\ {\sf L}_{22}-{\sf L}_{11}&{\sf
L}_{11}+{\sf L}_{22}-{\sf L}_{12}-{\sf L}_{21}\end{bmatrix},$ (90)

where ${\sf L}$ in terms of the hydrodynamic interaction tensor ${\sf H}$ is

${\sf L}=\begin{bmatrix}{\sf H}_{11}\cdot{\sf H}_{11}+{\sf H}_{12}\cdot{\sf
H}_{21}&{\sf H}_{11}\cdot{\sf H}_{12}+{\sf H}_{12}\cdot{\sf H}_{22}\\\ {\sf
H}_{21}\cdot{\sf H}_{11}+{\sf H}_{22}\cdot{\sf H}_{21}&{\sf H}_{12}\cdot{\sf
H}_{21}+{\sf H}_{22}\cdot{\sf H}_{22}\end{bmatrix}.$ (91)

Substituting in Eq. 90 ${\sf H}$ from Eq. B.2 and keeping terms only to order
$1/r$ gives

$\displaystyle{\sf D}_{11}$ $\displaystyle=$
$\displaystyle\left(\left(\frac{1}{6\pi\eta a}+\frac{1}{8\pi\eta
r}\right)^{2}+\left(\frac{1}{6\pi\eta b}+\frac{1}{8\pi\eta
r}\right)^{2}\right)\left({\sf I}-{\bf\hat{r}\hat{r}}\right)\,\,$ (92)
$\displaystyle+$ $\displaystyle\left(\left(\frac{1}{6\pi\eta
a}+\frac{1}{4\pi\eta r}\right)^{2}+\left(\frac{1}{6\pi\eta
b}+\frac{1}{4\pi\eta r}\right)^{2}\right){\bf\hat{r}\hat{r}},$
$\displaystyle{\sf D}_{12}$ $\displaystyle=$ $\displaystyle{\sf
D}_{21}=\left(\frac{1}{(6\pi\eta b)^{2}}-\frac{1}{(6\pi\eta
a)^{2}}\right)\left({\sf I}-{\bf\hat{r}\hat{r}}\right)$ (93) $\displaystyle+$
$\displaystyle\left(\frac{1}{(6\pi\eta b)^{2}}-\frac{1}{(6\pi\eta
a)^{2}}\right){\bf\hat{r}\hat{r}},$ $\displaystyle{\sf D}_{22}$
$\displaystyle=$ $\displaystyle\left(\left(\frac{1}{6\pi\eta
a}-\frac{1}{8\pi\eta r}\right)^{2}+\left(\frac{1}{6\pi\eta
b}-\frac{1}{8\pi\eta r}\right)^{2}\right)\left({\sf
I}-{\bf\hat{r}\hat{r}}\right)$ (94) $\displaystyle+$
$\displaystyle\left(\left(\frac{1}{6\pi\eta a}-\frac{1}{4\pi\eta
r}\right)^{2}+\left(\frac{1}{6\pi\eta b}-\frac{1}{4\pi\eta
r}\right)^{2}\right){\bf\hat{r}\hat{r}}.$

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7680114 |  5811$\pm$77 |  7.68${}_{-1.28}^{+1.45}$ |  1.06${}_{-0.05}^{+0.04}$ |  0.05$\pm$0.10 |  4.172${}_{-0.010}^{+0.008}$ |  0.26${}_{-0.04}^{+0.05}$ |  37.0${}_{-16.3}^{+34.3}$ |  27.3${}_{-13.5}^{+19.6}$ | 0 | MS | L
7771282 |  6248$\pm$77 |  3.24${}_{-0.32}^{+0.35}$ |  1.29${}_{-0.03}^{+0.03}$ |  -0.02$\pm$0.10 |  4.112${}_{-0.007}^{+0.007}$ |  1.01${}_{-0.18}^{+0.14}$ |  69.5${}_{-17.8}^{+14.6}$ |  10.4${}_{-1.7}^{+2.3}$ | 0 | MS | L
7871531 |  5501$\pm$77 |  9.96${}_{-1.77}^{+1.93}$ |  0.83${}_{-0.02}^{+0.03}$ |  -0.26$\pm$0.10 |  4.478${}_{-0.005}^{+0.007}$ |  0.33${}_{-0.03}^{+0.03}$ |  71.3${}_{-13.2}^{+12.0}$ |  33.1${}_{-4.1}^{+4.1}$ | 0 | MS | L
7940546 |  6235$\pm$77 |  2.33${}_{-0.08}^{+0.08}$ |  1.40${}_{-0.01}^{+0.03}$ |  -0.20$\pm$0.10 |  4.007${}_{-0.001}^{+0.003}$ |  1.14${}_{-0.03}^{+0.03}$ |  78.9${}_{-7.9}^{+7.4}$ |  9.9${}_{-0.4}^{+0.3}$ | 0 | MS | L
7970740 |  5309$\pm$77 |  12.98${}_{-2.00}^{+1.36}$ |  0.73${}_{-0.01}^{+0.03}$ |  -0.54$\pm$0.10 |  4.539${}_{-0.005}^{+0.004}$ |  0.26${}_{-0.02}^{+0.03}$ |  60.7${}_{-15.3}^{+17.4}$ |  39.2${}_{-9.0}^{+6.7}$ | 0 | MS | L
8006161 |  5488$\pm$77 |  3.59${}_{-1.45}^{+1.53}$ |  0.98${}_{-0.03}^{+0.03}$ |  0.34$\pm$0.10 |  4.494${}_{-0.007}^{+0.007}$ |  0.34${}_{-0.02}^{+0.02}$ |  37.0${}_{-3.4}^{+4.1}$ |  20.6${}_{-1.8}^{+2.2}$ | 0 | MS | L
8077137 |  6072$\pm$75 |  6.23${}_{-1.23}^{+0.56}$ |  1.12${}_{-0.05}^{+0.04}$ |  -0.09$\pm$0.10 |  4.056${}_{-0.013}^{+0.010}$ |  0.84${}_{-0.07}^{+0.06}$ |  72.8${}_{-12.6}^{+11.3}$ |  13.0${}_{-1.5}^{+1.3}$ | 0 | MS | K
8150065 |  6173$\pm$101 |  3.83${}_{-0.67}^{+0.99}$ |  1.19${}_{-0.05}^{+0.04}$ |  -0.13$\pm$0.15 |  4.220${}_{-0.008}^{+0.008}$ |  0.54${}_{-0.13}^{+0.11}$ |  64.0${}_{-21.1}^{+18.0}$ |  18.6${}_{-4.9}^{+6.4}$ | 0 | MS | L
8179536 |  6343$\pm$77 |  3.54${}_{-0.81}^{+1.04}$ |  1.16${}_{-0.06}^{+0.05}$ |  -0.03$\pm$0.10 |  4.255${}_{-0.010}^{+0.010}$ |  1.46${}_{-0.09}^{+0.10}$ |  55.7${}_{-7.4}^{+13.3}$ |  6.5${}_{-0.8}^{+1.2}$ | 0 | H | L
8228742 |  6122$\pm$77 |  2.89${}_{-0.18}^{+0.16}$ |  1.38${}_{-0.02}^{+0.02}$ |  -0.08$\pm$0.10 |  4.035${}_{-0.005}^{+0.007}$ |  0.64${}_{-0.04}^{+0.04}$ |  37.9${}_{-4.0}^{+6.2}$ |  11.0${}_{-1.3}^{+2.0}$ | 0 | MS | L
8292840 |  6239$\pm$94 |  3.85${}_{-0.75}^{+0.81}$ |  1.15${}_{-0.05}^{+0.05}$ |  -0.14$\pm$0.10 |  4.240${}_{-0.008}^{+0.008}$ |  1.45${}_{-0.07}^{+0.07}$ |  76.2${}_{-9.1}^{+8.9}$ |  7.7${}_{-0.5}^{+0.5}$ | 0 | MS | K
8349582 |  5699$\pm$74 |  8.03${}_{-0.70}^{+0.80}$ |  1.07${}_{-0.02}^{+0.02}$ |  0.30$\pm$0.10 |  4.163${}_{-0.003}^{+0.004}$ |  0.23${}_{-0.06}^{+0.07}$ |  59.6${}_{-24.1}^{+20.7}$ |  41.7${}_{-14.9}^{+19.6}$ | 1 | MS | K
8379927 |  6067$\pm$120 |  1.99${}_{-0.75}^{+0.85}$ |  1.12${}_{-0.04}^{+0.04}$ |  -0.10$\pm$0.15 |  4.388${}_{-0.007}^{+0.008}$ |  1.12${}_{-0.02}^{+0.02}$ |  63.3${}_{-2.3}^{+2.5}$ |  9.2${}_{-0.2}^{+0.3}$ | 0 | MS | L
8394589 |  6143$\pm$77 |  4.45${}_{-0.83}^{+0.94}$ |  1.04${}_{-0.03}^{+0.04}$ |  -0.29$\pm$0.10 |  4.322${}_{-0.008}^{+0.008}$ |  1.01${}_{-0.03}^{+0.03}$ |  71.1${}_{-5.9}^{+7.9}$ |  10.9${}_{-0.6}^{+0.6}$ | 0 | MS | L
8424992 |  5719$\pm$77 |  9.61${}_{-1.74}^{+1.92}$ |  0.92${}_{-0.04}^{+0.04}$ |  -0.12$\pm$0.10 |  4.359${}_{-0.007}^{+0.007}$ |  0.22${}_{-0.06}^{+0.06}$ |  59.1${}_{-30.3}^{+21.3}$ |  42.3${}_{-17.7}^{+19.4}$ | 1 | MS | L
8494142 |  6144$\pm$106 |  2.62${}_{-0.24}^{+0.26}$ |  1.42${}_{-0.02}^{+0.03}$ |  0.13$\pm$0.10 |  4.038${}_{-0.005}^{+0.005}$ |  0.67${}_{-0.28}^{+0.22}$ |  62.8${}_{-23.8}^{+18.5}$ |  14.5${}_{-4.5}^{+9.9}$ | 0 | MS | K
8554498 |  5945$\pm$60 |  5.60${}_{-0.42}^{+0.45}$ |  1.20${}_{-0.03}^{+0.02}$ |  0.17$\pm$0.05 |  4.007${}_{-0.003}^{+0.003}$ |  0.25${}_{-0.09}^{+0.21}$ |  48.1${}_{-36.6}^{+30.0}$ |  35.7${}_{-31.4}^{+25.6}$ | 0 | MS | K
8694723 |  6246$\pm$77 |  4.69${}_{-0.51}^{+0.48}$ |  1.14${}_{-0.02}^{+0.02}$ |  -0.42$\pm$0.10 |  4.113${}_{-0.009}^{+0.007}$ |  0.92${}_{-0.05}^{+0.05}$ |  34.7${}_{-2.7}^{+3.4}$ |  7.2${}_{-0.6}^{+0.8}$ | 0 | MS | L
8760414 |  5873$\pm$77 |  11.66${}_{-1.61}^{+1.28}$ |  0.81${}_{-0.02}^{+0.03}$ |  -0.92$\pm$0.10 |  4.329${}_{-0.005}^{+0.006}$ |  0.69${}_{-0.42}^{+0.26}$ |  7.6${}_{-1.7}^{+2.3}$ |  2.0${}_{-0.4}^{+4.5}$ | 2, 3 | MS | L
8866102 |  6325$\pm$75 |  2.60${}_{-0.53}^{+0.56}$ |  1.23${}_{-0.04}^{+0.04}$ |  0.01$\pm$0.10 |  4.262${}_{-0.007}^{+0.008}$ |  2.15${}_{-0.04}^{+0.04}$ |  78.2${}_{-4.7}^{+6.6}$ |  5.3${}_{-0.2}^{+0.2}$ | 0 | H | K
8938364 |  5677$\pm$77 |  10.25${}_{-0.65}^{+0.56}$ |  0.99${}_{-0.01}^{+0.01}$ |  -0.13$\pm$0.10 |  4.173${}_{-0.002}^{+0.007}$ |  0.61${}_{-0.50}^{+0.27}$ |  8.7${}_{-2.3}^{+57.5}$ |  2.0${}_{-0.2}^{+87.0}$ | 2 | MS | L
9025370 |  5270$\pm$180 |  6.55${}_{-1.13}^{+1.26}$ |  0.97${}_{-0.03}^{+0.03}$ |  -0.12$\pm$0.18 |  4.423${}_{-0.004}^{+0.007}$ |  0.43${}_{-0.04}^{+0.04}$ |  67.5${}_{-19.1}^{+15.2}$ |  24.7${}_{-4.7}^{+3.7}$ | 0 | MS | L
9098294 |  5852$\pm$77 |  8.08${}_{-0.73}^{+0.99}$ |  0.97${}_{-0.03}^{+0.02}$ |  -0.18$\pm$0.10 |  4.308${}_{-0.007}^{+0.005}$ |  0.36${}_{-0.04}^{+0.04}$ |  58.2${}_{-16.3}^{+21.0}$ |  27.2${}_{-7.0}^{+5.7}$ | 0 | MS | L
9139151 |  6302$\pm$77 |  1.32${}_{-0.75}^{+0.94}$ |  1.18${}_{-0.05}^{+0.04}$ |  0.10$\pm$0.10 |  4.382${}_{-0.008}^{+0.008}$ |  0.95${}_{-0.04}^{+0.04}$ |  73.5${}_{-11.0}^{+11.0}$ |  11.6${}_{-1.1}^{+0.8}$ | 0 | H | L
9139163 |  6400$\pm$84 |  1.60${}_{-0.22}^{+0.22}$ |  1.40${}_{-0.02}^{+0.03}$ |  0.15$\pm$0.09 |  4.200${}_{-0.008}^{+0.009}$ |  1.59${}_{-0.08}^{+0.07}$ |  33.5${}_{-3.0}^{+3.0}$ |  4.0${}_{-0.3}^{+0.3}$ | 1 | H | L
9206432 |  6538$\pm$77 |  1.53${}_{-0.30}^{+0.21}$ |  1.38${}_{-0.02}^{+0.04}$ |  0.16$\pm$0.10 |  4.220${}_{-0.007}^{+0.005}$ |  1.55${}_{-0.20}^{+0.17}$ |  34.3${}_{-4.2}^{+5.7}$ |  4.1${}_{-0.5}^{+1.0}$ | 0 | H | L
9353712 |  6278$\pm$77 |  2.15${}_{-0.13}^{+0.11}$ |  1.51${}_{-0.02}^{+0.03}$ |  -0.05$\pm$0.10 |  3.943${}_{-0.005}^{+0.007}$ |  0.75${}_{-0.17}^{+0.16}$ |  37.6${}_{-12.6}^{+29.1}$ |  9.5${}_{-3.9}^{+7.3}$ | 0 | H | L
9410862 |  6047$\pm$77 |  6.93${}_{-1.33}^{+1.49}$ |  0.97${}_{-0.04}^{+0.05}$ |  -0.31$\pm$0.10 |  4.300${}_{-0.008}^{+0.009}$ |  0.41${}_{-0.08}^{+0.09}$ |  46.4${}_{-16.7}^{+28.3}$ |  20.6${}_{-8.1}^{+9.8}$ | 1 | MS | L
9414417 |  6253$\pm$75 |  2.65${}_{-0.18}^{+0.16}$ |  1.40${}_{-0.03}^{+0.02}$ |  -0.13$\pm$0.10 |  4.016${}_{-0.005}^{+0.005}$ |  1.09${}_{-0.05}^{+0.05}$ |  58.1${}_{-6.9}^{+9.7}$ |  9.0${}_{-0.9}^{+1.1}$ | 0 | H | L
9592705 |  6174$\pm$92 |  2.33${}_{-0.16}^{+0.18}$ |  1.51${}_{-0.02}^{+0.03}$ |  0.22$\pm$0.10 |  3.961${}_{-0.004}^{+0.003}$ |  0.93${}_{-0.09}^{+0.09}$ |  59.3${}_{-12.1}^{+17.8}$ |  10.7${}_{-1.9}^{+2.1}$ | 0 | SG | K
9812850 |  6321$\pm$77 |  2.71${}_{-0.35}^{+0.46}$ |  1.37${}_{-0.05}^{+0.04}$ |  -0.07$\pm$0.10 |  4.053${}_{-0.009}^{+0.008}$ |  1.54${}_{-0.08}^{+0.09}$ |  81.0${}_{-10.0}^{+6.4}$ |  7.4${}_{-0.5}^{+0.4}$ | 0 | H | L
9955598 |  5457$\pm$77 |  6.29${}_{-1.84}^{+1.95}$ |  0.90${}_{-0.03}^{+0.04}$ |  0.05$\pm$0.10 |  4.497${}_{-0.005}^{+0.007}$ |  0.29${}_{-0.04}^{+0.04}$ |  53.4${}_{-11.9}^{+20.8}$ |  31.4${}_{-6.4}^{+9.1}$ | 0 | MS | L
9965715 |  5860$\pm$180 |  2.92${}_{-0.75}^{+0.86}$ |  1.21${}_{-0.05}^{+0.04}$ |  -0.44$\pm$0.18 |  4.272${}_{-0.009}^{+0.008}$ |  1.75${}_{-0.05}^{+0.05}$ |  58.3${}_{-3.2}^{+3.5}$ |  5.6${}_{-0.3}^{+0.3}$ | 0 | MS | L
10068307 |  6132$\pm$77 |  2.36${}_{-0.10}^{+0.08}$ |  1.47${}_{-0.02}^{+0.01}$ |  -0.23$\pm$0.10 |  3.967${}_{-0.004}^{+0.004}$ |  0.71${}_{-0.03}^{+0.03}$ |  41.7${}_{-4.1}^{+6.0}$ |  10.9${}_{-1.1}^{+1.5}$ | 0 | SG | L
10079226 |  5949$\pm$77 |  3.06${}_{-0.65}^{+0.70}$ |  1.12${}_{-0.03}^{+0.02}$ |  0.11$\pm$0.10 |  4.366${}_{-0.005}^{+0.005}$ |  0.65${}_{-0.08}^{+0.07}$ |  75.1${}_{-23.4}^{+10.6}$ |  16.8${}_{-3.0}^{+2.2}$ | 1 | MS | L
10162436 |  6146$\pm$77 |  2.46${}_{-0.11}^{+0.10}$ |  1.45${}_{-0.01}^{+0.02}$ |  -0.16$\pm$0.10 |  3.981${}_{-0.005}^{+0.005}$ |  0.84${}_{-0.05}^{+0.05}$ |  25.5${}_{-1.8}^{+2.1}$ |  5.9${}_{-0.4}^{+0.6}$ | 0 | SG | L
10454113 |  6177$\pm$77 |  2.89${}_{-0.53}^{+0.56}$ |  1.17${}_{-0.03}^{+0.02}$ |  -0.07$\pm$0.10 |  4.314${}_{-0.005}^{+0.005}$ |  0.76${}_{-0.07}^{+0.07}$ |  40.9${}_{-9.6}^{+26.9}$ |  10.0${}_{-2.5}^{+4.8}$ | 0 | MS | L
10514430 |  5784$\pm$98 |  7.84${}_{-0.91}^{+0.40}$ |  1.06${}_{-0.02}^{+0.04}$ |  -0.11$\pm$0.11 |  4.061${}_{-0.004}^{+0.004}$ |  0.18${}_{-0.05}^{+0.05}$ |  57.2${}_{-32.0}^{+23.5}$ |  53.6${}_{-27.2}^{+23.6}$ | 1 | MS | K
10516096 |  5964$\pm$77 |  7.01${}_{-1.45}^{+1.33}$ |  1.06${}_{-0.06}^{+0.05}$ |  -0.11$\pm$0.10 |  4.169${}_{-0.011}^{+0.010}$ |  0.48${}_{-0.03}^{+0.03}$ |  71.8${}_{-16.2}^{+12.5}$ |  22.6${}_{-3.1}^{+1.9}$ | 0 | MS | L
10586004 |  5770$\pm$83 |  6.43${}_{-0.61}^{+0.64}$ |  1.18${}_{-0.03}^{+0.02}$ |  0.29$\pm$0.10 |  4.071${}_{-0.005}^{+0.005}$ |  0.48${}_{-0.17}^{+0.16}$ |  59.3${}_{-22.4}^{+20.6}$ |  19.6${}_{-6.5}^{+11.1}$ | 1 | MS | K
10644253 |  6045$\pm$77 |  2.39${}_{-0.96}^{+1.12}$ |  1.10${}_{-0.04}^{+0.04}$ |  0.06$\pm$0.10 |  4.396${}_{-0.008}^{+0.007}$ |  0.24${}_{-0.08}^{+0.08}$ |  55.6${}_{-30.3}^{+24.0}$ |  38.0${}_{-19.1}^{+21.1}$ | 0 | MS | L
10666592 |  6350$\pm$80 |  2.11${}_{-0.24}^{+0.29}$ |  1.50${}_{-0.04}^{+0.04}$ |  0.26$\pm$0.08 |  4.017${}_{-0.007}^{+0.009}$ |  0.91${}_{-0.11}^{+0.11}$ |  47.2${}_{-14.8}^{+27.2}$ |  9.5${}_{-3.2}^{+3.6}$ | 0 | H | K
10730618 |  6150$\pm$180 |  3.05${}_{-0.29}^{+0.46}$ |  1.34${}_{-0.05}^{+0.04}$ |  -0.11$\pm$0.18 |  4.062${}_{-0.007}^{+0.008}$ |  0.56${}_{-0.25}^{+0.23}$ |  55.6${}_{-26.2}^{+23.7}$ |  16.1${}_{-7.2}^{+13.9}$ | 0 | MS | L
10963065 |  6140$\pm$77 |  7.15${}_{-1.61}^{+1.92}$ |  0.99${}_{-0.06}^{+0.06}$ |  -0.19$\pm$0.10 |  4.277${}_{-0.011}^{+0.011}$ |  0.67${}_{-0.03}^{+0.03}$ |  41.8${}_{-3.6}^{+4.7}$ |  11.5${}_{-1.0}^{+1.3}$ | 0 | MS | L
11081729 |  6548$\pm$82 |  1.88${}_{-0.42}^{+0.59}$ |  1.30${}_{-0.05}^{+0.04}$ |  0.11$\pm$0.10 |  4.245${}_{-0.009}^{+0.010}$ |  3.35${}_{-0.10}^{+0.10}$ |  82.9${}_{-5.9}^{+4.9}$ |  3.4${}_{-0.1}^{+0.1}$ | 0 | H | L
11133306 |  5982$\pm$82 |  5.14${}_{-0.88}^{+0.86}$ |  1.06${}_{-0.03}^{+0.04}$ |  -0.02$\pm$0.10 |  4.314${}_{-0.004}^{+0.007}$ |  0.39${}_{-0.13}^{+0.14}$ |  58.0${}_{-24.1}^{+21.9}$ |  24.6${}_{-10.0}^{+14.3}$ | 0 | MS | K
11253226 |  6642$\pm$77 |  1.60${}_{-0.13}^{+0.06}$ |  1.41${}_{-0.01}^{+0.02}$ |  -0.08$\pm$0.10 |  4.173${}_{-0.004}^{+0.005}$ |  2.55${}_{-0.12}^{+0.11}$ |  49.3${}_{-4.4}^{+6.3}$ |  3.4${}_{-0.3}^{+0.4}$ | 0 | H | L
11295426 |  5793$\pm$74 |  6.31${}_{-0.34}^{+0.32}$ |  1.07${}_{-0.02}^{+0.01}$ |  0.12$\pm$0.07 |  4.280${}_{-0.003}^{+0.003}$ |  0.22${}_{-0.03}^{+0.03}$ |  55.6${}_{-20.9}^{+22.9}$ |  42.6${}_{-14.5}^{+11.5}$ | 0 | MS | K
11401755 |  5911$\pm$66 |  7.10${}_{-0.59}^{+0.61}$ |  1.06${}_{-0.02}^{+0.03}$ |  -0.20$\pm$0.06 |  4.039${}_{-0.004}^{+0.004}$ |  0.55${}_{-0.10}^{+0.09}$ |  64.6${}_{-19.5}^{+17.4}$ |  18.5${}_{-4.3}^{+4.7}$ | 0 | MS | K
11772920 |  5180$\pm$180 |  10.67${}_{-2.97}^{+2.73}$ |  0.83${}_{-0.04}^{+0.04}$ |  -0.09$\pm$0.18 |  4.500${}_{-0.008}^{+0.005}$ |  0.31${}_{-0.04}^{+0.03}$ |  70.8${}_{-14.3}^{+12.6}$ |  35.1${}_{-4.7}^{+5.3}$ | 1 | MS | L
11807274 |  6225$\pm$75 |  3.59${}_{-0.45}^{+0.78}$ |  1.24${}_{-0.04}^{+0.04}$ |  0.00$\pm$0.08 |  4.135${}_{-0.007}^{+0.009}$ |  1.42${}_{-0.07}^{+0.07}$ |  76.8${}_{-9.4}^{+8.8}$ |  7.9${}_{-0.5}^{+0.5}$ | 0 | MS | K
11853905 |  5781$\pm$76 |  6.71${}_{-0.67}^{+0.77}$ |  1.12${}_{-0.03}^{+0.02}$ |  0.09$\pm$0.10 |  4.102${}_{-0.005}^{+0.004}$ |  0.27${}_{-0.09}^{+0.09}$ |  54.7${}_{-28.8}^{+24.5}$ |  34.0${}_{-17.2}^{+18.9}$ | 1 | MS | K
11904151 |  5647$\pm$74 |  10.23${}_{-0.67}^{+0.83}$ |  0.92${}_{-0.02}^{+0.01}$ |  -0.15$\pm$0.10 |  4.344${}_{-0.003}^{+0.003}$ |  0.24${}_{-0.07}^{+0.06}$ |  64.9${}_{-28.4}^{+18.1}$ |  40.9${}_{-12.8}^{+16.5}$ | 1 | MS | K
12009504 |  6179$\pm$77 |  3.97${}_{-0.43}^{+0.57}$ |  1.17${}_{-0.04}^{+0.02}$ |  -0.08$\pm$0.10 |  4.211${}_{-0.005}^{+0.007}$ |  1.16${}_{-0.04}^{+0.03}$ |  71.3${}_{-5.7}^{+8.0}$ |  9.4${}_{-0.5}^{+0.5}$ | 0 | MS | L
12069127 |  6276$\pm$77 |  2.01${}_{-0.13}^{+0.11}$ |  1.57${}_{-0.02}^{+0.03}$ |  0.08$\pm$0.10 |  3.912${}_{-0.004}^{+0.005}$ |  0.54${}_{-0.33}^{+1.26}$ |  16.6${}_{-6.3}^{+54.2}$ |  5.2${}_{-3.9}^{+39.9}$ | 1 | H | L
12069424 |  5825$\pm$50 |  6.67${}_{-0.77}^{+0.81}$ |  1.05${}_{-0.02}^{+0.02}$ |  0.10$\pm$0.03 |  4.287${}_{-0.007}^{+0.007}$ |  0.40${}_{-0.01}^{+0.01}$ |  44.7${}_{-2.9}^{+6.2}$ |  20.5${}_{-1.1}^{+2.0}$ | 1 | MS | L
12069449 |  5750$\pm$50 |  7.39${}_{-0.91}^{+0.89}$ |  0.99${}_{-0.02}^{+0.02}$ |  0.05$\pm$0.02 |  4.353${}_{-0.005}^{+0.007}$ |  0.31${}_{-0.01}^{+0.01}$ |  34.0${}_{-2.4}^{+3.0}$ |  21.2${}_{-1.5}^{+1.8}$ | 0 | MS | L
12258514 |  5964$\pm$77 |  4.05${}_{-0.16}^{+0.18}$ |  1.26${}_{-0.01}^{+0.01}$ |  0.00$\pm$0.10 |  4.126${}_{-0.003}^{+0.004}$ |  0.39${}_{-0.03}^{+0.04}$ |  35.0${}_{-6.6}^{+22.9}$ |  16.7${}_{-3.6}^{+10.0}$ | 1 | MS | L
12317678 |  6580$\pm$77 |  2.46${}_{-0.18}^{+0.22}$ |  1.34${}_{-0.01}^{+0.04}$ |  -0.28$\pm$0.10 |  4.048${}_{-0.009}^{+0.008}$ |  1.27${}_{-0.14}^{+0.13}$ |  35.3${}_{-5.2}^{+10.1}$ |  5.2${}_{-0.9}^{+1.8}$ | 0 | H | L
Supplementary Table 1: Continued from previous page.