††thanks: equally contributing authors††thanks: equally contributing authors

# Non-equilibrium fluctuations and nonlinear response of an active bath

Hunter Seyforth  Department of Physics, California State University Fullerton,
CA 92831 USA  Mauricio Gomez  Department of Physics, California State
University Fullerton, CA 92831 USA  W. Benjamin Rogers  Department of Physics,
Brandeis University, Waltham, MA 02452 USA  Jennifer L. Ross  Department of
Physics, Syracuse University, Syracuse, NY 13244 USA  Wylie W. Ahmed
<EMAIL_ADDRESS>Department of Physics, California State University
Fullerton, CA 92831 USA

###### Abstract

We analyze the dynamics of a passive colloidal probe immersed in an active
bath using an optical trap to study three physical processes: (1) the non-
equilibrium fluctuations transferred to the probe by the active bath, (2) the
friction experienced by the probe as it is driven through the active bath, and
(3) the force relaxation of the probe returning to its equilibrium position.
We measure the local force dynamics where all of the following characteristics
are of $\mathcal{O}(1)$: the size of the probe colloid relative to active bath
particle; the size of the probe colloid relative to the characteristic run-
length of an active particle; and the timescale of probe movement to the
persistence time of an active particle. We find at Péclet (Pe) $\ll 1$ the
active suspension exhibits shear thinning down to the solvent viscosity (but
not below); at $0.85<$ Pe $\leq 5.1$ the active bath shear thickens; and at Pe
$\geq 8.5$ the effective viscosity of the active bath shows a decreased effect
of thickening and plateaus. These results are in agreement with recent
modeling and simulations of the nonlinear rheology of an isotropic active
bath, providing experimental verification, and suggesting the model
predictions extends to moderately dense suspensions. Further, we observe that
the distribution of force fluctuations depends on Pe, unlike in passive
equilibrium baths. Lastly, we measure the energy transfer rate from the active
bath to the probe to be $\langle J\rangle\approx 10^{3}$ $k_{B}T/$s, which
leads to an increase in the effective diffusion of the probe by a factor of
$\sim 2$.

††preprint: APS/123-QED

##  I Introduction

Collections of self-propelled particles have become a cornerstone for
theoretical and experimental studies of active matter [1, 2, 3]. Model systems
(living and non-living) cover a wide range of length scales, from nanometer to
meter [4, 5, 6, 7, 8, 9, 10, 11, 12, 13], but they all share a common trait:
the individual objects that compose the system consume energy and generate
self-propulsion [14, 15, 1]. Consequently, these systems are far from
equilibrium and exhibit interesting dynamics, such as violation of the
fluctuation dissipation theorem (FDT) [16, 17, 18, 19], broken detailed
balance [20, 21, 22, 23], entropy production [24, 25, 26, 27], collective
motion [28, 29, 30, 31], giant density fluctuations [32, 33, 34], active self-
organization [35, 36, 37, 38], and novel rheology [39, 40, 41, 42, 43]; none
of which are observed in systems at thermodynamic equilibrium.

To obtain an understanding of the bulk properties of active baths,
investigations are often focused on length scales much larger than the
individual active particles using techniques such as microviscometers or
macroscopic rheometers [40]. These studies were the first to reveal the
intriguing observation of superfluidity in suspensions of swimming bacteria
[44, 45, 46]. This superfluid-like behavior results from a macroscopic balance
between viscous dissipation and the input energy of the swimming bacteria
[40], and have motivated a large number of theoretical studies [47, 48, 49,
50, 51, 52, 53, 54, 55, 56, 57, 58]. Some studies have investigated the local
dynamics at the microscopic scale and how they might give rise to novel bath
properties [59, 48, 47, 60, 49, 35]. And at the scale of individual swimmers,
investigations have revealed complex dynamics that depend on the local
environment [61, 62, 63, 64, 65, 66, 67], which could play a role in the bulk
active bath properties. Pioneering studies at the microscopic scale showed:
enhanced tracer diffusion [68, 69, 70] and force fluctuations [71], power-law
stress fluctuations and violation of FDT [16], and a memory-less friction
kernel [72].

Here, we use a well-established model system for creating a microscopic active
bath — a suspension of swimming _E. coli_ [73, 74] — and study the enhanced
local dynamics of an immersed probe particle due to active fluctuations. We
employ approaches from non-equilibrium statistical mechanics [17, 75, 16] and
colloid physics [76, 77, 78] to measure the local fluctuations and rheology at
the colloidal scale. Building on previous work [16, 72, 71], we study force
fluctuations by direct measurement of a passive colloidal probe in an active
bath using an optical trap and the photon momentum method (PMM). We use a
moderately dense concentration ($\phi_{\mathrm{eff}}=0.2$) to create an
isotropic active bath, where no long range structures, flows, or orientational
order are observed and test recent theoretical predictions for the non-
equilibrium properties [47]. Specifically we investigate the effect of the
active bath on the local fluctuations, microrheology, and relaxation of an
immersed probe particle in a regime that has not yet been explored: moderate
density and nonlinear response.

We find at Pe = 0 the probe experiences enhanced force fluctuations and the
active bath approaches the solvent viscosity, but not below; at intermediate
Pe (0.85 to 5.1) the active bath shear thickens to $\sim 3-5$X the viscosity
of a comparable passive colloidal suspension; and at high Pe (8.5 to 50.9) the
effective viscosity decreases and exhibits a plateau. Further, the amplitude
of force fluctuations in the active bath depend on Pe, a behavior that is
uniquely non-equilibrium.

##  II Materials and Methods

###  II.1 Sample Preparation

_Escherichia coli_ are a well-characterized model system for use as active
colloids [73]. _E. coli_ were purchased from Carolina Scientific (item
#155068) and used within 48 hrs of arrival. We determined the cell density in
our experiments by using optical density spectrophotometry. The measured OD600
= 0.8 corresponds to $n\sim 10^{9}$ cells/mL, which is equivalent to an
effective volume fraction of $\phi_{\mathrm{eff}}=0.2$. This effective volume
fraction accounts for cell body and flagella bundle. Randomly oriented cells
reach “overlap” at $\phi_{\mathrm{eff}}\sim 0.1$ [73]. We sandwiched a 20
$\mu$L droplet of solution containing _E. coli_ in a sample chamber made from
a glass slide and a coverslip (Fisher Scientific, 12-545F) with vacuum grease
(Dow Corning, Z273554) to seal the chamber. Throughout the paper, this active
suspension of swimming _E. coli_ is called the ‘active bath,’ and ‘passive
bath’ refers to a sample of only water. The inset of Fig. 1 shows a
representative image. Measurements were made in the middle of a sample chamber
of height $\sim$300 $\mu$m and thus hydrodynamic effects due to the confined
geometry have a small effect on measured viscosity ($<2\%$), as estimated by
Faxen’s correction for a microsphere near a boundary [79].

###  II.2 Optical Tweezer Measurements

For microscopy and optical trapping, we used a Nikon TE2000 with a 60x/1.2NA
water-immersion objective and Hamamatsu ORCA-Flash4.0 V2. The optical tweezer
system (Impetux Optics S.L.) includes the optical trap, piezo stage
positioning, and force detection. The 60x objective focuses the near-infrared
laser (1064 nm, IPG-YLR-10, IPG Photonics) to create the optical trap. The
photon momentum method (PMM) [80, 81] was implemented with a 1.4NA oil
immersion condenser and a position sensitive sensor, digitized at 50 kHz,
allows for force detection and laser tracking interferometry. For the force
calibration to be accurate, it is critical to use a condensing objective with
higher numerical aperture than the trapping objective and to minimize
scattering of light through the sample [80, 82]. The PMM approach provides
direct access to the optical trapping force, even in the nonlinear regime, and
does not depend on linear calibration of position and trap stiffness. A
drawback of this approach is we do not have direct access to position
information for trajectory analysis. Labview (National Instruments) was used
to control all experimental hardware and data acquisition.

We used a colloidal probe, $r=5$ $\mu$m,(Alfa Aesar, 42717) as our passive
tracer particle for all optical tweezer measurements. We chose a probe size
that allowed us to measure length scales larger than individual _E. coli_.
Force measurements were conducted separately on both active and passive baths.
There are three distinct stages in each measurement as shown in Fig. 1: stage
1, the spontaneous force fluctuations of the probe (piezo is stationary);
stage 2, nonlinear microrheology of the probe moving through the bath at
constant velocity, $\langle\mathbf{U}\rangle,$ covering a range from 2–120
$\mu$m/s; and stage 3, force relaxation of the probe as it recovers from stage
2 perturbation back to its equilibrium position. A representative example
experiment is shown in Fig. 1.

Figure 1:  Overview of experimental protocol: Spontaneous force fluctuations
are measured in stage 1 (left, yellow shaded), nonlinear mechanical response
in stage 2 (middle, green shaded), and force relaxation in stage 3 (right,
magenta shaded). The left vertical axis shows the optical trap force, measured
via PMM, in the direction parallel to stage motion for an active (black) and
passive (grey) bath. The right vertical axis shows the stage position (red),
indicating time periods of no motion (stage 1 and 3) and constant velocity
(stage 2). Data shown for constant velocity of $\langle\mathbf{U}\rangle=20$
$\mu$m/s (Pe = 8.5). Already evident in stage 2 (green shaded) is the
increased viscosity of the active suspension relative to water and the
viscoelastic-like relaxation in stage 3 (magenta shaded). Left inset show
representative image of a colloid optically trapped in the active bath (scale
bar = 10 $\mu$m). Note that duration of stage 1 and 3 are much longer, but not
shown for clarity.

###  II.3 Data Analysis

We calculate the force spectrum $\langle|\tilde{\mathbf{F}}|^{2}\rangle$ (Fig.
2a), by estimating the power spectrum of a finite force signal,
$\mathbf{F}(t)$, sampled at 50 kHz, using Welch’s method with a Hamming window
[83]. We fit the force spectrum to our analytic model using nonlinear least
squares [84]. The active energy spectrum (Fig. 2b) was calculated by taking
the ratio of force spectra from the active and passive bath and subtracting
thermal (passive) fluctuations,
$E_{\mathrm{act}}=\langle|\tilde{\mathbf{F}}_{\mathrm{active}}|^{2}\rangle/\langle|\tilde{\mathbf{F}}_{\mathrm{passive}}|^{2}\rangle-1$,
where $E_{\mathrm{act}}$ is a function of frequency in units of $k_{B}T$ [85].
All probability distributions were calculated assuming stationarity and
normalized such that, $\int\mathcal{P}(F)dF=1$, where $\mathcal{P}$ is the
probability density, and $F$ is the force of interest. For example, in the
inset of Fig. 2(a) we use the forces from stage 1 while in Fig. 4 we use the
force measured during stage 2. All data analysis was completed in MATLAB.

###  II.4 Microrheology

To measure the nonlinear response we use Pulling Active Microrheology (PAM)
[76, 77, 86], where the probe is pulled through the sample at constant
velocity for a duration of $0.5-1$ s. While these types of measurements are
often “mixed mode”, as in neither constant force nor velocity, our
experimental parameters (i.e. stage velocity, probe size, bath particle size)
put us well within the regime for constant velocity [78]. In PAM, it is common
to define a ‘generalized Stokes relation’, which relates the average force
$\langle\mathbf{F}\rangle$, taken from the finite force signal, on the probe
particle to its average velocity $\langle\mathbf{U}\rangle$, where
$\langle\mathbf{F}\rangle=6\pi r\eta_{\mathrm{eff}}\langle\mathbf{U}\rangle$,
$\eta_{\mathrm{eff}}$ is the effective viscosity of the suspension, and
$\mathbf{U}$ is the velocity of the stage, ranging from $2-120$ $\mu$m/s. We
use this relation to calculate the effective viscosity of our suspension as a
function of Péclet number (Pe). Pe is defined for active suspensions as,
$\mathrm{Pe}=\dot{\gamma}\tau_{r}$ [47], where
$\dot{\gamma}=3\langle|\mathbf{U}|\rangle/\sqrt{2}r$ is the shear rate and
$\tau_{r}$ is the persistence time of the active bath particle. For _E. coli_
, persistence times are roughly one second [65, 64], so we use $\tau_{r}=1$ s
for simplicity, such that $\mathrm{Pe}=\dot{\gamma}$. Therefore in our PAM
experiments we explore the regime, $0.85<\mathrm{Pe}<50.9$. This data is color
coded in figures as Pe = 0.85 (brown), 1.7 (red), 3.4 (orange), 5.1 (yellow),
8.5 (green), 17.0 (cyan), 33.9 (blue), 50.9 (royal). We refer to this as
nonlinear rheology for two reasons: (1) the forces measured (via PMM) are
outside the linear regime of the optical trap and (2) the measured viscosity
has a nonlinear relationship to shear rate.

To estimate the effective viscosity of our moderate volume fraction suspension
in the absence of activity, we employ the widely used Krieger-Dougherty
relation, $\eta_{\mathrm{eff}}/\eta_{0}=(1-\phi/\phi_{max})^{-2}$ [87, 88, 89,
90, 91], where $\phi$ is the volume fraction and $\eta_{0}$ is the viscosity
of the background solvent. Using our volume fraction
$\phi=\phi_{\mathrm{eff}}=0.2$ and a $\phi_{max}=0.63$ (for spherical packing)
we estimate the viscosity of an equivalent isotropic passive suspension to be
$\eta_{\mathrm{eff}}/\eta_{0}=2.15$. This value provides a baseline expected
viscosity of suspension without activity.

Figure 2:  Non-equilibrium fluctuations in an active bath: (a) Force spectra
calculated from stage 1 are shown for a colloid in an active (black) and
passive bath (grey). At low frequencies ($f<50$ Hz) the force spectra in the
active bath is greater. Theoretical fit to equation 2 (solid lines) estimates
the viscosity of the bath ($\eta=1$ mPa-s), and characterizes the active
process on average by it’s active burst velocity ($v=2.9\pm 0.06$ $\mu$m/s)
and timescale ($\tau=26\pm 0.7$ ms). Fit for the passive bath (solid grey)
corresponds the same viscosity as the active bath ($\eta=1$ mPa-s) and zero
activity ($v,\tau=0$). Inset shows the probability density of force
fluctuations for an active (black) and passive (grey) bath. (b) The active
energy spectrum quantifies the non-thermal energetic fluctuations of a colloid
in the active bath (shaded grey region indicates standard error of the mean).
Integrating this spectrum provides an estimate of the energy dissipation rate,
$\langle J\rangle$, via the Harada-Sasa equality [75]. Inset shows log-log
plot of active energy spectrum.

###  II.5 Theoretical model

As discussed previously [85], we model the stochastic forces of the optically
trapped colloidal probe subject to thermal and nonthermal forces with the
overdamped Langevin equation [3, 92]. That is, the position
$\mathbf{r}(t)\in\mathbb{R}^{2}$ of the colloidal particle is governed by

$\gamma\dot{\mathbf{r}}+\kappa\mathbf{r}=\gamma\mathbf{u}+\sqrt{2D}\gamma\bm{\xi},$
(1)

which balances the deterministic frictional and optical trap forces with the
random active and thermal forces. In equation (1), $\kappa$ is the optical
trap stiffness, $\gamma=6\pi R\eta$ is the friction coefficient of the Stokes’
drag where $R$ is the particle radius and $\eta$ is viscosity, and $D$ is the
thermal diffusion coefficient of the zero-mean, $\delta$-correlated Gaussian
white noise process $\bm{\xi}$. In this model, active bath particles collide
with the probe particle, transferring energy, which manifests in an active
burst velocity of the probe, $\mathbf{u}$. This active burst velocity,
$\mathbf{u}$, is modeled as an Active Ornstein-Uhlenbeck (AOUP) process with
characteristic strength, $v$, timescale, $\tau$, and correlations $\langle
u(t)u(s)\rangle=v^{2}\exp^{-|t-s|/\tau}$ [85, 3, 93]. This model allows for an
analytic form of the force spectrum, as derived previously [85],

$\begin{split}S\\!_{f\\!f}(\omega)=&\left(2\kappa^{2}D-\frac{2\tau\kappa^{2}v^{2}}{\mu^{2}\tau^{2}-1}\right)\frac{1}{\mu^{2}+\omega^{2}}\\\
&+\frac{2\tau\kappa^{2}v^{2}}{(\mu^{2}\tau^{2}-1)}\frac{1}{\tau^{-2}+\omega^{2}},\end{split}$
(2)

where $\omega$ is frequency in rad/s, $\mu=\kappa/\gamma$, and
$v=|\mathbf{u}|$. Equation (2) is fit to the experimentally measured force
spectrum measured in stage 1 to extract non-equilibrium activity. We note that
more complex and realistic models of the active process could be used
specifically for _E. coli_ [64, 94], however, we use the AOUP process for
generalizability to non-bacterial active baths and analytic tractability [93].

##  III Results and Discussion

###  III.1 Stage 1: Non-equilibrium force fluctuations

To extract the non-equilibrium force fluctuations on the colloid due to the
active bath we focus on stage 1 (Fig. 1, yellow) where the colloid is
fluctuating in the stationary trap due to forces from the surrounding media.
We use the force spectrum, a commonly used approach [17], to quantify the
force fluctuations on a colloid in both an active and passive bath. The
average force spectra, shown in Fig. 2a for active (black dots) and passive
(grey dots) baths, exhibit two notable features: (1) In the high frequency
regime ($f\gtrsim 50$ Hz) they collapse on one another; and (2) at lower
frequencies ($f\lesssim 50$ Hz) the two curves diverge. These measurements
show that in the active bath the high frequency fluctuations are dominated by
thermal forces and the low frequency fluctuations are dominated by non-thermal
forces due to activity, as seen previously [16, 72, 95].

By fitting the analytic equation for the force spectra (equation 2) we extract
several physical parameters from the model, specifically, $\eta$, $v$, and
$\tau$. Interestingly, the extracted viscosity was indistinguishable between
the active and passive bath, $\eta=1$ mPa-s, during stage 1. This suggests
that the swimming bacteria in the active bath do not contribute to the overall
suspension viscosity as experienced by the colloid but do contribute to
enhanced fluctuations [39, 42, 50, 46, 59, 48, 47]. This dichotomy highlights
the lack of a direct connection between fluctuation and dissipation.

The two parameters that characterize the microscopic activity of the bath are
the burst velocity and timescale (which are both zero for a passive bath). The
burst velocity, $\langle v\rangle=2.9\pm 0.06$ $\mu$m/s, and timescale,
$\langle\tau\rangle=26\pm 0.7$ ms, represent the average non-equilibrium
fluctuation transmitted to the probe colloid from the active bath. Both of
these values are smaller than those of a single swimmer [64], as expected
since the probe colloid is larger and its motion is likely due to many
collisions. The burst velocity allows estimation of the average non-
equilibrium force fluctuation to be approximately, $F=6\pi r\eta v\approx 0.3$
pN, sustained for an average time $\tau$. This activity is manifested in a
wider distribution of force fluctuations experienced by the probe during stage
1 in the active vs passive bath (Fig. 2, inset). This type of non-Gaussian
force fluctuations are common in non-equilibrium systems [96, 97].

To further characterize the non-equilibrium fluctuations we calculate the
active energy spectrum from the ratio of the force spectra [85, 98]. Fig. 2b
shows that low-frequency active fluctuations have energy scales on the order
of $k_{B}T$, which corresponds to a dissipation rate of $\langle
J\rangle\approx 10^{3}$ $k_{B}T/$s when integrated over all available
frequencies, _a la_ the Harada-Sasa equality [75, 85]. This value, $\langle
J\rangle$, estimates the average rate of energy transferred from the active
bath to the probe colloid that is manifested in translational fluctuations. It
is worth noting, that this value is remarkably close to the power dissipated
by an individual swimming bacteria [99, 100], but this is likely a coincidence
since the overall motion of the probe colloid is presumably due to a large
number of collisions. To quantitatively relate the average burst
velocity/timescale, force fluctuation, and dissipation rate of the active bath
to the individual swimming bacteria requires a detailed micromechanical model
that considers momentum exchange — a topic of future work.

Overall, analysis of spontaneous force fluctuations during stage 1 (stationary
piezo stage) allows characterization of the amplitude, timescale, and
energetics of non-equilibrium fluctuations experienced by the probe colloid in
an active bath. The parameters extracted from our theoretical fit provides an
estimate of the effective self-diffusion of the probe colloid in an active
bath where, $D_{\mathrm{eff}}=D_{\mathrm{thermal}}+D_{\mathrm{active}}$, where
$D_{\mathrm{thermal}}=k_{B}T/6\pi r\eta$ and $D_{\mathrm{active}}=v^{2}\tau/6$
[101]. We find that the thermal and active diffusion coefficients are
$4.4\times 10^{-14}$ m2/s and $3.6\times 10^{-14}$ m2/s, respectively,
indicating that the active bath almost doubles the effective diffusion of the
probe colloid at long timescales.

Figure 3: Nonlinear microrheology and effective viscosity: (a) Ensemble
averaged force as a function of time, $\langle|\mathbf{F}(t)|\rangle$, is
shown for measurements in stage 2 (constant velocity) for Pe = 0.85 (brown),
1.7 (red), 3.4 (orange), 5.1 (yellow), 8.5 (green), 17.0 (cyan), 33.9 (blue),
50.9 (royal). Plateau forces clearly increase with Pe. (b) Effective
viscosity, $\eta_{\mathrm{eff}}$, as a function of Pe. Dashed horizontal lines
show the viscosity of water ($\eta_{\mathrm{water}}$) and expected viscosity
of the passive suspension ($\eta_{\mathrm{suspension}}$) according to the
Krieger-Dougherty relation. Note the horizontal axis break (//) to allow a
data point at zero Pe, where the active bath has an effective viscosity
equivalent to water. Increasing Pe causes shear thickening with decreasing
strength.

###  III.2 Stage 2: Nonlinear microrheology and effective viscosity

To characterize the response of the active bath to an applied force we focus
on stage 2 (Fig. 1, green) where the colloid is pulled through the active
suspension at constant velocity (or Pe). The force response increases in
magnitude with increasing Pe, and shows an initial slope followed by a
plateau, as shown in Fig. 3a. To quantify the viscous response of the active
bath we calculate the effective viscosity ($\eta_{\mathrm{eff}}$, see methods)
which exhibits a strong dependence on Pe (Fig. 3b).

A recent theoretical study has highlighted that the effective viscosity is
strongly dependent on local length scales [47], but experimental comparisons
are lacking. For direct comparison to this recent theoretical work we
introduce two length scales: The distance moved by the colloidal probe during
a characteristic time, $L=U\tau_{L}$, where $U=\langle|\mathbf{U}|\rangle$ is
the probe speed maintained for a time $\tau_{L}$. The distance moved by the
active bath particle, $\ell=v\tau_{r}$, where $v$ is the self-propulsion speed
and $\tau_{r}$ is the persistence time. Here, since both timescales are of
$\mathcal{O}(1)$, then $L/\ell\sim U/v$. Further we introduce the center-to-
center separation distance of the probe and active particle upon contact to be
$R_{c}=r+a$ where $r$ is the size of the colloidal probe and $a$ is the size
of the active particle. In our experiments $\ell/R_{c}\sim 1$, meaning the
distance traveled by an active bath particle during its characteristic
reorientation time, $\tau_{r}$, is comparable the center-to-center distance of
the probe and active particle pair.

At “zero” Pe (or zero shear rate), where $L\ll\ell$, the viscosity is
equivalent to that of water even though it is actually a dense suspension.
This zero shear viscosity is deduced from the stage 1 fit, because
$\eta_{\mathrm{eff}}$ is not defined for $\mathbf{U}=0$. In this regime, the
active bath particles travel a much greater distance than the probe during a
time, $\tau_{r}$, and are able to transfer force to the probe from all
directions. According to the Krieger-Dougherty relation, a passive suspension
of $\phi_{\mathrm{eff}}=0.2$ the expected viscosity is
$\eta_{\mathrm{eff}}\approx 2$ mPa-s, whereas our measured viscosity is
roughly half that. This is consistent with several previous studies that have
found superfluid behavior of active suspensions due to an effective shear
thinning caused by the active swimmers [49, 47, 40, 46, 45, 59, 102]. However,
our data indicates that the effective microviscosity decreases to the solvent
viscosity but we do not observe further thinning. This discrepancy between
previous studies could be due to measurements that probe different length
scales and/or the absence of large scale ordering in our active bath.
Specifically, the microviscosity measured here characterizes the local
environment at the colloidal scale ($r=5$ $\mu$m), whereas previous
measurements were of bulk environments with length scales on the order of
$10^{2}$ $\mu$m for microfluidic viscometers [45, 60] or $10^{3}-10^{4}$
$\mu$m for macrorheometry [46, 102]. Our measured thinning of the active bath
down to the solvent viscosity, but not lower, is in agreement with recent
simulations of an isotropic active bath [47]. Superfluidization to levels
below the solvent viscosity may require large-scale shear to organize flow
fields [59, 44] and may only be evident on larger lengthscales.

At intermediate Pe (0.85 to 5.1), where $L/\ell\sim 1$, we see a large
increase in effective viscosity of three to five times the value expected for
a passive suspension. In this regime, the probe particle and the active bath
particles move comparable distances during a time, $\tau_{r}$. The proposed
explanation [47] is that active particles behind the probe have difficulty
pushing because they are moving at roughly the same speed, whereas the
opposite is true for active particles in front of the probe that are able to
push backward on the probe — leading to force thickening. It is interesting to
note that this mechanism for force thickening [47] is completely independent
of hydrodynamic lubrication interactions as occurs in passive colloidal
suspensions [103, 104]. Recent simulations further support the above mechanism
of force thickening, namely an inhomogeneous distribution of active particles
[105].

At large Pe (8.5 to 50.9), where $L>\ell$, we see a relative decrease in the
effective viscosity (or a decreasing effect of shear thickening) that seems to
plateau. In this regime, the probe particle moves much further than the active
bath particle during the reorientation time $\tau_{r}$. Here, in line with
Burkholder and Brady [47], we expect that the active particles behind the
probe are not able to fill in the wake left by the probe motion and active
particles in front are not able to escape and accumulate in the boundary
layer. Essentially, at high probe velocities the active bath cannot “keep up”
and the effective viscosity resembles that of a passive suspension exhibiting
a high Pe plateau. The measured plateau is roughly twice the expected
viscosity for a passive suspension of hard colloidal spheres estimated via the
Krieger-Dougherty relation for $\phi_{\mathrm{eff}}=0.2$ [87]. This larger
plateau viscosity could be due to non-spherical geometry or interactions
between the active bath particles (_E. coli_), which are unaccounted for in
this estimate.

Overall, our nonlinear microrheology results largely confirm theoretical
predictions [47]: At Pe $\ll 1$, the active bath particle motion dominates and
thins the suspension leading to a reduced zero-shear plateau viscosity
equivalent to that of the solvent (water). At intermediate Pe, where the
motion of the probe particle and active bath particle are comparable
($L\sim\ell$), the active bath particles push backward on the advancing probe
and the suspension thickens. At Pe $>8.5$, the probe motion dominates
($L>\ell$) and the active bath particles cannot keep up with its motion
leading to a plateau viscosity as seen in passive suspensions. This type of
non monotonic shear thickening at the single particle level is qualitatively
similar to previous theoretical predictions in dilute suspensions [50].

Figure 4: Force fluctuations depend on Péclet number: Colored symbols in
legend indicate Pe for each data set. (a) In a passive bath, the probability
density of force fluctuations shows no dependence on Pe. (b) In an active
bath, the distribution of forces clearly widens with increasing Pe. (c) The
effective energy of the active bath consistently exhibits fluctuations greater
than equilibrium and this effect increases with Pe. Dashed horizontal line
indicates thermal equilibrium, $E_{\mathrm{eff}}=k_{B}T$, (symbols
differentiate data for the passive ($\times$) and active bath ($\bullet$)).

###  III.3 Force fluctuations depend on Péclet number

An advantage of measuring the nonlinear response of the active bath using
optical tweezer microrheology is access to full information on the force
fluctuations experienced by the probe [106]. These fluctuations are related to
effective temperature relations and non-equilibrium work theorems [107, 76,
108]. To focus on the fluctuations, we analyze the direction orthogonal to PAM
to remove the direct influence of the trap motion and plot the probability
density of force fluctuations in Fig. 4. For the passive bath composed of a
Newtonian solvent (i.e water) the force fluctuations orthogonal to trap motion
do not depend on Pe (Fig. 4a).

For the active bath suspension, the fluctuations increase with Pe and become
non-Gaussian (Fig. 4b). We can characterize this by plotting the effective
energy, $E_{\mathrm{eff}}$, of the particle as a function of Pe (Fig. 4c). We
calculate $E_{\mathrm{eff}}$ from the variance of the force distributions in
Fig. 4a,b; assuming the fluctuations in the passive bath at Pe = 0 have an
energy of $k_{B}T$ (grey, $\times$). Specifically, the effective energy was
calculated as $E_{\mathrm{eff}}=\sigma^{2}/\sigma^{2}_{\mathrm{grey,}\times}$,
where $\sigma^{2}$ is the variance of the force distribution of interest and
$\sigma^{2}_{\mathrm{grey,}\times}$ is the variance of the force distribution
at Pe = 0 in a passive bath.

The $E_{\mathrm{eff}}$ plotted in Fig. 4c shows a clear dependence on Pe for
the active bath ($\bullet$) but not for the passive bath ($\times$). At low
Pe, this relationship is weak but the $E_{\mathrm{eff}}$ of the active bath is
always greater than the passive bath. As Pe increases, the $E_{\mathrm{eff}}$
of the active bath clearly increases. Enhanced fluctuations at high Pe (but
not low Pe) have also been observed in dense suspensions of passive colloids
[76]. Therefore, we interpret this as follows: At low Pe, enhanced
fluctuations come primarily from the activity of the bath particles. At high
Pe, enhanced fluctuations are a combination of activity and steric
interactions due to the probe pushing active bath particles at high shear
rate. This interpretation is consistent with our effective viscosity
measurements (Fig. 3b) where at Pe $<8.5$ activity of the bath plays an
important role, whereas at Pe $\geq 8.5$ a viscous plateau is observed,
qualitatively consistent with passive colloidal suspensions.

Figure 5: Force relaxation depends on Pe: For Pe $\geq 8.5$ the force
relaxation exhibits a rapid decay for $t\leq 50$ ms, followed by a slow
relaxation. For Pe $<8.5$ the initial rapid decay is not observed. Colored
symbols in legend indicate Pe for each data set. Inset shows the normalized
force relaxation.

###  III.4 Stage 3: Force relaxation

Force relaxation is observed in stage 3 (Fig. 1, magenta) where the optical
trap is stationary and the probe is relaxing from its perturbed state in stage
2 (Fig. 1, green). This force relaxation is challenging to interpret due to
the large number of physical processes occurring simultaneously, e.g. active
fluctuations, heterogeneous bath density, colloidal suspension dynamics,
viscoelastic effects. However, one clear observation is the dependence on Pe
as shown in Fig. 5. For Pe $\geq 8.5$, forces exhibit an initial rapid decay
during the first 50 ms (dashed vertical line in Fig. 5) followed by a slow
relaxation to the equilibrium position. For Pe $<8.5$, forces exhibit only the
slow relaxation to the equilibrium position. This is in stark contrast to the
nearly instantaneous relaxation for a Newtonian fluid in thermal equilibrium
(Fig. 1, grey).

Fig. 5 inset is the normalized force relaxation, which accentuates this effect
showing a rapid decay for Pe $\geq 8.5$ and a slow relaxation for Pe $<8.5$.
We interpret this as follows: At Pe $\geq 8.5$, the rapid decay is due mainly
to the passive properties of the bath, where large forces cause steric
rearrangements of the bath particles. This is followed by a slower force
relaxation due to the active fluctuations at $t>50$ ms. This is consistent
with measurements of the force spectra (Fig. 2a) where the timescale of the
active process was estimated to be $\langle\tau\rangle=26$ ms and thus the
effects of such activity is visible on timescales greater than that. At Pe
$<8.5$, the force relaxation is dominated by the bath activity, because steric
interactions due to large deformation are absent. Recent work suggests that
activity of the bath can have a strong effect on relaxation dynamics [109,
110]. To interpret the relaxation, extension of active viscoelastic models
[111] to the nonlinear regime and incorporating active elements into a
viscoelastic memory kernel are promising next steps. These models must
consider the microscopic dynamics (e. g. the detailed interactions between the
probe and bath particles) — a topic reserved for future work.

##  IV Conclusion

Altogether, our results show that an immersed micron-scale probe in a
moderately dense ($\phi_{\mathrm{eff}}=0.2$) active bath of _E. coli_
experiences, on average, active forces of $\langle F\rangle\sim 0.3$ pN for a
duration of $\langle\tau\rangle=26$ ms resulting in non-thermal energy
transfer of $\langle J\rangle\approx 10^{3}k_{B}T/$s. This results in enhanced
diffusion at long timescales and superfliud-like thinning down to the solvent
viscosity at Pe $\sim 0$. At intermediate Pe, the active bath shear thickens
to 3–5X the viscosity of a comparable passive suspension, exhibits increased
amplitude of force fluctuations and a slow relaxation back to equilibrium from
its perturbed state. At higher Pe, the active bath exhibits a viscous plateau
of 2X the viscosity of a comparable passive suspension, shows force
fluctuations that increase with Pe, and exhibits a rapid force decay followed
by a slow relaxation to equilibrium from its perturbed state. Our results
complement previous experimental [16, 72, 68] and theoretical studies [40,
47], and contribute to the emerging picture that when the distances traversed
in a characteristic time by the immersed probe and active bath particle are
small, $L/\ell\leq 1$, the active bath exhibits novel non-equilibrium
properties; and when $L/\ell>1$ that active bath behaves much like a passive
colloidal suspension. A natural important extension of this work would be to
systematically vary the activity of the active bath, controlling $L/\ell$, to
study the effect on active fluctuations and viscosity experienced by the
embedded probe. This could be accomplished using a heat exchanging fluid bath
[112] or localized laser heating [113, 114].

## Acknowledgements

This material is based upon work supported by the National Science Foundation
under Grant No. NSF DMR-2004566 to WWA, NSF DMR-2004417 to JLR, and NSF DMR-
2004400 to WBR. HS and MG were partially supported by the Dan Black Family
Trust Fellowship.

## References

  * [1] C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, “Active particles in complex and crowded environments,” Reviews of Modern Physics, vol. 88, no. 4, p. 045006, 2016.
  * [2] G. Popkin, “The physics of life,” Nature News, vol. 529, no. 7584, p. 16, 2016.
  * [3] É. Fodor and M. C. Marchetti, “The statistical physics of active matter: From self-catalytic colloids to living cells,” Physica A: Statistical Mechanics and its Applications, vol. 504, pp. 106–120, 2018.
  * [4] S. Ghosh, A. Somasundar, and A. Sen, “Enzymes as active matter,” Annual Review of Condensed Matter Physics, vol. 12, pp. 177–200, 2021.
  * [5] J. Elgeti, R. G. Winkler, and G. Gompper, “Physics of microswimmers—single particle motion and collective behavior: a review,” Reports on progress in physics, vol. 78, no. 5, p. 056601, 2015.
  * [6] R. Alert and X. Trepat, “Living cells on the move,” Physics Today, vol. 74, no. 6, p. 30, 2021.
  * [7] H. H. Wensink, J. Dunkel, S. Heidenreich, K. Drescher, R. E. Goldstein, H. Löwen, and J. M. Yeomans, “Meso-scale turbulence in living fluids,” Proceedings of the national academy of sciences, vol. 109, no. 36, pp. 14308–14313, 2012.
  * [8] A. Bricard, J.-B. Caussin, N. Desreumaux, O. Dauchot, and D. Bartolo, “Emergence of macroscopic directed motion in populations of motile colloids,” Nature, vol. 503, no. 7474, pp. 95–98, 2013.
  * [9] L. Giomi, N. Hawley-Weld, and L. Mahadevan, “Swarming, swirling and stasis in sequestered bristle-bots,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 469, no. 2151, p. 20120637, 2013.
  * [10] C. Scholz, S. Jahanshahi, A. Ldov, and H. Löwen, “Inertial delay of self-propelled particles,” Nature communications, vol. 9, no. 1, pp. 1–9, 2018.
  * [11] C. Gouiller, F. Raynal, L. Maquet, M. Bourgoin, C. Cottin-Bizonne, R. Volk, and C. Ybert, “Mixing and unmixing induced by active camphor particles,” Physical Review Fluids, vol. 6, no. 1, p. 014501, 2021.
  * [12] A. Cavagna and I. Giardina, “Bird flocks as condensed matter,” Annu. Rev. Condens. Matter Phys., vol. 5, no. 1, pp. 183–207, 2014.
  * [13] F. Ginelli, F. Peruani, M.-H. Pillot, H. Chaté, G. Theraulaz, and R. Bon, “Intermittent collective dynamics emerge from conflicting imperatives in sheep herds,” Proceedings of the National Academy of Sciences, vol. 112, no. 41, pp. 12729–12734, 2015.
  * [14] S. Ramaswamy, “The mechanics and statistics of active matter,” Annu. Rev. Condens. Matter Phys., vol. 1, no. 1, pp. 323–345, 2010.
  * [15] M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, “Hydrodynamics of soft active matter,” Reviews of Modern Physics, vol. 85, no. 3, p. 1143, 2013.
  * [16] D. T. Chen, A. Lau, L. A. Hough, M. F. Islam, M. Goulian, T. C. Lubensky, and A. G. Yodh, “Fluctuations and rheology in active bacterial suspensions,” Physical review letters, vol. 99, no. 14, p. 148302, 2007.
  * [17] W. W. Ahmed, É. Fodor, and T. Betz, “Active cell mechanics: measurement and theory,” Biochimica et Biophysica Acta (BBA)-Molecular Cell Research, vol. 1853, no. 11, pp. 3083–3094, 2015.
  * [18] D. Mizuno, C. Tardin, C. F. Schmidt, and F. C. MacKintosh, “Nonequilibrium mechanics of active cytoskeletal networks,” Science, vol. 315, no. 5810, pp. 370–373, 2007.
  * [19] P. Martin, A. Hudspeth, and F. Jülicher, “Comparison of a hair bundle’s spontaneous oscillations with its response to mechanical stimulation reveals the underlying active process,” Proceedings of the National Academy of Sciences, vol. 98, no. 25, pp. 14380–14385, 2001.
  * [20] C. Battle, C. P. Broedersz, N. Fakhri, V. F. Geyer, J. Howard, C. F. Schmidt, and F. C. MacKintosh, “Broken detailed balance at mesoscopic scales in active biological systems,” Science, vol. 352, no. 6285, pp. 604–607, 2016\.
  * [21] I. A. Martínez, G. Bisker, J. M. Horowitz, and J. M. Parrondo, “Inferring broken detailed balance in the absence of observable currents,” Nature communications, vol. 10, no. 1, pp. 1–10, 2019.
  * [22] F. Gnesotto, F. Mura, J. Gladrow, and C. P. Broedersz, “Broken detailed balance and non-equilibrium dynamics in living systems: a review,” Reports on Progress in Physics, vol. 81, no. 6, p. 066601, 2018.
  * [23] A. P. Tabatabai, D. S. Seara, J. Tibbs, V. Yadav, I. Linsmeier, and M. P. Murrell, “Detailed balance broken by catch bond kinetics enables mechanical-adaptation in active materials,” Advanced Functional Materials, vol. 31, no. 10, p. 2006745, 2021.
  * [24] C. Nardini, É. Fodor, E. Tjhung, F. Van Wijland, J. Tailleur, and M. E. Cates, “Entropy production in field theories without time-reversal symmetry: quantifying the non-equilibrium character of active matter,” Physical Review X, vol. 7, no. 2, p. 021007, 2017.
  * [25] L. Dabelow, S. Bo, and R. Eichhorn, “Irreversibility in active matter systems: Fluctuation theorem and mutual information,” Physical Review X, vol. 9, no. 2, p. 021009, 2019.
  * [26] S. Shankar and M. C. Marchetti, “Hidden entropy production and work fluctuations in an ideal active gas,” Physical Review E, vol. 98, no. 2, p. 020604, 2018.
  * [27] S. Pigolotti, I. Neri, É. Roldán, and F. Jülicher, “Generic properties of stochastic entropy production,” Physical review letters, vol. 119, no. 14, p. 140604, 2017.
  * [28] C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein, and J. O. Kessler, “Self-concentration and large-scale coherence in bacterial dynamics,” Physical review letters, vol. 93, no. 9, p. 098103, 2004.
  * [29] T. Vicsek and A. Zafeiris, “Collective motion,” Physics reports, vol. 517, no. 3-4, pp. 71–140, 2012.
  * [30] A. Sokolov and I. S. Aranson, “Physical properties of collective motion in suspensions of bacteria,” Physical review letters, vol. 109, no. 24, p. 248109, 2012.
  * [31] H. Wioland, E. Lushi, and R. E. Goldstein, “Directed collective motion of bacteria under channel confinement,” New Journal of Physics, vol. 18, no. 7, p. 075002, 2016.
  * [32] H.-P. Zhang, A. Be’er, E.-L. Florin, and H. L. Swinney, “Collective motion and density fluctuations in bacterial colonies,” Proceedings of the National Academy of Sciences, vol. 107, no. 31, pp. 13626–13630, 2010.
  * [33] S. Ramaswamy, R. A. Simha, and J. Toner, “Active nematics on a substrate: Giant number fluctuations and long-time tails,” EPL (Europhysics Letters), vol. 62, no. 2, p. 196, 2003.
  * [34] L. Berthier and J. Kurchan, “Non-equilibrium glass transitions in driven and active matter,” Nature Physics, vol. 9, no. 5, pp. 310–314, 2013.
  * [35] S. Gokhale, J. Li, A. Solon, J. Gore, and N. Fakhri, “Dynamic clustering of passive colloids in dense suspensions of motile bacteria,” arXiv preprint arXiv:2110.02294, 2021.
  * [36] M. F. Hagan and A. Baskaran, “Emergent self-organization in active materials,” Current opinion in cell biology, vol. 38, pp. 74–80, 2016\.
  * [37] G. S. Redner, M. F. Hagan, and A. Baskaran, “Structure and dynamics of a phase-separating active colloidal fluid,” Physical review letters, vol. 110, no. 5, p. 055701, 2013.
  * [38] S. Mishra, A. Baskaran, and M. C. Marchetti, “Fluctuations and pattern formation in self-propelled particles,” Physical Review E, vol. 81, no. 6, p. 061916, 2010.
  * [39] Y. Hatwalne, S. Ramaswamy, M. Rao, and R. A. Simha, “Rheology of active-particle suspensions,” Physical review letters, vol. 92, no. 11, p. 118101, 2004.
  * [40] D. Saintillan, “Rheology of active fluids,” Annual Review of Fluid Mechanics, vol. 50, pp. 563–592, 2018.
  * [41] G. I. Menon, “Active matter,” in Rheology of complex Fluids, pp. 193–218, Springer, 2010.
  * [42] B. M. Haines, I. S. Aranson, L. Berlyand, and D. A. Karpeev, “Effective viscosity of dilute bacterial suspensions: a two-dimensional model,” Physical biology, vol. 5, no. 4, p. 046003, 2008.
  * [43] V. Gyrya, K. Lipnikov, I. Aranson, and L. Berlyand, “Effective shear viscosity and dynamics of suspensions of micro-swimmers from small to moderate concentrations,” Journal of mathematical biology, vol. 62, no. 5, pp. 707–740, 2011.
  * [44] A. Sokolov and I. S. Aranson, “Reduction of viscosity in suspension of swimming bacteria,” Physical review letters, vol. 103, no. 14, p. 148101, 2009.
  * [45] J. Gachelin, G. Mino, H. Berthet, A. Lindner, A. Rousselet, and É. Clément, “Non-newtonian viscosity of escherichia coli suspensions,” Physical review letters, vol. 110, no. 26, p. 268103, 2013\.
  * [46] H. M. López, J. Gachelin, C. Douarche, H. Auradou, and E. Clément, “Turning bacteria suspensions into superfluids,” Physical review letters, vol. 115, no. 2, p. 028301, 2015.
  * [47] E. W. Burkholder and J. F. Brady, “Nonlinear microrheology of active brownian suspensions,” Soft matter, vol. 16, no. 4, pp. 1034–1046, 2020.
  * [48] E. W. Burkholder and J. F. Brady, “Fluctuation-dissipation in active matter,” The Journal of chemical physics, vol. 150, no. 18, p. 184901, 2019.
  * [49] S. Takatori and J. Brady, “Superfluid behavior of active suspensions from diffusive stretching,” Physical review letters, vol. 118, no. 1, p. 018003, 2017.
  * [50] D. Saintillan, “The dilute rheology of swimming suspensions: A simple kinetic model,” Experimental Mechanics, vol. 50, no. 9, pp. 1275–1281, 2010.
  * [51] C. Maes, “Fluctuating motion in an active environment,” Physical Review Letters, vol. 125, no. 20, p. 208001, 2020.
  * [52] S. Ye, P. Liu, F. Ye, K. Chen, and M. Yang, “Active noise experienced by a passive particle trapped in an active bath,” Soft matter, vol. 16, no. 19, pp. 4655–4660, 2020.
  * [53] S. Liu, S. Shankar, M. C. Marchetti, and Y. Wu, “Viscoelastic control of spatiotemporal order in bacterial active matter,” Nature, vol. 590, no. 7844, pp. 80–84, 2021.
  * [54] A. Puglisi and U. Marini Bettolo Marconi, “Clausius relation for active particles: What can we learn from fluctuations,” Entropy, vol. 19, no. 7, p. 356, 2017.
  * [55] S. Chaki and R. Chakrabarti, “Effects of active fluctuations on energetics of a colloidal particle: Superdiffusion, dissipation and entropy production,” Physica A: Statistical Mechanics and its Applications, vol. 530, p. 121574, 2019.
  * [56] S. Chaki and R. Chakrabarti, “Entropy production and work fluctuation relations for a single particle in active bath,” Physica A: Statistical Mechanics and its Applications, vol. 511, pp. 302–315, 2018.
  * [57] A. Lau and T. C. Lubensky, “Fluctuating hydrodynamics and microrheology of a dilute suspension of swimming bacteria,” Physical review E, vol. 80, no. 1, p. 011917, 2009.
  * [58] M. Knežević, L. E. Avilés Podgurski, and H. Stark, “Oscillatory active microrheology of active suspensions,” Scientific reports, vol. 11, no. 1, pp. 1–10, 2021.
  * [59] S. Guo, D. Samanta, Y. Peng, X. Xu, and X. Cheng, “Symmetric shear banding and swarming vortices in bacterial superfluids,” Proceedings of the National Academy of Sciences, vol. 115, no. 28, pp. 7212–7217, 2018.
  * [60] Z. Liu, K. Zhang, and X. Cheng, “Rheology of bacterial suspensions under confinement,” Rheologica Acta, vol. 58, no. 8, pp. 439–451, 2019.
  * [61] A. J. Mathijssen, N. Figueroa-Morales, G. Junot, É. Clément, A. Lindner, and A. Zöttl, “Oscillatory surface rheotaxis of swimming e. coli bacteria,” Nature communications, vol. 10, no. 1, pp. 1–12, 2019\.
  * [62] N. Figueroa-Morales, G. L. Mino, A. Rivera, R. Caballero, E. Clément, E. Altshuler, and A. Lindner, “Living on the edge: transfer and traffic of e. coli in a confined flow,” Soft matter, vol. 11, no. 31, pp. 6284–6293, 2015.
  * [63] G. Junot, N. Figueroa-Morales, T. Darnige, A. Lindner, R. Soto, H. Auradou, and E. Clément, “Swimming bacteria in poiseuille flow: The quest for active bretherton-jeffery trajectories,” EPL (Europhysics Letters), vol. 126, no. 4, p. 44003, 2019.
  * [64] N. Figueroa-Morales, R. Soto, G. Junot, T. Darnige, C. Douarche, V. A. Martinez, A. Lindner, and E. Clément, “3d spatial exploration by e. coli echoes motor temporal variability,” Physical Review X, vol. 10, no. 2, p. 021004, 2020.
  * [65] A. Patteson, A. Gopinath, M. Goulian, and P. Arratia, “Running and tumbling with e. coli in polymeric solutions,” Scientific reports, vol. 5, no. 1, pp. 1–11, 2015.
  * [66] E. P. Ipiña, S. Otte, R. Pontier-Bres, D. Czerucka, and F. Peruani, “Bacteria display optimal transport near surfaces,” Nature Physics, vol. 15, no. 6, pp. 610–615, 2019.
  * [67] A. Martínez-Calvo, C. Trenado-Yuste, and S. S. Datta, “Active transport in complex environments,” arXiv preprint arXiv:2108.07011, 2021.
  * [68] X.-L. Wu and A. Libchaber, “Particle diffusion in a quasi-two-dimensional bacterial bath,” Physical review letters, vol. 84, no. 13, p. 3017, 2000\.
  * [69] A. Jepson, V. A. Martinez, J. Schwarz-Linek, A. Morozov, and W. C. Poon, “Enhanced diffusion of nonswimmers in a three-dimensional bath of motile bacteria,” Physical Review E, vol. 88, no. 4, p. 041002, 2013.
  * [70] G. Grégoire, H. Chaté, and Y. Tu, “Active and passive particles: Modeling beads in a bacterial bath,” Physical Review E, vol. 64, no. 1, p. 011902, 2001.
  * [71] G. Soni, B. J. Ali, Y. Hatwalne, and G. Shivashankar, “Single particle tracking of correlated bacterial dynamics,” Biophysical journal, vol. 84, no. 4, pp. 2634–2637, 2003.
  * [72] C. Maggi, M. Paoluzzi, L. Angelani, and R. Di Leonardo, “Memory-less response and violation of the fluctuation-dissipation theorem in colloids suspended in an active bath,” Scientific reports, vol. 7, no. 1, pp. 1–7, 2017.
  * [73] J. Schwarz-Linek, J. Arlt, A. Jepson, A. Dawson, T. Vissers, D. Miroli, T. Pilizota, V. A. Martinez, and W. C. Poon, “Escherichia coli as a model active colloid: A practical introduction,” Colloids and Surfaces B: Biointerfaces, vol. 137, pp. 2–16, 2016.
  * [74] H. C. Berg, E. coli in Motion.  Springer Science & Business Media, 2008.
  * [75] T. Harada and S.-i. Sasa, “Equality connecting energy dissipation with a violation of the fluctuation-response relation,” Physical review letters, vol. 95, no. 13, p. 130602, 2005.
  * [76] L. Wilson, A. Harrison, W. Poon, and A. Puertas, “Microrheology and the fluctuation theorem in dense colloids,” EPL (Europhysics Letters), vol. 93, no. 5, p. 58007, 2011.
  * [77] L. G. Wilson and W. C. Poon, “Small-world rheology: an introduction to probe-based active microrheology,” Physical Chemistry Chemical Physics, vol. 13, no. 22, pp. 10617–10630, 2011.
  * [78] T. M. Squires and J. F. Brady, “A simple paradigm for active and nonlinear microrheology,” Physics of Fluids, vol. 17, no. 7, p. 073101, 2005.
  * [79] J. Leach, H. Mushfique, S. Keen, R. Di Leonardo, G. Ruocco, J. Cooper, and M. Padgett, “Comparison of faxén’s correction for a microsphere translating or rotating near a surface,” Physical Review E, vol. 79, no. 2, p. 026301, 2009.
  * [80] A. Farré and M. Montes-Usategui, “A force detection technique for single-beam optical traps based on direct measurement of light momentum changes,” Optics express, vol. 18, no. 11, pp. 11955–11968, 2010.
  * [81] J. Gieseler, J. R. Gomez-Solano, A. Magazzù, I. P. Castillo, L. P. García, M. Gironella-Torrent, X. Viader-Godoy, F. Ritort, G. Pesce, A. V. Arzola, et al., “Optical tweezers: A comprehensive tutorial from calibration to applications,” arXiv preprint arXiv:2004.05246, 2020.
  * [82] Y. Jun, S. K. Tripathy, B. R. Narayanareddy, M. K. Mattson-Hoss, and S. P. Gross, “Calibration of optical tweezers for in vivo force measurements: How do different approaches compare?,” Biophysical journal, vol. 107, no. 6, pp. 1474–1484, 2014.
  * [83] P. Welch, “The use of fast fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Transactions on audio and electroacoustics, vol. 15, no. 2, pp. 70–73, 1967.
  * [84] T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM Journal on optimization, vol. 6, no. 2, pp. 418–445, 1996.
  * [85] C. Jones, M. Gomez, R. M. Muoio, A. Vidal, R. A. Mcknight, N. D. Brubaker, and W. W. Ahmed, “Stochastic force dynamics of the model microswimmer chlamydomonas reinhardtii: Active forces and energetics,” Physical Review E, vol. 103, no. 3, p. 032403, 2021.
  * [86] R. M. Robertson-Anderson, “Optical tweezers microrheology: from the basics to advanced techniques and applications,” 2018.
  * [87] I. M. Krieger and T. J. Dougherty, “A mechanism for non-newtonian flow in suspensions of rigid spheres,” Transactions of the Society of Rheology, vol. 3, no. 1, pp. 137–152, 1959.
  * [88] D. Quemada, “Rheology of concentrated disperse systems and minimum energy dissipation principle,” Rheologica Acta, vol. 16, no. 1, pp. 82–94, 1977\.
  * [89] R. C. Ball and P. Richmond, “Dynamics of colloidal dispersions,” Physics and Chemistry of Liquids, vol. 9, no. 2, pp. 99–116, 1980.
  * [90] J. Van der Werff and C. De Kruif, “Hard-sphere colloidal dispersions: the scaling of rheological properties with particle size, volume fraction, and shear rate,” Journal of Rheology, vol. 33, no. 3, pp. 421–454, 1989.
  * [91] E. Boek, P. V. Coveney, H. Lekkerkerker, and P. van der Schoot, “Simulating the rheology of dense colloidal suspensions using dissipative particle dynamics,” Physical Review E, vol. 55, no. 3, p. 3124, 1997.
  * [92] K. Sekimoto, “Langevin equation and thermodynamics,” Progress of Theoretical Physics Supplement, vol. 130, pp. 17–27, 1998.
  * [93] D. Martin, J. O’Byrne, M. E. Cates, É. Fodor, C. Nardini, J. Tailleur, and F. van Wijland, “Statistical mechanics of active ornstein-uhlenbeck particles,” Physical Review E, vol. 103, no. 3, p. 032607, 2021.
  * [94] A. Baskaran and M. C. Marchetti, “Statistical mechanics and hydrodynamics of bacterial suspensions,” Proceedings of the National Academy of Sciences, vol. 106, no. 37, pp. 15567–15572, 2009.
  * [95] P. Bohec, J. Tailleur, F. van Wijland, A. Richert, and F. Gallet, “Distribution of active forces in the cell cortex,” Soft matter, vol. 15, no. 35, pp. 6952–6966, 2019.
  * [96] K. C. Leptos, J. S. Guasto, J. P. Gollub, A. I. Pesci, and R. E. Goldstein, “Dynamics of enhanced tracer diffusion in suspensions of swimming eukaryotic microorganisms,” Physical Review Letters, vol. 103, no. 19, p. 198103, 2009\.
  * [97] I. Rushkin, V. Kantsler, and R. E. Goldstein, “Fluid velocity fluctuations in a suspension of swimming protists,” Physical review letters, vol. 105, no. 18, p. 188101, 2010.
  * [98] S. Eldeen, R. Muoio, P. Blaisdell-Pijuan, N. La, M. Gomez, A. Vidal, and W. Ahmed, “Quantifying the non-equilibrium activity of an active colloid,” Soft matter, vol. 16, no. 31, pp. 7202–7209, 2020.
  * [99] T. Ishikawa, N. Yoshida, H. Ueno, M. Wiedeman, Y. Imai, and T. Yamaguchi, “Energy transport in a concentrated suspension of bacteria,” Physical review letters, vol. 107, no. 2, p. 028102, 2011.
  * [100] S. Chattopadhyay, R. Moldovan, C. Yeung, and X. Wu, “Swimming efficiency of bacterium escherichiacoli,” Proceedings of the National Academy of Sciences, vol. 103, no. 37, pp. 13712–13717, 2006.
  * [101] S. C. Takatori, W. Yan, and J. F. Brady, “Swim pressure: stress generation in active matter,” Physical review letters, vol. 113, no. 2, p. 028103, 2014\.
  * [102] J. Y. Chui, C. Douarche, H. Auradou, and R. Juanes, “Rheology of bacterial superfluids in viscous environments,” Soft Matter, 2021.
  * [103] X. Cheng, J. H. McCoy, J. N. Israelachvili, and I. Cohen, “Imaging the microscopic structure of shear thinning and thickening colloidal suspensions,” Science, vol. 333, no. 6047, pp. 1276–1279, 2011.
  * [104] S. Pednekar, J. Chun, and J. F. Morris, “Simulation of shear thickening in attractive colloidal suspensions,” Soft matter, vol. 13, no. 9, pp. 1773–1779, 2017.
  * [105] M. Knezevic, L. Podgurski, and H. Stark, “Oscillatory active microrheology of active suspensions,” arXiv preprint arXiv:2109.11218, 2021.
  * [106] L. Wilson, A. Harrison, A. Schofield, J. Arlt, and W. Poon, “Passive and active microrheology of hard-sphere colloids,” The Journal of Physical Chemistry B, vol. 113, no. 12, pp. 3806–3812, 2009.
  * [107] J. Kurchan, “Non-equilibrium work relations,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2007, no. 07, p. P07005, 2007.
  * [108] D. J. Evans and D. J. Searles, “The fluctuation theorem,” Advances in Physics, vol. 51, no. 7, pp. 1529–1585, 2002.
  * [109] L. Caprini, A. Puglisi, and A. Sarracino, “Fluctuation–dissipation relations in active matter systems,” Symmetry, vol. 13, no. 1, p. 81, 2021.
  * [110] L. Caprini, “Generalized fluctuation–dissipation relations holding in non-equilibrium dynamics,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2021, no. 6, p. 063202, 2021.
  * [111] D. Banerjee, V. Vitelli, F. Jülicher, and P. Surówka, “Active viscoelasticity of odd materials,” Physical Review Letters, vol. 126, no. 13, p. 138001, 2021.
  * [112] S. Krishnamurthy, S. Ghosh, D. Chatterji, R. Ganapathy, and A. Sood, “A micrometre-sized heat engine operating between bacterial reservoirs,” Nature Physics, vol. 12, no. 12, pp. 1134–1138, 2016.
  * [113] V. Blickle and C. Bechinger, “Realization of a micrometre-sized stochastic heat engine,” Nature Physics, vol. 8, no. 2, pp. 143–146, 2012.
  * [114] I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, J. M. Parrondo, and R. A. Rica, “Brownian carnot engine,” Nature physics, vol. 12, no. 1, pp. 67–70, 2016.