# When does an active bath behave as an equilibrium one?

Shubhendu Shekhar Khali<EMAIL_ADDRESS>Institute of Physics,
Sachivalaya Marg, Bhubaneswar 751005, India  Fernando Peruani
<EMAIL_ADDRESS>LPTM, CY Cergy Paris Université, 2 avenue A. Chauvin,
95302 Cergy-Pontoise cedex, France  Debasish Chaudhuri  communicating author:
<EMAIL_ADDRESS>Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005,
India  Max-Planck Institute for the Physics of Complex Systems, Nöthnitzer
Str. 38, 01187 Dresden, Germany

###### Abstract

Active baths are characterized by a non-Gaussian velocity distribution and a
quadratic dependence with active velocity $v_{0}$ of the kinetic temperature
and diffusion coefficient. While these results hold in over-damped active
systems, inertial effects lead to normal velocity distributions, with kinetic
temperature and diffusion coefficient increasing as $\sim v_{0}^{\alpha}$ with
$1<\alpha<2$. Remarkably, the late-time diffusivity and mobility decrease with
mass. Moreover, we show that the equilibrium Einstein relation is
asymptotically recovered with inertia. In summary, the inertial mass restores
an equilibrium-like behavior.

A fluid in equilibrium can be characterized as a heat bath in terms of its
temperature, viscous drag, and diffusivity, obeying the Einstein relation [1].
Active Brownian particles (ABP), under certain conditions, can constitute a
similar homogeneous and isotropic (as opposed to polar) active fluid – i.e.,
an active scalar fluid – in the presence of local energy dissipation and self-
propulsion [2, 3, 4, 5, 6, 7]. A question that naturally arises is to what
extent such a fluid can be characterized as an active heat bath. The active
nature of ABPs is determined by the self-propulsion speed $v_{0}$ and the
orientational diffusivity of the heading direction $D_{r}$. This leads to a
persistent random motion for individual ABPs, in dimension $d$, characterized
by a late-time active diffusivity that scales as
$\frac{v_{0}^{2}}{D_{r}d(d-1)}$. Early experiments on tracer particle dynamics
in a bacterial suspension showed enhanced active diffusion [8, 9]. A higher
density reduces diffusivity in equilibrium [10] but increases it in a non-
equilibrium bacterial bath [8]. Various theoretical techniques were used to
obtain the impact of an active bath on tracer particles [11, 12, 13, 14, 15,
16, 17]. Recent works have characterized ABP systems in terms of kinetic
temperature, effective diffusivity, and viscous drag as a function of changing
activity [18, 19, 20, 21, 22, 23, 24, 25]. While diffusivity and kinetic
temperature increase with activity, a non-monotonic variation of viscous drag
has been predicted [22]. At the motility-induced phase separation (MIPS) [26,
27, 28], it was shown that the kinetic temperature could vary across a steady-
state system with low (high) temperature characterizing the dense (dilute)
phase [20]. However, it remains unclear to what extent such a description can
be developed into a coherent self-consistent picture of active fluid, given,
e.g., the breaking of time-reversal symmetry and the absence of equilibrium
fluctuation-dissipation.

Despite tremendous progress in the study of active matter [3, 2, 4, 5, 6],
until recently, relatively little attention was paid to the impact of inertia
on the active matter except for [29, 20, 30, 31, 32, 33, 34]. One reason for
this is the extremely short time ($\sim 100$ ns) and length scales (Angstrom)
for the ballistic-diffusive crossover in colloidal particles. However, for
larger active elements, including birds, fish, and animals on the one hand,
and artificial macro-sized robots [35, 36, 37] on the other hand, inertial
effects can be substantial. This paper considers a homogeneous and isotropic
fluid of active Brownian particles (ABP) and probes its active bath-like
characteristics. A remarkable fact emerges: not only transient behaviors, even
the asymptotic properties, including the effective diffusivity and mobility at
the steady state, do depend on inertial mass, in sharp contrast to a non-
interacting ABP gas. Furthermore, the strong non-Gaussian distribution of
velocities returns towards equilibrium-like Gaussian for large mass. Finally,
with increasing mass, the deviation from the equilibrium fluctuation-
dissipation relation drops sharply. In summary, while activity amplifies non-
equilibrium features, increasing the inertial mass brings the fluid back to
equilibrium.

The model.– We consider $N$ active Brownian particles (ABP) of mass $m$, a
moment of inertia $I$, and diameter $\sigma$ moving in a two-dimensional (2d)
rectangular box of area $L_{x}\times L_{y}$ with periodic boundary conditions
(thus, density $\rho=N/L_{x}L_{y}$). The particles self-propel in directions
$\hat{n}_{i}=(\cos\theta_{i},\sin\theta_{i})$ with force ${\bf
F}_{A,i}=\gamma_{t}v_{0}\hat{n}_{i}$. Without interactions, this leads to a
propulsion speed $v_{0}$. The heading direction (i.e., $\theta_{i}$) undergoes
a long-time diffusion leading to an effective persistent motion. The inertial
active dynamics evolve as

$\displaystyle m\dot{\textbf{v}}_{i}$ $\displaystyle=$
$\displaystyle-\gamma_{t}\textbf{v}_{i}+\textbf{F}_{i}+\textbf{F}_{A,i}+\gamma_{t}\sqrt{2D_{t}}\,{\eta_{i}(t)}$
$\displaystyle I\ddot{\theta_{i}}$ $\displaystyle=$
$\displaystyle-\gamma_{r}\dot{\theta_{i}}+\gamma_{r}\sqrt{2D_{r}}\,{\zeta_{i}(t)},$
(1)

where $\textbf{F}_{i}=-\nabla_{i}\sum_{j}U(r_{ij})$, the symbols $\eta_{i}(t)$
and $\zeta_{i}(t)$ represent Gaussian white noises, and viscous drags,
associated with translation and rotation, are described by
$\gamma_{t}\textbf{v}_{i}$ and $\gamma_{r}\dot{\theta_{i}}$, respectively.
Interactions among particles are due to volume exclusion effects modeled via
the Weeks-Chandler-Anderson potential:
$U(r)=4\epsilon[(\sigma/r)^{12}-(\sigma/r)^{6}]+\epsilon$ if the inter-
particle separation $r<r_{c}=2^{1/6}\,\sigma$ and $U(r)=0$ otherwise. The
units of length and energy are set by $\sigma$ and $\epsilon$, respectively.
The rotational diffusivity $D_{r}$ can have a non-thermal active origin.
Inertial relaxations in translation and rotation take time scales
$\tau_{I}=m/\gamma_{t}$ and $\tau_{d}=I/\gamma_{r}$. Beyond $\tau_{d}$,
diffusion in the heading direction leads to a persistent motion with a bare
persistence time $\tau_{p}=1/D_{r}$, which sets the unit of time. Unless
specified otherwise, we use an equilibrium heat bath with
$D_{t}=1.0\,\sigma^{2}D_{r}$, and a small and constant $\tau_{d}=0.33\tau_{p}$
[20]. We particularly focus on the impact of changing activity in terms of the
Péclet number ${\rm Pe}=v_{0}/(D_{r}\sigma)$ and that of the reduced mass
using $M=\tau_{I}/\tau_{p}$. This system can undergo motility-induced phase
separation (MIPS) for low enough $M$, and high enough Pe at a moderate density
[20]. In order to fully characterize the properties of a homogeneous active
bath, we fix the density to a low value, $\rho=0.1$, corresponding to a
packing fraction $8\%$, such that the system does not phase-separate even in
the over-damped limit [7].

Figure 1: Velocity distributions in the particle frame. Probability
distributions of velocity components (a) parallel $P(v_{\parallel})$ and (b)
perpendicular $P(v_{\perp})$ to the heading direction at $Pe=100$ and for
different $M$ values indicated in the legend. Similarly, in $P(v_{\perp})$,
non-Gaussian long tails disappear at high inertia. For comparison, we present
the distribution functions for non-interacting particles at $M=0.01$ using the
solid black lines in both plots.

Results.– An increase in Pe, as expected, drives the system away from
equilibrium. In Fig.1, we show the change in velocity distribution to reveal
the impact of inertia. For this purpose, we consider the two components of the
velocity, in the heading direction $v_{\parallel}={{\bf v}\cdot\hat{n}}$ and
perpendicular to it $v_{\perp}=({\mathbb{1}}-\hat{n}\hat{n})\cdot{\bf v}$. In
2d, $v_{\perp}$ is a scalar. At small $M$, inertial lag is small. Despite
that, $P(v_{\parallel})$ at $M=0.01$ shows a long tail; see the red curve in
Fig.1(a). This tail, which is absent in a system of non-interacting active
particles, emerges due to enhanced frontal collision in the heading direction
and the resultant inertial recoil. At larger inertia, secondary back
collisions become prominent, symmetrizing the distribution to a Gaussian-like
profile, e.g., at $M=5$ 111In the Supplementary Information provided online,
we discuss further how $M$ equilibrates the system, analyze the pair
distribution function, show the calculation of mobility, diffusivity, and the
behavior of scaling exponents describing kinetic temperature, effective
diffusivity, and mobility.. The asymmetry in collisions will be scrutinized
using a pair distribution function in Fig. 2. The $P(v_{\perp})$ distribution
also shows inertial restoration of equilibrium-like behavior. The long non-
Gaussian tails are observable at small $M$ and disappear with an increase in
$M$. The comparison with distributions obtained in the absence of interaction,
black curves in Fig.1(a) and (b), shows that the non-Gaussian behaviors are
strongly dependent on the inter-particle collisions. At high inertia, bouncing
backward and forward from neighboring particles symmetrizes the distributions
rendering them equilibrium-like shapes.

Figure 2: Pair distribution in the heading direction: (a) $P(r_{\parallel})$
at $Pe=100$ and various $M$ values indicated in the legend. The fore-aft
asymmetry decreases with increasing $M$. One gets a fully symmetric
distribution at equilibrium, as shown in (c). (b) Variation of asymmetry
parameter $A_{p}$ as a function of Pe is shown for various $M$ values
indicated in the legend. (d) An approximate data collapse is obtained for
inertial systems using appropriate rescaling of (b). (e) The variation of
scale factors shown as a function of $M$. At large $M$, they show approximate
dependencies $\mathcal{E}\sim M^{0.4}$ and $\mathcal{F}\sim M^{0.6}$.  Figure
3: (a) The increase in kinetic temperature $k_{B}T_{\rm kin}=m\langle
v^{2}\rangle/2$ with Pe is shown at various $M$ values indicated in the
legend. At large Pe, $k_{B}T_{\rm kin}\sim{\rm Pe}^{\alpha_{k}}$ with
$\alpha_{k}=1.90,1.69,1.54,1.53$ at $M=0.01,0.1,1,5$. (b) The increase in
effective diffusivity $D_{\textrm{eff}}$ With Pe is shown at at various $M$
values listed in the legend. Asymptotically it grows as $D_{\rm eff}\sim{\rm
Pe}^{\alpha_{d}}$ with $\alpha_{d}=1.98,1.43,1.13,1.17$ at $M=0.01,0.1,1,5$.
(c) At large $M$, the kinetic temperature decreases as $k_{B}T_{\rm kin}\sim
M^{-\gamma_{k}}$ with $\gamma_{k}=1,0.8,0.75$ at Pe = 1, 10, and 100. (d)
$D_{\rm{eff}}$ reduces with $M$ to saturate, and follows a scaling form
$D_{\rm{eff}}={\cal B}+{\cal D}\,M^{-\gamma_{d}}$ with
$\gamma_{d}=0.43,0.52,0.81$ at Pe = 1, 10, and 100. (e) The mobility $\mu$ as
a function of Pe decreases as $\mu\sim{\rm Pe}^{-\alpha_{\mu}}$ where
$\alpha_{\mu}=0.02,0.13,0.31,0.41$ at $M=0.01,0.1,\,1,5$. (f) It decreases
with $M$ as $\mu\sim M^{-\gamma_{\mu}}$ at large $M$ with
$\gamma_{\mu}=0.82,0.77,0.52$ at $Pe=1,\,10,\,100$.

The impact of interaction manifests in the pair-distribution function in the
heading direction, i.e., the probability $P(r_{\parallel})$ of finding a
neighboring particle in direction $\hat{n}$; see Fig.2. Due to enhanced
frontal collision, more particles accumulate in front. Such accumulations in
front and associated depletion wakes have been analyzed recently in overdamped
dilute ABPs [39]. A peak in the back appears due to secondary back collisions
experienced by inertial ABPs after frontal recoil. With increasing $M$, recoil
increases, reducing the frontal accumulation and increasing the secondary
collisions from the back – these affect the symmetrization of the pair
distribution – restoring equilibrium-like behavior.

The asymmetry in the pair distribution is quantified in terms of the parameter
$A_{p}$ that measures the difference between the heights of the front and back
peaks (Fig.2(b) ). For fixed $M$, the asymmetry initially increases with Pe.
For overdamped systems (black solid line denoting $M=0$), the increase is
followed by saturation in the absence of recoil. In contrast, in the presence
of inertia, after the initial increase, $A_{p}$ decreases with Pe. The data
collapse in Fig.2(d) shows that at small Pe, the asymmetry increases with Pe
as $A_{p}\sim{\rm Pe}^{0.4}$ and then decreases as $A_{p}\sim{\rm
Pe}^{-0.35}$. Such scaling properties are common to all inertial ABPs. The
increase is due to enhanced frontal collisions associated with increased
activity. The decrease is due to inertial recoil and is the reason behind the
restoration of equilibrium-like behavior.

Another way of measuring how far the system is from equilibrium is given by
the extent to which the equilibrium Einstein relation is violated ${\cal
I}=\mid D_{\rm eff}-\mu\,k_{B}T_{\rm kin}\mid$, where $\mu$ is the particle
mobility and $T_{\rm kin}$ the so-called kinetic energy [40, 41]. Following a
tracer particle dynamics is a useful tool to characterize the properties of a
bath [11, 12, 14, 15, 16, 17]. In the presence of translational fluctuations,
the impact of the activity on diffusivity at low Pe gets overshadowed by
$D_{t}$. While these effects can be subtracted out, for simplicity and
intending to understand the impact of the activity, in the following, we set
$D_{t}=0$. From the late-time behavior of the mean-squared displacements
(MSD), we obtain the effective diffusivity $D_{\rm eff}$. The kinetic
temperature $k_{B}T_{\rm kin}$ is readily obtainable from the velocity
fluctuations. Using a separate numerical calculation of the change in velocity
$\langle v_{x}\rangle$ of a test particle under an external force $f_{x}$, we
obtain the mobility $\mu=(\partial\langle v_{x}\rangle/\partial
f_{x})_{f_{x}=0}$ around the non-equilibrium steady states of the ABPs. In the
presence of an external force on the test particle, a local statistical
reorganization of other ABPs follows. Such a reorganization depends on active
speed and inertia, resulting in mobility variation. Using these, we obtain
violation ${\cal I}$.

In Fig.3, we show the variations of kinetic temperature, diffusivity, and
mobility as a function of Pe and $M$. At large enough values of Pe and $M$,
and they show the following scaling forms: (i) $k_{B}T_{\rm kin}\sim{\rm
Pe}^{\alpha_{k}}$ and $k_{B}T_{\rm kin}\sim M^{-\gamma_{k}}$, (ii) $D_{\rm
eff}\sim{\rm Pe}^{\alpha_{d}}$ and $D_{\rm eff}\sim M^{-\gamma_{d}}$, (iii)
$\mu\sim{\rm Pe}^{-\alpha_{\mu}}$ and $\mu\sim M^{-\gamma_{\mu}}$. Note that
mobility decreases both with Pe and $M$, while the other quantities increase
with Pe.

The kinetic temperature $k_{B}T_{\rm kin}=m\langle v^{2}\rangle/2$ increases
as a function of Pe (Fig.3(a) ). The black solid line captures a Pe2 scaling
common to all $M$ values at small Pe. The asymptotic behavior $k_{B}T_{\rm
kin}\sim{\rm Pe}^{\alpha_{k}}$ at various $M$ values shows how the exponent
$\alpha_{k}$ decreases with $M$.

In Fig.3(b) we show how the effective diffusivity $D_{\textrm{eff}}\sim{\rm
Pe}^{\alpha_{d}}$ increases with Pe. At small Pe, $D_{\textrm{eff}}\sim{\rm
Pe}^{2}$ for all $M$. This behavior continues to all Pe in the limit of small
inertia, $M=0.01$, as in over-damped systems. However, with increasing
inertia, the value of the growth exponent $\alpha_{d}$ decreases, and finally,
$D_{\rm{eff}}\sim{\rm Pe}^{\alpha_{d}}$ becomes approximately linear with Pe
at $M\gtrsim 1$.

In Fig.3(c) and (d), we show the $M$-dependency of the kinetic temperature and
effective diffusivity. The kinetic temperature fits to the functional form
$k_{B}T_{\rm kin}={\cal A}/({\cal C}+M^{\gamma_{k}})$, such that at large $M$,
they show a scaling form $k_{B}T_{\rm kin}\sim M^{-\gamma_{k}}$ with the decay
exponent $\gamma_{k}$ decreasing with Pe. The effective diffusivity
$D_{\rm{eff}}$ decays with $M$ until it saturates, and follows a scaling form
$D_{\rm{eff}}={\cal B}+{\cal D}\,M^{-\gamma_{d}}$ with the decay exponent
$\gamma_{d}$ increasing with Pe.

Finally, we consider the mobility $\mu$. In the over-damped limit, $\mu$ is
approximately independent of activity Pe ($M=0.01$ in Fig.3(f) ). This
behavior agrees with the observation in Ref. [23] at low densities. However,
in the presence of inertia, mobility decreases with both Pe and $M$. At the
larger activity, an increase in frontal collision frequency in the direction
of external force increases the effective drag, leading to an effective
thickening of the active fluid. Such a thickening is in qualitative agreement
with Ref. [25], unlike the non-monotonic variation predicted in Ref. [22], and
oppose to the active thinning predicted in Ref. [42]. As a function of Pe, the
mobility decreases as $\mu\sim{\rm Pe}^{-\alpha_{\mu}}$ where the exponent
$\alpha_{\mu}$ itself increases with $M$ (Fig.3(f) ). The collision and recoil
in inertial ABPs further increase the drag coefficient for higher inertial
mass. The reduction of $\mu$ with $M$ follows a form similar to that of the
dependence of $k_{B}T_{\rm kin}$ on $M$ (Fig.3(c) ), such that at large $M$
$\mu\sim M^{-\gamma_{\mu}}$. Note that $\gamma_{\mu}$ also decreases with
$Pe$.

Figure 4:  The violation of equilibrium fluctuation-dissipation ${\cal I}=\mid
D_{\textrm{eff}}-\mu k_{B}T_{\rm{kin}}\mid$, a measure of how far the system
is from equilibrium as a function of $Pe$ and $M$, is shown in (a) and (b),
respectively. ${\cal I}$ increase with Pe as Pe${}^{\alpha_{d}}$ or
Pe${}^{(\alpha_{k}-\alpha_{\mu})}$. With $M$, it decreases to saturate at
${\cal B}$.

With the help of these results, we find that the violation ${\cal I}$
increases with Pe, but decreases with $M$. In particular, our findings
indicate that ${\cal I}\sim\mid{\rm Pe}^{\alpha_{d}}-{\rm
Pe}^{-\alpha_{\mu}+\alpha_{k}}\mid$, i.e., it can grow as ${\rm
Pe}^{\alpha_{d}}$ or ${\rm Pe}^{\alpha_{k}-\alpha_{\mu}}$ depending on whether
$\alpha_{d}>(\alpha_{k}-\alpha_{\mu})$ or not (Fig.4(a) ). On the other hand,
${\cal I}\sim\mid{\cal B}+{\cal
D}M^{-\gamma_{d}}-M^{-(\gamma_{\mu}+\gamma_{k})}\mid$ decreases with $M$ to
saturate at a Pe-dependent value ${\cal B}$ (Fig.4(b) ). Such a decrease is
more significant at higher Pe. For small Pe, the system remains close to
equilibrium for all $M$ values.

Concluding remarks.–  We have shown how the non-equilibrium properties of an
active fluid consisting of ABPs, at a density far away from the onset of MIPS
behave as a function of activity and inertial mass. While the departure from
equilibrium gets pronounced with increasing Pe, i.e., activity, we found that
inertial mass restores equilibrium-like properties. In particular, we showed
that the non-Gaussian velocity distributions, the fore-aft asymmetry in the
(heading-direction) pair distribution, and the absence of an Einstein
fluctuation-dissipation relation between diffusivity, mobility, and kinetic
temperature observed in active over-damped systems, crossover to their
equilibrium counterparts with inertial mass. In short, we found that the
inertial recoil can effectively thermalize the active fluid. Remarkably, the
late-time diffusivity and mobility of the bath depend on inertial mass, in
contrast to free ABPs. The effective diffusivity and temperature grow with
active velocity and decrease with inertial mass. In contrast, effective
mobility decreases with mass and activity. Together, these findings show a
reduction in the violation of equilibrium fluctuation-dissipation with
increasing inertia. In summary, inertia brings back equilibrium-like behavior
in the active fluid.

###### Acknowledgements.

Numerical calculations were supported by SAMKHYA, the high-performance
computing facility at the Institute of Physics, Bhubaneswar. D.C. thanks
Arghya Majee for critical reading of the manuscript, C.Y. Cergy Paris
Université for a Visiting Professorship which enabled the initiation of
current work, SERB, India, for financial support through Grant No.
MTR/2019/000750, and International Centre for Theoretical Sciences (ICTS-
TIFR), Bangalore, for an Associateship. F.P. acknowledges financial support
from C.Y. Initiative of Excellence (grant ’Investissements d’Avenir’
ANR-16-IDEX-0008), INEX 2021 Ambition Project CollInt and Labex MME-DII,
projects 2021-258 and 2021-297.

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##  Appendix A Supplemental Information: When does an active bath behave as an
equilibrium one?

Figure 5: Inertia brings back equilibrium-like features. The speed
distributions in heading direction at $Pe=10$ for different masses $M=0.01$ in
(a), $M=0.1$ in (b), $M=1$ in (c) and $M=5$ in (d). In each plot, blue and red
curves represent the results for $D_{t}$ = 1.0 and 0.0, respectively.

###  A.1 Inertial thermalization

As has been pointed out in the main text, with increasing $M$, the velocity
distribution gets thermalized due to collisions and recoil. Here we supplement
that finding with an independent measure in which we compare the velocity
distribution in the heading direction in the presence ($D_{t}=1$) and absence
($D_{t}=0$) of a thermal bath (Fig.5). For small $M$, the two distributions
are significantly different from each other. With increasing $M$, the
distribution in the absence of thermal bath starts to thermalize, aided by
relatively faster orientational relaxation of the heading direction during a
slow inertial relaxation, and collisions from the front and secondary
collisions from back after the frontal recoil. As a result, they start to come
close to each other to merge at $M\gtrsim 1$.

Figure 6:  Probability distribution of particle separation of interacting
neighbors projected along the heading direction $P(r_{\parallel})$ at the
system density $\rho=0.1$ for $Pe=2,50$ and $100$. The results In (a)
correspond to an over-damped system and in (b), (c), and (d) for an under-
damped system with $M$ values = 0.01, 0.1, and 1.0, respectively. The common
legend representing different $Pe$ values is placed inside plot (d).

###  A.2 Pair distribution

In Fig.6, we show the distribution of particles around a test particle in its
heading direction at Pe=1, 10, and 100 and inertia $M=0,0.01,0.1,1$. These
distributions are used to calculate the asymmetry parameter $A_{p}$ presented
in the main text.

Figure 7:  The mean velocity of a tagged particle in response to the external
force $F_{x}$ is shown for $Pe=1$, 10, and 100 in (a) at $M=0.01$, in (b) at
$M=0.1$, in (c) at $M=1.0$ and in (d) at $M=5$. The slope of the corresponding
linear fits at $F_{x}=0$ gives mobility $\mu=\lim_{F_{x}\to 0}\langle\delta
v_{x}\rangle/F_{x}$. By definition $\langle\delta v_{x}\rangle=0$ at
$F_{x}=0$. For the sake of clear visualization, each curve for Pe = 10 and 100
is shifted upwards.

###  A.3  Mobility

To measure mobility, we apply an additional force $F_{x}$ on the tagged
particle and calculate its mean velocity along the same direction $\langle
v_{x}\rangle$ for various $F_{x}$ values. The mean velocity $\langle\delta
v_{x}\rangle=\langle v_{x}\rangle|_{F_{x}}-\langle v_{x}\rangle|_{F_{x}=0}$ as
a function of $F_{x}$ is shown in the figure 7. The slope of the linear fit
for $\langle\delta v_{x}\rangle$ vs $F_{x}$ curve near $F_{x}=0$ gives
mobility. In the overdamped regime $M=0.01$, mobility does not show dependency
on Pe as its value is observed to be the same $\mu\sim 0.9$ for Pe = 1, 10,
and 100 Fig.7 (a). In the underdamped limit ($M\geq 0.1$), mobility decreases
with Pe, see Fig.7 (b),(c), (d).

Figure 8:  Mean square displacement for Pe = 1, 5, 20, and 70 in (a) at
$M=0.01$, in (b) at $M=0.1$, in (c) at $M=1.0$ and in (d) at $M=5$. The common
legend representing different Pe values is placed at the extreme right. These
calculations are done for $D_{t}=0$.

###  A.4 Mean-squared displacement

In Fig.8, we plot mean-squared displacement $\langle r^{2}\rangle$ scaled by
time $t$ for different $M$ and Pe values. All of them show ballistic to
diffusive crossover at a time scale determined by the orientational
persistence of the heading direction, a quantity that is kept constant in this
paper. The asymptotic diffusivities $D_{\rm eff}=\lim_{t\to\infty}\langle{\bf
r}^{2}\rangle/4t$ are obtained from these graphs and used in the main text.

Figure 9:  Scaling exponents $\gamma_{i}$ and $\alpha_{i}$ used in Fig.4 are
presented as a function of Pe (a) and $M$ (b), respectively. $i=k,d,\mu$
represent exponents of kinetic energy, diffusion, and mobility.

###  A.5 Scaling exponents

The variations of scaling exponents determining the Pe and $M$ dependence of
kinematic temperature, effective diffusivity, and mobility are shown in Fig.9.