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#  Hydrodynamic interactions in anomalous rheology of active suspensions

Haruki Hayano and Akira Furukawa  Institute of Industrial Science, University
of Tokyo, Meguro-ku, Tokyo 153-8505, Japan

###### Abstract

We explore a mechanism of the anomalous rheology of active suspensions by
hydrodynamic simulations using model pusher swimmers. Our simulations
demonstrate that hydrodynamic interactions under shear flow systematically
orient swimmers along the extension direction, which is responsible for
determining the global swimming states and the resulting significant viscosity
reduction. The present results indicate the essential role of hydrodynamic
interactions in the elementary processes controlling the rheological
properties in active suspensions. Furthermore, such processes may be the
substance of the previously proposed scenario for anomalous rheology based on
the interplay between the rotational diffusivities and the external shear
flow.

##  I Introduction

Anomalous rheology observed in the broad class of active suspensions is one of
the most typical phenomena highlighting distinctive differences from passive
systems Hatwalne ; Sokolov ; Gachelin ; Lopez ; Liu ; PNAS2020 ; Rafai ;
Review2 ; Marchetti ; Review3  . In particular, for rod-like extensile pusher
micro-swimmers (such as E. coli), a significant viscosity reduction has been
experimentally observed at lower shear rates and volume fractions Sokolov ;
Gachelin ; Lopez ; Liu ; PNAS2020 , which frequently leads to a superfluid
state with zero viscosity Lopez ; PNAS2020 . A seminal study by Hatwalne et
al. Hatwalne  predicted that if an orientational order along the extension
axis of the applied flow is somehow realized, the active dipolar forces
intensify the mean flow, reducing the resistive stress required to drive the
external flow and thus the viscosity. Following that, many theoretical
attempts have been made to predict or explain the anomalous rheology in active
suspensions (see papers Ishikawa_Pedray ; Cates ; Haines ; Giomi ; Saintillan1
; Ryan ; Moradi_Najafi ; Nechtel_Khair ; Takatori_Brady  and the references
therein).

In dilute suspensions of rod-like particles, the orientational distribution of
particles under shear flow are known to be enhanced along the extension axis
when (thermal or athermal) random rotational diffusion processes exist
Hinch_Leal1 ; Hinch_Leal2 . By taking such fluctuation effects into account,
the viscosity reduction was successfully modeled within the framework of
continuum kinetic theory Haines ; Saintillan1 .

In dilute/semidilute active suspensions, hydrodynamic interactions (HIs) are
expected to play a crucial role in couplings among constituents Review1 ;
LaugaB . In Ref. Ryan , it is theoretically demonstrated that the long-range
HIs induce marked density fluctuations that provide additional sources of the
effective rotational noise, resulting in a decrease in the viscosity. Indeed,
recent experiments PNAS2020  indicate a close link between collective many-
body properties and anomalous rheology. Nevertheless, due to the highly
nonlinear and nonequilibrium nature of HIs, our understanding of the extent to
which interactions among swimmers are involved in the rheological properties
is still lacking beyond the effective one-body theory.

In this study, we investigate the mechanism of the anomalous viscosity
reduction observed in active suspensions by revisiting the role of HIs. Our
analysis, along with a phenomenological explanation, elucidates that swimming
along the extension axis of the applied flow is hydrodynamically favorable,
resulting in a significant reduction of the viscosity. Furthermore, we argue
that in usual swimming bacteria, such as E. coli, the self-propulsive forces
are strong enough that the induced HIs can compete or dominate other effects
like thermal fluctuations even in dilute suspensions.

##  II Model swimmer system

For the present purpose, we perform hydrodynamic simulations of model active
suspensions composed of $N$ rod-like dumbbell swimmers with a prescribed force
dipole. Our model swimmer, schematically shown in Fig. 1(a), is composed of
body and flagellum parts. The body part is treated as a rigid body, while the
flagellum part is regarded as a massless “phantom” particle simply following
the body’s motions. This treatment always keeps the relative position of these
two parts unchanged. For the $\alpha$-th swimmer ($\alpha=1,\cdots,N$), it is
assumed that the force $F_{A}{\hat{\mbox{\boldmath$n$}}}_{\alpha}$ acting on
the (front) body is exerted by the (rear) flagellum and that the flagellum
also exerts the force $-F_{A}{\hat{\mbox{\boldmath$n$}}}_{\alpha}$ directly on
the solvent fluid. Here, ${\hat{\mbox{\boldmath$n$}}}_{\alpha}$ is the
direction of the $\alpha$-th swimmer, and these forces compose a dipolar force
(please refer to Appendix A for details). The present particle-base model is
essentially the same as those proposed in Refs. Graham1 ; Graham2  and used in
Refs. Haines ; Ryan ; Gyrya ; Decoene ; Furukawa_Marenduzzo_Cates . Continuum
kinetic models of hydrodynamically interacting rod-like swimmers with
prescribed stresses or forces were also developed Haines ; Ryan ; Saintillan-
Shelley ; Saintillan-Shelley2 ; Saintillan-Shelley3 ; Baskaran-Marchetti . In
Refs. Saintillan-Shelley ; Saintillan-Shelley2 ; Saintillan-Shelley3 , it was
demonstrated that nonlinear hydrodynamic effects can lead to larger-scale
correlated motions with marked density fluctuations.

As illustrated in Fig. 1(a), the body and flagellum parts are assumed to have
the same shape and are each described by a superposition of three spheres with
a common radius $R$. The spheres composing the body are located at the
positions
${{\mbox{\boldmath$R$}}}_{i,\alpha}^{(b)}={{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}+(2-i)R{\hat{\mbox{\boldmath$n$}}}_{\alpha}$
($i=1,2,3$), where ${{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}$ is the $\alpha$-th
swimmer’s center-of-mass position. Similarly, the spheres composing the
flagellum part are located at
${{\mbox{\boldmath$R$}}}_{i,\alpha}^{(f)}={{\mbox{\boldmath$R$}}}_{\alpha}^{(CF)}+(2-i)R{\hat{\mbox{\boldmath$n$}}}_{\alpha}$
($i=1,2,3$), where
${{\mbox{\boldmath$R$}}}_{\alpha}^{(CF)}={{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}-4.5R{\hat{\mbox{\boldmath$n$}}}_{\alpha}={{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}-\ell_{0}{\hat{\mbox{\boldmath$n$}}}_{\alpha}$
is the position of the center of the flagellum. The shape of the present model
swimmer shows the head-tail symmetry, and the mid point is thus given by
${{\mbox{\boldmath$R$}}}_{\alpha}=({{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}+{{\mbox{\boldmath$R$}}}_{\alpha}^{(CF)})/2$.
Although arbitrary shapes of swimmers with an imposed head-tail asymmetry can
be composed, we may obtain qualitatively the same results as long as these
swimmers have rod-like forms with the prescribed force dipoles.

Periodic boundary conditions are imposed in the $x$\- and $y$-directions with
the linear dimension $L$, and the planner top and bottom walls are placed at
$z=H/2$ and $-H/2$, respectively. The shear flow is imposed by moving the top
and bottom walls in the $x$-direction at constant velocities $V/2$ and $-V/2$,
respectively, whereby the mean shear rate is $\dot{\gamma}=V/H$. This
situation is illustrated in Fig. 1(b). Hydrodynamic interactions among the
swimmers are incorporated by adopting the smoothed profile method (SPM) SPM ;
SPM2 ; SPM3 , which is one of the mesoscopic simulation techniques LB1 ; LB2 ;
FPD ; FPD2 ; MPC ; DPD  . In the SPM SPM ; SPM2 ; SPM3 , the dynamics of rigid
particles and a host fluid can be considered simultaneously with vastly
reducing numerical costs by replacing particle-fluid boundaries with smoothed
ones and by taking particle rigidity into account through the body force term
in the Navier-Stokes equation. The details of the simulation methods are
presented in Appendix A and B.

Figure 1: (Color online) (a) Our model swimmer comprises “body” and
“flagellum” parts with symmetric shapes. Each parts are constituted by a
superposition of three spheres with radius $R$. We assume that the force
$F_{A}{\hat{\mbox{\boldmath$n$}}}_{\alpha}$ is exerted on the body, while
$-F_{A}{\hat{\mbox{\boldmath$n$}}}_{\alpha}$ is directly exerted on the
solvent through the flagellum part, with
${\hat{\mbox{\boldmath$n$}}}_{\alpha}$ being the orientation of the
$\alpha$-th swimmer. These forces constitute a force dipole with the magnitude
$F_{A}\ell_{0}$. Here, $\ell_{0}$ is the characteristic swimmer’s length and,
for the present model, is given as the separation distance between the body
and flagellum centers. (b) The periodic boundary conditions are imposed in the
$x$\- and $y$-directions with the linear dimension $L$. The shear flow is
imposed by moving the top and bottom walls in the $x$-direction at constant
velocities $V/2$ and $-V/2$, respectively. These two walls are separated in
the $z$-direction by the distance $H$.

##  III Results

###  III.1 Steady-state properties: weak alignment of the swimmers and the
resultant viscosity reduction

Figure 2: (Color online) (a) The $\dot{\gamma}$-dependence of the viscosity
$\eta$ for various $\phi$ at $H=128$. For smaller $\dot{\gamma}$ and larger
$\phi$, $\eta$ becomes smaller. The inset shows $H$-dependence of $\eta$ for
two different $\phi$ at $\dot{\gamma}=10^{-3}$. In the present range of $H$,
$\eta$ is significantly smaller for larger $H$. (b) The $\phi$-dependent
viscosity for active and passive rod-like particles at $H=128$ and
$\dot{\gamma}=10^{-3}$. Our preliminary results for passive ($F_{A}=0$) and
active-puller ($F_{A}=-20$) cases show that $\eta$ increases with increasing
$\phi$; the viscosity enhancement is much stronger in the active-puller case.
In both (a) and (b), the dashed lines indicate the solvent viscosity
$\eta_{s}(=1)$.

First, in Fig. 2, we show the viscosity $\eta$ for various conditions. In this
study, the viscosity is defined as

$\displaystyle\eta=\dfrac{1}{{\dot{\gamma}}L^{2}}\int{\rm d}x{\rm
d}y\langle\Sigma_{xz}(x,y,\pm H/2)\rangle,$ (1)

where $\Sigma_{xz}(x,y,\pm H/2)$ is the $xz$ component of the stress tensor at
the walls and $\langle\cdots\rangle$ hereafter denotes taking the time average
in a steady state. Here, the solvent viscosity is scaled to be $1$. At a
relatively low shear rate $\dot{\gamma}$, we find that $\eta$ takes lower
values than the solvent viscosity, which qualitatively agrees with the
experimental results Sokolov ; Gachelin ; Lopez ; Liu ; PNAS2020 . This
behavior strongly depends on $H$ and the volume fraction of the swimmers
defined as $\phi=N{\mathcal{V}}^{(b)}/L^{2}H$ with ${\mathcal{V}}^{(b)}$ being
the volume of the body part. The viscosity can be divided into three parts:
$\eta=\eta_{s}+\Delta\eta_{p}+\Delta\eta_{a}$, where $\eta_{s}(=1$ in this
study) is the solvent viscosity, $\Delta\eta_{p}$ and $\Delta\eta_{a}$ are the
passive and active contributions, respectively (see Appendix A for more
details). In the present framework, $\Delta\eta_{a}$ is given as

$\displaystyle\Delta\eta_{a}=-\dfrac{1}{{\dot{\gamma}}L^{2}H}F_{A}\ell_{0}\sum_{\alpha=1}^{N}\langle\hat{n}_{\alpha,x}(t)\hat{n}_{\alpha,z}(t)\rangle,$
(2)

where ${\hat{\mbox{\boldmath$n$}}}_{\alpha}(t)$ is the unit vector
representing the $\alpha$-th swimmer’s orientation at time $t$, and
${\hat{n}}_{\alpha,x}$ and ${\hat{n}}_{\alpha,z}$ are its $x$ and $z$
components, respectively. Essentially identical expressions of Eq. (2) were
previously derived (see Refs. Haines ; Saintillan1 ; Review3  for example).
From Eq. (2), when swimmers tend to align along the extension direction of the
flow field ($\langle\hat{n}_{\alpha,x}\hat{n}_{\alpha,z}\rangle>0$),
$\Delta\eta_{a}<0$. Since the contribution of $\Delta\eta_{p}$ to $\eta$ is
positive in general, a significant decrease in the viscosity occurs from the
negative $\Delta\eta_{a}$.

To further explore what swimming states are involved in the viscosity
reduction, we investigate the following steady-state quantities:
${\rho}(z)=\sum_{\alpha=1}^{N}\langle\delta[{\mbox{\boldmath$r$}}-{\mbox{\boldmath$R$}_{\alpha}}(t)]\rangle$,
${\mbox{\boldmath$p$}}(z)=\sum_{\alpha=1}^{N}\langle{\hat{\mbox{\boldmath$n$}}}_{\alpha}(t)\delta[{\mbox{\boldmath$r$}}-{\mbox{\boldmath$R$}_{\alpha}}(t)]\rangle$,
and
${\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$W$}}}}(z)=\sum_{\alpha=1}^{N}\langle[{\hat{\mbox{\boldmath$n$}}}_{\alpha}(t){\hat{\mbox{\boldmath$n$}}}_{\alpha}(t)-{\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$\delta$}}}}/3]\delta[{\mbox{\boldmath$r$}}-{\mbox{\boldmath$R$}_{\alpha}}(t)]\rangle$.
Here, ${\mbox{\boldmath$R$}_{\alpha}}(t)$ is the center of the force dipole
($\neq$ the center-of-mass position), $\rho(z)$ is the denisty, and
${\mbox{\boldmath$p$}}(z)/\rho(z)$ and
${\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$W$}}}}(z)/\rho(z)$
represent the polarization vector and the nematic order parameter tensor,
respectively Review2 . These quantities, which depend only on $z$ at steady
state, are shown in Figs. 3(a)-(h) for various conditions. In Figs. 3(a) and
(b), $\rho(z)$ has significant peaks near the boundary walls, and otherwise,
it is almost constant, indicating that the walls attract swimmers. Such
behaviors were already reported and discussed in the literature (for example,
Refs. Berke ; Ji-Tang ; Review1 ; LaugaB ; Figueroa-Morales ; Bianchi ;
Denissenko ; Ezhilan ; Ezhilan-Saintillan ; Yan-Brady ). In the present model,
without thermal fluctuations, when placing one swimmer near the wall, it
continues to swim along the wall, which suggests that the force-dipole
prescribed to the swimmer contributes to the wall attraction Berke . However,
in the many-swimmer case, significant disturbances are induced by interactions
among the swimmers. Such disturbances produce an outgoing flux from the wall
to the bulk region. Meanwhile, self-propulsive motions give incoming flux to
the wall from the bulk. Competition between these two flux terms should
determine the amount of accumulation of swimmers at the walls Ezhilan-
Saintillan ; Yan-Brady .

Figures 3(c)-(f) show ${\mbox{\boldmath$p$}}(z)/{\rho}(z)$. Due to the flow
and geometrical symmetries, $p_{y}(z)=0$ for all $z$. For $\phi\gtrsim 0.03$,
swimmers trapped at the walls tilt their “heads” to the walls Vigeant ;
Spagnolie_Lauga ; Sipos . Moreover, the tilting angle is greater for larger
$\phi$, which may be caused by HIs among the swimmers on the wall. These
issues will be studied elsewhere.

In terms of the viscosity reduction, among Figs. 3(a)-(h), of particular
interest are (g) and (h), exhibiting $W_{xz}(z)/\rho(z)$. At $\phi=0.01$ and
$H=128$, where the viscosity reduction is absent (see Fig. 2),
$W_{xz}(z)\lesssim 0$ as a whole. For larger $\phi$ and $H$, in contrast,
$W_{xz}(z)>0$ for all $z$. Within the present range of $\phi$ and $H$, by
increasing these parameters, the upward convex form of $W_{xz}(z)/\rho(z)$
tends to grow. Equation (2) is rewritten as

$\displaystyle\Delta\eta_{a}=-\dfrac{1}{{\dot{\gamma}}H}F_{A}\ell_{0}\int_{-H/2}^{H/2}{\rm
d}zW_{xz}(z),$ (3)

through which $W_{xz}(z)$ is directly related to the viscosity reduction.

Figure 3: (Color online) Subfigures (a), (c), (e), and (g) represent
$\rho(z)$, $p_{z}(z)/\rho(z)$, $p_{x}(z)/\rho(z)$, and $W_{xz}(z)/\rho(z)$,
respectively, for various $\phi$ at $H=128$ and $\dot{\gamma}=10^{-3}$, while
the results for various $H$ at $\phi=0.01$ and $\dot{\gamma}=10^{-3}$ are
shown in (b), (d), (f), and (h).

The reduced viscosity immediately indicates the reduced shear rate at the
walls. Here, we briefly review this behavior. For a swimmer with
${\hat{n}}_{\alpha,x}{\hat{n}}_{\alpha,z}>0$, the active-force reinforces the
applied flow. More specifically, in a small region including the swimmer, the
velocity gradient is intensified, while in outer regions, the opposite
happens. A superposition of such contributions gives the net effects on the
mean flow, and we observe a lower shear rate at the walls,
${\dot{\gamma}}_{w}$, in exchange for a greater shear rate in the interior
region. These situations are schematically illustrated in Figs. 4 (a) and (b).
Consequently, as shown in Fig. 4(c), the shear stress required to maintain the
applied shear rate $\dot{\gamma}=V/H$ is reduced Hatwalne , and the viscosity
is given by

$\displaystyle\eta=\eta_{s}\dfrac{\dot{\gamma}_{w}}{\dot{\gamma}},$ (4)

which is smaller than $\eta_{s}$ for $\dot{\gamma}_{w}<\dot{\gamma}$. Such a
modulation of the velocity field accompanying with the viscosity reduction was
certainly observed in experiments of E. coli suspensions PNAS2020 . Notably,
in contrast to active (pusher) suspensions, dispersed particles in the usual
passive system suppress the velocity gradient in the interior region, and the
observed viscosity is increased.

Figure 4: (Color online) (a) For a swimmer with
${\hat{n}}_{\alpha,x}{\hat{n}}_{\alpha,z}>0$, the surrounding velocity
gradient is intensified, while away from the swimmer, the opposite occurs. (b)
The net flow is determined by a superposition of the individual swimmers’
contributions. (c) The $x$ component of the velocity field averaged over the
steady state at $\phi=0.032$ and $H=256$. At the walls, the shear rate is
lower than $\dot{\gamma}$ as
${\dot{\gamma}}_{w}=(\eta/\eta_{s}){\dot{\gamma}}<{\dot{\gamma}}$, in exchange
for a greater shear rate in the interior region. (d) $W_{xz}(z)/\rho(z)$ at
the same condition as (c).

###  III.2 Key role of hydrodynamic interactions in determining the global
swimming states

Then, we investigate how such observed global steady states are realized. To
this end, taking the flow and geometrical symmetries into account, it is
useful to classify one-swimmer states into the following four states (see Fig.
5): state 1, $({\hat{n}}_{\alpha,x}>0,{\hat{n}}_{\alpha,z}>0)$, state 2,
$({\hat{n}}_{\alpha,x}<0,{\hat{n}}_{\alpha,z}>0)$, state 3,
$({\hat{n}}_{\alpha,x}<0,{\hat{n}}_{\alpha,z}<0)$, and state 4,
$({\hat{n}}_{\alpha,x}>0,{\hat{n}}_{\alpha,z}<0)$. States 1(2) and 3(4) are
equivalent; that is, they can be converted to each other by simply rotating
the coordinate frame about the $y$-axis by $\pi$. Figures 3(g) and (h) with
Eq. (3) indicate that, when $\Delta\eta_{a}<0$, states 1 and 3 with
${\hat{n}}_{\alpha,x}{\hat{n}}_{\alpha,z}>0$ are realized more favorably than
states 2 and 4 with ${\hat{n}}_{\alpha,x}{\hat{n}}_{\alpha,z}<0$. Thus, we may
further classify states 1 and 3 into the favorable $F$-state and states 2 and
4 into the unfavorable $UF$-state.

Figure 5: (Color online) Fourfold classification of (one-swimmer) swimming
states. These four states are further classified into the favorable $F$\- and
the unfavorable $UF$-states.

Now, the question is why states 1 and 3, which contribute to $W_{xz}>0$, are
more favorable than states 2 and 4. In our simulations, thermal effects are
absent, and excluded volume effects are almost irrelevant because the
suspensions mainly considered here are dilute. Instead, hydrodynamic effects
are expected to determine the overall swimming states. We expect swimmer’s
motions to be largely disturbed by HIs even without approaching the contact
distance to other swimmers; we regard such events as hydrodynamic collisions.
We support this perspective by analyzing the transition probabilities between
the swimming states: we pick up a pair of swimmers whose separation at time
$t$ is less than a certain close distance $d_{0}$; then, the transition
probabilities are determined by comparing their states at $t-\Delta t$ and
$t+\Delta t$. In this study, we set $d_{0}=0.7\ell_{0}$ and $\Delta
t=1.25\tau_{0}$. Here, $\tau_{0}=\ell_{0}/v_{s}$ is the time to travel the
distance of the swimmer size ($\sim\ell_{0}$), with $v_{s}$ being the average
swimming speed. In the present range of $\phi$, the average distance between
neighboring swimmers, $l_{N}=(\phi/{\mathcal{V}}^{(b)})^{-1/3}$ is 2 $\sim$ 3
times larger than $d_{0}$. Although quantitative evaluations of the transition
probabilities significantly depend on $d_{0}$ and $\Delta t$, the qualitative
discussion presented below is not affected as far as $d_{0}$ and $\Delta t$
are sufficiently smaller than $l_{N}$ and $l_{N}/v_{s}$, respectively. In
Appendix C, we discuss how the present definition can capture hydrodynamic
collisions with the settings of $d_{0}$ and $\Delta t$.

Table 1:  State probabilities $P_{\mu}$ and the transition probabilities ${\mathcal{W}}_{\mu|\nu}$ for various conditions $(10^{2}\phi,H,\dot{\gamma})$ | $P_{F}$ | $P_{UF}$ | ${\mathcal{W}}_{F|F}$ | ${\mathcal{W}}_{F|UF}$ | ${\mathcal{W}}_{UF|F}$ | ${\mathcal{W}}_{UF|UF}$
---|---|---|---|---|---|---
$(1,128,10^{-3})$ | 0.51 | 0.49 | 0.69 | 0.31 | 0.39 | 0.61
$(1,384,10^{-3})$ | 0.6 | 0.4 | 0.67 | 0.33 | 0.44 | 0.56
$(3.2,128,10^{-3})$ | 0.54 | 0.46 | 0.63 | 0.37 | 0.44 | 0.56
$(3.2,128,3\times 10^{-4})$ | 0.51 | 0.49 | 0.62 | 0.38 | 0.41 | 0.59
$(3.2,128,0)$ | 0.51 | 0.49 | 0.60 | 0.40 | 0.41 | 0.59

Table I shows the numerically obtained probability of the $\mu$-state,
$P_{\mu}$, and the transition probability from the $\mu$\- to $\nu$-states,
${\mathcal{W}}_{\mu|\nu}$, ($\mu,\nu$=$F,UF$) for various conditions. Here,
$P_{\mu}$ and ${\mathcal{W}}_{\mu|\nu}$ are calculated for swimmers in the
region $-0.3\leq z/H\leq 0.3$. At ${\dot{\gamma}}=0$, because the $F$\- and
$UF$-states are not distinguished,
${\mathcal{W}}_{F|UF}\cong{\mathcal{W}}_{UF|F}$, and $P_{F}\cong P_{UF}$.
However, for ${\dot{\gamma}}\neq 0$, we find that ${\mathcal{W}}_{F|UF}$ is
significantly smaller than ${\mathcal{W}}_{UF|F}$. As $H$ and $\phi$ increase
(in the dilute regime), the population of the $F$-state swimmers with
${\hat{n}}_{\alpha,x}{\hat{n}}_{\alpha,z}>0$ increases, indicating that an
increase in the collision frequency or time further promotes transitions.

For swimmers trapped at the walls, the hydrodynamic torques arising from the
applied flow weakly align them along the flow direction because their heads
are slightly tilted against the walls. Thus, for trapped swimmers, the
population of the $F$-state is slightly larger than that of the $UF$-state.
After longer-term traps, swimmers leave from the bottom (top) wall by raising
(dropping) their heads, which changes their states ($F\rightleftarrows UF$).
Reflecting such conditions, for swimmers just after leaving the walls, the
population of the $UF$-state is slightly larger than that of the $F$-state
(not shown here). As swimmers move inward from the boundary walls, transitions
from the $UF$\- to $F$-states are gradually promoted by collisions. Due to the
geometrical symmetry, the population of the $F$-state is maximized at $z=0$,
leading to the upward convex form of $W_{xz}(z)/\rho(z)$. In an ideal bulk
system or a system with periodic boundary conditions without walls, a detailed
balance between the $F$\- and $UF$-states,
$P_{F}{\mathcal{W}}_{F|UF}=P_{UF}{\mathcal{W}}_{UF|F}$, should be realized.
Such a detailed balance may nearly hold at larger $H$ and $\phi$ in the
present system, but that was not investigated in detail.

Table 2:  Transition probability from states 1 to $\mu$ through a hydrodynamic collision with another swimmer in state $\nu$, $w_{1|\mu}^{(\nu)}$, at $\phi=0.01$, $H=384$, and $\dot{\gamma}=10^{-3}$.  $\nu$ | $w_{1|1}^{(\nu)}$ | $w_{1|2}^{(\nu)}$ | $w_{1|3}^{(\nu)}$ | $w_{1|4}^{(\nu)}$
---|---|---|---|---
1 | 0.75 | 0.05 | 0.01 | 0.19
2 | 0.67 | 0.20 | 0.03 | 0.10
3 | 0.61 | 0.10 | 0.07 | 0.22
4 | 0.52 | 0.04 | 0.02 | 0.42
Table 3:  Transition probability from states 2 to $\mu$ through a collision with another swimmer in state $\nu$, $w_{2|\mu}^{(\nu)}$, at $\phi=0.01$, $H=384$, and $\dot{\gamma}=10^{-3}$.  $\nu$ | $w_{2|1}^{(\nu)}$ | $w_{2|2}^{(\nu)}$ | $w_{2|3}^{(\nu)}$ | $w_{2|4}^{(\nu)}$
---|---|---|---|---
1 | 0.37 | 0.51 | 0.06 | 0.05
2 | 0.19 | 0.66 | 0.15 | 0.00
3 | 0.14 | 0.46 | 0.37 | 0.03
4 | 0.25 | 0.51 | 0.19 | 0.05

Tables II and III show the numerically obtained transition probabilities of a
swimmer in states 1 and 2 before a collision, respectively, at $\phi=0.01$,
$H=384$, and $\dot{\gamma}=10^{-3}$. Here, $w_{\lambda|\mu}^{(\nu)}$
represents the transition probability from states $\lambda$ to $\mu$ through a
collision with another swimmer in state $\nu$. Note that similar results are
obtained at different parameters where negative $\Delta\eta_{a}$ is obtained.
We find significant differences between $w_{1|\mu}^{(\nu)}$ and
$w_{2|\mu}^{(\nu)}$. For both cases the majorities are $w_{\mu|\mu}^{(\nu)}$,
whereas a swimmer in state 1 is more likely to retain its state unchanged by a
collision than one in state 2.

We can understand the role of HIs in the elementary processes of these state
transitions through the following phenomenological arguments, which are
separately provided for different cases.

####  III.2.1 Hydrodynamic collisions between two swimmers in states 1 and 2,
and 2 and 3

Figure 6: (Color online) Schematic illustrations for two approaching swimmers
in the $F$\- and $UF$-states: case (A) where the two swimmers are in states 1
and 2 (a), and case (B) where those are in states 2 and 3 (b). The torques due
to HIs and the shear flow are described by the black and green curved arrows,
respectively.

Let us first consider that two swimmers in the $F$\- and $UF$-states are
approaching. There are essentially two different cases: case (A) where
swimmers in states 1 and 2 (equivalently, 3 and 4) are approaching, and case
(B) where swimmers in states 3 and 2 (equivalently, 1 and 4) are approaching.

For case (A), as schematically shown in Fig. 6(a), HIs tend to rotate the
swimmers in opposite directions LaugaB , while the externally applied shear
flow rotates them in the same direction. As the swimming directions become
parallel to each other and perpendicular to the flow direction, the torques
due to HIs grow weaker, but those arising from the shear flow grow stronger.
Furthermore, once two swimmers move nearly side by side (Fig. 6(a)), a
hydrodynamic attraction acts on them, which may make a swimmer in state 1 drag
one in state 2 into eventually moving in the same direction in the collision
process. These hydrodynamic effects are expected to promote the transition
from states 2 to 1, responsible for $w_{1|2}^{(2)}<w_{2|1}^{(1)}$ and
$w_{1|1}^{(2)}>w_{2|2}^{(1)}$.

In contrast, in case (B), due to similar asymmetry in the net torques,
$w_{3|2}^{(2)}$ (not shown here but equivalent to $w_{1|4}^{(4)}$) is slightly
larger than $w_{2|3}^{(3)}$. In this case, the torques both due to HIs and the
shear flow are reduced as their swimming directions become parallel along with
the flow direction (see Fig. 6(b) for a schematic), and therefore, the
difference between $w_{3|2}^{(2)}$ ($\cong w_{1|4}^{(4)}$) and $w_{2|3}^{(3)}$
is less notable than that between $w_{1|2}^{(2)}$ and $w_{2|1}^{(1)}$: namely,
hydrodynamic collisions of case (A) predominantly contribute to the transition
from the $UF$ to $F$ states, whereas those of case (B) are marginal.

####  III.2.2 Hydrodynamic collisions between two swimmers in the same states

Here, we consider the following two cases: case (A’) where two swimmers are
both in state 1 (equivalently, both in state 3), and case (B’) where those are
both in state 2 (equivalently, both in state 4). For these cases, schematics
are shown in Figs. 7(a) and (b).

For both cases (A’) and (B’), the torques caused by HIs rotate the swimmers in
opposite directions and grow weaker as the swimmers become parallel to each
other. On the other hand, for the torques caused by the shear flow, in case
(A’), as the collision proceeds, the torque resisting the transition to state
2 grows stronger, whereas the other torque, which helps the transition to
state 4, grows weaker. In (B’), the opposite occurs: one torque due to the
shear flow promoting the transition to state 1 grows stronger, while the other
one resisting the transition to state 3 grows weaker. The difference in how
the torques contribute to the transition is expected to be responsible for the
measured difference in the transition probabilities. That is, as shown in
Tables II and III, $w_{1|2}^{(1)}<w_{2|1}^{(2)}$ and $w_{2|3}^{(2)}\lesssim
w_{1|4}^{(1)}$, resulting in $w_{1|1}^{(1)}>w_{2|2}^{(2)}$.

Figure 7: (Color online) Schematic illustrations of the two cases: case (A’)
where the two swimmers are both in state 1 (a) and case (B’) where those are
both in state 2 (b). The torques due to HIs and the shear flow are described
by the black and green curved arrows, respectively.

####  III.2.3 Hydrodynamic collisions between two swimmers in states 1 and 3,
and 2 and 4

Figure 8: (Color online) Schematic illustrations of cases (A”1) and (A”2),
shown in (a) and (c), respectively, where the two swimmers are in states 1 and
3, and those of cases (B”1) and (B”2), shown in (b) and (d), respectively,
where the two swimmers are in states 2 and 4. The torques due to HIs and the
shear flow are described by the black and green curved arrows, respectively.

When two approaching swimmers are in states 1 and 3, there are essentially two
different cases, (A”1) and (A”2), which are illustrated in Figs. 8(a) and (c),
respectively. As torques arising from HIs rotate the two swimmers in opposite
directions, the swimming directions become perpendicular and parallel to the
flow in cases (A”1) and (A”2), respectively. In case (A”1), the torques due to
the shear flow, which prevent the swimmers from changing from the $F$\- to
$UF$-states, grow stronger. On the other hand, in case (A”2), such torques
promoting the changes to the $UF$-state grow weaker.

When swimmers in states 2 and 4 are approaching, there are two different
cases, (B”1) and (B”2), as schematically shown in Figs. 8(b) and (d),
respectively. With similar arguments, in case (B”1), the torques due to the
shear flow, which promote the transition to the $F$-state, grow stronger,
while, in case (B”2), those preventing the changes to the $F$-state grow
weaker.

These differences may result in the difference between $w_{1|\mu}^{(3)}$ and
$w_{2|\mu}^{(4)}$. That is, when two swimmers in states 1 and 3 are
approaching each other, the swimmers tend to remain their states unchanged
more than when they are in states 2 and 4: $w_{1|2}^{(3)}<w_{2|1}^{(4)}$ and
$w_{1|4}^{(3)}\gtrsim w_{2|3}^{(4)}$, resulting in
$w_{1|1}^{(3)}>w_{2|2}^{(4)}$. Note that hydrodynamic collisions of cases
(A”1) and (B”1) predominantly contribute to the transition from the $UF$ to
$F$ states, whereas those of cases (A”2) and (B”2) are marginal.

The realistic collision processes are more complicated; thus, the present
arguments are oversimplified. However, they qualitatively explains why HIs
systematically promote the transition from the $UF$ to $F$ states.

##  IV Discussion and Concluding remarks

It has been known that rod-like particles in a shear flow tend to orientate to
the extension direction due to an interplay between flow and rotational
diffusivities Hinch_Leal1 ; Hinch_Leal2 : for a rod-like particle, although
the torque due to shear flow becomes unidirectional and stronger as its
orientation becomes perpendicular to the flow direction, the torque due to
thermal rotational diffusivities is bidirectional and does not depend on the
rod orientation. By considering such an effect, an explanation for the
viscosity reduction was provided Haines ; Saintillan1 . Moreover, it was
proposed that the activity-induced HIs provide a source of random rotations in
addition to thermal fluctuations and tumbling Ryan . The present study further
illuminates the role of HIs: even starting from a random state, our results
suggest that steady global states where the swimmers are weakly aligned along
the extension axis may form as self-organization by repeated hydrodynamic
collisions. A study of this issue would be an interesting task for future
studies.

In typical microorganisms systems, the propulsive forces are sufficiently
strong that hydrodynamic effects may dominate over thermal fluctuations.
Below, we validate this condition by considering a typical experimental
situation PNAS2020 : an E. coli suspension at a volume fraction
$\phi(={\mathcal{V}}^{(b)}/l_{N}^{3})=$0.01 at room temperature ($\sim 300$K),
for which the average separation distance is $l_{N}\sim 5\mu$m and the thermal
rotational diffusion coefficient is $D_{T}\lesssim$1s-1. Hereafter, we assume
that the swimming speed is $v_{s}\sim 10\mu$m/s, the magnitude of the force
dipole is $P\sim 10^{-18}$N$\cdot$m, the cell size is $\ell_{0}\sim 1\mu$m,
the cell volume is ${\mathcal{V}}^{(b)}\sim 1\mu$m3, and the solvent viscosity
is $\eta_{s}\sim 10^{-3}$Pa$\cdot$s. For a duration $\sim 1/D_{T}\gtrsim 1$s,
a swimmer may at least once approach another swimmer closer than $l_{c}\sim
2\mu$m, estimated by $\pi l_{c}^{2}v_{s}\times(1/D_{T})\times(1/l_{N})^{3}\sim
1$. The magnitude of the rotational flow field, $\omega$, induced at a
distance $r$ from a swimmer is approximately given as $\omega\sim
0.1P/(\eta_{s}r^{3})$ LaugaB . Therefore, at $r=l_{N}$, $\omega\sim 1$s-1,
while at $r=l_{c}$, $\omega\sim 10$s-1. By such a hydrodynamic “collision”
process, which lasts for approximately $\ell_{0}/v_{s}\sim 0.1$s, swimming
motions can be largely affected more than by thermal fluctuations. In other
words, reorientation due to HIs may be a faster process than thermal
rotational diffusion.

In this study, we have explored a mechanism of the anomalous rheology of
active suspensions, focusing on the role of HIs. Before closing, we present
the following remarks. (1) Our pusher model is transformed into a puller model
by simply changing the sign of the active forces. Our preliminary results
shown in Fig. 2(b) suggest that the viscosity of the puller model is increased
more than that in the passive systems, which agrees with experimental
observations for motile and immotile puller bacterial suspensions Rafai . (2)
In Ref. Liu , under Poiseuille flow, lower viscosity is observed for smaller
separation between the walls. This contrasts with the present result, where
the viscosity reduction is enhanced by increasing $H$ under simple shear. In
addition, the viscosity reduction occurs at larger shear rates than those of
the experiments of Refs. Lopez ; PNAS2020 . These differences may be
attributed to the difference in the flow geometry. We plan to investigate
these issues further elsewhere.

###### Acknowledgements.

This work was supported by KAKENHI (Grants No. 26103507, 25000002, and
20H05619) and the special fund of Institute of the Industrial Science, The
University of Tokyo.

##  Appendix A simulation method

In our simulations, we use the smoothed-profile method (SPM) SPM ; SPM2 ; SPM3
to accomodate many-body hydrodynamic interactions (HIs) among the constituent
swimmers. In Ref. SPM3 , it is found that the SPM can quantitatively reproduce
far-field and intermediate-field aspects of HIs, whereas the near-field HIs
are slightly underestimated at closer distances. Furthermore, like many other
methods, the SPM cannot also resolve the singular lubrication forces. For more
details of the qualitative evaluations on the SPM, please refer to Refs. SPM2
; SPM3 .

Figure 9: (Color online) In this study, to incorporate the present model
swimmer into the SPM, the body and flagellum parts are represented through the
field variables, $\Psi_{\alpha}^{(b)}({{\mbox{\boldmath$r$}}})$ and
$\Psi_{\alpha}^{(f)}({{\mbox{\boldmath$r$}}})$, respectively. We plot
$\Psi_{\alpha}^{(b)}({{\mbox{\boldmath$r$}}})+\Psi_{\alpha}^{(f)}({{\mbox{\boldmath$r$}}})$
in the $xy$-plane, where both
${{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}=(2.25R,0,0)$ and
${{\mbox{\boldmath$R$}}}_{\alpha}^{(CF)}=(-2.25R,0,0)$ are included. The
discretized mesh size $h$ is the same as that used in practical simulations
($h=0.3125R$ and $\xi=0.5h$). Here, $\xi$ is the interface thickeness
controlling the degree of smoothness of
$\Psi_{\alpha}^{(b)}({{\mbox{\boldmath$r$}}})$ and
$\Psi_{\alpha}^{(f)}({{\mbox{\boldmath$r$}}})$.

For this purpose, the body and flagellum parts described above are represented
through the field variables, $\Psi_{\alpha}^{(b)}({{\mbox{\boldmath$r$}}})$
and $\Psi_{\alpha}^{(f)}({{\mbox{\boldmath$r$}}})$, respectively:

$\displaystyle\Psi_{\alpha}^{(b)}({{\mbox{\boldmath$r$}}})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\psi[{{\mbox{\boldmath$r$}}},{{\mbox{\boldmath$R$}}}_{i,\alpha}^{(b)}]~{}~{}~{}~{}\biggl{(}{\rm
for}~{}~{}\sum_{i=1}^{3}\psi[{{\mbox{\boldmath$r$}}},{{\mbox{\boldmath$R$}}}_{i,\alpha}^{(b)}]\leq
1\biggr{)}$ (6) $\displaystyle=$ $\displaystyle 1~{}~{}~{}~{}\biggl{(}{\rm
for}~{}~{}\sum_{i=1}^{3}\psi[{{\mbox{\boldmath$r$}}},{{\mbox{\boldmath$R$}}}_{i,\alpha}^{(b)}]>1\biggr{)}.$

and

$\displaystyle\Psi_{\alpha}^{(f)}({{\mbox{\boldmath$r$}}})$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{3}\psi[{{\mbox{\boldmath$r$}}},{{\mbox{\boldmath$R$}}}_{i,\alpha}^{(f)}]~{}~{}~{}~{}\biggl{(}{\rm
for}~{}~{}\sum_{i=1}^{3}\psi[{{\mbox{\boldmath$r$}}},{{\mbox{\boldmath$R$}}}_{i,\alpha}^{(f)}]\leq
1\biggr{)}$ (8) $\displaystyle=$ $\displaystyle-1~{}~{}~{}~{}\biggl{(}{\rm
for}~{}~{}\sum_{i=1}^{3}\psi[{{\mbox{\boldmath$r$}}},{{\mbox{\boldmath$R$}}}_{i,\alpha}^{(f)}]>1\biggr{)}.$

In this study, we adopt the following function to $\psi$ as

$\displaystyle\psi[{{\mbox{\boldmath$r$}}},{{\mbox{\boldmath$R$}}}_{i,\alpha}^{(\mu)}]=\dfrac{1}{2}\biggl{\\{}\tanh\biggl{[}\dfrac{1}{\xi}(R-|{{\mbox{\boldmath$r$}}}-{{\mbox{\boldmath$R$}}}_{i,\alpha}^{(\mu)}|)\biggr{]}+1\biggr{\\}},$
(9)

where $\mu=b,f$, and $\xi$ is the interface thickness controlling the degree
of smoothness. In Fig. 9, we show the cross section of the model swimmer
described by $\Psi_{\alpha}^{(b)}({{\mbox{\boldmath$r$}}})$ and
$\Psi_{\alpha}^{(f)}({{\mbox{\boldmath$r$}}})$ including both
${{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}$ and
${{\mbox{\boldmath$R$}}}_{\alpha}^{(CF)}$ in the same plane.

The working equations for the velocity field
${{\mbox{\boldmath$v$}}}({{\mbox{\boldmath$r$}}},t)$ are given as

$\displaystyle\rho\biggl{(}\dfrac{\partial}{\partial
t}+{{\mbox{\boldmath$v$}}}\cdot{{\mbox{\boldmath$\nabla$}}}\biggr{)}{{\mbox{\boldmath$v$}}}$
$\displaystyle=$
$\displaystyle{{\mbox{\boldmath$\nabla$}}}\cdot{\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$\Sigma$}}}_{vis}}-{{\mbox{\boldmath$\nabla$}}}p+{{\mbox{\boldmath$f$}}}_{H}+{{\mbox{\boldmath$f$}}}_{A}^{(f)},$
$\displaystyle{\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$\Sigma$}}}_{vis}}$
$\displaystyle=$
$\displaystyle\eta_{s}\bigl{[}{{\mbox{\boldmath$\nabla$}}}{{\mbox{\boldmath$v$}}}+({{\mbox{\boldmath$\nabla$}}}{{\mbox{\boldmath$v$}}})^{\dagger}\bigr{]},$
(11) $\displaystyle{{\mbox{\boldmath$\nabla$}}}\cdot{{\mbox{\boldmath$v$}}}$
$\displaystyle=$ $\displaystyle 0.$ (12)

Equation (LABEL:Navier_Stokes) is the usual Navier-Stokes equation
Landau_LifshitzB . Here,
${\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$\Sigma$}}}_{vis}}$
given in Eq. (11) is the viscous stress tensor with $\eta_{s}$ being the
solvent viscosity, and the hydrostatic pressure $p$ is determined by the
incompressibility condition, Eq. (12). In addition,
${{\mbox{\boldmath$f$}}}_{H}$ is the body force required to satisfy the rigid
body condition, and ${{\mbox{\boldmath$f$}}}_{A}^{(f)}$ is the active force
directly exerted by the flagellum part to the fluid:

$\displaystyle{{\mbox{\boldmath$f$}}}_{A}^{(f)}({{\mbox{\boldmath$r$}}})=\dfrac{1}{{\mathcal{V}}_{\alpha}^{(f)}}\sum_{\alpha=1}^{N}\Psi_{\alpha}^{(f)}({{\mbox{\boldmath$r$}}}){\hat{\mbox{\boldmath$n$}}}_{\alpha}F_{A},$
(13)

where ${\mathcal{V}}_{\alpha}^{(f)}=\int{\rm
d}{{\mbox{\boldmath$r$}}}\Psi_{\alpha}^{(f)}({{\mbox{\boldmath$r$}}})$ is the
volume of the flagellum part. In addition, the volume of the body part is give
as ${\mathcal{V}}_{\alpha}^{(b)}=\int{\rm
d}{{\mbox{\boldmath$r$}}}\Psi_{\alpha}^{(b)}({{\mbox{\boldmath$r$}}})$. In
this study, because the shapes of the body and flagellum parts are assumed to
be the same, ${\mathcal{V}}_{\alpha}^{(b)}={\mathcal{V}}_{\alpha}^{(f)}$.

As described in the main text, the periodic boundary conditions are imposed in
the $x$\- and $y$-directions with the linear dimension $L$, and the planar top
and bottom walls are placed at $z=H/2$ and $-H/2$, respectively, with $H$
being the separation distance. The shear flow is imposed by moving the top and
bottom walls in the $x$-direction at constant velocities $V/2$ and $-V/2$,
respectively, whereby the mean shear rate is given as $\dot{\gamma}=V/H$. We
impose no-slip boundary conditions at the top and bottom walls:
${{\mbox{\boldmath$v$}}}(x,y,H/2)=(V/2){\hat{\mbox{\boldmath$x$}}}$ and
${{\mbox{\boldmath$v$}}}(x,y,-H/2)=-(V/2){\hat{\mbox{\boldmath$x$}}}$.

The equations of motions of the center-of-mass velocity,
${{\mbox{\boldmath$V$}}}_{\alpha}^{(G)}$, and the angular velocity with
respect to the center-of-mass, ${{\mbox{\boldmath$\Omega$}}}_{\alpha}^{(G)}$,
are

$\displaystyle M_{\alpha}\dfrac{d{{\mbox{\boldmath$V$}}}_{\alpha}^{(G)}}{dt}$
$\displaystyle=$
$\displaystyle{{\mbox{\boldmath$F$}}}_{\alpha,{H}}+{{\mbox{\boldmath$F$}}}_{\alpha,{int}}+{{\mbox{\boldmath$F$}}}_{\alpha,{A}}^{(b)}+{{\mbox{\boldmath$F$}}}_{\alpha,{ex}},$
$\displaystyle{\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$I$}}}}_{\alpha}\cdot\dfrac{d{{\mbox{\boldmath$\Omega$}}}_{\alpha}^{(G)}}{dt}$
$\displaystyle=$
$\displaystyle{{\mbox{\boldmath$N$}}}_{\alpha,{H}}+{{\mbox{\boldmath$N$}}}_{\alpha,{int}}+{{\mbox{\boldmath$N$}}}_{\alpha,{ex}},$
(15)

where

$\displaystyle M_{\alpha}=\rho{\mathcal{V}}_{\alpha}^{(b)}$ (16)

and

$\displaystyle{\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$I$}}}}_{\alpha}$
$\displaystyle=$ $\displaystyle\int{\rm
d}{{\mbox{\boldmath$r$}}}\rho\Psi_{\alpha}^{(b)}({{\mbox{\boldmath$r$}}})\biggl{[}|\Delta{{\mbox{\boldmath$r$}}}_{\alpha}|^{2}{\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$\delta$}}}}-\Delta{{\mbox{\boldmath$r$}}}_{\alpha}\Delta{{\mbox{\boldmath$r$}}}_{\alpha}\biggr{]}$
(17)

are the mass and the moment of inertia of the $\alpha$-th swimmer’s body,
respectively. Here,
$\Delta{{\mbox{\boldmath$r$}}}_{\alpha}={{\mbox{\boldmath$r$}}}-{{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}$.
In this study, the swimmer’s density is assumed to be the same as the solvent
density. In Eqs. (LABEL:VG) and (15),
${{\mbox{\boldmath$F$}}}_{\alpha,{int}}^{(G)}$ and
${{\mbox{\boldmath$N$}}}_{\alpha,{int}}^{(G)}$ are the force and torque acting
on the $\alpha$-th swimmer’s body, respectively, due to the particle-particle
and particle-wall potential interactions:

$\displaystyle{{\mbox{\boldmath$F$}}}_{\alpha,{int}}$ $\displaystyle=$
$\displaystyle-\sum_{\beta\neq\alpha}\sum_{i,\mu\in\alpha}\sum_{j,\nu\in\beta}\dfrac{\partial}{\partial{{\mbox{\boldmath$R$}}}_{i\alpha}^{(\mu)}}U^{\mu\nu}(|{{\mbox{\boldmath$R$}}}_{i\alpha}^{(\mu)}-{{\mbox{\boldmath$R$}}}_{j\beta}^{(\nu)}|)-\sum_{i,\mu\in\alpha}\dfrac{\partial}{\partial{{\mbox{\boldmath$R$}}}_{i\alpha}^{(\mu)}}\biggl{[}W^{\mu}\biggl{(}|z_{i\alpha}^{(\mu)}-\frac{H}{2}|\biggr{)}+W^{\mu}\biggl{(}|z_{i\alpha}^{(\mu)}+\frac{H}{2}|\biggr{)}\biggr{]}$
$\displaystyle{{\mbox{\boldmath$N$}}}_{\alpha,{int}}$ $\displaystyle=$
$\displaystyle-\sum_{\beta\neq\alpha}\sum_{i,\mu\in\alpha}\sum_{j,\nu\in\beta}({{\mbox{\boldmath$R$}}}_{i,\alpha}^{(\mu)}-{{\mbox{\boldmath$R$}}}_{\alpha}^{(G)})\times\dfrac{\partial}{\partial{{\mbox{\boldmath$R$}}}_{i\alpha}^{(\mu)}}U^{\mu\nu}(|{{\mbox{\boldmath$R$}}}_{i\alpha}^{(\mu)}-{{\mbox{\boldmath$R$}}}_{j\beta}^{(\nu)}|)$
(19)
$\displaystyle-\sum_{i,\mu\in\alpha}({{\mbox{\boldmath$R$}}}_{i,\alpha}^{(\mu)}-{{\mbox{\boldmath$R$}}}_{\alpha}^{(G)})\times\dfrac{\partial}{\partial{{\mbox{\boldmath$R$}}}_{i\alpha}^{(\mu)}}\biggl{[}W^{\mu}\biggl{(}|z_{i\alpha}^{(\mu)}-\frac{H}{2}|\biggr{)}+W^{\mu}\biggl{(}|z_{i\alpha}^{(\mu)}+\frac{H}{2}|\biggr{)}\biggr{]},$

where $i,j=1,2,3$ and $\mu,\nu=b,f$. Here, $U^{\mu\nu}$ is the interaction
potential between two spheres which each comprise the body or the flagellum
part of different swimmers, and $W^{\mu}$ is the interaction potential between
such a sphere and the planar wall. The explicit forms of $U^{\mu\nu}$ and
$W^{\mu}$ are provided below. In Eqs. (LABEL:VG) and (15),
${{\mbox{\boldmath$F$}}}_{\alpha,{ext}}$ and
${{\mbox{\boldmath$N$}}}_{\alpha,{ext}}$ are the force and torque exerted on
the $\alpha$-th swimmer due to the external field, which are absent in the
present study. The active force acting on the body part,
${{\mbox{\boldmath$F$}}}_{\alpha,{A}}^{(b)}$, is given as

$\displaystyle{{\mbox{\boldmath$F$}}}_{\alpha,{A}}^{(b)}=F_{A}{\hat{\mbox{\boldmath$n$}}}_{\alpha}.$
(20)

Eqs. (13) and (20) prescribe a force dipole
$F_{A}\ell_{0}{\hat{\mbox{\boldmath$n$}}}_{\alpha}$ with
$\ell_{0}{\hat{\mbox{\boldmath$n$}}}_{\alpha}={{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}-{{\mbox{\boldmath$R$}}}_{\alpha}^{(CF)}$
[see also Eq. (23)]. Finally, ${{\mbox{\boldmath$F$}}}_{\alpha,{H}}$ and
${{\mbox{\boldmath$N$}}}_{\alpha,{H}}$ are the force and torque exerted on the
$\alpha$-th swimmer due to HIs. The explicit forms of
${{\mbox{\boldmath$F$}}}_{\alpha,{H}}$,
${{\mbox{\boldmath$N$}}}_{\alpha,{H}}$, and the body force
${{\mbox{\boldmath$f$}}}_{H}$ can be given in the discretized equations of
motion as Eqs. (29), (30), and (32), respectively in the next section.

We assume the following form of the interparticle potential:

$\displaystyle
U^{\mu\nu}(r)=\epsilon(1-\delta_{\mu,f}\delta_{\nu,f})\biggl{(}\dfrac{2R}{r}\biggr{)}^{12},$
(21)

where $\epsilon$ is a positive energy constant and $\delta_{\mu,f}$ is the
Kronecker delta. This form prevents the body part of a swimmer from
overlapping on different swimmers but allows overlaps among the flagellum
parts. The wall-particle interaction potential $W^{\mu}$ is introduced to
prevent the penetration of particles through the boundary walls and is assumed
to be given as

$\displaystyle W^{\mu}(z)=\epsilon\biggl{(}\dfrac{2R}{z}\biggr{)}^{12},$ (22)

where we assume the same energy constant as that of $U^{\mu\nu}$. In Eqs. (21)
and (22), $\mu,\nu=b,f$.

In our simulations, we make the equations dimensionless by measuring space and
time in units of $h$, which is the discretization mesh size used when solving
Eqs. (LABEL:Navier_Stokes)-(12), and $t_{0}=\rho h^{2}/\eta_{s}$, which is the
momentum diffusion time across the unit length. Accordingly, the scaled
solvent viscosity is $1$, and the units of velocity, stress, force, and energy
are chosen to be $h/t_{0}$, $\rho h^{2}/t_{0}^{2}$, $\rho h^{4}/t_{0}^{2}$ and
$\rho h^{5}/t_{0}^{2}$, respectively. In our simulations, we set $\epsilon=30$
and $F_{A}=20$. The parameters determining the swimmer’s shape are set to be
$R=3.2$,
$\ell_{0}=|{{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}-{{\mbox{\boldmath$R$}}}_{\alpha}^{(CF)}|=4.5R$
and $\xi=0.5$. In this study, the swimmers’ volume fraction is identified as
that of the rigid body particles given by
$\phi=N{\mathcal{V}}_{\alpha}^{(b)}/HL^{2}$.

In Ref. SPM2 , the general scheme deriving the volume-average stress tensor,
${\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$S$}}}}$, in the
framework of the SPM is provided:
${\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$S$}}}}$ is divided
into the three parts,
${\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$s$}}}}_{s}$,
${\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$s$}}}}_{p}$, and
${\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$s$}}}}_{a}$, due
to the solvent, passive, and active contributions, respectively. The passive
part is further divided into two parts arising from HIs and the potentials
($U^{\mu\nu}$ and $W^{\mu}$). Such sources to the stress tensor exist without
active forces, so we call
${\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$s$}}}}_{p}$ the
“passive” stress. In this Appendix, according to a similar procedure given in
Ref. SPM2 , we derive an expression of the active stress as follows.

$\displaystyle{\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$s$}}}}_{a}$
$\displaystyle=$
$\displaystyle-\dfrac{1}{L^{2}H}\sum_{\alpha=1}^{N}F_{A}{\hat{\mbox{\boldmath$n$}}}_{\alpha}\int{\rm
d}{{\mbox{\boldmath$r$}}}{{\mbox{\boldmath$r$}}}\biggl{[}\dfrac{\Psi^{b}_{\alpha}({{\mbox{\boldmath$r$}}})}{{\mathcal{V}}_{\alpha}^{(b)}}-\dfrac{\Psi^{f}_{\alpha}({{\mbox{\boldmath$r$}}})}{{\mathcal{V}}_{\alpha}^{(f)}}\biggr{]}$
(23) $\displaystyle=$
$\displaystyle-\dfrac{1}{L^{2}H}\sum_{\alpha=1}^{N}F_{A}{\hat{\mbox{\boldmath$n$}}}_{\alpha}\biggl{(}{{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}-{{\mbox{\boldmath$R$}}}_{\alpha}^{(CF)}\biggr{)}$
$\displaystyle=$
$\displaystyle-\dfrac{1}{L^{2}H}\sum_{\alpha=1}^{N}F_{A}\ell_{0}{\hat{\mbox{\boldmath$n$}}}_{\alpha}{\hat{\mbox{\boldmath$n$}}}_{\alpha}.$

As usual, we may redefine the active stress by making
${\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$s$}}}}_{a}$
traceless by substituting
$-({1}/{L^{2}H})\sum_{\alpha=1}^{N}F_{A}\ell_{0}{\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$\delta$}}}}/3$

In the main text, the viscosity is determined as
$\eta=(1/{\dot{\gamma}}L^{2})\int{\rm d}x{\rm d}y\langle\Sigma_{xz}(x,y,\pm
H/2)\rangle$, where $\Sigma_{xz}(x,y,\pm H/2)$ is the $xz$ component of the
stress tensor at the walls located at $z=\pm H/2$ and $\langle\cdots\rangle$
denotes taking the time average in a steady state. Because the relation
$\langle S_{xz}\rangle=\langle\Sigma_{xz}\rangle$ holds, $\eta=\langle
S_{xz}\rangle/{\dot{\gamma}}$. As denoted above,
${\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$S$}}}}$ are
divided into three parts, and then, we have $\eta=\eta_{s}+\langle
s_{p,xz}\rangle/\dot{\gamma}+\langle s_{a,xz}\rangle/\dot{\gamma}$. The
passive part $<s_{p,xz}>/\dot{\gamma}$ positively contributes to the
viscosity, and therefore, the viscosity reduction is entirely due to (weak)
alignment of the force dipoles along with the extension direction,
$\langle{\hat{n}}_{\alpha,x}{\hat{n}}_{\alpha,z}\rangle>0$. In the main text,
we also discuss how such an alignment of the swimmers in the interior regions
reduces the velocity gradient at the boundaries, which is observed as a
viscosity reduction.

We may define the swimmer’s Reynolds number as $Re=\rho
v_{s}\ell_{0}/\eta_{s}$ with $v_{s}$ being the average swimming speed. In the
present simulations, $v_{s}\sim 0.1$, giving $Re\sim 1$, which is
unrealistically large, but may not be problematic for the following reason. It
is known that unless $Re$ is much greater than unity, the induced flow
patterns around a moving particle do not change enough for the inertia effect
to significantly change the resultant dynamics and transport properties. In
Ref. LB2 , an excellent discussion is presented for colloidal simulations. In
practice, by using a smaller value of $\rho$, it is possible to reduce $Re$ as
much as possible. For this treatment, a smaller time increment is required to
ensure the stability of the time integration of the equations of motion,
whereas it leads to longer simulation times. In our preliminary simulations at
$Re\sim 0.1$, which is still unrealistically large, we confirm that the main
results obtained in the present study remain almost unchanged.

##  Appendix B Explicit time-integration algorithm

Here, following Refs. SPM ; SPM2 , we describe an explicit time-integration
scheme for solving model equations as follows.

The set of physical variables is assumed to be clearly defined at the discrete
time step $t_{n}=n\Delta t$.

First, we solve Eq. (LABEL:Navier_Stokes) without including
${{\mbox{\boldmath$f$}}}_{H}$ as

$\displaystyle{{\mbox{\boldmath$v$}}}^{*}={{\mbox{\boldmath$v$}}}_{n}+\dfrac{1}{\rho}\int_{t_{n}}^{t_{n+1}}{\rm
d}s\biggl{(}-\rho{{\mbox{\boldmath$v$}}}\cdot{{\mbox{\boldmath$\nabla$}}}{{\mbox{\boldmath$v$}}}+{{\mbox{\boldmath$\nabla$}}}\cdot{\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$\Sigma$}}}_{vis}}+{{\mbox{\boldmath$f$}}}_{A}^{(f)}\biggr{)}^{\bot},$

where ${{\mbox{\boldmath$v$}}}_{n}$ is the velocity field at $t=t_{n}$ and
$(\cdots)^{\bot}$ denotes taking the transverse part.

Second, we update ${{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}$ and
${\hat{\mbox{\boldmath$n$}}}_{\alpha}$ as

$\displaystyle{{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}(t_{n+1})$
$\displaystyle=$
$\displaystyle{{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}(t_{n})+\int_{t_{n}}^{t_{n+1}}{\rm
d}s{{\mbox{\boldmath$V$}}}_{\alpha}^{(G)}$ (25)
$\displaystyle{\hat{\mbox{\boldmath$n$}}}_{\alpha}(t_{n+1})$ $\displaystyle=$
$\displaystyle{\hat{\mbox{\boldmath$n$}}}_{\alpha}(t_{n})+\int_{t_{n}}^{t_{n+1}}{\rm
d}s{{\mbox{\boldmath$\Omega$}}}_{\alpha}^{(G)}\times{\hat{\mbox{\boldmath$n$}}}_{\alpha}.$
(26)

With these updated ${{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}$ and
${\hat{\mbox{\boldmath$n$}}}_{\alpha}$, we also update the variables that
determine the swimmer’s shapes.

Third, the particle velocities and angular velocities are updated by solving
Eqs. (LABEL:VG) and (15) as

$\displaystyle{{\mbox{\boldmath$V$}}}_{\alpha}^{(G)}(t_{n+1})={{\mbox{\boldmath$V$}}}_{\alpha}^{(G)}(t_{n})+\dfrac{1}{M_{\alpha}}\int_{t_{n}}^{t_{n+1}}{\rm
d}s({{\mbox{\boldmath$F$}}}_{\alpha,{H}}+{{\mbox{\boldmath$F$}}}_{\alpha,{int}}+{{\mbox{\boldmath$F$}}}_{\alpha,{A}}^{(b)}+{{\mbox{\boldmath$F$}}}_{\alpha,{ex}}),$
(27)
$\displaystyle{{\mbox{\boldmath$\Omega$}}}_{\alpha}^{(G)}(t_{n+1})={{\mbox{\boldmath$\Omega$}}}_{\alpha}^{(G)}(t_{n})+{\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$I$}}}^{-1}_{\alpha}}\cdot\biggl{[}\int_{t_{n}}^{t_{n+1}}{\rm
d}s({{\mbox{\boldmath$N$}}}_{\alpha,{H}}+{{\mbox{\boldmath$N$}}}_{\alpha,{int}}+{{\mbox{\boldmath$N$}}}_{\alpha,{ex}})\biggr{]}.$
(28)

Here, the explicit forms of $\int_{t_{n}}^{t_{n+1}}{\rm
d}s{{\mbox{\boldmath$F$}}}_{\alpha,{H}}$ and $\int_{t_{n}}^{t_{n+1}}{\rm
d}s{{\mbox{\boldmath$N$}}}_{\alpha,{H}}$ are given as

$\displaystyle\int_{t_{n}}^{t_{n+1}}{\rm
d}s{{\mbox{\boldmath$F$}}}_{\alpha,{H}}=\int{\rm
d}{{\mbox{\boldmath$r$}}}\rho\Psi_{\alpha,n+1}^{(b)}\biggl{\\{}{{\mbox{\boldmath$v$}}}^{*}-\biggl{[}{{\mbox{\boldmath$V$}}}_{\alpha}^{(G)}(t_{n})+{{\mbox{\boldmath$\Omega$}}}_{\alpha}^{(G)}(t_{n})\times\bigl{(}{{\mbox{\boldmath$r$}}}-{{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}(t_{n+1})\bigr{)}\biggr{]}\biggr{\\}}$
(29)

and

$\displaystyle\int_{t_{n}}^{t_{n+1}}{\rm
d}s{{\mbox{\boldmath$N$}}}_{\alpha,{H}}=\int{\rm
d}{{\mbox{\boldmath$r$}}}\rho\Psi_{\alpha,n+1}^{(b)}\bigl{(}{{\mbox{\boldmath$r$}}}-{{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}(t_{n+1})\bigr{)}\times\biggl{\\{}{{\mbox{\boldmath$v$}}}^{*}-\biggl{[}{{\mbox{\boldmath$V$}}}_{\alpha}^{(G)}(t_{n})+{{\mbox{\boldmath$\Omega$}}}_{\alpha}^{(G)}(t_{n})\times\bigl{(}{{\mbox{\boldmath$r$}}}-{{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}(t_{n+1})\bigr{)}\biggr{]}\biggr{\\}},$
(30)

where $\Psi_{\alpha,n+1}^{(b)}$ denotes
$\Psi_{\alpha}^{(b)}({{\mbox{\boldmath$r$}}})$ at $t=t_{n+1}$.

Finally, we update the velocity field by embedding the rigid body motions in
${{\mbox{\boldmath$v$}}}^{*}$ through the body force
${{\mbox{\boldmath$f$}}}_{H}$ as

$\displaystyle{{\mbox{\boldmath$v$}}}_{n+1}={{\mbox{\boldmath$v$}}}^{*}+\dfrac{1}{\rho}\int_{t_{n}}^{t_{n+1}}{\rm
d}s{{\mbox{\boldmath$f$}}}_{H}^{\bot}.$ (31)

The explicit form of $\int_{t_{n}}^{t_{n+1}}{\rm
d}s{{\mbox{\boldmath$f$}}}_{H}$ is determined to approximately fulfill the
rigid body condition inside the swimmers’ body region, and it is given by

$\displaystyle\int_{t_{n}}^{t_{n+1}}{\rm
d}s{{\mbox{\boldmath$f$}}}_{H}=-\sum_{\alpha=1}^{N}\rho\Psi_{\alpha,n+1}^{(b)}\biggl{\\{}{{\mbox{\boldmath$v$}}}^{*}-\biggl{[}{{\mbox{\boldmath$V$}}}_{\alpha}^{(G)}(t_{n+1})+{{\mbox{\boldmath$\Omega$}}}_{\alpha}^{(G)}(t_{n+1})\times\bigl{(}{{\mbox{\boldmath$r$}}}-{{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}(t_{n+1})\bigr{)}\biggr{]}\biggr{\\}}.$
(32)

Equations (29), (30), and (32) enforce the momentum and angular momentum
exchanges between solvent and swimmer’s body. The velocity field at the new
time step is

$\displaystyle{{\mbox{\boldmath$v$}}}_{n+1}={{\mbox{\boldmath$v$}}}^{*}\bigl{[}1-\sum_{\alpha=1}^{N}\Psi_{\alpha,n+1}^{(b)}\bigr{]}+\sum_{\alpha=1}^{N}\Psi_{\alpha,n+1}^{(b)}\biggl{[}{{\mbox{\boldmath$V$}}}_{\alpha}^{(G)}(t_{n+1})+{{\mbox{\boldmath$\Omega$}}}_{\alpha}^{(G)}(t_{n+1})\times\bigl{(}{{\mbox{\boldmath$r$}}}-{{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}(t_{n+1})\bigr{)}\biggr{]}$
(33)

with

$\displaystyle\nabla\cdot{{\mbox{\boldmath$v$}}}_{n+1}$ $\displaystyle=$
$\displaystyle 0.$ (34)

Therefore, within the particle domain ($\Psi_{\alpha,n+1}^{(b)}=1$) the
velocity field coincides with the particle velocity as
${{\mbox{\boldmath$v$}}}_{n+1}={{\mbox{\boldmath$V$}}}_{\alpha}^{(G)}(t_{n+1})+{{\mbox{\boldmath$\Omega$}}}_{\alpha}^{(G)}(t_{n+1})\times({{\mbox{\boldmath$r$}}}-{{\mbox{\boldmath$R$}}}_{\alpha}^{(G)}(t_{n+1}))$,
while within the fluid domain ($\Psi_{\alpha,n+1}^{(b)}=0$)
${{\mbox{\boldmath$v$}}}_{n+1}={{\mbox{\boldmath$v$}}}^{*}$. In the interface
domain ($0<\Psi_{\alpha,n+1}^{(b)}<1$) the particle velocity smoothly matches
the solvent velocity ${{\mbox{\boldmath$v$}}}^{*}$. That is, the fluid is
prevented from penetrating inside the particle domain.

##  Appendix C Evaluation of the hydrodynamic effects on the collision process
in our simulations

As denoted in the main text, when the separation distance between two swimmers
is less than $d_{0}$, these swimmers are assumed to be undergoing a collision;
in the present study, we set $d_{0}=0.7\ell_{0}$, and the collision time
$\Delta t$ is set to be $\Delta t=1.25\ell_{0}/v_{s}$, where $\ell_{0}$ is the
swimmer size and $v_{s}$ is the average swimming speed. This definition of
collision is somewhat arbitrary, but we show below that it (with the above-
presented sets of $d_{0}$ and $\Delta t$) can appropriately describe the
hydrodynamic collisions. In the following, we make use of the parameter values
of $F_{A}=20$, $\ell_{0}=14.4$, $v_{s}=0.11$, and ${\mathcal{V}}^{(b)}=355$.

The magnitude of the rotational flow field, $\omega$, induced at a distance
$r$ from a swimmer is approximately $\omega\sim
0.1F_{A}\ell_{0}/\eta_{s}r^{3}$ LaugaB . At the average distance between
neighboring swimmers, $r=l_{N}=({\mathcal{V}}^{(b)}/\phi)^{1/3}$,
$\omega\sim(10^{-1}\phi)\sim 10^{-3}$ for $\phi\sim 0.01$: on average,
swimming motions are disturbed by random flows induced by other swimmers at a
distance of $\sim l_{N}$. However, when approaching a specific swimmer, the
flow field created by those swimmers deterministically influences their
swimming motions. With a similar argument, at $r=d_{0}$, $\omega\sim 10^{-2}$,
and therefore during a collision ($\sim\ell_{0}/v_{s}\sim 10^{2}$), the
swimmer’s trajectory is largely disturbed by HIs as
$(\ell_{0}/v_{s})\times\omega\gtrsim 1$. In a typical situation with
$\dot{\gamma}=10^{-3}$, for a duration of $\dot{\gamma}^{-1}$, because $(\pi
d_{0}^{2}v_{s}/{\dot{\gamma}})\times(1/l_{N})^{3}\sim 10^{2}\phi\gtrsim 1$ for
$\phi\geq 0.01$, at least one “collision” may occur. Therefore, in our
simulations, the effects of HIs on the swimming motions surpass (or at least
compete with) the mean-flow effects for $\dot{\gamma}\lesssim 10^{-3}$.

We note that because $(\pi d_{0}^{2}v_{s}\Delta t)\times(1/l_{N})^{3}\cong
10\phi\lesssim 1$ in our simulations ($\phi\leq 0.05$), it is rare that for a
duration of $\Delta t$, three or more swimmers are at distances closer than
$d_{0}$. In other words, a hydrodynamic collision can be considered a single
event.

## References

  * (1) Y. Hatwalne, S. Ramaswamy, M. Rao, and R. A. Simha, Rheology of Active-particle suspensions, Phys. Rev. Lett. 92, 118101 (2004).
  * (2) A. Sokolov and I. S. Aranson, Reduction of viscosity in suspension of swimming bacteria, Phys. Rev. Lett. 103, 148101 (2009).
  * (3) J. Gachelin, G. Miño, H. Berthet, A. Lindner, A. Rousselet, and E. Clément, Non-Newtonian viscosity of Escherichia coli suspensions, Phys. Rev. Lett. 110, 268103 (2013).
  * (4) H. M. López, J. Gachelin, C. Douarche, H. Auradou, and E. Clément, Turning bacteria suspensions into superfluids, Phys. Rev. Lett. 115, 028301 (2015).
  * (5) A. Liu, K. Zhang, and X. Cheng, Rheology of bacterial suspensions under confinement, Rheologica Acta 58, 439 (2019).
  * (6) V.A. Martinez, E. Clément, J. Arlt, C. Douarche, A. Dawson, J. Schwarz-Linek, A.K. Creppy, V. S̆kultéty, A.N. Morozov, H. Auradou, and W.C.K. Poon, Symmetric shear banding and swarming vortices in bacterial superfluids, Proc. Natl. Acad. Sci. U.S.A. 117 2326 (2020).
  * (7) S. Rafaï, L. Jibuti, and P. Peyla, Effective viscosity of microswimmer suspensions, Phys. Rev. Lett. 104, 098102 (2010).
  * (8) M.C. Marchetti, J.F. Joanny, S. Ramaswamy, T.B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys. 85, 1143 (2013).
  * (9) M.C. Marchetti, Frictionless fluids from bacterial teamwork, Nature 525, 37 (2015).
  * (10) D. Saintillan, Rheology of active fluids, Annu. Rev. Fluid Mech. 50, 563 (2018).
  * (11) T. Ishikawa and T.J. Pedray, The rheology of a semi-dilute suspension of swimming model micro-organisms, J. Fluid Mech., 588, 300 (2007).
  * (12) M.E. Cates, S.M. Fielding, D. Marenduzzo, E. Orlandini, and J.M. Yeomans, Shearing active gels close to the Isotropic-Nematic transition, Phys. Rev. Lett. 101, 068102 (2008).
  * (13) B.M. Haines, A. Sokolov, I.S. Aranson, L. Berlyand, and D.A. Karpeev, Three-dimensional model for the effective viscosity of bacterial suspensions, Phys. Rev. E 80, 041922 (2009).
  * (14) L. Giomi, T.B. Liverpool, and M.C. Marchetti, Sheared active fluids: Thickening, thinning, and vanishing viscosity, Phys. Rev. E 81, 051908 (2010).
  * (15) D. Saintillan, The dilute rheology of swimming suspensions: A simple kinetic model, Exp Mech 50, 1275 (2010).
  * (16) S.D. Ryan, B.M. Haines, L. Berlyand, F. Ziebert, and I.S. Aranson, Viscosity of bacterial suspensions: Hydrodynamic interactions and self-induced noise, Phys. Rev. E 83, 050904(R) (2011).
  * (17) M. Moradi and A. Najafi, Rheological properties of a dilute suspension of self-propelled particles, EPL 109, 24001 (2015).
  * (18) T.M. Bechtel and A.S. Khair, Linear viscoelasticity of a dilute active suspension, Rheol Acta 56, 149 (2017).
  * (19) S. C. Takatori and J.F. Brady, Superfluid Behavior of Active Suspensions from Diffusive Stretching, Phys. Rev. Lett. 118, 018003 (2017).
  * (20) E.J. Hinch and L.G. Leal, The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles, J. Fluid Mech. 52, 683 (1972).
  * (21) E.J. Hinch and L.G. Leal, Constitutive equations in suspension mechanics. Part 2. Approximate forms for a suspension of rigid particles affected by Brownian rotations, J. Fluid. mech. 76, 187 (1976).
  * (22) E. Lauga and T.R. Powers, The hydrodynamics of swimming microorganisms, Rep. Prog. Phys. 72, 096601 (2009).
  * (23) E. Lauga, The Fluid Dynamics of Cell Motility (Cambridge University Press, Cambridge 2020).
  * (24) J. P. Hernandez-Ortiz, C. G. Stoltz, and M. D. Graham, Transport and Collective Dynamics in Suspensions of Confined Swimming Particles, Phys. Rev. Lett. 95, 204501 (2005).
  * (25) P. T. Underhill, J. P. Hernandez-Ortiz, and M. D. Graham, Diffusion and Spatial Correlations in Suspensions of Swimming Particles, Phys. Rev. Lett. 100, 248101 (2008).
  * (26) D. Saintillan and M. Shelley, Orientational Order and Instabilities in Suspensions of Self-Locomoting Rods, Phys. Rev. Lett. 99 (2007) 058102.
  * (27) V. Gyrya, I. S. Aranson, L. V. Berlyand, and D. Karpeev, A Model of Hydrodynamic Interaction Between Swimming Bacteria, Bull. Math. Biol. 72, 148 (2010).
  * (28) A. Decoene, S. Martin, and B. Maury, Microscopic modeling of active bacterial suspensions, Math. Model. Nat. Phenom. 6, 98 (2011).
  * (29) A. Furukawa, D. Marenduzzo, and M.E. Cates, Activity-induced clustering in model dumbbell swimmers: The role of hydrodynamic interactions, Phys. Rev. E 90, 022303 (2014).
  * (30) D. Saintillan and M. Shelley, Instabilities and pattern formation in active particle suspensions: Kinetic theory and continuum simulations, Phys. Rev. Lett. 100 (2008) 178103.
  * (31) D. Saintillan and M. Shelley, Instabilities, pattern formation and mixing in active suspensions, Phys. Fluids 20 (2008) 123304.
  * (32) D. Saintillan and M. J. Shelley, Active suspensions and their nonlinear models, C. R. Phys. 14, 497 (2013).
  * (33) A. Baskaran and M. C. Marchetti, Statistical mechanics and hydrodynamics of bacterial suspensions, Proc. Natl. Acad. Sci. USA 106, 15567 (2009).
  * (34) Y. Nakayama and R. Yamamoto, Simulation method to resolve hydrodynamic interactions in colloidal dispersions, Phys. Rev. E. 71, 036707 (2005).
  * (35) J. J. Molina, K. Otomura, H. Shiba, H. Kobayashi, M. Sano, and R. Yamamoto, Rheological evaluation of colloidal dispersions using the smoothed profile method: formulation and applications, J. Fluid Mech. 792, 590 (2016).
  * (36) R. Yamamoto, J. J. Molina, and Y. Nakayama, Smoothed profile method for direct numerical simulations of hydrodynamically interacting particles, Soft Matter, 17, 4226 (2021).
  * (37) A.J.C. Ladd, Short-time motion of colloidal particles - numerical-simulation via a fluctuating lattice-Boltzmann equation. Phys. Rev. Lett. 70, 1339 (1993).
  * (38) M. E. Cates, K. Stratford, R. Adhikari, P. Stainsell, J.-C. Desplat, I. Pagonabarraga, and A.J. Wagner, Simulating colloid hydrodynamics with lattice Boltzmann methods. J. Phys.: Condens. Matter 16, S3903 (2004).
  * (39) H. Tanaka and T. Araki, Simulation Method of Colloidal Suspensions with Hydrodynamic Interactions: Fluid Particle Dynamics, Phys. Rev. Lett. 85, 1338 (2000).
  * (40) A. Furukawa, M. Tateno, and H. Tanaka, Physical foundation of the fluid particle dynamics method for colloid dynamics simulation, Soft Matter, 14, 3738 (2018).
  * (41) P. Español and P. Warren, Statistical Mechanics of Dissipative Particle Dynamics, Europhys. Lett. 30, 191 (1995).
  * (42) S. JI, R. Jiang, R.G. Winkler, and G. Gompper, Mesoscale hydrodynamic modeling of a colloid in shear-thinning viscoelastic fluids under shear flow, J. Chem. Phys. 135 134116 (2011).
  * (43) A.P. Berke, L. Turner, H.C. Berg, and E. Lauga, Hydrodynamic Attraction of Swimming Microorganisms by Surfaces, Phys. Rev. Lett. 101, 038102 (2008).
  * (44) G. Li and J.X. Tang, Phys. Rev. Lett. 103, 078101 (2009). Accumulation of Microswimmers near a Surface Mediated by Collision and Rotational Brownian Motion
  * (45) N. Figueroa-Morales, G. L. Miño, A. Rivera, R. Caballero, E. Cł’ement, E. Altshuler, and A. Lindner, Living on the edge: transfer and traffic of E. coli in a confined flow, Soft Matter 11, 6284 (2015).
  * (46) S. Bianchi, F. Saglimbeni, and R. Di Leonardo, Holographic Imaging Reveals the Mechanism of Wall Entrapment in Swimming Bacteria, Phys. Rev. X 7, 011010 (2017).
  * (47) P. Denissenko, V. Kantsler, D.J. Smith, and J. Kirkman-Brown, Human spermatozoa migration in microchannels reveals boundary-following navigation, Proc. Natl. Acad. Sci. U.S.A. 109, 8007 (2012).
  * (48) B. Ezhilan, R. Alonso-Matilla, and D. Saintillan D (2015) On the distribution and swim pressure of run-and-tumble particles in 50 confinement. J. Fluid Mech. 781, R4 (2015).
  * (49) B. Ezhilan and D. Saintillan, Transport of a dilute active suspension in pressure-driven channel flow. J. Fluid Mech. 777. 482 (2015).
  * (50) W. Yan and J.F. Brady The force on a boundary in active matter. J. Fluid Mech. 785, R1 (2015).
  * (51) M. A.-S. Vigeant, R. M. Ford, M. Wagner, and L. K. Tamm, Simplified modeling of E. coli mortality after genome damage induced by UV-C light exposure, Appl. Environ. Microbiol. 68, 2794 (2002).
  * (52) S.E. Spagnolie and E. Lauga, Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations, J. Fluid Mech. 700, 105 (2012).
  * (53) O. Sipos, K. Nagy, R. Di Leonardo, and P. Galajda, Hydrodynamic Trapping of Swimming Bacteria by Convex Walls, Phys. Rev. Lett. 114, 258104 (2015).
  * (54) L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon Press, New York, 1959).


Copyright is held by the author/owner(s).

# Science Bots:
a Model for the Future of Scientific Computation?

Tobias Kuhn

Department of Humanities, Social and Political Sciences, ETH Zurich,
Switzerland<EMAIL_ADDRESS>
###### Abstract

As a response to the trends of the increasing importance of computational
approaches and the accelerating pace in science, I propose in this position
paper to establish the concept of “science bots” that autonomously perform
programmed tasks on input data they encounter and immediately publish the
results. We can let such bots participate in a reputation system together with
human users, meaning that bots and humans get positive or negative feedback by
other participants. Positive reputation given to these bots would also shine
on their owners, motivating them to contribute to this system, while negative
reputation will allow us to filter out low-quality data, which is inevitable
in an open and decentralized system.

###### category:

K.4.2 Computers and Society Organizational Impacts

###### keywords:

Automation

###### keywords:

autonomous agent, bot, nanopublication, scholarly communication, semantic
publishing

††conference: WWW 2015 Companion, May 18–22, 2015, Florence, Italy.

##  1 Introduction

As datasets become increasingly important in all branches of science, many
have proposed methods and tools to publish data [3] [12] [1]. Nanopublications
[5] are an approach to bundle atomic data snippets (in RDF) in small packages
together with their provenance and metadata. Such nanopublications can be
manually created by scientists and linked to their articles, but they can also
be automatically extracted from existing datasets or be directly created by
programs that implement scientific methods.

In general, computer programs form a third kind of scientific contribution,
besides narrative articles and datasets. While many such programs are openly
available, there are no conventions or standards of how to reliably link data
to the software that produced it, including the version of the software and
the input it received. Moreover, due to the focus of the scientific life cycle
on the publication of articles, scientific software is typically applied only
to the data available at the time of writing a paper. It is often not the case
that new output data is made public when new input data becomes available. To
tackle these problems, I argue that we can encapsulate certain types of
scientific algorithms as small independent agents that take inputs of a given
type and produce, for example, nanopublications, and they could do this in a
real-time and automatic manner as new input data becomes available.

##  2 Background

I borrow the term “bot” from Wikipedia, where bots are applied, for example,
to revert edits that are the results of vandalism [7] [14]. A prominent
example is a bot that has created around 454 000 articles for the Swedish
Wikipedia [6]. The fact that bots can be powerful also in a negative sense has
become apparent with the rise of botnets [13], and with the increasing problem
of "social bots" that pretend to be humans [4]. I argue here that the power of
bots could also be harnessed in a positive way for scientific computation. In
contrast to the agents in the original Semantic Web paper [2], such bots would
not propose or make decisions as a kind of personal assistant, but they would
only publish data snippets and they would do that without any further
interaction with humans.

In previous work, I showed how the concept of nanopublications can be extended
and I mentioned the possible use of bots to create them [9]. I also presented
an approach to attach cryptographic hash values to nanopublication identifiers
to make them verifiable and immutable [11]. Based on that work, I have started
to establish a nanopublication server network, with which nanopublications can
be published, retrieved, and archived in a reliable, trustworthy, and
decentralized manner [10], which could serve as the basis for the
communication for bots.

##  3 Approach

In this position paper, I propose bots as a general concept for scientific
computation. For example, a bot could apply text mining to extract relations
from the abstracts of the constantly growing PubMed database, another bot
could regularly measure the temperature at a given location and publish the
results, and yet another one could infer new facts from existing
nanopublications by applying specified rules or heuristics (e.g. if disease X
is related to gene Y, which is targeted by drug Z then Z might help to treat
X). Importantly, these bots can automatically publish the obtained data
without double-checking or direct supervision by their creators, and these
data can be made immediately accessible to everybody (including other bots).

In a system that treats bots as first-class citizens, we have to expect that
some bots (and humans for that matter) will produce low-quality contributions,
and we have to make sure that this does not affect the reliability and
trustworthiness of the system. I argue that we can achieve that without
introducing a central authority, without making concessions with respect to
the openness of the system, and without delaying the publishing of results. We
simply need a sufficiently accurate automatic method to discern good
contributions from bad ones, which can be achieved by a reputation system. We
can let scientists and bots participate in the same reputation system, where
they would increase their reputation by receiving positive feedback by other
participants on the usefulness and quality of their contributions. Positive
reputation of a bot, in turn, would give credit and reputation to the
scientist who created it.

To arrive at a simple exemplary model to explain the approach, we can define a
relation "is contributed by", where bots can occur on either side: They are
contributions, as they were programmed and created by somebody, but they are
also contributors, as they can create new digital entities on their own. We
can define a second type of relation to represent assessments. For the sake of
simplicity, we model here only positive assessments and strip them of all
granularity and detail, and we can call the resulting relation “gives positive
assessment for”.

Figure 1:  A simple example of a graph of contributors and contributions with
edges for creatorship and assessments.

Fig. 1 shows a simple example of such a graph with two kinds of edges,
representing creatorship and assessments. To determine the reputation or
importance of the nodes, we can in the simplest case treat the two types of
edges identically and rank the nodes by applying a network measure such as
Eigenvector centrality (which is closely related to Google’s PageRank
algorithm to rank websites), as shown by the red numbers. The person at the
top-left has a high reputation because he is endorsed by the person in the
middle. The latter has a high reputation because her direct and indirect
contributions were positively assessed by others (even though she has not
received a direct assessment herself). The third person to the right, however,
has not contributed anything that was positively assessed by others (only by
his own bot), and therefore his reputation is low. Of course, there are many
possible variations and extensions, such as bidirectional contribution edges
for the Eigenvector calculation, as indicated by the gray numbers. In general,
as one cannot influence incoming links from the part of the network that is
not under one’s control, there is no way to efficiently game the system. The
scalability of such algorithms in open and decentralized systems is
demonstrated by their successful application by search engines and peer-to-
peer systems [8].

Bots could free scientists from routine tasks and therefore allow them to
focus on the interesting questions. Furthermore, this approach could increase
the value and appreciation of datasets and software as research products, and
give due credit to their creators. With appropriate reputation mechanisms,
this can be achieved in a fully open and decentralized environment.

## References

  * [1] S. Bechhofer, D. De Roure, M. Gamble, C. Goble, I. Buchan. Research objects: Towards exchange and reuse of digital knowledge. FWCS 2010.
  * [2] T. Berners-Lee, J. Hendler, O. Lassila. The Semantic Web . Scientific American 284:5.
  * [3] J. Brase. DataCite - A global registration agency for research data. COINFO 2009.
  * [4] E. Ferrara, O. Varol, C. Davis, F. Menczer, A. Flammini. The Rise of Social Bots. arXiv preprint 1407.5225, 2014.
  * [5] P. Groth, A. Gibson, J. Velterop. The anatomy of a nanopublication. Information Services and Use 30:1, 2010.
  * [6] L. Guldbrandsson. Swedish Wikipedia surpasses 1 million articles with aid of article creation bot. Wikimedia blog, 17 June 2013.
  * [7] A. Halfaker, J. Riedl. Bots and Cyborgs: Wikipedia’s Immune System. Social Computing 45:3. 2012.
  * [8] S. D. Kamvar, M. T. Schlosser, H. Garcia-Molina. The EigenTrust algorithm for reputation management in P2P networks. WWW 2003.
  * [9] T. Kuhn, P. E. Barbano, M. Levente Nagy, M. Krauthammer. Broadening the Scope of Nanopublications. ESWC 2013.
  * [10] T. Kuhn, C. Chichester, M. Dumontier, M. Krauthammer. Publishing without Publishers: a Decentralized Approach to Dissemination, Retrieval, and Archiving of Data. arXiv preprint 1411.2749, 2014.
  * [11] T. Kuhn, M. Dumontier. Trusty URIs: Verifiable, Immutable, and Permanent Digital Artifacts for Linked Data. ESWC 2014.
  * [12] M. A. Parsons, R. Duerr, J.-B. Minster. Data Citation and Peer Review. Eos, Transactions American Geophysical Union 91:34, 2010.
  * [13] M. A. Rajab, J. Zarfoss, F. Monrose, A. Terzis. A multifaceted approach to understanding the botnet phenomenon. IMC 2006.
  * [14] T. Steiner. Bots vs. Wikipedians, Anons vs. Logged-Ins. WWW Companion 2014.


11institutetext: Institute of Physics, Jan Kochanowski University, ul.
Uniwersytecka 7, 25-406, Kielce, Poland

# Decays of the tensor glueball in a chiral approach

Arthur<EMAIL_ADDRESS>11

###### Abstract

Glueballs remain an experimentally undiscovered prediction of QCD. Lattice QCD
predicts a spectrum of glueballs, with the tensor $(J^{PC}=2^{++})$ glueball
being the second lightest, behind the scalar glueball. From an effective
hadronic model based on spontaneous and explicit chiral symmetry breaking, we
compute decay ratios of the tensor glueball into various meson decay channels.
We find the tensor glueball to primarily decay into 2 vector mesons, dominated
by $\rho\rho$ and $K^{*}K^{*}$. These results are compared to experimental
data of decay rates of spin 2 mesons. Based on this comparison we make
statements on the eligibility of these mesons as potential tensor glueball
candidates.

##  1 Introduction

The experimental verification of glueballs has been, and still is, a long-
standing open issue in QCD Gross:2022hyw . Numerous theoretical Giacosa:2005bw
; Brunner:2015oqa  and experimental Klempt:2022qjf  approaches have made
headway, yet the situation is still not completely clear Mathieu:2008me ;
Crede:2008vw ; Llanes-Estrada:2021evz ; Chen:2022asf . The different
theoretical methods agree on the mass hierarchy of the lowest lying glueball
states, with the scalar $(J^{PC}=0^{++})$ being the lightest and the tensor
$(J^{PC}=2^{++})$ the second lightest glueball. In this work we will focus on
the tensor glueball, for which there are many experimentally observed
isoscalar-tensor candidate resonances. We will present results on the tensor
glueball Vereijken:2023jor  in the extended Linear Sigma Model
Parganlija:2012fy , which is an extension of earlier works on axial-tensor
mesons in the same model Jafarzade:2022uqo . Different glueballs have been
studied before in the same type of model, such as the scalar Janowski:2014ppa
and the pseudoscalar glueballEshraim:2012jv .

##  2 Chiral model

The meson resonances are gathered into the nonets $V^{\mu}$ $(J^{PC}=1^{--})$
containing vector mesons, $A_{1}^{\mu}$ $(J^{PC}=1^{++})$ containing axial-
vector mesons, $P$ $(J^{PC}=0^{-+})$ containing pseudoscalar mesons, $S$
$(J^{PC}=0^{++})$ containing scalar mesons, $T^{\mu\nu}$ $(J^{PC}=2^{++})$
containing tensor mesons, and $A_{2}^{\mu\nu}$ $(J^{PC}=2^{--})$ containing
axial-tensor mesons. For details on the resonance assignment of the nonets see
Vereijken:2023jor ; Jafarzade:2022uqo . The tensor glueball itself is a flavor
blind object $G_{2,\mu\nu}$.

The chiral invariant Lagrangians relevant to us are as follows

$\displaystyle\mathcal{L}_{\lambda}=\frac{\lambda}{\sqrt{6}}G_{2,\mu\nu}\Big{(}\text{Tr}\Big{[}\\{L^{\mu},L^{\nu}\\}\Big{]}+\text{Tr}\Big{[}\\{R^{\mu},R^{\nu}\\}\Big{]}\Big{)}\text{
,}$ (1)
$\displaystyle\mathcal{L}_{\alpha}=\frac{\alpha}{\sqrt{6}}G_{2,\mu\nu}\Big{(}\text{Tr}\Big{[}\Phi\textbf{R}^{\mu\nu}\Phi^{\dagger}\Big{]}+\text{Tr}\Big{[}\Phi^{\dagger}\textbf{L}^{\mu\nu}\Phi\Big{]}\Big{)}\text{
.}$ (2)

These are the leading terms in large-$N_{c}$ expansion, where $N_{c}$ is the
number of colors of underlying the gauge group. The nonets of chiral partners
are grouped together and are given by:

$\displaystyle L^{\mu}$ $\displaystyle:=V^{\mu}+A_{1}^{\mu}\ \text{ ,
}R^{\mu}:=V^{\mu}-A_{1}^{\mu}\text{ , }\Phi=S+iP$
$\displaystyle\mathbf{L}^{\mu\nu}$
$\displaystyle=T^{\mu\nu}+A_{2}^{\mu\nu}\text{ ,
}\mathbf{R}^{\mu\nu}=T^{\mu\nu}-A_{2}^{\mu\nu},$ (3)

such that they obey the transformation rules $L^{\mu}\to
U_{L}L^{\mu}U_{L}^{\dagger}$, $R^{\mu}\to U_{R}R^{\mu}U^{\dagger}_{R}$,
$\Phi\to U_{L}\Phi U^{\dagger}_{R},\mathbf{R}^{\mu\nu}\to
U_{R}\mathbf{R}^{\mu\nu}U^{\dagger}_{R},\mathbf{L}^{\mu\nu}\to
U_{L}\mathbf{L}^{\mu\nu}U^{\dagger}_{L}$ under the chiral transformations of
$U_{L}(3)\times U_{R}(3)$. The first Lagrangian (1) models the 2 body decays
of the tensor glueball into 2 vector mesons, into 2 pseudoscalar mesons, and
into an axial-vector and a pseudoscalar meson. The second Lagrangian (2) leads
to the decay into a tensor and a pseudoscalar meson. Since the coupling
constants $\alpha$ and $\lambda$ are not a priori known and cannot be fitted
to experimental data, we are limited to computing decay ratios, seperately for
each Lagrangian. Lattice calculations in Athenodorou:2020ani  find a tensor
glueball mass of $2369$ MeV. The decay ratios of the first Lagrangian with
respect to $\pi\pi$ for this mass are shown in table 1. As evident from the
results in table 1, the 2-vector decay channel is dominant, in particular the
decays into $\rho\rho$ and $K^{*}\bar{K}^{*}$. A similar dominance of the
2-vector channel was recently found in Hechenberger:2023ljn  with the
holographic Witten-Sakai-Sugimoto model.

Decay Ratio | theory | Decay Ratio | theory | Decay Ratio | theory
---|---|---|---|---|---
$\frac{G_{2}(2369)\longrightarrow\bar{K}\,K}{G_{2}(2369)\longrightarrow\pi\,\pi}$ | $0.4$ | $\frac{G_{2}(2369)\longrightarrow\rho(770)\,\rho(770)}{G_{2}(2369)\longrightarrow\pi\,\pi}$ | $51$ | $\frac{G_{2}(2369)\longrightarrow a_{1}(1260)\,\pi}{G_{2}(2369)\longrightarrow\pi\,\pi}$ | $0.26$
$\frac{G_{2}(2369)\longrightarrow\eta\,\eta}{G_{2}(2369)\longrightarrow\pi\,\pi}$ | $0.1$ | $\frac{G_{2}(2369)\longrightarrow\bar{K}^{\ast}(892)\,\bar{K}^{\ast}(892)}{G_{2}(2369)\longrightarrow\pi\,\pi}$ | $44$ | $\frac{G_{2}(2369)\longrightarrow K_{1\,,A}\,K}{G_{2}(2369)\longrightarrow\pi\,\pi}$ | $0.12$
$\frac{G_{2}(2369)\longrightarrow\eta\,\eta^{\prime}}{G_{2}(2369)\longrightarrow\pi\,\pi}$ | $0.005$ | $\frac{G_{2}(2369)\longrightarrow\omega(782)\,\omega(782)}{G_{2}(2369)\longrightarrow\pi\,\pi}$ | $17$ | $\frac{G_{2}(2369)\longrightarrow f_{1}(1285)\,\eta}{G_{2}(2369)\longrightarrow\pi\,\pi}$ | $0.03$
$\frac{G_{2}(2369)\longrightarrow\eta^{\prime}\,\eta^{\prime}}{G_{2}(2369)\longrightarrow\pi\,\pi}$ | $0.01$ | $\frac{G_{2}(2369)\longrightarrow\phi(1020)\,\phi(1020)}{G_{2}(2369)\longrightarrow\pi\,\pi}$ | $7$ | $\frac{G_{2}(2369)\longrightarrow f_{1}(1420)\,\eta}{G_{2}(2369)\longrightarrow\pi\,\pi}$ | $0.008$
Table 1: Decay ratios of $G_{2}$ w.r.t. $\pi\pi$ for a mass of $2369$ MeV. The
columns are sorted as $PP$ on the left, $VV$ in the middle, and $AP$ on the
right.

##  3 Results & Data Comparison

Resonances | Decay Ratios | PDG Workman:2022ynf  | Model Prediction
---|---|---|---
$f_{2}(1910)$ | $\rho(770)\rho(770)/\omega(782)\omega(782)$ | $2.6\pm 0.4$ | $3.1$
$f_{2}(1910)$ | $f_{2}(1270)\eta/a_{2}(1320)\pi$ | $0.09\pm 0.05$ | $0.07$
$f_{2}(1910)$ | $\eta\eta/\eta\eta^{\prime}(958)$ | $<0.05$ | $\sim 8$
$f_{2}(1910)$ | $\omega(782)\omega(782)/\eta\eta\prime(958)$ | $2.6\pm 0.6$ | $\sim 200$
$f_{2}(1950)$ | $\eta\eta/\pi\pi$ | $0.14\pm 0.05$ | $0.081$
$f_{2}(1950)$ | $K\overline{K}/\pi\pi$ | $\sim 0.8$ | $0.32$
$f_{2}(1950)$ | $4\pi/\eta\eta$ | $>200$ | $>700$
$f_{2}(2150)$ | $f_{2}(1270)\eta/a_{2}(1320)\pi$ | $0.79\pm 0.11$ | $0.1$
$f_{2}(2150)$ | $K\overline{K}/\eta\eta$ | $1.28\pm 0.23$ | $\sim 4$
$f_{2}(2150)$ | $\pi\pi/\eta\eta$ | $<0.33$ | $\sim 10$
$f_{J}(2220)$ | $\pi\pi/K\overline{K}$ | $1.0\pm 0.5$ | $\sim 2.5$
Table 2: Decay ratios for the decay channels with available data.

We compare results to available data of spin-2 isoscalar resonances
($J^{PC}=2^{++}$, I = 0) with masses of 1.9 GeV and upwards. These are the
$f_{2}(1910),f_{2}(1950),f_{2}(2010),f_{2}(2150),f_{J}(2220),f_{2}(2300),$ and
the $f_{2}(2340)$. In table 2 decay ratios are computed and compared with PDG
data Workman:2022ynf  where available, revealing how well they fit as glueball
candidates. We see that every glueball candidate other than the $f_{2}(1950)$
has disagreement with experimental data, and that the $f_{2}(1950)$ fits
reasonably well, given uncertainties on both sides. Therefore we interpret
$f_{2}(1950)$ to be the best candidate for the lightest tensor glueball, which
has not been the first time it has been proposed as the tensor glueball, see
e.g. Godizov:2016vuw .

##  4 Conclusion

In this note we have computed decays of the tensor glueball using a chiral
hadronic model. We find that the decay to two vectors, in particular
$\rho\rho$ and $K^{*}\bar{K}^{*}$, is dominant. Upon comparing with
experimental data, we find that the $f_{2}(1950)$ is the most suitable
candidate (even if some deviations are present) for the tensor glueball based
on the known decay ratios. In the future, one should investigate why there is
up to a $400$ MeV mass difference between this resonance and predicted masses
from lattice methods. One possible explanation could be the role of mesonic
loops and/or the mixing with nearby quark-antiquark states.

## Acknowledgements

We thank Francesco Giacosa and Shahriyar Jafarzade for useful discussions. We
also acknowledge financial support from the Polish National Science Centre
(NCN) via the OPUS project 2019/33/B/ST2/00613.

## References

  * (1) F. Gross, E. Klempt, S. J. Brodsky, A. J. Buras, V. D. Burkert, G. Heinrich, K. Jakobs, C. A. Meyer, K. Orginos and M. Strickland, et al. [arXiv:2212.11107 [hep-ph]].
  * (2) F. Giacosa, T. Gutsche, V. E. Lyubovitskij and A. Faessler, Phys. Rev. D 72, 114021 (2005) doi:10.1103/PhysRevD.72.114021 [arXiv:hep-ph/0511171 [hep-ph]].
  * (3) F. Brünner, D. Parganlija and A. Rebhan, Phys. Rev. D 91, no.10, 106002 (2015) [erratum: Phys. Rev. D 93, no.10, 109903 (2016)] doi:10.1103/PhysRevD.91.106002 [arXiv:1501.07906 [hep-ph]].
  * (4) E. Klempt, K. V. Nikonov, A. V. Sarantsev and I. Denisenko, Phys. Lett. B 830, 137171 (2022) doi:10.1016/j.physletb.2022.137171 [arXiv:2205.07239 [hep-ph]].
  * (5) V. Crede and C. A. Meyer, Prog. Part. Nucl. Phys. 63, 74-116 (2009) doi:10.1016/j.ppnp.2009.03.001 [arXiv:0812.0600 [hep-ex]].
  * (6) V. Mathieu, N. Kochelev and V. Vento, Int. J. Mod. Phys. E 18, 1-49 (2009) doi:10.1142/S0218301309012124 [arXiv:0810.4453 [hep-ph]].
  * (7) F. J. Llanes-Estrada, Eur. Phys. J. ST 230, no.6, 1575-1592 (2021) doi:10.1140/epjs/s11734-021-00143-8 [arXiv:2101.05366 [hep-ph]].
  * (8) H. X. Chen, W. Chen, X. Liu, Y. R. Liu and S. L. Zhu, Rept. Prog. Phys. 86, no.2, 026201 (2023) doi:10.1088/1361-6633/aca3b6 [arXiv:2204.02649 [hep-ph]].
  * (9) R. L. Workman et al. [Particle Data Group], PTEP 2022, 083C01 (2022) and 2023 update doi:10.1093/ptep/ptac097
  * (10) A. Vereijken, S. Jafarzade, M. Piotrowska and F. Giacosa, Phys. Rev. D 108, no.1, 014023 (2023) doi:10.1103/PhysRevD.108.014023 [arXiv:2304.05225 [hep-ph]].
  * (11) D. Parganlija, P. Kovacs, G. Wolf, F. Giacosa and D. H. Rischke, Phys. Rev. D 87, no.1, 014011 (2013) doi:10.1103/PhysRevD.87.014011 [arXiv:1208.0585 [hep-ph]].
  * (12) S. Jafarzade, A. Vereijken, M. Piotrowska and F. Giacosa, Phys. Rev. D 106, no.3, 036008 (2022) doi:10.1103/PhysRevD.106.036008 [arXiv:2203.16585 [hep-ph]].
  * (13) S. Janowski, F. Giacosa and D. H. Rischke, Phys. Rev. D 90, no.11, 114005 (2014) doi:10.1103/PhysRevD.90.114005 [arXiv:1408.4921 [hep-ph]].
  * (14) W. I. Eshraim, S. Janowski, F. Giacosa and D. H. Rischke, Phys. Rev. D 87, no.5, 054036 (2013) doi:10.1103/PhysRevD.87.054036 [arXiv:1208.6474 [hep-ph]].
  * (15) A. Athenodorou and M. Teper, JHEP 11, 172 (2020) doi:10.1007/JHEP11(2020)172 [arXiv:2007.06422 [hep-lat]].
  * (16) F. Hechenberger, J. Leutgeb and A. Rebhan, Phys. Rev. D 107, no.11, 114020 (2023) doi:10.1103/PhysRevD.107.114020 [arXiv:2302.13379 [hep-ph]].
  * (17) A. A. Godizov, Eur. Phys. J. C 76, no.7, 361 (2016) doi:10.1140/epjc/s10052-016-4229-z [arXiv:1604.01689 [hep-ph]].