Unjamming and emergent nonreciprocity in active ploughing through a compressible viscoelastic fluid
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nature communications
Article https://doi.org/10.1038/s41467-022-31984-z
Unjamming and emergent nonreciprocity in
active ploughing through a compressible
viscoelastic fluid
Received: 16 September 2021 Jyoti Prasad Banerjee1,5, Rituparno Mandal 2,5
, Deb Sankar Banerjee3,
Shashi Thutupalli 1,4 & Madan Rao 1
Accepted: 8 July 2022
A dilute suspension of active Brownian particles in a dense compressible vis-
Check for updates coelastic fluid, forms a natural setting to study the emergence of non-
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reciprocity during a dynamical phase transition. At these densities, the
transport of active particles is strongly influenced by the passive medium and
shows a dynamical jamming transition as a function of activity and medium
density. In the process, the compressible medium is actively churned up – for
low activity, the active particle gets self-trapped in a cavity of its own making,
while for large activity, the active particle ploughs through the medium, either
accompanied by a moving anisotropic wake, or leaving a porous trail. A
hydrodynamic approach makes it evident that the active particle generates a
long-range density wake which breaks fore-aft symmetry, consistent with the
simulations. Accounting for the back-reaction of the compressible medium
leads to (i) dynamical jamming of the active particle, and (ii) a dynamical non-
reciprocal attraction between two active particles moving along the same
direction, with the trailing particle catching up with the leading one in finite
time. We emphasize that these nonreciprocal effects appear only when the
active particles are moving and so manifest in the vicinity of the jamming-
unjamming transition.
There is a lot of interest in the effective long-range interactions that dynamical arrest. Our results are based on (a) numerical simulations of
emerge amongst active particles moving through a dynamically an agent-based model, and (b) analytical and numerical treatments of
responsive medium. Examples include motile particles embedded in a hydrodynamic equations. We find that there is a dynamical feedback
Stokesian fluid1 or on an elastic substrate2 and diffusiophoretic flows in between the motility of the active particles and the corresponding
a viscous suspension of chemically active particles3. The absence of slow remodelling of the passive compressible medium. Such active
time reversal symmetry manifests in large density fluctuations at particles are ploughers, as opposed to cruisers whose motility speed is
steady state4,5, and in the appearance of nonreciprocal interactions unaffected by the medium, e.g., ref. 2. As a result, ploughers exhibit a
between particles2,3,6–12. jamming-unjamming transition at fixed medium density. We find that
In this paper, we ask whether nonreciprocal interactions could in the unjammed phase, the moving active particles develop a dynamic
emerge as a result of a dynamical phase transition. To realise such nonreciprocal interaction with each other arising from the compres-
emergent nonreciprocity, we study a dilute collection of motile active sibility of the passive medium. We emphasise that this long-range
particles embedded in a dense compressible fluid suspension close to nonreciprocal sensing appears only when the active particles are
1
Simons Centre for the Study of Living Machines, National Centre for Biological Sciences (TIFR), Bangalore, India. 2Institute for Theoretical Physics, Georg-
August-Universität Göttingen, 37077 Göttingen, Germany. 3Department of Physics, Carnegie Mellon University, Pittsburgh, PA, USA. 4International Centre for
Theoretical Sciences (TIFR), Bangalore, India. 5These authors contributed equally: Jyoti Prasad Banerjee, Rituparno Mandal. e-mail: shashi@ncbs.res.in;
madan@ncbs.res.in
Nature Communications | (2022)13:4533 1
Article https://doi.org/10.1038/s41467-022-31984-z
moving through a momentum non-conserving medium, and conse- taken to be purely repulsive (inverse power law) with particle diameter
quently shows up in the neighbourhood of the jamming-unjamming σ (details of the simulation appear in Supplementary Note 1). We work
transition. in the high friction limit, where particle inertia can be ignored.
Our study reveals a hitherto unappreciated facet in this intensely Starting from homogeneous and isotropic initial conditions, we
researched field of dense active assemblies13–21, where the focus has evolve the system to its steady state by integrating over a time of order
been on fluidisation15, intermittency16 and jamming16,17 close to glass 102 τα, where τα is the density relaxation time, also called α-relaxation
transition. The current work should be relevant to a variety of cellular time (Supplementary Note 2). Throughout, we work in the low-
and non-cellular contexts, where the medium is dense but compres- temperature regime T = 0.5, 10−1, 10−3, over a wide range of densities
sible, pliable but slow to relax. Such situations can occur in the (i) ϕ 2 ½0:08; 0:97, and scan through a broad range of the active para-
transport of constituent or embedded particles in the cytoplasm22,23, meters f and τ.
(ii) facilitated transport of transcription factories and exogenous par-
ticles embedded within the cell nucleus24, (iii) movement of bacteria Interplay between active self-propulsion and viscoelasticity of
and cancer cells in fabricated soft porous media or in tissues25–29, (iv) the medium
burrowing movement of ants and worms in dense soil30–33, and (v) Figure 1A shows a schematic of a dilute suspension (ϕa ≪ ϕ) of self-
intrusion of active particles in a disordered bubble raft or a dense propelled particles moving through a dense compressible medium.
suspension of solid colloidal particles34–36. While the macroscopic structural properties of such dense assemblies
are rather innocuous, their dynamical features display characteristic
Results slow relaxation, aging43 and dynamical arrest15 as the density ϕ is
Dynamics of active Brownian particles in a passive medium increased. The dynamics of the medium at large space and time scales,
Our model of the two-dimensional background passive medium is is summarised in a phase diagram (Fig. 1B) in the f − ϕ plane (for fixed T,
similar to the Kob-Andersen37,38 binary mixture of soft spheres with a ϕa and τ). The phase diagram is constructed by computing the α-
volume fraction ϕ so as to be able to tune it across a glass transition at relaxation time τα from the decay of the density overlap function Q(t)
constant temperature T. To this passive medium, we add a dilute (Supplementary Note 2) using the definition Q(τα) = 1/e. This phase
amount ϕa ≪ ϕ of active Brownian soft particles (ABPs)39–42, which are diagram clearly shows macroscopic liquid and solid (glass) phases
made motile by assigning to them independent random forces f = f n, adjoining a cage-hopping “super-cooled” regime; fitting τα to a Vogel-
whose orientation n ðcos θ; sin θÞ is exponentially correlated over a Fulcher form (Supplementary Note 2) provides an estimate for the
persistence time τ. The dynamics of all the interacting particles label- glass transition density ϕVFT(f) (Fig. 1B)15.
led i are described by a Langevin equation subject to a thermal noise ϑ Typical of an approach to a glass, the mean square displacement
of zero mean and variance equal to 2γkBT, while the subset i 2 A of (MSD) averaged over all the passive particles shows a plateauing and
ABPs are subject to additional active stochastic forces, cage-hopping dynamics, as the density ϕ is increased (Fig. S1)34,44–46.
From these graphs we extract the long time diffusivity D∞ (Supple-
N mentary Note 2). In the limit ϕa ≪ ϕ and of small τ, we may deduce the
€ i = γx_ i ∂i ∑ V ij + f ni 1ði2AÞ + ϑi ;
mx
j≠i ð1Þ linear microrheological properties of the passive medium from the
Fourier transform of the MSD47, with an effective temperature
θ_ i = ξ i for i 2 A:
obtained from the mean kinetic energy of the passive particles. Fig. 1C
where 1ði2AÞ is the indicator function which ensures that the active clearly shows that the medium is a viscoelastic Maxwell fluid, with the
forces are restricted to particles i in the active set A. The orientation elastic response G0 ~ ω2 and the viscous response G″ ~ ω, for small ω,
angle θi undergoes rotational diffusion described by an athermal noise where the crossover timescale τM increases exponentially with the
ξi, with zero mean and correlation hξ i ðtÞξ j ðt 0 Þi = 2τ 1 δ ij δðt t 0 Þ. Its increase in area fraction ϕ close to the glass transition.
effect on the xi-dynamics is an exponentially correlated vectorial noise We now turn our attention to the minority component, the small
with correlation time τ, which being unrelated to the drag γ, violates fraction of motile active particles—Fig. 2A–D show typical trajectories
the fluctuation-dissipation relation. The inter-particle potential Vij is of the active motile particles at increasing values of ϕ, keeping f and τ
Fig. 1 | Active motile particles in a dense medium—approach to glass and vis- (open circles) is obtained by fitting τα to a Vogel-Fulcher form (Supplementary
coelasticity. A Schematic of dilute suspension of self-propelled particles of area Note 2). The black squares represent state-points where the simulations have been
fraction ϕa (red particles with arrows showing instantaneous direction n of pro- performed. C Frequency dependence of the elastic G0 and viscous G″ responses, at
pulsive force f n) moving through a dense compressible passive fluid of area frac- different values of ϕ, shows that the passive system (i.e., f = 0) behaves as a vis-
tion ϕ (grey particles). B Dynamical phase diagram in the f − ϕ plane for fixed coelastic Maxwell fluid with relaxation time τM. (inset) τM as a function of area
T = 0.5, ϕa = 0.017 and τ = 50, showing macroscopic liquid and solid (glass) phases fraction ϕ increases exponentially close to the glass transition. These quantities are
adjoining the cage-hopping “super-cooled liquid” regime, as determined from the time averaged (over the time origin) and ensemble averaged (over 64 independent
α-relaxation time, τα (Supplementary Note 2). The glass transition at density ϕVFT(f) simulations). The numerical errors are very small, less than 1% of the actual values.
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Fig. 2 | Crossover in transport characteristics of minority active particles. errors are very small, less than 1% of the actual values. F Crossover scaling collapse
A–D Typical trajectories of the active particle as a function of ϕ at fixed f = 1 and of the late time diffusion coefficient D∞(ϕ, f), described in (2), in the scaling variable
τ = 50, showing (i) activity-dominated transport, (ii) glass dominated cage-hopping y ≡ f/δϕν, where δϕ (0.026 ≤ δϕ ≤ 0.770) is the deviation from the MCT value,
transport and (iii) dynamical arrest, recorded over a time t = 500. E Mean square ϕMCT(f). The scaling exponents are found to be μ ≈ 6.5 and ν ≈ 2.5. The dashed lines
displacement (scaled by time) computed from active particle trajectories suggests suggest a crossover of the scaling function from DðyÞ ! y2 to DðyÞ ! y, as y
a crossover as a function of ϕ. These quantities are time averaged (over the time increases.
origin) and ensemble averaged (over 64 independent simulations). The numerical
fixed. The density of the passive medium affects the transport of the function of porosity25. In as much as our study applies to this bacterial
active particles—thus at low density ϕ, the motile particles show an motility context, we suggest that the reported crossover in ref. 25
activity-dominated transport (Fig. 2A, B), which crosses over to a cage- reflects a phenotypic change arising from a coupling of the normal
hopping dominated transport (Fig. 2C), as ϕ increases. As ϕ increases bacterial movement to the physical properties of the dense passive
further, while still being less than ϕVFT(f), the active particles get medium.
dynamically arrested, (Fig. 2D). The plot of the MSD of the active
particles for the different values of ϕ (Fig. 2E), suggests a crossover Remodelling of the compressible viscoelastic medium by the
collapse from activity-dominated diffusion proportional to f2τ to a motile particles
glass dominated cage-hopping diffusion with a Vogel-Fulcher form to, We see that there is a strong feedback between the nature of active
finally, dynamical arrest. We verify this using a crossover scaling form particle transport and the dynamical remodelling of the passive
for the late time diffusion coefficient (Fig. 2F) medium by the active particles48. This is especially prominent in
the “super-cooled” liquid regime above the glass transition, where the
f active motile particles churn up the medium, inducing large density
D1 ðϕ; f Þ = δϕμ D y ν ð2Þ
δϕ fluctuations that result in long-lived density modulations that back-
react on the transport of the active particles. For a fixed active force f
with δϕ = ϕ*(f) − ϕ, the deviation of ϕ from its value where the late time and temperature T, the physical characteristics of the under-dense
diffusion coefficient goes to zero. There are two possible choices for regions are a result of the interplay between the active driving time τ
ϕ*(f)—the Vogel-Fulcher glass transition density ϕVFT (from an and the ϕ-dependent density relaxation time, τα(ϕ).
exponential fit) used to define the phase diagram in Fig. 1B and Associated with a typical trajectory of the active particles shown in
ϕMCT, which is the mode-coupling estimate (having a power law form) Fig. 3, we generate a density map of the medium in the vicinity of the
of the glass transition. The reason for choosing ϕMCT = ϕ*(f) rather than active particle, as a function of τ and τα(ϕ), keeping f large (f = 3.0) and
ϕVFT is because the latter, being greater than ϕMCT, is very difficult to T low (T = 10−3). The geometry and dynamics of the under-dense
approach either experimentally or in a simulation. The excellent regions created by the active particles, show striking variations—(i) a
collapse with exponents μ ≈ 6.5 and ν ≈ 2.5, suggests ‘critical behaviour’ halo (density wake) that moves with the motile particle, (ii) a static
at the mode-coupling transition, ϕMCT(f). The asymptotic behaviour of cavity that traps the active particle and (iii) a long-lived porous and
the crossover scaling function DðyÞ at small y (Fig. 2F), suggests that tortuous trail as the active particle ploughs through the medium. In
D∞ ≈ f2 δϕ3/2, which crosses over to D∞ ≈ f δϕ4 as y = fδϕ−2.5 goes to ∞. Fig. 4A, we show how the shape of the under-dense region sharply
We remark on the connection between the crossover scaling of changes from circular to elongated as a function of τ. This geometrical
the MSD of active particles as a function of the density of the passive transition appears to coincide with a dynamical transition in the active
medium with recent observations on the crossover behaviour of particle transport—Fig. 4B shows that the speed of the active particle
bacterial motility in a three dimensional porous medium as a _ goes from being non-zero (where the active particle ploughs
∣hRi∣
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Article https://doi.org/10.1038/s41467-022-31984-z
Fig. 3 | Remodelling of the compressible medium by the active particles. Under- porous trail. The associated trajectories of the active particles at τα(ϕ) = 168 (left of
dense regions in the compressible medium (darker blue colours indicate low ρ, and the phase diagram) and τα(ϕ) = 3960 (right of the phase diagram) for the five values
the yellow colours indicate a high ρ where ρ is the local density) that is remodelled of τ corresponding to the phase diagram. The background colours outlining the
by the motile particles, as a function of the persistence time τ and density relaxation particle trajectories corresponding to the respective regions marked on the phase
time τα(ϕ), for fixed (large) f. The geometry of the under-dense regions goes from diagram.
being a static cavity to a moving wake around the motile particle, to a long-lived
Fig. 4 | Dynamical transition in the active particle transport. A The geometry of it gets self-trapped in a cavity of its own making. C Dynamical response of the
the under-dense regions is characterised by a shape parameter, ψ = λλ + λ
, where λ± passive medium recorded at different time points, to a step active force from a
+ + λ
are the eigenvalues of the moment of gyration tensor (Supplementary Note 3), and single active particle (shown at right), measured in the frame of reference of the
goes from being circular (ψ ≈ 0) to elongated (ψ ≈ 1) as τ increases. B Mean of the moving particle. The response is fore-aft asymmetric and relaxes slowly on
magnitude of the velocity (over a time interval Δt = 20) of the active particle, as a switching off the active force.
function of the persistence time τ shows a dynamical transition at τ ≈ 1, below which
through the medium) to zero (where the active particle is self-trapped The back-reaction from the passive medium, can either facilitate
in a quasi-circular cavity of its own making), as τ decreases. movement of the active particle (in the ‘moving wake’ and ‘porous’
The profile and lifetime of the under-dense regions upon active regimes) or trap the active particle (in the active ‘self-trapping’ regime
remodelling, is a dynamical imprint of the transiting active particle —a similar self-trapping regime has been described in ref. 49, although
(both its magnitude and direction) on the medium. In Fig. 4C, we it must be noted that the active particles considered there are squir-
activate only one of the particles of the medium, by imposing a step mers, carrying a force-dipole in a momentum conserving background
active force for a fixed duration. We measure the dynamical response fluid. Self-propelled droplets have also recently been shown to get
of the passive medium (the change in ρ(x, t), the local density from its chemotactically caged in chemical trails of their own making50). A
initial uniform profile) from the start of the activity, in the frame of striking example of such facilitated transport of active particles in a
reference of the moving active particle. We see that the density compressible gel is the the ATP-dependent movement of transcription
response δρ(x, y) is fore-aft asymmetric and that this asymmetric factories that move through the dense nuclear medium of cells24.
profile relaxes slowly on switching off the active force. This demon-
strates that the passive medium (i) is a compressible fluid, and (ii) Hydrodynamics of active particles moving in a compressible
retains a memory of the moving event (its magnitude and direction) viscoelastic fluid
for some time. For a deeper understanding of the interplay between the movement of
In summary, we find that the active particles remodel the passive the active particles and the asymmetric dynamical response of the
compressible medium and that the remodelled compressible medium compressible passive medium, we construct a set of active hydro-
reacts back on the active particle affecting its large scale movement. dynamic equations5 and analyse their solutions in simple situations.
Nature Communications | (2022)13:4533 4Article https://doi.org/10.1038/s41467-022-31984-z
The passive compressible fluid is described by the local density ρ density and velocity fields of the passive fluid from our simulation
and velocity v fields, while the dilute collection of active particles i with trajectories, using an interpolation and smoothing scheme (Supple-
position Ri(t) are propelled by an active force of magnitude f along mentary Note 3).
their orientations ni(t). The density of the medium obeys a continuity Take the case of a single particle, with no orientational fluctua-
equation, tions—Fig. 5A shows a simulation snapshot of an active particle sur-
rounded by the compressible medium. We compute the coarse-
∂t ρ + ∇ ðρvÞ = 0 ð3Þ grained fields, ρ(r, t), v(r, t) and their spatial derivatives—these results
appear in Fig. 5B–D. To verify (4), we plot vy(r, t) vs. ρ∂yρ to obtain B/Γ,
Since the dynamics is overdamped, the local velocity of the medium is from which we compute vx(r, t). The relation between the local velocity
obtained by local force balance, of the medium and the pile up of the density embodied in (4) is shown
to hold up in Fig. 5B, even making allowance for a possible density
Γv = η∇2 v Bρ∇ρ + ∑ f ni ðtÞδ r Ri ðtÞ ð4Þ dependent coefficient B/Γ (the contribution from viscous dissipation is
i2A significantly lower than the rest and so we drop it). We find that there is
a dynamical transition between self-trapping at low f and movement
where the velocity of the compressible fluid is driven by the body- (Fig. 5C). From this, we obtain the value of γ from the slope using (5)
forces f ni2A imposed by the moving active particles. Note that this and hence C/γ (Fig. 5D). From this we see that the form of the back-
contribution is present, since the dynamics takes place in a medium reaction embodied in (5) is also borne out in Fig. 5D, albeit with a
that does not conserve momentum. density (or force) dependent C/γ (inset Fig. 5D). Note that, consistent
The second term on the right represents the forces that oppose with our discussion above, C/γ is an order of magnitude larger than B/Γ.
the movement of the passive particles that are being pushed by the The fact that C/γ drops suddenly beyond f ≈ 2.5, would suggest that the
active body-forces. These come from an active pressure, which, to friction experienced by the active particle increases with increasing f
leading order, arises from a local compressibility of the passive fluid, and then saturates to a constant value. In principle, if this drop is large
P ∝ ρ2 + …, due to inter-particle interactions (at low T, the linear con- enough, this could lead to an active discontinuous shear thickening51.
tribution is insignificant). The “compressibility” B, with units of
force × length, is positive and can in principle depend on ρ. Linearised hydrodynamics of a single active particle moving in a
The other terms correspond to the usual momentum dissipation, compressible medium
a viscous contribution η coming from collisions with the passive par- We first look at the dynamics of a single active particle in the com-
ticles and friction Γ arising from both collisions with other particles pressible medium. Let us take the limit of large τ, and so over the
and from the ambient medium. Importantly, in the high density regime timescale of interest, the orientation n is fixed, say along the x^ direction.
approaching the glass transition, these kinetic coefficients η and Γ may As the active particle moves through the medium, it creates density
be strongly dependent on the local density ρ. inhomogeneities, which relax over time. Using (3) and (4) we get,
The dynamics of the active particles in the overdamped limit is
also given by force balance. In the dilute limit, low ϕa, when there are B f
∂t ρ + ∇ ρ2 ∇ρ + ∂x ρδ ð2Þ ðx X ðtÞ; yÞ = 0: ð7Þ
no direct interactions between active particles, the balance is again Γ Γ
between the propulsive body-forces and the local pressure due to the
compressible fluid, This nonlinear equation resembles an anisotropic Burgers equation
with a source52, and so one might expect travelling pulse solutions. To
γR_ i = f ni Cρ∇ρ∣at Ri ð5Þ see this explicitly, we perform a linear analysis about the uniform
density of the ploughed medium. In this limit, we take Γ and B to be
independent of ρ. Again in this limit, we ignore the back-reaction term
θ_ i = ξ i ðtÞ ð6Þ Cρ ∇ ρ in (5); we will say that the active velocity, v0, is reduced from its
bare value f/γ in a ρ dependent manner.
for all i 2 A. Note that ni ðcos θi ; sin θi Þ, and the athermal orienta- After initial transients, the density excess of the medium can then
tional noise ξ i has zero mean and is delta-correlated, be written in terms of the collective coordinate,
hξ i ðtÞξ j ðt 0 Þi ¼ 2τ 1 δ ij δðt t 0 Þ. Equation (5) accounts for the back-
reaction of the medium on the dynamics of the active particles, which ρðr; tÞ = ρðr RðtÞÞ
feels a block due to particle pile up ahead of it. We have assumed in (6)
that the direction of the propulsion force is set by some internal Analytical solutions of the resulting linearised equation can be easily
detailed-balance violating mechanism, independent of the passive obtained by transforming the equation to coordinates in the moving
medium. Note that the active compressibility C is positive, dependent frame of the active particle, u = x − X(t) and w = y, followed by Fourier
on ρ, and could be different from B. Further, in the limit of low ϕa, one transforming in (u, w) (see Supplementary Note 4). The excess density
expects γ to be a single particle friction, while Γ to be a collective profile ρ(u, w) in the co-moving frame takes the form,
frictional dissipation; the latter could be high when ρ is large.
We will refer to such polar active particles as ploughers, as I ðu; wÞ
ρðu; wÞ = 2DðuÞ I 1 ðu; wÞ + 2 ð8Þ
opposed to cruisers, whose speed is unaffected by the medium, e.g., ξ
ref. 2.
Note that although the passive and active particles have com- where,
parable sizes, we treat the passive medium using a coarse-grained 2 1 3
density, but the active particles as “point-particulate”. Thus, our u=ξ u 2
+ w 2 2
hydrodynamic description should be valid over scales larger than a few I 1 ðu; wÞ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 1 4 5
u2 + w 2 ξ
particle sizes. 2 1 3
We now check whether the continuum hydrodynamic equations, u2 + w 2 2
(3)–(5), describe, in a coarse-grained sense, the agent-based dynamics I 2 ðu; wÞ = K 0 4 5
ξ
represented by (17). For this we compute the local coarse-grained
Nature Communications | (2022)13:4533 5Article https://doi.org/10.1038/s41467-022-31984-z
Fig. 5 | Verifying hydrodynamic equations by coarse-graining agent-based C Mean velocity of the active particle X_ (red and green dots) vs. active force, f,
simulations. A Simulation snapshot of the x–y configurations showing an active showing the dynamical transition between self-trapping at low f and movement.
particle at the origin (black dot) surrounded by particles comprising the com- The mean is obtained by averaging over different initial realisations of the passive
pressible medium. We compute the coarse-grained fields, ρ(r, t), v(r, t) and their medium. Following (5), the value of γ can be extracted from the slope at large f.
spatial derivatives in the shaded (blue) annular region and the thin shaded (black) D To verify (5), we plot f =γ X_ and ρ∂xρ (just in front of the active particle) vs. f.
rectangular region (Supplementary Note 3). B To verify (4), we plot vy(r, t) vs. ρ∂yρ This allows us to compute the parameter C/γ (inset), a measure of the back-reaction
in the blue-shaded region in A at a given time; the best-fit line (dashed line) gives the of the medium on the motile particle. The value of C/γ ≈ 150 is an order of magni-
parameter B/Γ = 4.29 ± 0.093. (inset) Using this value of B/Γ, we compute vx(r, t) and tude larger than B/Γ and drops down to 100 at larger f. For C and D the smaller
ρ∂xρ vs. x along the thin black rectangular region in A and find good agreement. background symbols indicate data from individual simulations.
with K0 and K1 being the modified Bessel functions of the second kind, Knowing the density profile to linear order, we use (4) and (5) to
and compute the velocity flow field of the passive fluid and the velocity of
the active particle. A comparison of the density profiles and the velo-
ðρ0 + ρð0; 0ÞÞf uξ city flows with the simulation results is shown in Fig. 6A–F. The
DðuÞ = e ð9Þ
2πv0 ξΓ agreement is satisfying; in particular the demonstration in Fig. 6C that
the excess density profile behind the moving active particle decays as
The decay length ξ is given by, the advertised power-law. One may, in principle, improve on the linear
theory by setting up a diagrammatic perturbation expansion. How-
2Bρ20 ever, since the linear theory compares well with the numerical simu-
ξ = : ð10Þ
Γv0 lation of (17) and with the “exact” numerical solution of the nonlinear
equation (7) in d = 1 (next section), we do not take this up here.
Since the fixed direction of motility breaks rotational invariance, it is
natural to expect an anisotropy in the density profile. However, what Accuracy of linear theory—comparison with “exact” numerical
comes as a surprise is that moving density profile breaks fore-aft analysis of nonlinear equation in d = 1
symmetry. This is most apparent when we set w = 0+, and use the A linear analysis in d = 1, shows that the density profile is,
asymptotic expansion53
ðρ0 + ρð0; 0ÞÞf uξ
rffiffiffiffiffiffiffiffi ρðuÞ = e for u > 0
π z 1 v0 ξΓ ð13Þ
K 0 ðzÞ ~ e 1 + ... ð11Þ
2∣z∣ 8z =0 for u < 0;
and where the length scale is given by ξ = Bρ20 =Γv0 (Supplementary Note 5).
The density piles up in front of the active particle and decays expo-
rffiffiffiffiffiffiffiffi
π z 3 nentially ahead of it, while behind the active particle there is no wake.
K 1 ðzÞ ~ e 1+ + ... ð12Þ
2∣z∣ 8z We now check to see how this calculated profile compares with an
exact numerical solution of the nonlinear equation (7). The accurate
for large z. We see immediately that in the moving frame, the density numerical solution of this nonlinear PDE requires some care due to
profile in front to the motile particle is piled up and decays exponen- shock forming tendencies in the convective term (Supplementary
tially over a scale ξ from the pile up. The larger the active force f, the Note 6). The result for the density profile of the travelling pulse is
smaller is ξ, implying a sharper pile up. However, behind the active shown in Supplementary Fig. 5. The comparison with the linear theory
particle there is a long-range under-dense region, which decays as a is quite good, the absence of the wake is vividly apparent in the one-
power-law ∣u∣−3/2 (Supplementary Note 4). dimensional exact numerical solution (Supplementary Fig. 5).
Nature Communications | (2022)13:4533 6Article https://doi.org/10.1038/s41467-022-31984-z
Fig. 6 | Density profile and flow field surrounding a single active particle borne out by the simulations. The dark solid symbols show averages from multiple
moving through the passive medium. A Density profile (heat map) and velocity simulation runs, together with standard deviation. D–F The profiles of excess
flow field (arrows, scaled for better visualisation) of the medium from simulations density and velocity components in the moving frame along u at w/ξ = 0+ (red) and
(in the co-moving frame (u, w) scaled by the decay length ξ ≈ 14) at ϕ = 0.45, T = 0.1, w/ξ = 0− (blue). Insets are the results from the linearised hydrodynamic equations.
f = 2.0, and τ → ∞. B Corresponding density profile (heat map) and velocity flow field The dark solid symbols show averages from multiple simulation runs, together with
(arrows, scaled for better visualisation) of the medium obtained from the linearised standard deviation. The data presented here are averaged over time and 256
hydrodynamic theory. C The linearised theory (inset) predicts that the excess independent simulations and the errorbars, denoting the standard deviations, are
density of the passive medium is fore-aft asymmetric and shows an exponential estimated from this ensemble).
decay in front and an algebraic decay behind the moving active particle, which is
(Fig. 7A), i.e., X_ rel < 0, starting from the initial value, till it reaches
Two active particles moving through the compressible medium
Xrel ≈ 1.5064ξ, which is a stable fixed point of the dynamics. This
The fore-aft asymmetric long-range density wake around the motile
corresponds to a bound state of the two particles in this
particle has an unusual effect on the interactions between two or more
linearised theory, which appears to be consistent with the
motile particles. This is best illustrated by considering the dynamics of
simulations (Fig. 7C). The speed of approach of the particle 2 to
two motile particles in a simplifying geometry where both particles
particle 1 first increases slowly and then rapidly decreases to
move in the same direction with their separation vector being parallel
zero as the bound state is reached (Fig. 7C, inset). This
or perpendicular to the direction of motion.
nonreciprocal sensing2,3,6, is a consequence of the fore-aft
Let the trajectories of the two active particles be represented by
asymmetric wake and causes the trailing particle to catch up
R1(t), R2(t), in the limit of large persistence time, so that we can take the
with the leading one in finite time (Fig. 7B).
orientations n1 and n2 to be time independent. To OðδρÞ, these parti-
A dominant balance analysis of the equation for Xrel shows an
cles, individually, leave a time dependent anisotropic and fore-aft
early time scaling of the form Xrel ∝ ∣t − t*∣2/7 as the particles
asymmetric wake described by ρ(x, y, t), whose back-reaction on the
approach each other (Fig. 7C).
movement of the active particles themselves, is easily estimated
2. The separation vector between the particle 1 and the particle 2 is
along the ^y direction, perpendicular to their direction of
γR_ 1 = f n1 Cρ0 ∇ρ∣self Cρ0 ∇ρ∣1 2 ð14Þ motion (Fig. 7D).
1
Here again the centre of mass velocity X_ cm > 0 (and is the same
as the single particle speed, Fig. 7E), and the inter-particle
γR_ 2 = f n2 Cρ0 ∇ρ∣self Cρ0 ∇ρ∣2 ð15Þ
2
1
separation Yrel decreases in time (Fig. 7D), i.e., Y_ rel < 0, starting
from the initial value. This leads to the trajectories of particles 1
where i ← j denotes the effect of particle j on particle i. These equations
and 2 converging towards each other in a symmetrical
may be cast in terms of the relative coordinate Rrel = R1 − R2 and the
manner (Fig. 7D).
centre of mass Rcm = (R1 + R2)/2. For the two geometries under con-
An asymptotic analysis shows that the relative position Yrel ∝ ∣
sideration, we find the following (details in Supplementary Note 4).
t − t*∣1/2 as t → t* (Fig. 7F).
1. The separation vector between the leading particle 1 and the
trailing particle 2 is along the x^ direction, parallel to their direction The second case resembles the magnetic force between two
of motion (Fig. 7A). parallel wires carrying current in the same direction, and is a con-
We find that while the centre of mass velocity X_ cm > 0 (Fig. 7B sequence of the breaking of time reversal symmetry. Likewise, a pair of
inset), the inter-particle separation Xrel decreases in time active particles initially moving towards each other, will scatter off
Nature Communications | (2022)13:4533 7Article https://doi.org/10.1038/s41467-022-31984-z
Fig. 7 | Dynamics of two particles moving through the passive medium. A–C. stable fixed point at Xrel ≈ 1.5064ξ (inset), verified in the the numerical simulations
Two active particles moving along the x-axis (the direction of their active forces as a flattening of Xrel as t approaches t*. D–F Two active particles moving along the x-
whose magnitude f = 2, and large persistence time τ) with separation vectors par- axis with separation vectors perpendicular to the direction of motion (rest same as
allel to the direction of motion. A Time dependence of the positions of the pair of above). D Time dependence of y-positions of the active particles showing con-
active particles (after subtracting the measured single particle displacement Xt), _ vergent trajectories. E The x-component of centre of mass velocity is positive and
along x^. B The ratio of the particle velocities remains greater than one, indicating a constant, solid line is the prediction from theory. Inset shows y-component centre
speed up of the trailing particle towards the leading particle, a manifestation of of mass velocity is zero, indicating symmetric approach. F The two active particles
nonreciprocity. Inset: the centre of mass velocity X_ cm ðtÞ remains positive and con- approach each other in finite time t*. The solid line, a prediction from theory,
stant (solid line is the prediction from theory). C The two active particles approach suggests that the relative position has a scaling form Yrel ∝ ∣t − t*∣α, where α = 1/2
each other in finite time t*. The solid line, a prediction from theory, suggests a as t → t*. The data presented here are averaged over 256 independent simulations
scaling form Xrel ∝ ∣t − t*∣α, where α = 2/7. As the second particle approaches the first, and the errorbars, denoting the standard deviations, are estimated from this
it forms a bound state with the first particle, characterised by the existence of a ensemble.
(repel) and will slow down as they move away from each other. The (largest) density gradient ∇ ρ—this will lead to both taxis and phoresis,
scattering of active particles in other geometries can also be worked features that have been explored in refs. 3, 6 in other contexts. Such
out to this order. considerations lead to a simple physical version of sense-and-capture,
even in the absence of any kind of chemical sensing.
Discussion The success of our hydrodynamic analysis motivates us to go
In this paper, we have studied the dynamics of a dilute suspension of beyond the study of one and two active particles and look at many-
active Brownian particles moving through a dense compressible pas- body effects54. In ref. 15, we had seen how the minority component self-
sive fluid that dissipates momentum through friction. The dynamical propelled particles cluster on account of activity; the long-range
interplay between the active particles and the passive medium, not nonreciprocal interaction observed here, will translate to a new kind of
only results in a remodelling of the passive medium, but also in a back- nonreciprocal motility induced clustering42 of active particles mediated
reaction on the movement of the active particles themselves. Such by the passive medium. This and its relationship with the anisotropic
active ploughers show a jamming transition at fixed density of the Burgers equation with coloured noise52 will be taken up later.
medium. In the unjammed phase, a moving active plougher generates
a fore-aft asymmetric density wake, which is the source of the long- Methods
range nonreciprocal interaction between moving active particles Agent-based simulations
mediated dynamically through the passive compressible medium. This We work with a modified binary mixture (see Supplementary Note 1 for
emergent nonreciprocal interaction is a consequence of a dynamical details), at a fixed area fraction, ϕ, where particles interact via a
phase transition to a state with finite current. This leads to a non- potential,
reciprocal sensing wherein a trailing particle senses and catches up
!12 !2 !4
with a leading particle moving ahead of it. Further, the movement of σ ij σ ij σ ij
the active particle leaves a dynamical trace on the responsive medium, V ij = 4ϵij + v 0 + v2 + v4 ð16Þ
r ij r ij r ij
these effects of nonreciprocity are more indelibly manifest in the
vicinity of the jamming-unjamming transition.
We recall that in (6) we have assumed that the direction of the if rij < rc,ij or 0 otherwise. We fix the energy and length scales to be in
propulsion force is set by some mechanism internal to the active the units of ϵAA and σAA, respectively.
particle and therefore independent of the passive medium. To Of these a small fraction of particles ϕa is made active—their
experience the full scope of nonreciprocal effects possible here, one dynamics is described by active Brownian particles (ABP) (see Sup-
needs to extend the hydrodynamic equations (5), (6), to include an plementary Note 1 for details) immersed in a background of passive
active torque that drives ni to align along the direction of the smallest particles. All particles are subject to a thermal noise ϑ of zero mean and
Nature Communications | (2022)13:4533 8Article https://doi.org/10.1038/s41467-022-31984-z
variance equal to 2γT (setting kB = 1), obeying FDT. The subset i 2 A of 5. Marchetti, M. C. et al. Hydrodynamics of soft active matter. Rev.
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