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HomeScienceVol. 371, No. 6524Low rattling: A predictive principle for
self-organization in active collectives
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LOW RATTLING: A PREDICTIVE PRINCIPLE FOR SELF-ORGANIZATION IN ACTIVE COLLECTIVES

Pavel Chvykov https://orcid.org/0000-0001-6850-5994, Thomas A. Berrueta
https://orcid.org/0000-0002-3781-0934, [...] , Akash Vardhan
https://orcid.org/0000-0002-6392-3086, William Savoie
https://orcid.org/0000-0003-4638-3599, [...] , Alexander Samland
https://orcid.org/0000-0001-7788-6039, Todd D. Murphey
https://orcid.org/0000-0003-2262-8176, Kurt Wiesenfeld
https://orcid.org/0000-0002-7758-1005, Daniel I. Goldman
https://orcid.org/0000-0002-6954-9857, and Jeremy L. England
https://orcid.org/0000-0002-3331-2583 j@englandlab.com+6 authors +4 authors
fewerAuthors Info & Affiliations
Science
1 Jan 2021
Vol 371, Issue 6524
pp. 90-95
DOI: 10.1126/science.abc6182

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 * Contents
    * Shake, rattle, and help each other along
    * Abstract
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SHAKE, RATTLE, AND HELP EACH OTHER ALONG

In classical statistical mechanics, the deterministic dynamics of a many-body
system are replaced by a probabilistic description. Chvykov et al. work toward a
similar description for the nonequilibrium self-organization of collectives of
active particles. In these systems, continuously input energy drives localized
fluctuations, but larger-scale ordering can emerge, such as in the flight of a
flock of birds. A key concept in their theory is the importance of rattling,
whereby ordered patterns emerge through local collisions between neighbors at
specific frequencies. The authors demonstrate this behavior using a set of
flapping robots and produce related simulations of the robot behavior.
Science, this issue p. 90


ABSTRACT

Self-organization is frequently observed in active collectives as varied as ant
rafts and molecular motor assemblies. General principles describing
self-organization away from equilibrium have been challenging to identify. We
offer a unifying framework that models the behavior of complex systems as
largely random while capturing their configuration-dependent response to
external forcing. This allows derivation of a Boltzmann-like principle for
understanding and manipulating driven self-organization. We validate our
predictions experimentally, with the use of shape-changing robotic active
matter, and outline a methodology for controlling collective behavior. Our
findings highlight how emergent order depends sensitively on the matching
between external patterns of forcing and internal dynamical response properties,
pointing toward future approaches for the design and control of active particle
mixtures and metamaterials.

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Self-organization in nature is surprising because getting a large group of
separate particles to act in an organized way is often difficult. By definition,
arrangements of matter we call “orderly” are special, making up a tiny minority
of all allowed configurations. For example, we find each unique, symmetrical
shape of a snowflake visually striking, unlike any randomly rearranged clump of
the same water molecules. Thus, any theory of emergent order in many-particle
collectives must explain how a small subset of configurations are spontaneously
selected among the vast set of disorganized arrangements.
Spontaneous many-body order is well understood in thermal equilibrium cases such
as crystalline solids or DNA origami (1), where the assembling matter is allowed
to sit unperturbed for a long time at constant temperature T. The statistical
mechanical approach proceeds by approximating the complex deterministic dynamics
of the particles with a probabilistic “molecular chaos,” positing that the law
of conservation of energy governs otherwise random behavior (2). What follows is
the Boltzmann distribution for the steady-state probabilities, pss(q) ∝
exp[–E(q)/T], which shows that the degree to which special configurations q of
low energy E(q) have a high probability pss(q) in the long term depends on the
amplitude of the thermal noise. Orderly configurations can assemble and remain
stable, so long as interparticle attractions are strong enough to overcome the
randomizing effects of thermal fluctuations.
However, there are also many examples of emergent order outside of thermal
equilibrium. These include “random organization” in sheared colloids (3), phase
separation in multitemperature particle mixtures (4), and dynamic vortices in
protein filaments (5). A variety of ordered behaviors arise far from equilibrium
that cannot be explained in terms of simple interparticle attraction or energy
gradients (6–9).
In all of these examples, the energy flux from external sources allows different
system configurations to experience fluctuations of different magnitude (10,
11). We suggest that the emergence of such configuration-dependent fluctuations,
which cannot happen in equilibrium, may be key to understanding many
nonequilibrium self-organization phenomena. In particular, we introduce a
measure of driving-induced random fluctuations, which we term rattling R(q), and
argue that it could play a role in many far-from-equilibrium systems similar to
the role of energy in equilibrium. We test this in a number of systems,
including a flexible active matter system of simple robots we call “smarticles”
(smart active particles) (12) as a convenient test platform (see movie S1)
inspired by similar robo-physical emulators of collective behavior (13–15).
Despite their purely repulsive inter-robot interactions, we find that smarticles
spontaneously self-organize into collective “dances,” whose shape and motions
are matched to the temporal pattern of external driving forces (movies S2 and
S3). This platform and others (16–18), including the nonequilibrium ordering
examples mentioned above, all exhibit low-rattling ordered behaviors that echo
low-energy structures emergent at equilibrium. We thus motivate and test a
predictive theory based on rattling that may explain a broad class of
nonequilibrium ordering phenomena.
In devising our approach, we take inspiration from the phenomenon of
thermophoresis, which is the simplest example of purely nonequilibrium
self-organization. Thermophoresis is characterized by the diffusion of colloidal
particles from hot regions to cold regions (19). If noninteracting particles in
a viscous fluid are subject to a temperature T(q) that varies over position q,
their resulting density in the steady-state pss(q) will concentrate in the
regions of low temperature. Particles diffuse to regions where thermal noise is
weaker, and they become trapped there. With the diffusivity landscape set by
thermal noise locally according to the fluctuation-dissipation relation D(q) ∝
T(q) (20), the steady-state diffusion equation ∇2[D(q)pss(q)] = 0 is satisfied
by the probability density pss(q) ∝ 1/D(q). Hence, a low-entropy, “ordered”
arrangement of particles can be stable when the diffusivity landscape has a few
locations q that are strongly selected by their extremely low D(q) values.
We seek to extend this intuition to explain nonequilibrium self-organization
more broadly. However, a straightforward mathematical extension of the idea
encounters challenges in only slightly more complicated scenarios. For an
arbitrary diffusion tensor landscape D(q), in which diffusivity can depend on
the direction of motion, one can no longer find general solutions for the steady
state. Moreover, the steady-state density pss(q) at configuration q may depend
on the diffusivity D(q˜) at arbitrarily distant configurations q˜. Nonetheless,
we suggest that for most typical diffusion landscapes, the local magnitude of
fluctuations |D(q)| should statistically bias pss(q) and hence should be
approximately predictive of it. This insight, which is central to our theory, is
illustrated to hold numerically in Fig. 1A for a randomly constructed
two-dimensional anisotropic landscape, and in fig. S3 for higher dimensions.
Although contrived counterexamples that break the relationship may be
constructed, they require specific fine-tuning (see fig. S4).
Fig. 1 Rattling R is predictive of steady-state likelihood across
far-from-equilibrium systems.
(A) Inhomogeneous anisotropic diffusion in two dimensions, where the
steady-state density pss(q) is seen to be approximately given by the magnitude
of local fluctuations log|D(q)| ∝ R(q) (where |D| is the determinant of the
diffusion tensor). (B) A random walk on a large random graph (1000 states),
where Pss, the probability at a state, is approximately given by E, that state’s
exit rate. (C) An active matter system of shape-changing agents: an enclosed
ensemble of 15 “smarticles” in simulation. (D) Experimental realization of
similar agents with an enclosed three-robot smarticle ensemble. The middle row
shows that relaxation to the steady state of a uniform initial distribution is
accompanied by monotonic decay in the average rattling value in all cases,
analogous to free energy in equilibrium systems. The bottom row shows the
validity of the nonequilibrium Boltzmann-like principle in Eq. 3, where the
black lines in (A), (B), and (C) illustrate the theoretical correlation slope
for a sufficiently large and complex system (see supplementary materials). The
mesoscopic regime in (D) provides the most stringent test of rattling theory
(where we observe deviations in γ from 1), while also exhibiting global
self-organization. In the middle row, time units are arbitrary in (A) and (B);
time is in seconds in (C) and (D), where the drive period is 2 s.
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The key assumption underlying our approach is that the complex system dynamics
are so messy that only the amplitude of local drive-induced fluctuations governs
the otherwise random behavior—an assumption inspired by molecular chaos at
equilibrium. We expect this to apply when the system dynamics are so complex,
nonlinear, and high-dimensional that no global symmetry or constraint can be
found for its simplification. Although one cannot predict a configuration’s
nonequilibrium steady-state probability from its local properties in the general
case (21, 22), the feat becomes achievable in practice for “messy” systems. To
illustrate this point explicitly, we consider a discrete dynamical system with
random transition rates between a large number of states. Here, we can show
analytically that the net rate at which we exit any given state predicts its
long-term probability approximately, even though the exact result requires
global system knowledge (see Fig. 1B and supplementary materials for
derivation). This result may be related to the above discussion of
thermophoresis by noting that the discrete state exit rates are determined by
the continuum diffusivity if our dynamics are built by discretizing the domain
of a diffusion process.
To formulate our random dynamics assumption explicitly, we represent the complex
system evolution as a trajectory in time q(t), where the configuration vector q
captures the properties of the entire many-particle system. Our messiness
assumption amounts to approximating the full complex dynamics between two points
q(t) and q(t + δt) by a random diffusion process. To this end, we take the
amplitude of the noise fluctuations D(q) to locally reflect the amplitude of the
true configuration dynamics: |q(t + δt) – q(t)|2 ∝ D(q)δt for short rollouts q(t
→ t + δt) (i.e., samples of system trajectories) of duration δt initialized in
configuration q(t) = q (see supplementary materials for details). Through this
approximation, our dynamics are effectively reduced to diffusion in q-space,
which then allows us to locally estimate the steady-state probability of system
configurations from D(q) as in thermophoresis. Hence, the global steady-state
distribution may be predicted from the properties of short-time, local system
rollouts.
For rare orderly configurations to be strongly selected in a messy dynamical
system, the landscape of local fluctuations must vary in magnitude over a large
range of values. Whereas in thermophoresis these fluctuations are directly
imposed by an external temperature profile, in driven dynamical systems the
range of magnitudes results from the way a given pattern of driving can have a
different effect on different system configurations. The D(q) landscape is
emergent from the interplay between the pattern of driving and the library of
possible q-dependent system response properties. In practice, we observe that
the amplitudes of system responses to driving do often vary over several orders
of magnitude (Fig. 1). We see this phenomenology in many well-known examples of
active matter self-organization (3, 11, 23). For example, the crystals that form
in suspensions of self-propelled colloids in (24) may be seen as the collective
configurations that respond least diffusively to driving by precisely balancing
the propulsive forces among individual particles. This illustrates how the
low-D(q) configurations are selected in the steady state by an exceptional
matching of their response properties to the way the system is driven.
We apply these ideas in real complex driven systems whose response to driving we
cannot predict analytically, such as our robotic swarm of smarticles. In this
case, we require an estimator for the local value of D(q) based on observations
of short rollouts of system behavior. The estimator of the local diffusion
tensor that we choose here is the covariance matrix
C(q)=cov[v˜q,v˜q]
(1)
(25), where v˜q is seen as a random variable with samples drawn from
{(q˜(t)−q˜(0))/t}q¯(0)=q at various time points t along one or several short
system trajectories q˜(t) rolled out from q˜(0)=q. We assume these rollouts
q˜(t) to be long enough to capture fluctuations in the configuration variables
under the influence of a drive, but short enough to have q˜(t) stay near q (see
supplementary materials for details).
Although the covariance matrix reflects the amplitude of local fluctuations, we
are instead interested in a measure of their disorder if we want to estimate the
effective diffusivity. This follows from the observation that high-amplitude
ordered oscillations do not contribute to the rate of stochastic diffusion (10).
We suggest that the degree of disorder of fluctuations may be captured by the
entropy of the distribution of v˜q vectors, which is how we define rattling
R(q). Physically, vectors v˜q capture the statistics of the force fluctuations
experienced in configuration q, and so rattling measures the disorder in the
system’s driven response properties at that point. By approximating the
distribution of v˜q as Gaussian, we can express its entropy (up to a constant
offset) simply in terms of C(q) as
R(q)=12logdetC(q)
(2)
With this definition, we generalize the thermophoretic expression for the
steady-state density pss(q) ∝ 1/D(q) and express it in a Boltzmann-like form:
pss(q)∝exp[−γR(q)]
(3)
where γ is a system-specific constant of order 1 (see supplementary materials
for derivations). We note that when energy varies on the same scale as rattling,
the interaction between the two landscapes can generate strong steady-state
currents and may break this relation (10). Thus, rattling enables us to predict
the long-term global steady-state distribution based on empirical measurements
of short-term local system behavior, which suggests that probability density
accumulates over time in low-rattling configurations.
We study the collective behavior of a simple ensemble of smarticles, aligning
ourselves within the tradition of using robotic systems as flexible, physical
emulators for self-organizing natural systems (13–16). Each smarticle (Fig. 2A)
is composed of three 5.2-cm links, with two hinges actuated by motors programmed
to follow a driving pattern specified by a microcontroller. When a smarticle
sits on a flat surface, its arms do not touch the ground, so an individual robot
cannot move. However, a group of them can achieve complex motion by pushing and
pulling each other (movie S1) (26). The relative coordinates of the middle link
of each robot in the ensemble (x, y, θ) may be thought of as the internal system
configurations that dynamically respond to an externally determined driving
force arising from the time variation of arm angles (α1, α2) (27).
Fig. 2 Self-organization in a smarticle robotic ensemble.
(A) Front, back, and top views of a single smarticle. Of its five degrees of
freedom, we consider the time-varying arm angles (α1, α2) as “external” driving,
because these are controlled by a preprogrammed microcontroller, whereas the
robot coordinates (x, y, θ) are seen as an “internal” system configuration,
because these respond interdependently to the arms. (B) An example of a periodic
arm motion pattern. (C) Top view of three smarticles confined in a fixed ring,
all programmed to synchronously execute the driving pattern shown in (B). The
video frames, aligned on the time axis of (B), show one example of dynamically
ordered collective “dance” that can spontaneously emerge under this drive [see
(E) and movie S3 for others]. (D) Simulation video showing agreement with
experiment in (C). We color-code simulated states periodically in time and
overlay them for three periods to illustrate the dynamical order over time. (E)
The system’s configuration space, built from nonlinear functions of the three
robots’ body coordinates (x, y, θ). The steady-state distribution (blue)
illustrates the few ordered configurations that are spontaneously selected by
the driving out of all accessible system states (orange). Simulation data are
shown; see fig. S5B for experimental data and fig. S1 for details of how the
configuration space coordinates (q1, q2, q3) in (E) are constructed from the 3 ×
(x, y, θ) coordinates.
Expand for more
Open in viewer
This robotic active matter system offers substantial flexibility in choosing the
programmed patterns of driving as well as the properties of internal system
dynamics (friction coefficients, weights, etc.). Additionally, the smarticle
system has a flat potential energy landscape, allowing one to focus on the
contributions of the drive-induced fluctuations to the collective behavior,
which makes our findings broadly applicable to other strongly driven systems.
When the smarticles are within contact range (as ensured by a confining ring;
Fig. 1D), the forces experienced throughout the collective for a given pattern
of arm movement are an emergent function of all system coordinates. This
configuration-dependent forcing gives rise to varying rattling values, which we
refer to as the “rattling landscape,” and which we see to be a hallmark property
in many far-from-equilibrium examples. The rattling landscape then leads to some
system configurations being dynamically selected over others and allowing for
self-organization, just as the diffusivity landscape does in thermophoresis.
Finally, the combined effects of impulsive inter-robot collisions, nonlinear
boundary interactions, and static friction lead to a large degree of
quasi-random motion (26), making this a promising candidate system for exploring
our theory.
Reasoning that our fundamental assumption of quasi-random configuration dynamics
would be most valid in systems with many degrees of freedom, we also built a
simulation that would allow us to study the properties of larger smarticle
groups and explore different system parameters (fig. S9). In this regime, we
used simulations to gather enough data to sample the high-dimensional
probability distributions for our analysis. In a simulation of 15 smarticles, we
observed the tendency of the ensemble to reduce average rattling over time after
a random initialization. For this 45-dimensional system (x, y, θ for 15 robots),
the configuration-space dynamics are well approximated by diffusion, and so Eq.
3 holds, as seen in Fig. 1C. In addition, we noted the emergence of metastable
pockets of local order when groups of three or four nearby smarticles
self-organized into regular motion patterns for several drive cycles (movie S2).
A signature of such dynamical heterogeneity can be seen in the spectrum of the
covariance matrix C(q) from Eq. 1, as described in the supplementary materials
and fig. S10.
The transient appearance of dynamical order in subsets of smarticle collectives
raises the question of whether our rattling theory continues to hold for smaller
ensembles. For the remainder of this paper, we focus on ensembles of three
smarticles (as in Fig. 1D), which allows for exhaustive sampling of
configurations experimentally, as well as easier visualization of the
configuration space (as in Fig. 2E). Both in simulation and experiment, we found
that this regime exhibits a variety of low-rattling behaviors that manifest as
distinct, orderly collective “dances” (Fig. 2, C and D, and movie S3). Despite
its small size, this system is well described by rattling theory, as evidenced
by the empirical correlation between rattling and the steady-state likelihood of
configurations (Fig. 1D, bottom).
We consider self-organization as a consequence of a system’s landscape of
rattling values over configuration space. This rattling landscape is specific to
the particular drive forcing the system out of equilibrium, because different
drives will generally produce different dynamical responses in the same system
configuration. When the three-smarticle ensemble is driven (under the pattern in
Fig. 2B), the range of observed rattling values is so large that the
lowest-rattling configurations—and consequently those with the highest
likelihood—account for most of the steady-state probability mass. More than 99%
of probability accumulates in these spontaneously selected configurations, which
represent only 0.1% of all accessible system states (Fig. 2E). Moreover, in
these configurations the smarticles exhibit an orderly response to driving (Fig.
2, C and D, and movie S4). In practice, the ensemble spends most of its time in
or nearly in one of several distinct dances, with occasional interruptions by
stochastic flights from one such dynamical attractor to another (movie S5).
From the above observations, we can begin to understand self-organization in
driven collectives. In equilibrium, order arises when its entropic cost is
outweighed by the available reduction of energy. Analogously, a sufficiently
large reduction in rattling can lead to dynamical organization in a driven
system. Moreover, such a reduction can require matching between the system
dynamics and the drive pattern.
Through rattling theory we can predict how self-organized states are affected by
changes in the features of the drive. We expect the structure of the
self-organized dynamical attractors to be specific to the driving pattern, as
each drive induces its own rattling landscape. To test this, we programmed the
three smarticles with two distinct driving patterns (Fig. 3, A and B, top),
which we ran separately. The two resulting steady-state distributions, although
each is highly localized to a few configurations, are largely non-overlapping
(Fig. 3, A and B, bottom). This indicates that by tuning the drive pattern, it
may be possible to design the structure of the resulting steady state, and hence
to control the self-organized dynamics [see also (28–30)].
Fig. 3 Self-organized behaviors are fine-tuned to drive pattern.
(A and B) Changing the arm motion pattern slightly (top) affects which
configurations self-organize in the steady state (bottom, same 3D configuration
space as in Fig. 2E). (C) By mixing drives A and B as shown (top), we can
isolate only those configurations selected in both the steady states (circled in
purple; see movie S6), which follows as an analytical prediction of the theory.
(D) This prediction (Eq. 4) is quantitatively verified. All data shown are
experimental and are reproduced in simulation in fig. S7, along with derivations
in the supplementary materials.
Open in viewer
As a proof of principle for such control, we developed a methodology for
selecting particular steady-state behaviors by combining drives. By randomly
switching back and forth between drives A and B in Fig. 3, we define a compound
drive A+B (Fig. 3C and movie S6). We predicted that this drive would select only
those configurations common to both A and B steady states (Fig. 3, A and B,
bottom), because having low rattling under this mixed drive requires having low
rattling under both constituent drives. Our experiments confirmed this (Fig.
3C), and we were further able to quantitatively predict the probability that a
configuration would appear under the mixed drive on the basis of its likelihood
in each constituent steady state according to
1pssA+B∝1pssA+1pssB
(4)
as shown in Fig. 3D and fig. S7 (see supplementary materials for derivation).
This simple relationship suggests that by composing different drives in time,
one can single out desired configurations for the system steady state.
Moreover, we show that we can analytically predict and control the degree of
order in the system by tuning drive randomness (Fig. 4) as well as internal
system friction (movie S7, fig. S8, and supplementary materials). Because driven
self-organization arises when the system has access to a broad range of rattling
values, tuning it requires modulating the rattling of the most ordered behaviors
relative to the background high-rattling states.
Fig. 4 Tuning self-organization by modulating drive randomness.
Self-organization relies on the degree of predictability in its driving forces,
in a way that we can quantify and compute analytically. (A) As the drive becomes
less predictable (left to right in all panels), low-rattling configurations
gradually disappear. (B) The corresponding steady states, reflecting the
low-rattling regions of (A), become accordingly more diffuse. [(A) and (B) show
simulation data and use the same 3D configuration space as Fig. 2E]. (C) All
three correlations fall along the same line (blue, simulation; black,
experiment), verifying that our central predictive relation (Eq. 3) holds for
all drives here. The diminishing range of rattling values thus precludes strong
aggregation of probability, and with it self-organization. (D) Our theoretical
prediction (solid black line) indicating how the most likely configurations are
destabilized by drive randomness. Colored lines track the probability pss at 100
representative configurations q in simulation, and dashed black lines
analytically predict their trends. (movie S8; see supplementary materials for
derivation). Two specific configurations marked by pink and purple crosses are
tracked across analyses.
Open in viewer
We can directly manipulate the rattling landscape by modulating the entropy of
the drive pattern. This is done by introducing a probabilistic element to the
programmed arm motion. At each move, we introduce a probability of making a
random arm movement not included in the prescribed drive pattern. Increasing
this probability results in flattening the rattling landscape: Ordered states
experience an increase in rattling due to drive entropy, whereas states whose
rattling is already high do not (Fig. 4A). Correspondingly, the steady-state
distributions become progressively more diffuse (Fig. 4B), causing localized
pockets of order to give way to entropy and “melt” away—just as crystals might
in equilibrium physics [movie S8; see also (31)].
Even as the range of accessible rattling values in the system shrinks, the
predictive relation of Eq. 3 continues to hold (Fig. 4C), enabling quantitative
prediction of how self-organized configurations are destabilized. By calculating
the entropy of the drive pattern as we tune its randomness, we derive a lower
bound on rattling for the system. Thus, we can analytically predict how
steady-state probabilities change as a function of drive randomness, as shown in
Fig. 4D (up to normalization and γ; see supplementary materials for derivation).
This result confirms the simple intuition that more predictably patterned
driving forces offer greater opportunity for the system to find low-rattling
configurations and self-organize (see also fig. S6).
Our findings suggest that the complex dynamics of a driven collective of
nonlinearly interacting particles may give rise to a situation in which a new
kind of simplicity emerges. We have shown that when quasi-random transitions
among configurations dominate the dynamics, the steady-state likelihood can be
predicted from the entropy of local force fluctuations, which we refer to as
rattling. In what we term a “low-rattling selection principle,” configurations
are selected in the steady state according to their rattling values under a
given drive.
Low rattling provides the basis for self-organized dynamical order that is
specifically selected by the choice of driving pattern. We see analytically and
experimentally that the degree of order in the steady-state distribution
reflects the predictability of patterns in driving forces. Thus, driving
patterns with low entropy pick out fine-tuned configurations and dynamical
trajectories to stabilize. This makes it possible for one collective to exhibit
different modes of ordered motion depending on the fingerprint of the external
driving. These modes differ in their emergent collective properties, which
suggests “top-down” alternatives to control of active matter and metamaterial
design, where ensemble behaviors, rather than being microscopically engineered,
are dynamically self-selected by the choice of driving (30, 32).


ACKNOWLEDGMENTS

We thank P. Umbanhowar, H. Kedia, and J. Owen for helpful discussions. Funding:
Supported by ARO grant W911NF-18-1-0101 and James S. McDonnell Foundation
Scholar grant 220020476 (P.C. and J.L.E.); ARO MURI award W911NF-19-1-0233 and
NSF grant CBET-1637764 (T.A.B., A.S., and T.D.M.); NSF grants PoLS-0957659,
PHY-1205878, and DMR-1551095 and ARO grant W911NF-13-1-0347 (A.V., W.S., and
D.I.G.); and NSF grant PHY-1205878 (K.W.). Author contributions: P.C. derived
all theoretical results, performed simulations, data analysis, and contributed
to writing; T.A.B. performed all experiments in the main text and contributed to
writing and data analysis; A.V. and W.S. performed supplementary experiments;
A.S. aided in robot hardware and software fabrication; and J.L.E., D.I.G., K.W.,
and T.D.M. secured funding and provided guidance throughout. Competing
interests: The authors declare no competing interests. Data and materials
availability: All files needed for fabricating smarticles, as well as
representative data, can be found in (33).


SUPPLEMENTARY MATERIAL


SUMMARY

Materials and Methods
Supplementary Text
Figs. S1 to S10
References (34–46)
Movies S1 to S8


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Science
Volume 371 | Issue 6524
1 January 2021

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Copyright © 2021, American Association for the Advancement of Science.
https://www.sciencemag.org/about/science-licenses-journal-article-reuse
This is an article distributed under the terms of the Science Journals Default
License.

SUBMISSION HISTORY

Received: 6 May 2020
Accepted: 27 November 2020
Published in print: 1 January 2021

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ACKNOWLEDGMENTS

We thank P. Umbanhowar, H. Kedia, and J. Owen for helpful discussions. Funding:
Supported by ARO grant W911NF-18-1-0101 and James S. McDonnell Foundation
Scholar grant 220020476 (P.C. and J.L.E.); ARO MURI award W911NF-19-1-0233 and
NSF grant CBET-1637764 (T.A.B., A.S., and T.D.M.); NSF grants PoLS-0957659,
PHY-1205878, and DMR-1551095 and ARO grant W911NF-13-1-0347 (A.V., W.S., and
D.I.G.); and NSF grant PHY-1205878 (K.W.). Author contributions: P.C. derived
all theoretical results, performed simulations, data analysis, and contributed
to writing; T.A.B. performed all experiments in the main text and contributed to
writing and data analysis; A.V. and W.S. performed supplementary experiments;
A.S. aided in robot hardware and software fabrication; and J.L.E., D.I.G., K.W.,
and T.D.M. secured funding and provided guidance throughout. Competing
interests: The authors declare no competing interests. Data and materials
availability: All files needed for fabricating smarticles, as well as
representative data, can be found in (33).


AUTHORS

AFFILIATIONSEXPAND ALL

PAVEL CHVYKOV HTTPS://ORCID.ORG/0000-0001-6850-5994

Physics of Living Systems, Massachusetts Institute of Technology, Cambridge, MA
02139, USA.
View all articles by this author

THOMAS A. BERRUETA HTTPS://ORCID.ORG/0000-0002-3781-0934

Department of Mechanical Engineering, Northwestern University, Evanston, IL
60208, USA.
View all articles by this author

AKASH VARDHAN HTTPS://ORCID.ORG/0000-0002-6392-3086

School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA.
View all articles by this author

WILLIAM SAVOIE HTTPS://ORCID.ORG/0000-0003-4638-3599

School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA.
View all articles by this author

ALEXANDER SAMLAND HTTPS://ORCID.ORG/0000-0001-7788-6039

Department of Mechanical Engineering, Northwestern University, Evanston, IL
60208, USA.
View all articles by this author

TODD D. MURPHEY HTTPS://ORCID.ORG/0000-0003-2262-8176

Department of Mechanical Engineering, Northwestern University, Evanston, IL
60208, USA.
View all articles by this author

KURT WIESENFELD HTTPS://ORCID.ORG/0000-0002-7758-1005

School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA.
View all articles by this author

DANIEL I. GOLDMAN HTTPS://ORCID.ORG/0000-0002-6954-9857

School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA.
View all articles by this author

JEREMY L. ENGLAND* HTTPS://ORCID.ORG/0000-0002-3331-2583 J@ENGLANDLAB.COM

School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA.
GlaxoSmithKline AI/ML, 200 Cambridgepark Drive, Cambridge, MA 02140, USA.
View all articles by this author

FUNDING INFORMATION

National Science Foundation: CBET-1637764
National Science Foundation: CBET-1637764
National Science Foundation: CBET-1637764
National Science Foundation: PoLS-0957659
National Science Foundation: PoLS-0957659
National Science Foundation: PoLS-0957659
Army Research Office: W911NF-18-1-0101
National Science Foundation: DMR-1551095
National Science Foundation: PHY-1205878
Army Research Office: W911NF-19-1-0233
Army Research Office: W911NF-19-1-0233
National Science Foundation: DMR-1551095
National Science Foundation: PHY-1205878
National Science Foundation: PHY-1205878
Army Research Office: W911NF-18-1-0101
Army Research Office: W911NF-19-1-0233
National Science Foundation: DMR-1551095
National Science Foundation: PHY-1205878
James S. McDonnell Foundation: 220020476
Army Research Office: W911NF-13-1-0347
Army Research Office: W911NF-13-1-0347
James S. McDonnell Foundation: 220020476
Army Research Office: W911NF-13-1-0347

NOTES

*
Corresponding author. Email: j@englandlab.com


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Low rattling: A predictive principle for self-organization in active
collectives.Science371,90-95(2021).DOI:10.1126/science.abc6182

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 1.  * William Savoie,
     * Harry Tuazon,
     * Ishant Tiwari,
     * M. Saad Bhamla,
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     Amorphous entangled active matter, Soft Matter, 19, 10, (1952-1965),
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     Crossref
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     * Michael Levin,
     Synthetic morphology with agential materials, Nature Reviews
     Bioengineering, 1, 1, (46-59),
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     Crossref
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     * Lucio Mariniello,
     Unusual Mathematical Approaches Untangle Nervous Dynamics, Biomedicines,
     10, 10, (2581), (2022).https://doi.org/10.3390/biomedicines10102581
     Crossref
 4.  * G. Negro,
     * C. B. Caporusso,
     * P. Digregorio,
     * G. Gonnella,
     * A. Lamura,
     * A. Suma,
     Hydrodynamic effects on the liquid-hexatic transition of active colloids,
     The European Physical Journal E, 45, 9,
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 7.  * Jeremy Shen,
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     (6232-6239), (2022).https://doi.org/10.1109/ICRA46639.2022.9811965
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     * Jaquelin Dezha Peralta,
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     Crossref
 9.  * Stephen Freeland,
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     The Royal Society Interface, 19, 187,
     (2022).https://doi.org/10.1098/rsif.2021.0814
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 10. * Harry Tuazon,
     * Emily Kaufman,
     * Daniel I Goldman,
     * M Saad Bhamla,
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     Integrative and Comparative Biology, 62, 4, (890-896),
     (2022).https://doi.org/10.1093/icb/icac089
     Crossref
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MEDIA

FiguresMultimedia


FIGURES

Fig. 1 Rattling R is predictive of steady-state likelihood across
far-from-equilibrium systems.
(A) Inhomogeneous anisotropic diffusion in two dimensions, where the
steady-state density pss(q) is seen to be approximately given by the magnitude
of local fluctuations log|D(q)| ∝ R(q) (where |D| is the determinant of the
diffusion tensor). (B) A random walk on a large random graph (1000 states),
where Pss, the probability at a state, is approximately given by E, that state’s
exit rate. (C) An active matter system of shape-changing agents: an enclosed
ensemble of 15 “smarticles” in simulation. (D) Experimental realization of
similar agents with an enclosed three-robot smarticle ensemble. The middle row
shows that relaxation to the steady state of a uniform initial distribution is
accompanied by monotonic decay in the average rattling value in all cases,
analogous to free energy in equilibrium systems. The bottom row shows the
validity of the nonequilibrium Boltzmann-like principle in Eq. 3, where the
black lines in (A), (B), and (C) illustrate the theoretical correlation slope
for a sufficiently large and complex system (see supplementary materials). The
mesoscopic regime in (D) provides the most stringent test of rattling theory
(where we observe deviations in γ from 1), while also exhibiting global
self-organization. In the middle row, time units are arbitrary in (A) and (B);
time is in seconds in (C) and (D), where the drive period is 2 s.
GO TO FIGUREOPEN IN VIEWER
Fig. 2 Self-organization in a smarticle robotic ensemble.
(A) Front, back, and top views of a single smarticle. Of its five degrees of
freedom, we consider the time-varying arm angles (α1, α2) as “external” driving,
because these are controlled by a preprogrammed microcontroller, whereas the
robot coordinates (x, y, θ) are seen as an “internal” system configuration,
because these respond interdependently to the arms. (B) An example of a periodic
arm motion pattern. (C) Top view of three smarticles confined in a fixed ring,
all programmed to synchronously execute the driving pattern shown in (B). The
video frames, aligned on the time axis of (B), show one example of dynamically
ordered collective “dance” that can spontaneously emerge under this drive [see
(E) and movie S3 for others]. (D) Simulation video showing agreement with
experiment in (C). We color-code simulated states periodically in time and
overlay them for three periods to illustrate the dynamical order over time. (E)
The system’s configuration space, built from nonlinear functions of the three
robots’ body coordinates (x, y, θ). The steady-state distribution (blue)
illustrates the few ordered configurations that are spontaneously selected by
the driving out of all accessible system states (orange). Simulation data are
shown; see fig. S5B for experimental data and fig. S1 for details of how the
configuration space coordinates (q1, q2, q3) in (E) are constructed from the 3 ×
(x, y, θ) coordinates.
GO TO FIGUREOPEN IN VIEWER
Fig. 3 Self-organized behaviors are fine-tuned to drive pattern.
(A and B) Changing the arm motion pattern slightly (top) affects which
configurations self-organize in the steady state (bottom, same 3D configuration
space as in Fig. 2E). (C) By mixing drives A and B as shown (top), we can
isolate only those configurations selected in both the steady states (circled in
purple; see movie S6), which follows as an analytical prediction of the theory.
(D) This prediction (Eq. 4) is quantitatively verified. All data shown are
experimental and are reproduced in simulation in fig. S7, along with derivations
in the supplementary materials.
GO TO FIGUREOPEN IN VIEWER
Fig. 4 Tuning self-organization by modulating drive randomness.
Self-organization relies on the degree of predictability in its driving forces,
in a way that we can quantify and compute analytically. (A) As the drive becomes
less predictable (left to right in all panels), low-rattling configurations
gradually disappear. (B) The corresponding steady states, reflecting the
low-rattling regions of (A), become accordingly more diffuse. [(A) and (B) show
simulation data and use the same 3D configuration space as Fig. 2E]. (C) All
three correlations fall along the same line (blue, simulation; black,
experiment), verifying that our central predictive relation (Eq. 3) holds for
all drives here. The diminishing range of rattling values thus precludes strong
aggregation of probability, and with it self-organization. (D) Our theoretical
prediction (solid black line) indicating how the most likely configurations are
destabilized by drive randomness. Colored lines track the probability pss at 100
representative configurations q in simulation, and dashed black lines
analytically predict their trends. (movie S8; see supplementary materials for
derivation). Two specific configurations marked by pink and purple crosses are
tracked across analyses.
GO TO FIGUREOPEN IN VIEWER


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FiguresTables
View figure
Fig. 1
Fig. 1 Rattling R is predictive of steady-state likelihood across
far-from-equilibrium systems.
(A) Inhomogeneous anisotropic diffusion in two dimensions, where the
steady-state density pss(q) is seen to be approximately given by the magnitude
of local fluctuations log|D(q)| ∝ R(q) (where |D| is the determinant of the
diffusion tensor). (B) A random walk on a large random graph (1000 states),
where Pss, the probability at a state, is approximately given by E, that state’s
exit rate. (C) An active matter system of shape-changing agents: an enclosed
ensemble of 15 “smarticles” in simulation. (D) Experimental realization of
similar agents with an enclosed three-robot smarticle ensemble. The middle row
shows that relaxation to the steady state of a uniform initial distribution is
accompanied by monotonic decay in the average rattling value in all cases,
analogous to free energy in equilibrium systems. The bottom row shows the
validity of the nonequilibrium Boltzmann-like principle in Eq. 3, where the
black lines in (A), (B), and (C) illustrate the theoretical correlation slope
for a sufficiently large and complex system (see supplementary materials). The
mesoscopic regime in (D) provides the most stringent test of rattling theory
(where we observe deviations in γ from 1), while also exhibiting global
self-organization. In the middle row, time units are arbitrary in (A) and (B);
time is in seconds in (C) and (D), where the drive period is 2 s.
View figure
Fig. 2
Fig. 2 Self-organization in a smarticle robotic ensemble.
(A) Front, back, and top views of a single smarticle. Of its five degrees of
freedom, we consider the time-varying arm angles (α1, α2) as “external” driving,
because these are controlled by a preprogrammed microcontroller, whereas the
robot coordinates (x, y, θ) are seen as an “internal” system configuration,
because these respond interdependently to the arms. (B) An example of a periodic
arm motion pattern. (C) Top view of three smarticles confined in a fixed ring,
all programmed to synchronously execute the driving pattern shown in (B). The
video frames, aligned on the time axis of (B), show one example of dynamically
ordered collective “dance” that can spontaneously emerge under this drive [see
(E) and movie S3 for others]. (D) Simulation video showing agreement with
experiment in (C). We color-code simulated states periodically in time and
overlay them for three periods to illustrate the dynamical order over time. (E)
The system’s configuration space, built from nonlinear functions of the three
robots’ body coordinates (x, y, θ). The steady-state distribution (blue)
illustrates the few ordered configurations that are spontaneously selected by
the driving out of all accessible system states (orange). Simulation data are
shown; see fig. S5B for experimental data and fig. S1 for details of how the
configuration space coordinates (q1, q2, q3) in (E) are constructed from the 3 ×
(x, y, θ) coordinates.
View figure
Fig. 3
Fig. 3 Self-organized behaviors are fine-tuned to drive pattern.
(A and B) Changing the arm motion pattern slightly (top) affects which
configurations self-organize in the steady state (bottom, same 3D configuration
space as in Fig. 2E). (C) By mixing drives A and B as shown (top), we can
isolate only those configurations selected in both the steady states (circled in
purple; see movie S6), which follows as an analytical prediction of the theory.
(D) This prediction (Eq. 4) is quantitatively verified. All data shown are
experimental and are reproduced in simulation in fig. S7, along with derivations
in the supplementary materials.
View figure
Fig. 4
Fig. 4 Tuning self-organization by modulating drive randomness.
Self-organization relies on the degree of predictability in its driving forces,
in a way that we can quantify and compute analytically. (A) As the drive becomes
less predictable (left to right in all panels), low-rattling configurations
gradually disappear. (B) The corresponding steady states, reflecting the
low-rattling regions of (A), become accordingly more diffuse. [(A) and (B) show
simulation data and use the same 3D configuration space as Fig. 2E]. (C) All
three correlations fall along the same line (blue, simulation; black,
experiment), verifying that our central predictive relation (Eq. 3) holds for
all drives here. The diminishing range of rattling values thus precludes strong
aggregation of probability, and with it self-organization. (D) Our theoretical
prediction (solid black line) indicating how the most likely configurations are
destabilized by drive randomness. Colored lines track the probability pss at 100
representative configurations q in simulation, and dashed black lines
analytically predict their trends. (movie S8; see supplementary materials for
derivation). Two specific configurations marked by pink and purple crosses are
tracked across analyses.

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