Two-dimensional nonequilibrium nematic steady states, as found in agitated
granular-rod monolayers or films of orientable amoeboid cells, were predicted
[Europhys. Lett. {\bf 62} (2003) 196] to have giant number fluctuations, with
standard deviation proportional to the mean. We show numerically that the
steady state of such systems is {\em macroscopically phase-separated}, yet
dominated by fluctuations, as in the Das-Barma model [PRL {\bf 85} (2000)
1602]. We suggest experimental tests of our findings in granular and
living-cell systems.Comment: 4 pages, 6 .eps figures, accepted for publication in PRL 3 Aug 0

Mishra, Shradha

Ramaswamy, Sriram

English

arXiv

Two-dimensional nonequilibrium nematic steady states, as found in agitated granular-rod monolayers or films of orientable amoeboid cells, were predicted [Europhys. Lett. 62 (2003) 196] to have giant number fluctuations, with standard deviation proportional to the mean. We show numerically that the steady state of such systems is macroscopically phase separated, yet dominated by fluctuations, as in the Das-Barma model [PRL 85 (2000) 1602]. We suggest experimental tests of our findings in granular and living-cell systems

Publications of the IAS Fellows

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Active nematics are intrinsically phase-separated
Shradha Mishra and Sriram Ramaswamy∗†
Centre for Condensed Matter Theory, Department of Physics,
Indian Institute of Science, Bangalore 560 012 INDIA
(Dated: accepted for publication in Phys Rev Lett 3 Aug 06)
Two-dimensional nonequilibrium nematic steady states, as found in agitated granular-rod mono-
layers or films of orientable amoeboid cells, were predicted [Europhys. Lett. 62 (2003) 196] to have
giant number fluctuations, with standard deviation proportional to the mean. We show numerically
that the steady state of such systems is macroscopically phase-separated, yet dominated by fluctu-
ations, as in the Das-Barma model [PRL 85 (2000) 1602]. We suggest experimental tests of our
findings in granular and living-cell systems.
PACS numbers: 05.70.Ln Nonequilibrium and irreversible thermodynamics. 87.18.Ed Aggregation and
other collective behavior of motile cells. 45.70.-n Granular systems.
The ordering or “flocking” [1, 2, 3] of self-propelled
particles obeys laws strikingly different from those
governing thermal equilibrium systems of the same
spatial symmetry. Even in two dimensions, the ve-
locities of particles in such flocks show true long-
range order [1, 2], despite the spontaneous breaking
of continuous rotational invariance. Density fluctua-
tions in the ordered phase are anomalously large [2],
and the onset of the ordered phase is discontinuous
[4]. The ultimate origin of these nonequilibrium phe-
nomena is that the order parameter is not simply an
orientation but a macroscopic velocity. It is thus in-
triguing that even the nematic phase of a collection
of self-driven particles, which is apolar and hence has
zero macroscopic velocity, shows [5, 6] giant number
fluctuations [7], as a result of the manner in which
orientational fluctuations drive mass currents. This
Letter takes a closer look at these fluctuations and
shows that they offer a physical realisation of the
remarkable nonequilibrium phenomenon known as
fluctuation-dominated phase separation [8], hitherto
a theoretical curiosity.
Before presenting our results, we make precise the
term active nematic. An active particle extracts en-
ergy from sources in the ambient medium or an in-
ternal fuel tank, dissipates it by the cyclical motion
of an internal “motor” coordinate, and moves as a
consequence. For the anisotropic particles that con-
cern us here, the direction of motion is determined
predominantly by the orientation. Our definition
encompasses self-propelled organisms, living cells,
molecular motors, and macroscopic rods on a ver-
tically vibrated substrate (where the tilt of the rod
∗Also with CMTU, JNCASR, Bangalore 560064, India
†Electronic address: sriram@physics.iisc.ernet.in
serves as the motor coordinate). An active nematic
is a collection of such particles with axes on average
spontaneously aligned in a direction nˆ, with invari-
ance under nˆ→ −nˆ. We know of two realisations of
active nematics: collections of living amoeboid cells
[10] and granular-rod monolayers [11, 12].
100 101 102 103 104
100
101
102
103
104
<N> average N
∆ N
L=256
L=128
x
x0.5
FIG. 1: Number standard deviation ∆N scales roughly
as the mean N¯ , for system sizes L = 128, 256
We study active nematics in a simple numerical
model described in detail below. Our results confirm
(see Fig. 1) the giant number fluctuations (standard
deviation ∝ mean) [6] predicted by the linearised
analysis of [5], but are far richer: (i) A statisti-
cally uniform initial distribution of particles, on a
well-ordered nematic background, undergoes a del-
icate “fluctuation-dominated” [8] phase separation,
where the system explores many statistically sim-
ilar segregated configurations. (ii) The equal-time
two-point density correlator C(r), Fig. 5, shows a
collapse when plotted as a function of r/L(t), where
L(t) is the location of the first zero-crossing of C(r),
with a cusp at small r/L(t) signalling a departure
from Porod’s Law, i.e., the absence of sharp inter-
20 50 100 150 200 250 300
5
10
15
20
25
30
35
L (system size)
st
ea
dy
 s
ta
te
 L
(t)
100 105
100
101
102
t (time)
L(t
)
L=128
0.4t0.49
t1/3
FIG. 2: Coarsening length L(t) saturates to a value pro-
portional to system size. Inset: L(t) consistent with t1/2
despite conservation law.
faces. (iii) We confirm that the large density in-
homogeneities are indeed best thought of as phase
separation, by showing (1) that the saturation value
of L(t) is proportional to the linear size of the sys-
tem and (2) that the phase-separation order param-
eter – the time-averaged first spatial Fourier compo-
nent of the particle density – approaches a nonzero
value in the limit of large system size (Fig. 6). (iv)
L(t) grows clearly faster than the t1/3 for normal
conserved-order-parameter coarsening (see inset to
Fig. 2), and is consistent with t1/2 as expected by
analogy to [8]. Below we show how these results were
obtained, discuss them in detail, explain the analogy
to the work of [8], and suggest experimental tests for
these striking phenomena.
We begin with some background information. An
apolar, uniaxial, compressible nematic liquid crystal
is described by director and number-density fields
nˆ(r) and c(r), with fluctuations δn(r) and δc(r)
about their uniform mean values nˆ0 and c0. Let
us review first what happens at thermal equilibrium.
The system is then governed by an extended Frank
[13] free-energy F [nˆ, c] = (1/2)
∫
ddr[K(∇nˆ)2 +
A(δc)2/c0 + C1nˆ · ∇c∇ · nˆ + C2nˆ × ∇c · ∇ × nˆ],
where K is an elastic tensor, A the compressional
modulus at constant orientation, and C1,2 couple
orientation and density in the simplest symmetry-
allowed fashion. Equipartition applied to F implies
that the static structure factor Sq ≡
∫
ddr exp(−iq ·
r)〈δc(0)δc(r)〉/c0 is finite for q → 0; i.e., the mean
N¯ and standard deviation ∆N of the number N of
particles obey ∆N ∝
√
N¯ at equilibrium even when
C1, C2 6= 0.
An active nematic is a steady state away from ther-
mal equilibrium, in which not only dynamic correla-
tors but equal-time quantities like Sq or ∆N as well
must be inferred from equations of motion for nˆ and
c. As shown in [5], the equation of motion for nˆ is
qualitatively the same as for equilibrium nematics.
The feature [5] that distinguishes active nematics
crucially from their equilibrium counterparts is that
the current j in the continuity equation ∂tc = −∇ · j
for the density has a contribution ∝ ∇ · c(nˆnˆ) [14].
This term, which is ruled out at thermal equilib-
rium, has a simple, physically appealing origin: spa-
tial variation in the director field nˆ defines a curve;
the normal to this curve defines a local vectorial
asymmetry; for a driven system, such an asymme-
try implies a current [15]. If nˆ = (cos θ, sin θ) then
inhomogeneities in θ give a curvature-induced cur-
rent j = (jx, jz) ∝ (∂θ/∂z, ∂θ/∂x) (see Fig. 3),
analogous to [8] where particles slide with velocity
∝ ∇h on a fluctuating interface with height field
h, except that our current is not a gradient. the
Since nematic order is a spontaneous breaking of
rotation-invariance, large fluctuations in θ at long
wavelengths are expected to be present in abun-
dance, and to decay slowly, in any nematic, equi-
librium or otherwise. We showed in the previous
paragraph that the effect of these broken-symmetry
modes on the density field was benign in an equi-
librium nematic. In an active nematic, however,
the same orientational fluctuations, because of the
curvature-induced current we just mentioned, will
affect the density fluctuations substantially. A lin-
earised small-fluctuations analysis [5] showed that
they to lead to giant fluctuations in the number of
particles: ∆N/
√
N¯ ∝ N¯1/d in d dimensions, i.e.,
∆N ∝ N¯ for d = 2.
Such large fluctuations prompt the suspicion that an
analysis beyond Gaussian fluctuations would reveal
that the system is in fact phase-separated, as in [8].
There are two issues here: (i) whether the nonequi-
librium coupling mentioned above inevitably arises
in an active nematic; and (ii) whether it leads to
phase separation. Ref. [6] effectively answers the
first question in the affirmative; we focus on the sec-
ond.
We find it convenient to separate the density and
orientation degrees of freedom, and employ a dis-
crete model of lattice-gas particles coupled to an an-
gle field, incorporating explicitly the the nonequi-
librium curvature-induced current j ∝ (∂zθ, ∂xθ)
mentioned above, via a suitable choice of particle-
hopping rates. We consider a two-dimensional lat-
tice with angles θi ǫ [0, π] and noninteracting lattice-
gas occupancy variables ni = 0, 1 at each site i.
The angles evolve by Metropolis Monte Carlo up-
dates governed by the Lebwohl-Lasher [16] hamilto-
31
2
3 4
FIG. 3: Director variation from the bottom to the top of
the picture gives rise to a current in a transverse direction
nian H = −K ∑<ij> cos 2(θi − θj) yielding a ne-
matic phase at low temperature. Particle motion
is nonequilibrium: Hops of a particle at site i to a
nearest-neighbour site in the ±x direction are at-
tempted with probability 1/4 ± α(θ1 − θ2), and in
the ±z direction with probability 1/4 ± α(θ3 − θ4),
where θi are the angles at sites i = 1 to 4 as in
Fig. 3, and α encodes the strength [17] of the ac-
tive curvature-current coupling of [5]. In [5], the
0 10 20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
60
70
80
90
100
FIG. 4: Anisotropic clustering of particles, 20000 time
steps, system size 100, 50 % occupancy
effect of the concentration field on the dynamics of
the angle field is shown to play an insignificant role
in the giant number fluctuations. Accordingly, we
neglect it here, so that the particles are advected
passively by an autonomous angle field. This simpli-
fying approximation should not make a qualitative
difference deep in the nematic phase. Accordingly,
we work with K = 10.0, and let the angle field equi-
librate until a well-ordered nematic is formed. An
initial statistically uniform distribution of particles,
(mean occupancy =50 %) clusters and coarsens (Fig.
4), most strongly at ±45◦ to the mean direction of
nematic ordering. The anisotropic two-point den-
sity correlator C(r, t) shows a scaling collapse for a
given direction for all t, if plotted as a function of
r/L(t) for a coarsening length L(t), defined by the
value of r at the first zero-crossing, whose scaling
is consistent with t1/2. (see Fig. 5 and Fig. 2).
The value at which L(t) saturates is proportional
to the system size L (see Fig. 2), which makes a
strong case for true phase separation. The value of
the saturation length is numerically small compared
to L, probably because of the poorly-defined clus-
ters (Fig. 4) of fluctuation-dominated phase separa-
tion. That we work with hard-core particles, and on
timescales on which the macroscopic variation of the
mean nematic orientation is very small, also proba-
bly contributes. The exponent of 1/2 is because the
particles aggregate not by diffusion plus short-range
capture, but rather, by analogy with [8] (see also
[18]), because the broken-symmetry mode of the ne-
matic order sweeps the particles over large distances
via curvature-induced drift. A nematic fluctuation
on a scale ℓ collects particles in a time of order ℓ. For
two such domains to coalesce requires the nematic
director field on that scale to turn over, which is a
time ∼ ℓz where z = 2 is the dynamic exponent of
transverse fluctuations of the nematic director.
0 0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
r/L(t)
C(
r/L
(t)
)
5 10 15 20 25 30
−0.2
0
0.5
1
r
C(
r,t)
FIG. 5: Plots of the equal-time density correlator C(r, t)
at different times t collapse onto a single curve when r is
scaled by the zero-crossing coarsening length L(t); data
shown for direction transverse to nematic ordering Inset:
C(r, t) vs. r.
At long times a steady state is reached, and
C(r/L(t → ∞)) shows a cusp at small argument
(C(x) ∝ xa, a ≃ 0.33), which can be seen in Fig.
5 as well, signalling the absence of sharp interfaces
between regions rich and poor in particles, and a
power-law distribution of cluster sizes. For steady
state in the largest system, we also measured the
standard deviation ∆N in the number of particles
in an observation containing N¯ particles on average.
The plot of ∆N vs N¯ , Fig. 1, shows precisely the
linear dependence predicted by [5]. Faced with these
results, we ask: is this phase separation or a single
phase with large fluctuations? This is answered by
measuring the magnitude of the time-averaged low-
est spatial Fourier-component Q(1, 1) of the density,
shown in Fig. 6. Although the data are not con-
clusive, the flattening of the semilog plot as a func-
tion of system size rules out an exponential decay
to zero. Together with the proportionality of the
4coarsening length to the system size, and the nature
of the mechanism, this suggests strongly that active
nematics offer the most natural physical realisation
of macroscopic fluctuation-dominated phase separa-
tion [8]. As in [8], we find that the time-series of
0 20 40 60 80 100 120 140
10−1.3
10−1.2
10−1.1
L (system size)
<
Q(
1,1
)> 2 3 4 5
x 104
0
0.05
0.1
0.15
t (time)
Q(
1,1
) L=128 
FIG. 6: Time-averaged fourier component of the density
profile vs. wave vector k for wave vector direction (1,1),
for system sizes L = 16, 32, 64, 128; flattening in semilog
plot at largest size suggests nonzero value for L → ∞.
Inset: Time series of Q for (1,1) direction.
Q(1, 1) shows enormous fluctuations, (Fig. 6, inset),
with “crashes” during which other nearby fourier-
components gain weight. Thus, as in [8], the system
lurches from one macroscopically phase-separated
configuration to another, spending very little time in
non-phase-separated states. Lastly, the velocity au-
tocorrelation of tagged particles at low (15 %) con-
centration agrees qualitatively with the 1/t tail (plot
not shown) predicted by [5], over the range in which
a given particle moves unimpeded by others.
What experiments can test these results? The best
would be agitated layers of granular rods, for which
nematic phases have been reported [12]. Although
many features of these systems can be rationalized
in terms of equilibrium hard-rod theories [19], some
properties such as global circulation and swirls [11,
12] are clearly very nonequilibrium. These systems
as well as the living melanocyte nematic of Gruler
et al. [10] remain the most promising candidates for
experimental tests of the rich range of results made
here and in [5]. The confirmation of giant number
fluctuations in the numerical experiments of [6] is
encouraging in this regard.
The DST, India supports the Centre for Condensed
Matter Theory. SR thanks the IFCPAR (project
3504-2) and SM the CSIR, India for support. We
acknowledge valuable discussions with M. Barma, S.
Puri, C. Dasgupta and especially H. Chate´, who also
generously shared his results [6] before publication.
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[6] The numerical experiments of H. Chate´ et al., Phys.
Rev. Lett. 96, 180602 (2006), which we received in
preprint form while preparing this manuscript, con-
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nematic modification of the Vicsek [1] model.
[7] Large density inhomogeneities seem inevitable in
a wider variety of driven systems, such as traffic
[D. Chowdhury et al., Phys. Rep. 329, 199 (2000)],
granular fluids [D.R.M. Williams and F.C. MacKin-
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term from microscopics.
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the sign of this coupling: the arrow in Fig. 3 could
point to the left instead.
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also, the statics of nematic ordering in [6] is essen-
tially as in an equilibrium system.


Active nematics are intrinsically phase separated

Two-dimensional nonequilibrium nematic steady states, as found in agitated granular-rod monolayers or films of orientable amoeboid cells, were predicted [Europhys. Lett. 62, 196 (2003)] to have giant number fluctuations, with the standard deviation proportional to the mean. We show numerically that the steady state of such systems is macroscopically phase separated, yet dominated by fluctuations, as in the Das-Barma model [Phys. Rev. Lett. 85, 1602 (2000)]. We suggest experimental tests of our findings in granular and living-cell systems

Active nematics are intrinsically phase-separated

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