Novel critical phenomena in compressible polar active uids Dynamical and Functional Renormalization Group Studies Patrick Jentschand Chiu Fan Leey

2025-05-02 0 0 964.49KB 27 页 10玖币
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Novel critical phenomena in compressible polar active fluids: Dynamical and
Functional Renormalization Group Studies
Patrick Jentsch∗and Chiu Fan Lee†
Department of Bioengineering, Imperial College London,
South Kensington Campus, London SW7 2AZ, U.K.
(Dated: April 12, 2023)
Active matter is not only relevant to living matter and diverse nonequilibrium systems, but also
constitutes a fertile ground for novel physics. Indeed, dynamic renormalization group (DRG) analy-
ses have uncovered many new universality classes (UCs) in polar active fluids (PAFs) - an archetype
of active matter systems. However, due to the inherent technical difficulties in the DRG methodol-
ogy, almost all previous studies have been restricted to polar active fluids in the incompressible or
infinitely compressible (i.e., Malthusian) limits, and, when the -expansion was used in conjunction,
to the one-loop level. Here, we use functional renormalization group (FRG) methods to bypass
some of these difficulties and unveil for the first time novel critical behavior in compressible polar
active fluids, and calculate the corresponding critical exponents beyond the one-loop level. Specifi-
cally, we investigate the multicritical point of compressible PAFs, where the critical order-disorder
transition coincides with critical phase separation. We first study the critical phenomenon using a
DRG analysis and find that it is insufficient since two-loop effects are important to obtain a non-
trivial correction to the scaling exponents. We then remedy this defect by using a FRG analysis.
We find three novel universality classes and obtain their critical exponents, which we then use to
show that at least two of these universality classes are out of equilibrium because they violate the
fluctuation-dissipation relation.
I. INTRODUCTION
Active matter refers to many-body systems in which
the microscopic constituents can exert forces or stresses
on their surroundings and, as such, detailed balance is
broken at the microscopic level [1, 2]. However, even
if the microscopic dynamics are fundamentally different
from more traditional systems considered in physics, it
remains unclear whether novel behavior will emerge in
the hydrodynamic limits (i.e., the long time and large
distance limits [3]). One unambiguous way to settle this
question is to identify whether the system’s dynamical
and temporal statistics are governed by a new universal-
ity class (UC), typically characterized by a set of scal-
ing exponents [4–6]. These exponents can in principle
be determined using either simulation or renormaliza-
tion group (RG) methods. However, simulation studies
can be severely plagued by finite-size effects (e.g., two
recent controversies concern the scaling behavior of ac-
tive polymer networks [7, 8] and critical motility-induced
phase separation [9–11]). Therefore, RG analyses remain
as of today the gold standard in the categorization of
dynamical systems into distinct UCs. This perspective
has been particularly fruitful in biological physics, where
many new nonequilibrium universality classes have been
discovered in biology inspired systems [12–15]. Specifi-
cally, for polar active fluids (PAFs) [16–18], an archetype
of active matter systems, the use of dynamic renormaliza-
tion group (DRG) [19] analyses have led to, on one hand,
surprising realizations that certain types of PAFs are no
∗p.jentsch20@imperial.ac.uk
†c.lee@imperial.ac.uk
different from thermal systems in the hydrodynamic limit
[20, 21], and on the other hand discoveries of diverse
novel phases [17, 18, 22–30], critical phenomena [31–33]
and discontinuous phase transitions [34]. However, due
to the inherent technical difficulties in DRG methods,
all of these studies have been restricted to PAFs in the
incompressible or infinitely compressible (i.e., Malthu-
sian) limits except for rare exceptions [23, 24]. Further,
when a DRG analysis was used in conjunction with the
-expansion method, which was typically the case, it has
always been restricted to the one-loop level.
In this work, we apply for the first time functional
RG (FRG) methods on compressible PAFs and overcome
some of these technical challenges. Specifically, we inves-
tigate a multicritical region of dry compressible PAFs.
Although experimentally less accessible than simple crit-
ical points, multicritical points (MCPs) can offer sur-
prising new physics, even in models that are thought to
be well understood. For instance, nonperturbative fixed
points have been discovered in the extensively studied
O(N) model [35], and in systems where two order param-
eters compete, whose individual critical points belong to
equilibrium universality classes, the multicritical region
where both critical points coincide can be manifestly out
of equilibrium and demonstrate very interesting, spiral
phase diagrams [36].
We will first apply a traditional one-loop DRG ap-
proach to the MCP of our interest, demonstrating how it
is insufficient to capture its universal physics and then,
for the first time for PAFs, apply a FRG [37–44] analy-
sis that goes beyond the equivalent perturbative one-loop
level.
FRG analyses are intrinsically non-perturbative and
are based on an exact RG flow equation to which ap-
arXiv:2210.03830v2 [cond-mat.soft] 11 Apr 2023
2
proximate solutions can be readily obtained numerically.
Recent successes in the applications of FRG include the
elucidation of scaling behavior in, e.g., critical and multi-
critical N-component ferromagnets [35, 45–47], reaction-
diffusion systems [48–52], the Kardar-Parisi-Zhang model
[53–55], and turbulence [56–60], as well as non-universal
observables far from scaling regimes [61, 62]. Using
FRG, we uncover here three novel nonequilibrium UCs by
studying a multicritical region of dry compressible PAFs
and quantify the associate scaling behaviors beyond the
one-loop level.
The outline of this paper is as follows. In Sec. II, we
introduce the hydrodynamic theory of compressible polar
active matter and discuss salient features in its phase di-
agram, which enables us to define the multicritical point
of interest. We then show how general scaling invariance
of the equations of motion leads to powerlaw behavior
in the correlation functions in Sec. III. For the multi-
critical point, we show this first in the linear regime in
Sec. IV and then turn to the nonlinear regime in Sec. V.
In Sec. VI, we perform the one-loop DRG calculation
and argue why it is not sufficient to take into account
the nonlinearities and then present the FRG approach
in Sec. VII that we use instead. Using this method we
find three RG fixed points which represent three novel
nonequilibrium UCs. We discuss them and their scaling
behavior in Sec. VIII. Finally, we summarize our findings
and give an outlook on future work in Sec. IX.
II. COMPRESSIBLE POLAR ACTIVE FLUIDS
A. Equations of motion from symmetry and
conservation laws
Polar active matter aims to describe the collective be-
havior of swarming animals, e.g., flocks of birds, schools
of fish or bacterial swarms. In the fluid state, as in
passive fluids that are describable by the Navier-Stokes
equations, the relevant dynamical variables are the mo-
mentum density and density fields, denoted by gand ρ
respectively. Without any assumptions about the mi-
croscopic realization, one can then, based on symmetry
and conservation laws, construct a generic set of hydro-
dynamic equations of motion (EOM) for these variables.
Here, we assume that the particle number is conserved
(as opposed to, e.g., a Malthusian system in which birth
and death of particles can occur [22, 26, 27]). We thus
arrive at a continuity equation as the EOM of the density
field ρ:
∂tρ+∇ · g= 0 .(1)
For the momentum density field, we assume tempo-
ral, translational, rotational and chiral invariance. In
addition, we focused on active systems in which the con-
stituents move on a fixed frictional substrate, i.e., dry
active matter systems (as opposed to wet active systems
such as active suspensions) [2]. This symmetry consider-
ation leads to the following generic hydrodynamic EOM
of g[17, 18, 63]:
∂tg+λ1∇(|g|2) + λ2(g· ∇)g+λ3g(∇ · g) = µ1∇2g+µ2∇(∇ · g)−αg−β|g|2g−κ∇ρ+... +f,(2)
where the ellipsis represents the omitted higher-order
terms (i.e. terms of higher order in spatial derivatives
and the momentum density field).
In the EOM above, all coefficients are functions of ρ,
and the noise term f(r, t) is a zero mean Gaussian white
noise of the form
hfi(r, t)fj(r0, t0)i= 2Dδij δd(r−r0)δ(t−t0).(3)
The above hydrodynamic EOM (1,2) are termed the
Toner-Tu EOM [17, 18]. However, in contrast to the
original Toner-Tu formulation, we have chosen to use the
momentum field as the hydrodynamic variable instead
of the velocity field so that there is a linear relationship
between ρand g, which facilitates our discussion later.
B. Mean-field theory and homogeneous phases
Given the hydrodynamic EOM, one of the first, and
simplest, question to ask is: what are the mean-field ho-
mogeneous solutions to the EOM? Answering this ques-
tion amounts to focusing on temporally invariant, spa-
tially homogeneous, and noise-free solutions to the EOM.
These solutions are readily seen to be
|g|=(q|α|
β,if α < 0
0,otherwise ,(4)
where βis taken to be always positive for reasons of sta-
bility. Since gcan point in any direction, the |g|>0
state implies a spontaneous symmetry breaking of the ro-
tational symmetry, and corresponds to the homogeneous
ordered phase of polar active matter where collective mo-
tion emerges. In contrast, the |g|= 0 state corresponds
to the homogeneous disordered phase, i.e., there is no
3
collective motion.
C. Phase diagram: phase separations, critical and
multicritical phenomena
The homogeneous phases discerned from the previous
mean-field analysis are, however, not always stable, even
in the absence of the noise term f(3). The standard way
to ascertain the in/stability of the homogeneous phases in
this noiseless regime is to use a linear stability analysis.
Here, the temporal evolution of an initially small per-
turbation to a homogeneous solution is studied and the
growth or decay of its amplitude signifies whether the ho-
mogeneous state is unstable or stable, respectively. Since
the nature of the inhomogeneous states does not follow
from linear stability analysis alone and inhomogoneous,
analytic solutions of the noiseless mean-field equations
are most often very difficult, it is typically explored via
simulations. Using this method, complex phase diagrams
of polar active fluids have been uncovered [64–66]. In par-
ticular, distinct types of bulk phase separations, i.e., an
inhomogeneous state where two different phases co-exist,
have been demonstrated. As the large length and time-
scale-limit of all these different models, it is expected that
the hydrodynamic EOM captures the same phenomenol-
ogy in an encompassing phase diagram, schematically de-
picted in Fig. 1.
In particular, expressing αand κin Eq. (2) as
α=X
n≥0
αnδρn, κ =X
n≥0
κnδρn,(5)
where δρ =ρ−ρ0with ρ0being the average particle
density in the system, two disordered phases (with dis-
tinct densities) can co-exist if α0>0 and κ0<0 (blue
region in Fig. 1(a)) [10], while an ordered phase can co-
exist with a disordered phase if κ0>0 and α0<0 (green
region)[65]. Further, the system can become critical upon
fine-tuning: if α0>0 and κ0=κ1= 0, the resulting crit-
ical behavior belongs to the Ising universality class (UC)
(blue triangle) [10], while if α0=α1= 0 and κ0>0,
the associate critical behavior corresponds to a yet to
be characterized UC (yellow inverted triangle) [65]. Re-
cently, a third type of critical behavior was identified [66],
which corresponds to the merging of these two distinct
critical points by simultaneously fine-tuning α0,α1,κ0
and κ1to zero (red circle in Fig. 1(b)). The universal be-
havior of this new multicritical point is the focus of this
paper.
It is interesting to note that apart from these, either
homogeneous or bulk phase-separated, states that join at
the MCP, different states of microphase separation have
been observed as well [67–69].
O
D
ρ0
α0
O
D
ρ0
α0
a)b)
FIG. 1. Polar active fluids admit diverse phase transitions
and phase separations. These figures show qualitatively two
possible instances already discussed in [66]. (a) Depending
on the model parameter, e.g., α0, and the average density
ρ0, the system can be in the homogeneous disordered phase
(white region denoted by D) or the polar ordered phase (yel-
low region denoted by O); It can also phase separate into two
disordered phases with different densities (blue region), or
into one ordered phase and one disordered phase, again with
different densities (green region flanking the homogeneous or-
dered phase). The critical behavior associated with the first
type of phase separation is generically described by the Ising
UC (blue triangle) [10], and that associated with the second
type is described by a putatively novel UC yet to be described
(yellow inverted triangle) [65]. (b) Upon further fine-tuning,
these two critical points can coincide (red circle) [66], and the
resulting critical point is described by a novel UC uncovered
in the present work.
III. SCALE INVARIANT EQUATIONS OF
MOTION
It is generally expected that the EOM of general sys-
tems at critical points become invariant under rescaling
of lengths, time and fields [4–6]. Hence, at the multi-
critical point (MCP), we expect that the EOM (1,2) are
invariant under the rescaling
r→re`, t →tez`, ρ →ρeχρ`,g→geχg`,(6)
for some exponents z,χρand χg, that are a priori not
known. If this is the case, however, this defines a rescal-
ing symmetry of the theory which the correlation func-
tions have to obey as well. Take for example the density-
density correlation function:
Cρ(r, t) = hρ(r, t)ρ(0,0)i= e−2χρ`hρ(re`, tez`)ρ(0,0)i.
(7)
Choosing `=−ln r,rbeing measured against some ref-
erence scale, we see immediately that
Cρ(r, t) = r2χρSρρ t
rz,(8)
where Sρρ(.) is a scaling function that only depends on
the ratio t/rzwhich is invariant under rescaling (6). So
we immediately see that, if the EOM are invariant under
4
a rescaling transformation, the correlation functions will
generally express powerlaw behavior.
Likewise, this argument can be applied to the
momentum-momentum correlation function
Cg(r, t) = hg(r, t)g(0,0)i=r2χgSgg t
rz,(9)
where Sgg is again a scaling function with similar prop-
erties as Sρρ.
Ultimately, we will demonstrate that the EOM does
become scale invariant and determine the scaling expo-
nents using a FRG analysis, but first, we will illustrate
the scale invariance discussed here using the simple, but
quantitatively incorrect, linear theory.
IV. LINEAR REGIME
In the linear regime, i.e., when the non-linear terms
in Eq. (2) are neglected, the scaling behavior discussed
above can readily be seen. Around the MCP at which
critical disordered phase separation (blue triangle in
Fig. 1a)) merges with critical disorder-order of a generic
compressible PAF (yellow inverted triangle), |g| ≈ 0,
such that the linearized EOM are
∂tρ=−∇ · g,(10a)
∂tg=µ1∇2g+µ2∇(∇ · g) + ζ∇2∇ρ+f,(10b)
where we have introduced the term characterized by ζ
since, when κ0is fine-tuned to zero, this term is now
the leading order term linear in ρ. In Eq. (10), we have
redefined ρto be δρ to ease notation, and we will continue
to do so from now on.
A. Scaling exponents
1. Correlation functions
Upon rescaling time, lengths, and fields according to
Eq. (6), the linearized EOM (10) become
e(χρ−z)`∂tρ=−e(χg−1)`∇ · g,(11a)
e(χg−z)`∂tg= e(χg−2)`µ1∇2g+µ2∇(∇ · g)
+ e(χρ−3)`ζ∇2∇ρ+ e−(z+d)`/2f.(11b)
They thus remain unchanged if
zlin = 2, χlin
ρ=4−d
2, χlin
g=2−d
2.(12)
At the linear level we can therefore directly conclude
that
Cρ(r, t) = r2χlin
ρSlin
ρρ t
rzlin ,(13a)
Cg(r, t) = r2χlin
gSlin
gg t
rzlin ,(13b)
using the argument from Sec. III.
Since the linearized EOM (10) are solvable analytically
by performing a spatio-temporal Fourier transform, the
scaling behavior of the correlation functions can in fact
be demonstrated explicitly. This has the added advan-
tage that the expressions of the aforementioned scaling
functions, Sρρ and Sgg, can be obtained in the form of
integrals.
Specifically, by performing a spatiotemporal Fourier
transform, the linear EOM can be written as
ρ(˜
q) = q
ωGk(˜
q)fk(˜
q),(14a)
gk(˜
q) = Gk(˜
q)fk(˜
q),(14b)
g⊥(˜
q) = G⊥(˜
q)f⊥(˜
q),(14c)
where gk(˜
q) = g(˜
q)·ˆ
q,g⊥=g−gkˆ
q, with ˆ
qbeing the
unit vector in the direction of q,q=|q|, ˜q= (q, ω), and
the G’s in (14), or the “propagators”, are:
Gk(˜
q) = ω
−i(ω2−ζq4) + ωµkq2,(15a)
G⊥(˜
q) = 1
−iω+µ1q2,(15b)
where µk≡µ1+µ2.
Given the above expressions, the correlation functions
are now obtained straightforwardly:
Cρ(r, t) = Z˜
q
ei˜
q·˜
r2Dq2
(ω2−ζq4)2+ω2µ2
kq4,(16a)
Cg=C⊥
g+Ck
g,(16b)
C⊥
g(r, t) = Z˜
q
ei˜
q·˜
r2DP⊥(q)
ω2+µ2
1q4,(16c)
Ck
g(r, t) = Z˜
q
ei˜
q·˜
r2Dω2Pk(q)
(ω2−ζq4)2+ω2µ2
kq4,(16d)
where R˜
q≡Rddqdω/(2π)(d+1),˜
q·˜
r=q·r−ωt, and
Pk
ij (q)≡qiqj/q2and P⊥
ij (q)≡δij −qiqj/q2are the pro-
jectors parallel and transverse to qrespectively.
Focusing on Cρ(r, t) (16a) as an example, the substi-
tutions ω= Ω/r2and q=Q/r in the integral lead to
Cρ(r, t) = r4−dZddQdΩ
(2π)(d+1)
2DQ2exp iQ·ˆ
r−Ωt
r2
(Ω2−ζQ4)2+ Ω2µ2
kQ4,
(17)
which demonstrates the scaling form (8) with the scaling
exponents from the linear theory (12), and
Slin
ρρ (y) = ZddQdΩ
(2π)(d+1)
2DQ2exp [i (Q·ˆ
r−Ωy)]
(Ω2−ζQ4)2+ Ω2µ2
kQ4.(18)
As we will show later, all the scaling exponents from
the linear theory (12) are in fact incorrect for describing
the hydrodynamic behavior around the MCP due to the
nonlinearities in the EOM.
5
2. Divergence of correlation length
Besides the scaling exponents in the correlation func-
tions right at the MCP, the divergence of the correlation
length, as one approaches the MCP, is also governed by
another set of scaling exponents. For the Ising model,
this is the temperature. For the present MCP however,
this divergence is associated to two parameters α0,κ0.
The other two relevant parameters, α1and κ1, take a
role akin to the magnetic field in the Ising model.
Since κ0appears in the EOM as a speed of sound for
density wave, it is not immediately clear how it might
be related to the correlation length. However one can
show by rederiving the correlation functions (16) in the
presence of these two couplings, that the equal-time cor-
relation functions are
Cρ(r,0) = Zq
eiq·rD
(α0+µkq2)(κ0+ζq2),(19a)
C⊥
g(r,0) = Zq
eiq·rDP⊥(q)
α0+µ1q2,(19b)
Ck
g(r,0) = Zq
eiq·rDω2Pk(q)
α0+µkq2.(19c)
From this standard form it is clear that, in the linear
theory, both α−1/2
0and κ−1/2
0define a crossover scale,
and the larger of the two a correlation length for density
correlations, while α−1/2
0is always the correlation length
for momentum correlations.
As the divergence of the correlation length is described
by these two parameters, α0and κ0, there are also two
exponents which we call y1and y2. At the linear level,
these exponents correspond expectedly to their mean-
field values:
ylin
1=ylin
2= 2 .(20)
We note that these scaling exponents are again expected
to be modified by the nonlinearities, as we shall see in
the next section.
Experimentally, these exponents define a relationship
between the correlation length ξand the distances t1and
t2in the phase diagram (see the inset of Fig. 2) from
the critical points of disordered phase separation (blue
upwards triangle in Fig. 1 and blue line in the inset of
Fig. 2) and the critical order-disorder transition (yellow
downwards triangle in Fig. 1 and yellow line in in the
inset of Fig. 2),
ξ∼t−1
y1
1∼t−1
y2
2.(21)
This means that, upon approaching the MCP, one has
to enforce the relationship given by the right proportion-
ality in Eq. (21) to see a clear scaling behavior in the
correlation length. This relationship is also visualized in
Fig. 2.
1
2χρ
t
1
t
2
microscopics
log
(
)
FIG. 2. Scaling behavior of the correlation length when ap-
proaching the MCP. The inset shows the phase diagram in
terms of the couplings α0and κ0under the assumption that
α1=κ1= 0, and the main figure shows the equal-time den-
sity correlation function Cρ(r,0) in log-log scale (black lines)
which, if sufficiently close to the MCP (red circle in the inset)
shows the critical scaling behavior characterized by the scaling
exponent χρ(slope triangle). In reality though, the system
will never be exactly at the critical point. This can be char-
acterized by the distances t1and t2(gray lines in inset) from
the critical point of disordered phase separation (blue line in
inset) and from the critical order-disorder transition (yellow
line in inset) respectively. This manifests in a finite correla-
tion length ξabove which the scaling behavior breaks down
(gray lines in main figure). As one approaches the fixed point,
by decreasing t1and t2, the correlation length diverges. If this
is done carefully, such that the second relation in Eq. (21) re-
mains unchanged, for example by rescaling t1and t2by a
factor sy1and sy2respectively (gray arrow in inset), the cor-
relation length rescales according to Eq. (21), i.e. by a factor
s−1(gray arrow in main figure).
V. NONLINEAR REGIME
While the scaling behavior described in Sec. IV is quali-
tatively expected generally, the critical exponents (12,20)
obtained in the linear theory are only expected to be ex-
act when the spatial dimension dis high enough. Below
a certain upper critical dimension dc, nonlinear terms be-
come important which modify the scaling and correlation
length exponents. The linear exponents (12) can however
be used to gauge the importance of various nonlinearities
in the EOM (2) as dis lowered.
We now turn to the full EOM of g(2), and perform
a rescaling (6) with the linear exponents (12). If the
spatial dimension dis large enough, all nonlinear terms
are irrelevant, i.e., they vanish as `→ ∞. As ddecreases
from, say infinity, the nonlinear terms that first become
relevant (and are not fine-tuned to zero), i.e., terms that
diverge as `→ ∞, are
α2ρ2gand κ2ρ2∇ρ , (22)
摘要：

Novelcriticalphenomenaincompressiblepolaractiveuids:DynamicalandFunctionalRenormalizationGroupStudiesPatrickJentschandChiuFanLeeyDepartmentofBioengineering,ImperialCollegeLondon,SouthKensingtonCampus,LondonSW72AZ,U.K.(Dated:April12,2023)Activematterisnotonlyrelevanttolivingmatteranddiversenonequili...

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