Table Of ContentActive Particles on Curved Surfaces
Yaouen Fily, Aparna Baskaran, Michael F. Hagan
Martin Fisher School of Physics, Brandeis University, Waltham, MA 02453, USA
Recentstudieshavehighlightedthesensitivityofactivemattertoboundariesandtheirgeometries.
Here we develop a general theory for the dynamics and statistics of active particles on curved
surfaces and illustrate it on two examples. We first show that active particles moving on a surface
with no ability to probe its curvature only exhibit steady-state inhomogeneities in the presence
of orientational order. We then consider a strongly confined 3D ideal active gas and compute its
steady-state density distribution in a box of arbitrary convex shape.
6 Initially introduced as a theory of animal flocking, the box. In case a), we show that, at steady state, parti-
1 paradigm of active materials has grown into a robust cles in the isotropic phase have spatially uniform den-
0 framework to describe systems whose building blocks sity, independent of curvature. However, upon acquiring
2 have the ability to move autonomously by extracting en- orientationalorder,particlesexhibitcurvature-drivenin-
n ergy from their surroundings [1]. This makes it a natu- homogeneities in density and polarization. We compute
a ral starting point to understand the mechanics of living these inhomogeneities explicitly for weakly curved sur-
J
systems or engineer energy-consuming biomimetic mate- faces, and show that they are nonlocal and quadratic in
4 rials. the height of the surface’s deformations. In case b), we
1
One important class of systems in this paradigm con- show that strongly confined particles (particles in a con-
] sistsofactiveobjectsmovingonasurface[2]. Suchcases tainer much smaller than the persistence length of their
t abound in biology, and the boundaries are rarely flat. motion) never leave the boundary, that their orientation
f
o For example, the migration of cells along curved tissues is nearly aligned with the boundary’s normal, and that
s occurs on the gut’s lining [3, 4], on the surface of an their density is proportional to the local Gaussian cur-
.
t organoid [5], in a gastrulating embryo [6], and in the vature of the boundary, thus extending a previous result
a
m growing cornea [7]. At the macroscale, animals run and obtained in 2D [29, 30].
flock along uneven terrains. In the synthetic realm, ac-
-
d tive microtubule bundles adsorbed at a liquid-liquid in-
n terface [8] or on a lipid membrane [9] also exhibit strong
o curvature effects.
c
[ Another important class of systems is confined active
materials. Introducing flat walls suffices to prompt un-
2
usual physics [10–14], e.g. the breakdown of the concept
v
of equation of state [14]. Curved walls lead to even more
4
dramatic behaviors, including spontaneous flow [15–25],
2
3 concentrationofparticlesincertainregionsofspace[25–
0 31], and propulsion of passive objects [32, 33]. In cells,
0 active processes can interact with the membrane’s cur-
1. vature to help the cell change its shape [34, 35] or locate FIG. 1. (Color online) Left: Geometric notations. nˆ is
0 its own center [36]. the unit normal to the wall. (e1,e2) is the local basis of the
6 Despitetheprevalenceofcurvatureinsystemsofinter- tangent plane associated with the coordinates (cid:0)x1,x2(cid:1). The
1 particle(greensphere)movesalongtheprojectionofitsorien-
est, existing theories of active matter have been limited
: tationuˆ onto the surface. Right: Colormap of thepredicted
v to specific geometries of surfaces and confining bound-
particle density for a strongly confined 3D nonaligning ideal
Xi aries. In this work we address this limitation by consid- activegasinaprolatespheroidwithaspectratio2(seetext).
ering a system of self-propelled particles and i) describ-
r
a ing the generic effects of curvature on the dynamics of Statistical Framework: Let us consider a collection
such particles at the level of a Langevin equation and ii)
of self-propelled particles on a smooth curved surface
deriving a generic covariant statistical theory for the dy- parametrized by s(x1,x2). Each particle is characterized
namics of self-propelled particles that accounts for cur- by its position coordinates x = (cid:0)x1,x2(cid:1) and the compo-
vature induced effects for arbitrary surfaces. This for- nents u = (cid:0)u1,u2(cid:1) of its orientation in the local basis of
malism can be applied to a wide variety of problems,
the tangent plane {e } where e = ∂s/∂xa = ∂ s
from particles embedded in a curved surface, to parti- a a=1,2 a a
(superscripts/subscripts denote contravariant/covariant
cles whose path is temporarily blocked by a wall. We
indices) [37]. The equations of motion have the form
illustratethispointbystudyingtwocasesofinterestcor-
responding to these two situations: a) mutually align- dx a/dt=v a, Du a/dt=w a+σ a ξ a (1)
α α α α α b α
ing particles moving on a curved surface and b) a 3D
idealgasofactiveparticlesstronglyconfinedinaconvex where Greek indices are particle labels, ξ is a Gaussian
2
noise with zero mean and correlations (cid:104)ξ a(t)ξ (t(cid:48))(cid:105) = theverycoreofthetheoreticaldescriptionofactivemat-
α βb
2δ δaδ(t−t(cid:48))andv,w,σ arearbitraryfunctionsofthe ter. Hence Eq. (2) is central to understanding active
αβ b
positions and orientations of the particles. This allows systems on curved surfaces. We now illustrate this point
forparticleswitharbitraryinteractions(forcesthroughv by examining two concrete examples of interest.
and torques through w) and angular noise. The intrinsic Flocking on a curved surface: Let us begin with
time derivative Dua = dua/dt+Γa (dxb/dt)uc, where self-propelled particles that are entirely restricted to a
Dt bc
Γa =ea·∂ e are the Christoffel symbols of the second curvedsurface,suchascolloidstrappedataninterface,or
bc b c
kind, ensures that the equations of motion are explicitly animalsontheground. Inthiscasetheactiveforceisal-
covariant, i.e., they are by construction independent of waystangenttothesurface,andwecanchooseuau =1.
a
the parameterization of the curved surface. Wefocusonparticlesthatare“unaware”ofthethirddi-
To unfold the influence of curvature on emergent be- mension, i.e., whose dynamics never refers explicitly to
havior in diverse cases, we seek to develop a statistical the fact that the surface is embedded in 3D space, and
description that is also explicitly covariant. We begin by showthatcurvaturestillaffectstheircollectivebehavior.
defining the one-particle scalar probability density Even in the absence of noise or interactions, curvature
enters the dynamics through the intrinsic time deriva-
1(cid:42)(cid:88)(cid:89)2 (cid:43) tive Du in the equation of motion for the orientation
f(x,u,t)= δ[xa−X a(t)]δ[ua−U a(t)] dt
g α α [Eqs. (1)]. Similar to free-falling objects in general rela-
α i=1 tivity, such particles follow geodesics of the surface.
wheregisthedeterminantofthemetric,andtheaverage Clearly, these deflections have the potential to dis-
is over the particle trajectories {(X (t),U (t))} . f rupt the collective dynamics of flocking particles, as il-
α α α
obeys a Fokker-Planck (FP) equation of the form (see lustrated by the band-shaped flocks recently observed
appendix) on the surface of a sphere [2]. With this in mind, we
study a model of mutually aligning self-propelled par-
∂ f =−∇˘ (vaf)−∂ (waf)+∂ (cid:2)σac∂ (cid:0)σ bf(cid:1)(cid:3) ticles obeying the equations of motion (1) with v a =
t xa ua ua ub c (2) √v0uαa, wαa = (cid:80)βK(xαβ)(δab−uαauαb)uaβ and σαaαb =
D (δa −u au ). Here w corresponds to a pairwise
where v, w, σ are the mean-field versions of v, w, σ. r b α αb
aligningtorquereminiscentoftheXYmodelofferromag-
∇˘ isthephasespacecovariantspatialderivative,which
xa netism, and σ models the noise from the environment as
transforms like a covariant vector, whereas ∂ does not.
xa the projection of an uncorrelated Gaussian white noise
For a contravariant vector field za it reads
ontothecurvedsurface. ThekernelK controlstherange
∇˘ za =∂ za+Γa zc−Γd uc∂ za (3) of the aligning interaction. In the following we assume
xb xb bc bc ud that this range is much shorter than the radius of cur-
The first two terms form the usual covariant deriva- vature everywhere on the surface; i.e., the interparticle
tive ∇ za, which accounts for the fact that z(x,u) and distance x ≈(cid:2)(x a−x a)(x −x )(cid:3)1/2 wherever the
b αβ α β αa βa
z(x + dx,u) live in different tangent planes with dif- kernel is nonzero.
ferent local bases, therefore z can be locally uniform To understand the emergent behavior of this system,
(∇ za = 0) when its components are not (∂ za (cid:54)= 0). we apply standard coarse graining techniques (see ap-
xb xb
Likewise, local basis changes along the surface affect the pendix)toEq.(2)toobtainhydrodynamicequationsfor
orientationcoordinatesua,whichinturnaffectzaascap- the density ρ(x,t) = (cid:82) duf(x,u,t) and the polarization
turedbythesecondΓterm. Forfunctionsoftheposition pa(x,t) = (cid:82) duf(x,u,t)ua. In dimensionless units such
only, the last term vanishes and ∇˘ reduces to ∇. that v =D =1, these equations read
0 r
We stress that the covariance we discuss here is differ-
ent from the one studied by Graham [38] but similar to ∂tρ=−∇apa
the one discussed by Debbasch and Moreau [39] in the 1 (cid:18) κ2 (cid:19)
∂ pa =− ∇aρ+ κρ−1− pbp pa
contextofpassivediffusiononacurvedsurface(inwhich t 2 2 b (4)
cisastehaut iascthhoeicpeaortficploe’ssitmionomceonotrudmin)a.tesThimepclriuesciaalcphooiincet + κ8 (cid:2)5pb∇apb−5pa∇bpb−3pb∇bpa(cid:3)
of orientation coordinates through the definition of the
(cid:82)
local basis {e }. Therefore, we are not interested in co- where κ = drK(r)/2. They are formally identical to
a
varianceunderarbitrarychangesofthephasespacecoor- the equations reported in Ref. [40] in the flat case, ex-
dinates(asin[38]),butratherunderarbitrarychangesof cept for one important difference: the spatial derivatives
thepositioncoordinatesalongwiththeimpliedchangeof become covariant derivatives.
orientationcoordinates. Suchasituationarisesnaturally In the absence of the aligning interaction (κ=0), the
when the phase space of a particle consists of a position density evolves according to (cid:0)∂2+∂ (cid:1)ρ = ∆ρ/2, where
t t
on a curved surface and a vector tangent to the surface ∆ is the curved-space Laplacian. Therefore, barring
(orientation, momentum...). constraints from the boundary, the system diffusively
Most importantly, Eq. (2) is a good starting point to relaxes to homogeneity, with a diffusion constant 1/2.
derive (covariant) hydrodynamic equations, which are at The polarization obeys ∂ pa =−∇aρ/2−pa and relaxes
t
3
to zero as the density becomes homogeneous. In other order, the multipolar expansion of Eqs. (5)-(7) reads
words, in spite of the curvature-induced deflections, the p (cid:0)f −βf (cid:1)(cid:0)βx2−y2(cid:1)+4βf xy
steady state of a 2D ideal active gas is homogeneous and π (r)=− √ xx yy xy
isotropic regardless of the shape of the surface on which x 4π β (βx2+y2)2
it moves. (cid:15)(r)
=
κp
remAasinins ltihneeaflralyt sctaasbe,lethinisthhoemporgeesennecoeusofisaoltirgonpmicensttaates π (r)=−√βp (cid:0)fxx−βfyy(cid:1)xy−fxy(cid:0)βx2−y2(cid:1)
long as κρ < 1. Above this threshold, the system spon- y 2π (βx2+y2)2
taneously develops polar order. Unlike the flat case,
(cid:82)
however, there is no uniform polar solution, and finding where fab = drfab. The effect of a generic surface
a steady-state solution with polar order is highly non- irregularity on the steady state density and polarization
trivial. Linearstabilityinturncannotbestudiedwithout thus decays like r−2; i.e., it is long ranged.
a base state to linearize about. Strongly confined ideal active gas: For our sec-
ond example, let us consider a 3D self-propelled parti-
To obtain a general steady-state solution with polar cle whose path is blocked by a hard frictionless wall.
order to Eqs. (4), we now restrict ourselves to nearly The wall simply cancels the normal component of the
flat surfaces and expand the density ρ and polariza- velocity of the particle. The equation of motion for
tion pa about the flat-surface homogeneous polar solu- the orientation of the particle in this case is Dua =
√ √ Dt
tion (ρ,px = 2κρ−2/κ,py = 0). We assume a height −v u⊥La ub+ D [δa −uau /(1+u⊥)]ξb (seeappendix).
0 b r b b
profile h(x,y) whose irregularities have typical height H Here uau may take any value between 0 (orientation
a
andwidthW withH/W (cid:28)1,thusametrictensorgab = normal to th√e wall) and 1 (orientation tangent to the
δab+fab withfab =(∂ah)(∂bh)∼(H/W)2 (cid:28)1. Tolead- wall), u⊥ = 1−uaua is the component of the orienta-
ing order in H/W, the density deviation (cid:15) = ρ−ρ (cid:28) ρ tion normal to the wall, La =ea∂ nˆ is the shape tensor
b b
and the polarization deviation πa = pa − pa (cid:28) |p| ofthewallatthepositionoftheparticle,nˆ istheoutward
obey a set of third order, linear partial differential equa- pointingnormaltothewall,andξisthesamenormalized
tions with constant coefficients and curvature dependent Gaussian noise as before.
sourceterms. Deepinthepolarregion(Wκp(cid:29)1),these The effect of the wall is best explicated in the zero
equations reduce to a set of decoupled anisotropic Pois- noise limit. The eigenvalues of La are the principal cur-
b
sonequations,forwhichtheGreen’sfunctionGisknown. vaturesofthewall. Thecorrespondingtermintheequa-
Thus (cid:15) and πa can be written explicitly in terms of the tionofmotionforudescribesthewaythesurfacetiltsin
shape of the surface (see appendix): the particle’s frame of reference. Thus, the dynamics of
ua is controlled by the sign of the principal curvatures.
Forconvexwalls(positivecurvatures), ua relaxestozero
as the particle moves towards a location on the bound-
ακp2
(cid:15)(r)=− (∂2F +∂2F −2∂ F ) (5) ary where the normal is aligned with its orientation. If
2 y xx x yy xy xy the wall is concave (negative curvatures), ua grows until
πx(r)= ∂p(cid:15)(r)− pf (r) (6) uaua = 1 after which the particle points away from the
∂ρ 2 xx wall and leaves it. Additionally, La ub in general pro-
b
πy(r)=−p(cid:0)−∂ F +β∂ F +2∂2F (cid:1) (7) duces an effective torque on the particle that aligns ua
2 xy xx xy yy x xy with the direction in which the curvature is smallest.
When the box is convex and the noise is weak (when
the persistence length v /D is much larger than the
0 r
box), we expect the orientation to relax toward the
where F (r) = (cid:82) dr(cid:48)f (r(cid:48))G(r − r(cid:48)), G(x,y) =
ab √ ab normal to the boundary much faster than it fluctuates
log(cid:0)βx2+y2(cid:1)/(4π β), and β = 1/3. In the tran- (ua (cid:28) 1), and thus the particle never leaves the wall
sition region, where polar order is present but weak (whichwouldrequireuau =1)[29]. Weshowbelowthat
a
(0 < Wκp (cid:28) 1), the same expressions hold except with
thisapproximation,whichdefinesthestrongconfinement
β =1. Clearly,curvatureaffectsboththedensityandthe
regime,isself-consistent. Wefirstlinearizethedynamics
polarization, and this effect is nonlocal and linear in the around ua = 0 to obtain va = v ua, wa = −v La ub,
metric deviation fab (i.e. quadratic in H/W). Eq. (6) and σa =√D δa. We then consid0er an ideal acti0ve gbas,
shows that πx only changes i) by following the changes b r b
i.e., acollectionofN particleswithnegligibleinterparti-
of ρ the same way the flat-space solution would and ii)
cleinteractions. ExpandingtheFPequationinmoments
tomakeupforthecoordinatesnotbeingnormalizedany of ua and neglecting third order moments then yields a
more. Asaresult,thedirectionofpolarorderismodified,
closedsystemforthemomentsuptosecondorder,which
but its strength |(cid:104)u(cid:105)| = |p|/ρ remains unchanged (equal
we solve to get (see appendix)
to |p|/ρ), at least to leading order. To assess the non-
locality, we compute the far field deviation caused by a ρ= NK, (cid:104)ua(cid:105)=0, (cid:10)uaub(cid:11)= DrRab (8)
localized surface deformation near the origin. To leading 4π v0
4
where K is the determinant of the shape tensor, i.e. the In addition to establishing the ability of the theory
Gaussian curvature, and Ra is the inverse of the shape to solve diverse problems involving active particles on
b
tensor (its eigenvalues are the principal curvature radii). curved surfaces, these two examples highlight the vari-
Inotherwords,densityisproportionaltothelocalGaus- ety of ways in which curvature can affect active systems.
sian curvature, particles orient on average normal to the In our 2D model, the particles have no means to probe
surface, and the fluctuations of the orientation around the geometry of the surface beyond what is required to
the normal in a given direction are controlled by the lo- make the theory self-consistent (i.e., independent of the
cal radius of curvature along that direction. When the choice of coordinates). This in turn severely limits their
active persistence length v /D is larger than both radii responsetothesurfacegeometry. Atsteadystate,curva-
0 r
of curvature, ua remains small and particles stay on the ture only manifests itself when the system exhibits polar
boundary. This in turn proves the self-consistency of order. Additionally, the particles, not being aware the
our approach. Most importantly, the expression for ρ surface is embedded in 3D space, only respond to intrin-
can be used to predict the density of a 3D nonaligning sic curvature (i.e., the metric tensor and its derivatives).
ideal active gas on the boundary of its small confining Confined3Dparticles,ontheotherhand,aresensitiveto
box for arbitrary convex shapes (see the right panel of theangletheirorientationmakeswiththetangentplane.
Fig. 1 for an illustration). This is a key step toward By responding to the changes of this angle as they move
designing active devices whose geometries are tuned to along the surface, they can probe its curvature in spe-
achieve specific functionalities. A more detailed treat- cific directions – in other words, they can sense both
ment of this problem, including a comparison between intrinsic and extrinsic curvature [41]. The importance
our theory and the results of brownian dynamics simula- of the intrinsic/extrinsic distinction to classifying active
tions, will be published elsewhere. Our approach should behaviors on curved surfaces comes from the fact that,
alsobeapplicabletonon-convexcontainersfollowingthe likethesymmetries(polar/nematic/isotropic)ofactivity
methods in Ref. [30], however the transition from 2D to and interparticle interactions [1], it propagates from the
3D will make the mathematics more involved. individualtothecollectivedynamics. Ourworkprovides
a framework to study how it does so.
Conclusion: In this paper we combine the tools
of Riemannian geometry and nonequilibrium statistical
mechanics to propose a statistical framework for self- ACKNOWLEDGMENTS
propelled particles on arbitrary curved surfaces, which
canbeusedtoderivethestatisticalpropertiesofthesys- We thank Rastko Sknepnek and Silke Henkes for dis-
tem from the microscopic dynamics and the geometry cussions. This work was supported by the NSF, DMR-
of the surface. We motivate the framework and demon- 1149266, and the Brandeis Center for Bioinspired Soft
strate its use by determining the effect of surface curva- Materials, an NSF MRSEC, DMR-1420382. Compu-
tureontwosystemsofinterest: a2Dflockingmodel,and tational resources were provided by the NSF through
a 3D ideal active gas under strong convex confinement. XSEDE computing resources and the Brandeis HPCC.
APPENDIX
I. ONE-PARTICLE FOKKER-PLANCK EQUATION
We start from the equations of motion Eqs. (1) for N particles on the curved surface:
∀a∈{1,2}, ∀α∈[1,N], dx a/dt=v a, dua/dt+Γa (dxb/dt)uc =w a+σ a ξ a (9)
α α bc α α b α
where Greek indices are particle labels, latin indices are coordinate labels, ξ is a Gaussian noise with zero mean and
correlations (cid:104)ξ a(t)ξ (t(cid:48))(cid:105) = 2δ δaδ(t−t(cid:48)), Γa are the Christoffel symbols of the second kind, and v, w, σ are
α βb αβ b bc
arbitrary functions of the positions and orientations of the particles. The usual (noncovariant) one-particle phase
space density
f˜(x,u,t)=(cid:68)(cid:88)(cid:89)δ[xa−X a(t)]δ[ua−U a(t)](cid:69) (10)
α α
α a
obeys the usual one-particle Fokker-Planck equation
∂ f˜=−∂ (cid:16)vaf˜(cid:17)−∂ (cid:104)(cid:0)−Γa vbuc+wa(cid:1)f˜(cid:105)+∂ (cid:104)σac∂ (cid:16)σ bf˜(cid:17)(cid:105) (11)
t xa ua bc ua ub c
where overbars denote averages over the trajectories {(X (t),U (t))} of all the other particles. The equation is
α α α
closed by a mean-field approximation wherein the multi-particle distribution functions appearing in these averages
5
are written as products of one-particle distribution functions, i.e., correlations between particles are neglected:
(cid:90)
∀z ∈{v,w,σ}, za(x,u,t)≈ (cid:89)dxbβdubβ(cid:89)f˜(xβ,uβ,t)(cid:88)zαa({xβ},{uβ})(cid:89)δ(xb−xαb)δ(ub−uαb) (12)
β,b β α b
The scalar (i.e. coordinate independent) density f =f˜/g where g(x) is the determinant of the metric thus obeys
∂ f =−∂ (vaf)−Γa vcf +Γd uc∂ (vaf)−∂ (waf)+∂ (cid:2)σac∂ (cid:0)σ bf(cid:1)(cid:3) (13)
t xb bc bc ud ua ua ub c
where we have used the relationship ∂ g =ggbc∂ g =2gΓb and the fact that g doesn’t depend on u. The first
xa xa bc ba ab
three terms on the right-hand side form minus the phase space covariant derivative ∇˘(vaf) defined by Eq. (3).
II. MOMENT EXPANSION
The hydrodynamic equations are obtained from the first few moments in the orientation u of Eq. (13), the first of
which is the density field ρ:
(cid:90) (cid:42)(cid:89)n (cid:43) 1 (cid:90) (cid:89)n
ρ(x,t)≡ duf(x,u,t), uam (x,t)≡ duf(x,u,t) uam (14)
ρ(x,t)
m=1 m=1
where du=(cid:112)g(x)(cid:81) dua is the covariant orientational volume element. Multiplying the Fokker-Planck equation by
(cid:81)n uam, integratinag over u, and integrating by part as needed to eliminate derivatives of f leads to
m=1
(cid:32) (cid:42) n (cid:43)(cid:33) (cid:32) (cid:42) n (cid:43)(cid:33) n (cid:42) n (cid:43)
(cid:89) (cid:89) (cid:88) (cid:89)
∀n, ∂ ρ uam =−∇ ρ vb uam + ρ wap uam
t b
m=1 m=1 p=1 m=1
m(cid:54)=p
n (cid:42) n (cid:43) n (cid:42) n (cid:43)
(cid:88) (cid:89) (cid:88) (cid:89)
+ ρ σb (∂ σcap) uam + ρ σapσcaq uam (15)
c vb c
p=1 m=1 p,q=1 m=1
m(cid:54)=p p(cid:54)=q m(cid:54)=p,q
where ∇ is the usual covariant derivative. Since the moments are by construction contravariant tensors of order n,
Eq. (15) is explicitly covariant. It is not in general a closed set of equations for the moments, however it is possible
to make it one by expanding va, wa, and σab in powers of ua.
III. FLYING XY MODEL
A. Coarse graining
Apart from curvature, our equations of motion are identical to those from Ref. [40], and we use approximations
that are equivalent to theirs. We, however, work with the coordinates of u in the surface coordinate system rather
than its angle with a reference axis as the latter would be tricky to define on a curved surface. Let pa(x,t) = ρ(cid:104)ua(cid:105)
and qab(x,t)=ρ(cid:0)(cid:10)uaub(cid:11)− 1gab(cid:1) be the polarization field and nematic tensor field, respectively. To leading order in
2
the short range of the aligning interaction, the mean-field torque is
(cid:90)
wa = dx(cid:48)du(cid:48)(δa −uau )u(cid:48)bK(|x−x(cid:48)|)f˜(x,u,t)f˜(x(cid:48),u(cid:48),t)≈K (δa −uau )f˜(x,u,t)pb(x,t) (16)
b b 0 b b
where K =(cid:82) dxK(x) and dx=(cid:112)g(x)(cid:81) dxa is the covariant volume element. Substituting this expression as well
0 √ a
as va = v ua and σa = D (δa −uau ) in Eqs. (15) and using the covariant form of the standard 2D third order
moment c0losure ρ(cid:10)uabubuc(cid:11)=r (gabbpc+gbacpb+gbcpa)/4 [42] yields
∂ ρ+v ∇ pa =0 (17)
t 0 a
(cid:18) (cid:19)
v 1
∂ pa+ 0∇aρ+v ∇ qab =−D pa+K ρpa−qabp (18)
t 2 0 b r 0 2 b
∂ qab+ v0 (cid:0)∇apb+∇bpa−gab∇ pc(cid:1)=−4D qab+K (cid:0)papb−gabpcp (cid:1) (19)
t 4 c r 0 c
6
The hydrodynamic equations are then obtained by assuming that qab is a fast variable (∂ qab ≈ 0) and that the
t
hydrodynamic fields vary slowly (neglecting spatial derivatives of order two and higher):
∂ ρ=−∇ pa
t a
∂ pa =−v0∇aρ+(cid:18)K0ρ−D − K02 pbp (cid:19)pa+ K0v0 (cid:2)5p ∇apb−5pa∇ pb−3pb∇ pa(cid:3) (20)
t 2 2 r 2D b 16D b b b
r r
Finally we rescale times and lengths using the reorientation time D−1 and the persistence length v /D to get the
r 0 r
dimensionless hydrodynamic equations Eqs. (4).
B. Polar steady-state on a weakly curved surface
Here we assume the surface is described by a height profile h(x,y) whose irregularities have typical height H and
width W with H/W (cid:28) 1. Then the metric tensor is g = δ +f with f = (∂ h)(∂ h) ∼ (H/W)2 (cid:28) 1 and
ab ab ab ab a b
the Christoffel symbols are Γ ∼ ∂ f ∼ H2/W3. Let (cid:15) = ρ−ρ (cid:28) ρ and πa = pa−pa (cid:28) |p| be the density and
abc a bc
polarization deviations from the flat-space mean-field steady-state. We assume we are at steady-state in the polar
√
phase (κρ > 1) and that the flat-space base state is polarized along the x axis: px = 2κρ−2/κ, py = 0. We first
note that the density conservation equation 0 = ∇ pa ≈ ∇ πa +Γb pb implies πa ∼ (H/W)2p, then expand the
a a ba
dimensionless hydrodynamic equations Eqs. (4) to leading order in H/W:
2∂ π +2∂ π =−p∂ (f +f ) (21)
x x y y x xx yy
4(2κp−∂ )(cid:15)−2κp(4κp−∂ )π =κp2(4κp−∂ )f (22)
x x x x xx
−4∂ (cid:15)+κp(5∂ π −3∂ π )=κp2(3∂ f −4∂ f ) (23)
y y x x y x xy y xx
wherealltheChristoffelsymbolshavebeenexpressedintermsofthemetricusingΓ = 1(∂ g +∂ g −∂ g ). We
abc 2 c ab b ac a bc
further simplify (22) by assuming either strong polar order (κpW (cid:29)1 thus κp(cid:29)∂ ) or weak polar order (κpW (cid:28)1
x
thus κp(cid:28)∂ ). In both cases Eqs. (21)-(23) reduce to a set of anisotropic Poisson equations:
x
(cid:0)∂2+β∂2(cid:1)(cid:15)=−κp2 (cid:0)∂2f +∂2f −2∂ f (cid:1) (24)
x y 2 y xx x yy xy xy
(cid:0)∂2+β∂2(cid:1)πx =−p(cid:16)(cid:2)∂2+(1+β)∂2(cid:3)f +∂2f −2∂ f (cid:17) (25)
x y 2 x y xx x yy xy xy
(cid:0)∂2+β∂2(cid:1)πy =−p(cid:16)−∂ f +β∂ f +2∂2f (cid:17) (26)
x y 2 xy xx xy yy x xy
where β = 1 if κpW (cid:28) 1 and β = 1/3 if κpW (cid:29) 1. The solution can then be given an explicit expression in terms
of the source terms, hence the metric, using the well known Green’s function for (cid:0)∂2+β∂2(cid:1) in 2D unbounded space
√ x y
G(x,y)=log(cid:0)βx2+y2(cid:1)/(4π β), which satisfies (cid:0)∂2+β∂2(cid:1)G=δ(x)δ(y):
x y
(cid:90) log(cid:0)βx(cid:48)2+y(cid:48)2(cid:1)
u(x,y)= dx(cid:48)dy(cid:48)U(x−x(cid:48),y−y(cid:48)) √ (27)
4π β
where u is either (cid:15) or πa and U is the corresponding source. Finally, taking the derivatives appearing in the source
terms out of the integral yields Eqs. (5)-(7).
IV. CONFINED 3D ACTIVE PARTICLES
A. Equations of motion on the boundary
We consider a single 3D active particle with position r and unit orientation vector uˆ moving along a wall that
cancels the normal component of its velocity whenever it points toward the wall. The equations of motion are
dr duˆ
=v (1−nˆ⊗nˆ)uˆ, =uˆ×(η×uˆ) (28)
dt 0 dt
where v is the self-propulsion speed, nˆ is the normal to the wall at r, η is a white Gaussian noise with zero mean
0
and correlations (cid:104)η (t)η (t(cid:48))(cid:105) = 2D δ δ(t−t(cid:48)), and latin letters from the middle of the alphabet (i, j,...) label 3D
i j r ij
7
cartesian coordinates (whereas latin letters from the beginning of the alphabet label coordinates on the surface). We
now project Eqs. (28) onto the local basis (e ,e ) of the plane tangent to the surface:
1 2
dx dxa
ea· = (29)
dt dt
ea·v uˆ =v ua (30)
0 0
duˆ (cid:18)dub de du⊥ dnˆ(cid:19) dua
ea· =ea· e +ub b + nˆ+u⊥ = +v Γa ubuc+v u⊥La ub (31)
dt dt b dt dt dt dt 0 bc 0 b
ea·(cid:2)uˆ×(η×uˆ)(cid:3)=(cid:2)(δa −uau )eb−u⊥uanˆ(cid:3)·η ≡Ξa (32)
b b
√
where u⊥ = uˆ ·nˆ = 1−u ua and we have used ea·∂ e = Γa and ea·∂ nˆ = −nˆ ·∂ ea = La . The noise Ξa(t)
a b c bc b b b
defined by Eq. (32) is Gaussian with zero mean and correlations
(cid:10)Ξa(t)Ξb(t(cid:48))(cid:11)=2D (cid:0)gab−uaub(cid:1)δ(t−t(cid:48)) (33)
r
whichwerewriteasΞa =σaξbwhereξisaGaussiannoisewithzeromeanandcorrelations(cid:10)ξa(t)ξb(t(cid:48))(cid:11)=2gabδ(t−t(cid:48))
b
and
σa =(cid:112)D (cid:18)δa − uaub (cid:19) (34)
b r b 1+u⊥
Therefore
Dua = dua +Γa dxbuc =−v u⊥La ub+(cid:112)D (cid:18)δa − uaub (cid:19)ξb (35)
Dt dt bc dt 0 b r b 1+u⊥
B. Steady-state in the strong confinement regime
In the strong confinement regime uau (cid:28)1 and we linearize Eq. (35) in u (except for the intrinsic time derivative
a
whose linearization would break covariance):
dxa Dua (cid:112)
=v ua, =−v La ub+ D ξa (36)
dt 0 Dt 0 b r
Toobtainthesteady-statedensitywesubstitutethecorrespondingexpressionforv, w, σ inEqs.(15), set∂ =0, and
t
neglect third order moments ((cid:10)uaubuc(cid:11)≈0) [29]. The first three equations then read
0=−∇ (ρ(cid:104)ua(cid:105)), 0=−∇ (cid:0)ρ(cid:10)uaub(cid:11)(cid:1)+v La ρ(cid:10)ub(cid:11), 0=La (cid:10)ucub(cid:11)+Lb (cid:104)ucua(cid:105)+2Drgab (37)
a b 0 b c c v
0
Sincegab,Lab,(cid:10)uaub(cid:11)areallsymmetricandgab andLab haveonlypositiveeigenvaluesandcommutewitheachother,
Lab also commutes with (cid:10)uaub(cid:11) and the third equation reduces to
(cid:10)uaub(cid:11)= DrRab (38)
v
0
where Ra is the inverse of the shape tensor La . Next we look for a fluxless ((cid:104)ua(cid:105) = 0) solution to Eqs. (37).
b b
Substituting Eq. (38) into the middle equation and using the fluxless assumption leads to
∇ (cid:0)Rabρ(cid:1)=0 (39)
b
Wenowpickapointxontheboundaryandgotoitslocalnormalcoordinates,inwhich∇ =∂ andRab isdiagonal.
a a
In the vicinity of x the radius of curvature tensor can be written as
(cid:18) (cid:19) (cid:18) (cid:19)
R (x) 0 cosφ(x) sinφ(x)
R(x)=P−1· 1 ·P, P =
0 R (x) −sinφ(x) cosφ(x)
2
8
where R are the principal curvature radii and φ is the angle between the curvature eigenbasis at x and that at x,
1,2
which by construction verifies φ(x)=0. With these notations, we have
(cid:18) (cid:19)
∂ R +(R −R )∂ φ
∇·R(x)= 1 1 1 2 2 (40)
∂ R +(R −R )∂ φ
2 2 1 2 1
We use the Mainardi-Codazzi equations, which follow from ∂ e =∂ e , to eliminate φ:
bc a cb a
R
(R1−R2)∂2φ= R1∂1R2
∇ L =∇ L =⇒ 2 (41)
c ab b ac R
(R1−R2)∂1φ= R2∂2R1
1
Substituting in Eq. (40), we find
(cid:18) (cid:19)
R ∂ [ln(R R )]
∇·R(x)= 1 1 1 2 =−R·∇(lnK) (42)
R ∂ [ln(R R )]
2 2 1 2
where K ≡ 1/(R R ) = detL is the Gaussian curvature. The final result is explicitly covariant and holds in any
1 2
coordinate system. Applying it to Eq. (39) yields
0=ρ(cid:0)∇ Rij(cid:1)+Rij(∇ ρ)=ρRij∇ ln(cid:16) ρ (cid:17) (43)
j j j K
Integrating and using the normalization constraint ρdS =N where N is the number of (independent) particles on
(cid:118)
the boundary and the Gauss-Bonnet theorem KdS =4π for a closed surface with no handles leads to
(cid:118)
NK
ρ= (44)
4π
Plugging this result back into Eqs. (37) proves the existence of a fluxless steady-state that verifies Eqs. (8).
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