Table Of ContentSwitching of Nonequilibrium Depletion Force Caused by Blockade Effect
S.P. Lukyanets∗ and O.V. Kliushnychenko
Institute of Physics, National Academy of Sciences, Prospect Nauky 46, 03028 Kyiv, Ukraine
Theconcentration-dependentswitchingofthenon-equilibriumdepletionforcesbetweenobstacles
in an interacting Brownian gas flow is presented. It is shown that this switching is caused by
the blockade effect for the gas particles. With increasing equilibrium gas concentration, the gas
5 particles blockade causes the obstacle wake inversion (trace profile “turn-over”)that, in turn, leads
1 to the change of sign of dissipative interaction. Some non-linear effects such as formation of a
0 cavity-like sparse wake behind the obstacle and the dissipative pairing effect are discussed briefly.
2 Theresults are obtained within the lattice gas model in themean-field approximation.
r
a
M I. INTRODUCTION powerlaw,e.g.,[7–11]. However,for anobstaclein agas
flow,thegasperturbationprofilecantakereversedstruc-
1 turewithanextendeddenseregionaheadoftheobstacle
The motion of impurities in a Brownian gas is accom-
and a localized low-density region in its wake [22]. It
panied by its density perturbation (impurity trace, or
] should be noted that similar effect in one-dimensionwas
h wake) that, in turn, induces additional non-equilibrium
c correlations between the impurities. These correlations, discussedin[23]. Thisswitchingofthe profiledirectivity
e is caused by the effect of particles blockade ahead of the
called dissipative or non-equilibrium depletion forces [1–
m obstacle. The blockade is due to hard-core-like interac-
4], are responsible for the coherent part of the collective
- frictionforceaswellasforpossibledissipativestructures tion between the particles and is possible if the flowing
at of impurity ensemble. gas density is high enough.
t As the inversion of density perturbation takes place
s In equilibrium, the depletion interaction is usually
. for arbitrary number and configuration of obstacles, see
t short-range;thespatialrangeisoftheorderofthedeple-
a [22], we may expect that this inversion has to affect the
tion agent characteristic length scale [5, 6]. In contrast,
m dissipative interaction between the obstacles.
the non-equilibrium forces between the impurities may
- exhibitlong-rangebehaviorduetoalong-livingdiffusion The aim of this paper is to show that trace inver-
d
n trace induced by their motion [7–11]. In addition, such sion is responsible for the change in sign of the induced
o forces often have unusual properties, e.g., they violate non-equilibrium interaction between the obstacles, e.g.,
c the Newton’s third law [1, 2, 12, 13]. switching of the effective attraction to repulsion.
[ It is clear that specific properties of the dissipative in- We consider the simplest model of a lattice gas with
2 teraction are determined by the structure and the shape twoobstacleswhosesizesaremuchlargerthanthelattice
v of the perturbation profile, which, in turn, significantly constant. Weusethemean-fieldandthelong-wavelength
5 depends on the interaction between the gas particles. approximations, neglecting short-range correlations and
7 fluctuations in the gas, see [22].
The presence of inter-particle interaction, even short-
5
7 range one (as in a lattice gas, when a lattice site can be
0 occupied by only one particle), is responsible for a num-
. berofunexpectedkineticeffects,e.g.,“backcorrelations” II. TRACE INVERSION IN THE LATTICE GAS
1 MODEL
effect [14], drifting spatial structures [15–17], effects of
0
5 “negative” mass transport [18–20], dissipative pairing ef-
1 fect for tracers passing through a lattice gas [21]. As was shown in [18, 22], the problem of an obstacle
: Animportantmechanismoftenunderlyingsuchkinetic in a lattice gas flow can be considered as a limiting case
v
i phenomena is a blockade effect. As was shown in [22], ofatwo-componentgas,oneofthecomponentsisatrest
X the blockade effect implicates significant changes in the while another one is flowing in a uniform external field.
r form of an obstacle trace. The trace structures for the We resort to the simplest model of a two-component
a
casesofimpurity movinginalattice gasandfixedimpu- latticegas,wheneachlatticesitecanbeoccupiedonlyby
rity or obstacle in a gas flow do not coincide in general. one particle, see [14]. The kinetics of multi-component
In particular, the obstacle trace can take atypical form lattice gas is defined by the jumps of particles to the
characterized by switching of its direction depending on neighboring vacant sites. The variation of the i-th site
the equilibriumgasconcentration. Usually,the profileof occupancy by the particles of α sort during the time in-
the gas perturbation induced by moving inclusion has a terval ∆t, τ0 ≪∆t≪τl (τ0 is the duration of a particle
dense localized region ahead of the inclusion and a long jumptoaneighboringsite, τl beingthe lifetime ofapar-
low-density tail behind it, which damps according to a ticle on a site), is described by the standard continuity
equation (see, e.g., [14, 24])
nα(t+∆t)−nα(t)= Jα −Jα +δJα, (1)
i i ji ij i
∗Emailaddress: [email protected] j
X(cid:0) (cid:1)
2
wherenα =0,1arethelocaloccupationnumbersofpar-
i
ticlesαatthei-thsite,Jα =ναnα 1− nβ ∆tgives
ij ij i β j
the mean number of jumps (of α p(cid:16)articles from(cid:17)site i to
P
a neighboring site j per time ∆t, β(α) indexes particle
species), να =να is the mean frequency of these jumps.
ij
ThetermδJα = (δJα−δJα)standsfortheLangevin
i j ji ij
sourcethatisdefinedbythefluctuationsδJα ofthenum-
ji
P
ber ofjumps betweensites j andi during∆t [24]. These
fluctuations are caused by the fast, as compared to the
timescale∆t,processesandwillbeneglectedforsimplic-
ity. It means that we neglect fluctuation-induced forces
that may be appreciable for closely located inclusions.
In what follows we consider only two components, de-
noted by n and v. In the absence of external fields for
the regular lattice we suppose that νn = ν = const for
ji
the mobile component n, while the component v is as-
sumed to be at rest, νv = 0. The presence of driv-
ji
ing field leads to asymmetry of particle jumps. Suppos-
ing the activation mechanism of jumps and the driving
field G to be weak, the frequency may be written as
νn ≈ν[1+(G,r −r )/(2kT)],orν± ≈ν±δν,whereν+
ji i j
denotes the jump frequency along the field, ν− – against
it, δν = νa|G|/(2kT) (a is the lattice constant), condi-
tion a|G|/(2kT)<1 being satisfied.
Equations for the average local occupation numbers
canbeobtainedfromEqs.(1)usingthelocalequilibrium
approximation (Zubarev approach) [24, 25] which coin-
cides,inourcase,withthemeanfieldapproximation[26].
Introducing time derivatives [27], in the long-wavelength
approximation, see [15–18], macroscopic kinetics of the
mobile component n is given by the equation
∂ n=∇2n−∇(v∇n−n∇v)−(g,∇)[n(1−v−n)], (2)
τ
where n = n(r,τ) and v = v(r) are the average occu-
pation numbers of the two components at the point r,
g = aG/(2kT). Here, we have introduced dimensionless
spatial coordinate r/a → r and time τ = νt, and ∂
τ Figure1: Thetwo-dimensionalequilibratedconcentrationdis-
stands for the partial time derivative. tributionsδn(x,y)=n(x,y)−n0(meanoccupationnumbers)
Second term on the right-hand side of Eq. (2) is a re- of the lattice gas particles near the inclusion at the different
ducednonlinearmixingflowwhichdescribesmutualdrag values of equilibrium concentration n0 are presented on con-
ofonecomponentbyanotherandarisesasaresultofthe tourplots: (a)n0 =0.2,(b)n0 =0.5,(c)n0 =0.8. Theexter-
particledistinguishabilityandthelocalrepulsion(due to nalfield g (|g|=0.5) is directed along thex-axis;theimper-
excluded volume constraint) in a lattice gas [18]. This meable (n0 =1) circular inclusions of radius R=7 (in units
of a) are placed at the origin (denoted by circles). The gray
flow leads to the array of anomalous diffusive transport
backgroundcorrespondstotheequilibriumgasconcentration
effects,e.g.,thedrageffect,formationofthedriftingspa-
n0 for every contour plot, in consistence with the colorbar.
tial structures [15–17], effects of “negative” mass trans-
Dashed contour lines underline the exact inversion between
port [18–20], and induced long-time correlations [28].
profiles (a) and (c),while thecorresponding one-dimensional
Inordertodemonstratethetraceinversioneffectmen- profiles δn(x) = n(x,y = 0) −n0 show the “switching” of
tioned above we now consider a single macroscopic ob- asymptotic behavior.
stacle of a circular form in a flow of driven lattice gas
particles. Fig. 1 presents the two-dimensional numerical
solutions of Eq. (1) in the mean-field approximation for At high concentrations (n > 1/2), the profile takes
0
different equilibrium concentrations n . unconventional form with an extended dense region in
0
For low concentrations (n < 1/2), the structure of front of the inclusion and a localized depleted one in its
0
theprofileistypical[1,3,4,11]andischaracterizedbya wake, resembling the form of a cavity, Fig. 1c or Fig. 2.
lengthy depleted region behind the inclusion (wake) and Hence, the main part of the density perturbation is now
a localized dense region in front of it, Fig. 1a. shiftedto theregioninfrontofthe obstacleanddirected
3
upstream.
Thisbehavioriscausedbytheblockadeeffectofthegas
particles flow due to the short-range repulsion between
them (the condition that a lattice site can be occupied
byonlyoneparticle). Forsufficientlyhighconcentrations
n , gas particles have no time to leave the blockade zone
0
aheadoftheobstaclevialateraldiffusionand,asaresult,
the dense region ahead of the obstacle has to grow. The
blockadeeffectisnon-linearandbecomessignificantnear
theobstaclesurfaceand/orforlargeobstacleswhosesize
is much larger than the lattice constant.
Thetraceinversiondirectlyfollowsfromthesymmetry
of Eq. (2) that can be represented in the form
∂ n=∇(h∇n−n∇h−gnh), (3)
τ
where h = 1−m−v is the vacancy concentration, i.e.,
concentration of the empty lattice sites. Equation (3) is
invariant under transformation n↔h, g ↔−g and can
Figure 2: The example stationary wake (numerical result)
easily be rewritten as
whoseprofilem(x,y=0)isoftheformofBlochwall,describ-
ingthecavitybehindtheinclusion(blackcircle);equilibrium
∂ h=∇(n∇h−h∇n+gnh). (4)
τ medium concentration m0 = 0.63, field |g| = 0.5, inclusion
radius R=7.
Equation(3) describesthe kinetics of the gasparticles
in sweeping field g at equilibrium gas concentration n
0
while Eq. (4) corresponds to the vacancy kinetics in the
opposite field −g at the equilibriumconcentrationof va-
where ϑ is the angle between g and r, r =|r|; constants
cancies h =1−n . The tail of the obstacle trace in the
0 0 α = α (R) are determined from boundary conditions
n n
gasflow is characterizedby a depleted gasregionbehind
on the obstacle surface, β =(1/2−n )/|1/2−n |=±1,
0 0
the obstacle, whereas the trace in the corresponding va-
and q =|1/2−n ||g|.
0
cancy flow (which has opposite direction) has a depleted
tailofvacanciesthatcorrespondstoadenseregionofgas In the particular case of n = 1/2, δn is an odd func-
0
particles. tion of x, see (5), and the density perturbation becomes
For distribution v(r) with mirror symmetry along the asymmetrical and extended in the ±x directions, see
field direction r k g, i.e., v(r ) = v(−r ), each density Fig. 1b. As was shown in [22], its asymptotic behavior
k k k
perturbation profile δn(r,n ) = n(r,n )−n at a given damps according to the power law δn(r) ∼ −(g,r)R/r2
0 0 0
concentrationn canbe obtainedby the inversiontrans- that is similar to the scatteredelectrostatic potential for
0
formation of the profile δn(r,1 − n ) at concentration a dielectric particle in a uniform electric field.
0
1−n , that is expressed by relation
0 Thelinearflowapproximationgivesthequalitativede-
scription of the trace inversion effect and the character
δn(r +r ;1−n )=−δn(−r +r ;n ), (5)
k ⊥ 0 k ⊥ 0
of the asymptotic behavior of δn. In the particular case
r andr arelongitudinalandtransversecomponentsof of the point-like inclusion the method gives δn ∼ r−3/2
k ⊥
r, with respect to the external field g. that is in satisfactory agreement with numerical results
andcoincideswithasymptoticbehavioroftherelaxation
For relatively small obstacle with size R, the asymp-
of the trace induced by a moving intruder presented in
totic behavior of the gas density perturbation δn(r) at
[9].
|r| ≫ R as well as the trace inversion effect can be
obtained in the linear approximation, see [22]. Far However, as was mentioned above, non-linear effects
from the obstacle, one may assume that the distribution maybesignificantneartheobstaclesurface. Fig.2shows
n=n +δnshouldweaklydifferfromtheequilibriumone the role of this non-linear effect in the formation of the
0
n . Thus, it is possible to linearize Eq. (3) by neglecting density perturbation profile δn. The behavior δn near
0
term with (δn)2. For the case of two-dimensionallattice the surfacehasapronouncedstep-likecharacterthatde-
gas the trace asymptotic δn(r) for a single obstacle of scribes a localized dense region ahead of the obstacle at
radius R is given by (see [22]) n <1/2ora localizeddepletedregionbehindit(resem-
0
bling the form of a cavity) at n >1/2.
0
δn(r)∼e−qr(1−βcosϑ)×
Intuitively,suchbehaviorcanbe expected. Inthepar-
× αn (qr)−21 + 4n2−1(qr)−32 +··· cosnϑ, (6) ticularcaseofalatticegaswithnoobstacleEq.(3),with
8 v ≡0, admits the kink-like solution.
n=0 (cid:20) (cid:21)
X
4
III. NON-EQUILIBRIUM FORCE IN A form for the equilibrium depletion force that can be ex-
LATTICE GAS MODEL pressed in terms of the free energy of a lattice gas with
the volume occupied by impurities being excluded [1, 2].
In the non-equilibrium case, there are various ways to
In the non-equilibrium case the dissipative or non-
introduce dissipative force or interaction between inclu-
equilibriumforcecanbewritteninaformsimilartothat
sions via Brownian gas environment. The ways are not
of (7), see [1, 2]
equivalenttoeachotherandmayleadtodifferentresults
in general, see [2]. In particular, there is the Smolu-
chowski approach, the approaches based on a master Fj(t)=− ∇RjV(ri−Rj) δhniit
equation, free energy, etc., see [1, 2, 29, 30]. In this pa- Xi (cid:2) (cid:3)
per, we use the approach based on a master equation, →− n(r)δhn(r)i dr, (12)
t
see[2],thatiscloseto the oftenexploitedmethodoffast ZS
variableelimination,see,e.g.,[31–33]. Consideringinclu-
whereδhn i =hn i −hn iis the densityperturbationof
sions as particles of a heavy gas component with given i t i t i
the lattice gas and
interaction potential between them and the original gas
particles(thelightcomponent),theforce(effectiveinter-
actionpotential) between the inclusions canbe obtained hniit = {niρ({n},t)}. (13)
by averaging it over the light component. X{n}
In the equilibrium case, the average force acting on
ρ({n},t) is the probability of finding a given occupancy
the jth inclusion centered at the lattice site R can be
j configuration{n}atthetimet,whichsatisfiesthemaster
formally written as (see [2] for details)
equation for hopping process (see, e.g., [34]) with initial
conditions corresponding to the equilibrium distribution
Fejq =− ni∇RjV(ri−Rj) (9).
* + The total dissipative force, Eq. (12), exerted on an
i
X
inclusion is defined by the induced density perturbation
=− hnii∇RjV(ri−Rj) (7) (obstacletrace)andcanbeconventionallyrepresentedby
Xi two parts. The first part is individual friction force on
where n =0,1 is the occupation number of site r . The theisolatedinclusion. Thesecondparttakesintoaccount
i i
jth inclusion is modeled by the interaction potential V collective,orcoherent,contributionthatisrelatedtothe
between its center and the gas particles. dissipative interaction between the obstacles, caused by
their mutual influence via their traces.
hnii= {niρ({n},0)} (8) In what follows, we will be interested in the case of
{Xni} steady-state non-equilibrium interaction, i.e., the limit-
ing case t→∞.
istheaveragedoccupationnumber(gasconcentration)of To describe the time behavior of averaged occupation
the ith site and ρ({n},0) is the equilibrium probability, numberhn i weexploitanapproachbasedonthekinetic
i t
or statistical operator in matrix representation [24], of equation for dynamical variable n , Eq.(1), and use the
i
finding a given occupancy configuration {n} results for the average occupation number n obtained in
the previous section.
ρ({n},0)=Z−1exp(−H{n}/kT), (9)
Z = exp(−H{n}/kT), where H{n} is the lattice
{n} IV. FORCE SWITCHING
gas-inclusions Hamiltonian
P
H{n}=H {n}+ V(r −R )n , (10) Inthis sectionwe willshowthe changeofsignofdissi-
0 i j i
pative interaction between two obstacles, that is caused
i,j
X bytheirtraceinversion. First,wequalitativelyshowthat
and H {n} is the lattice gas Hamiltonian without inclu- suchinteractionswitchingwithincreasinggasconcentra-
0
sions. tion n0 is a direct consequence of the system symmetry
In the continuum limit and for inclusions given by a (5). Todemonstratethispropertyweresorttothecaseof
hard-core-type potential, the expression for the average two inclusions located perpendicular to the gas flow, see
force takes the custom form (see, e.g., [29]) Fig.3. Forsimplicity,theinclusionsaregivenbysmooth
distributions with compact carrier v (r) = v(r − R ),
j j
Feq =− n(r)hn(r)idr, (11) the inclusions’ centers are located at points R1 =(0,y0)
j and R = (0,−y ). In this case, the force exerted
ZSj 2 0
on the jth inclusion by gas perturbation δn(r) reads
whereS isthe jthinclusionsurfaceand n(r)is its exte- F = δn∇v dr. Toshowthepossibilityoftheforcesign
j j j
riornormalatthepointr. Expression(7)givesstandard changing, it is enough to consider only the y-component
R
5
Figure3: Transversealignment. Theparticleconcentrationdistributionsatthreedifferentregimesofthedissipativeinteraction:
(a)n0 =0.2—effectiverepulsion(F1y2 >0,F2y1 <0,|F2y1|=|F1y2|),(b)n0 =0.5—nointeraction(F2y1 =F1y2 =0),(c)n0 =0.8
— effective attraction (F2y1 > F1y2). The external field g (|g| = 0.5) is directed along the x-axis; the impermeable (n0 = 1)
circular inclusions are of the radius R = 7 (in units of a); the distance between inclusion centers d = 2y0 = 4R. The gray
background corresponds to theequilibrium gas concentration n0 for everycontour plot, in consistence with thecolorbars.
of the force Fy(n ) exerted on an inclusion at equilib-
j 0
rium gas concentration n . Taking into account (5) one
0
can obtain
Fy(n )= δn(r;n )∂ v (r)dr=−Fy(1−n ). (14)
j 0 0 y j j 0
Z
In the particular case of half-filled medium (n = 1/2)
0
Fy(1/2) = −Fy(1/2) ≡ 0, i.e., the effective interaction
j j
between the obstacles vanishes.
Wenextconsidernumericallythewake-mediatedforce
between two obstacles for two configurations: the ob-
stacles are placed parallel and perpendicular to the gas
flow. Thetotalforceexertedonagivenobstacleincludes
thepartassociatedwithindividualfrictionforceandthat
associatedwiththeinfluenceofanotherobstacle. Tosep- Figure 4: The dissipative forces F1y2(n0) and F2y1(n0) as a
arate out the inter-obstacle contribution from the total functions of the equilibrium medium concentration n0 in the
case of transverse alignment. The values marked with circles
dissipative force we consider the quantity [1]
correspond to theprofiles presented on Fig. 3.
F =F −F0 = [δn(r,R ,R )−δn(r,R )]∇v (r)dr,
ij i i i j i i
Z (15) blocksthegasflow,sothattheregionbetweentheobsta-
where F is the total force acting on the ith obstacle in cles becomes depleted. As Fig. 4 suggests,this collective
i
the presence of jth one and F0 is its individual friction blockade effect of gas particles leads to the effective at-
force. i traction between obstacles in a dense medium, F2y1 > 0
and Fy <0.
12
Transverse alignment (Fig. 3). One can see from Thus,whenthegasconcentrationn increasesthe dis-
0
the symmetry of this configuration that the forces two sipative interaction between the obstacles switches from
obstacles exert on each other are equal and opposite, repulsiontoattraction. Inaddition,the non-linearinter-
Fy = −Fy, i.e., their interaction is always Newtonian, obstacleattractionmaycharacterizethepairingeffectac-
12 21
see Fig. 4. companiedby the creationofcommonperturbationcoat
Atlowequilibriumconcentrations(n0 <1/2),Fig.3a, around the obstacles. The effect of a similar nature was
the dissipative interaction manifests itself as an effec- obtained earlier in [11] for two driven intruders.
tFi1vy2e>re0p,uslseieoFnibg.et4w.eQenuatlhiteatoivbesltya,ctlehsi,ssrienpcuelsFio2yn1i<ss0imapnldy theInintchleusnio0n=s 1v/a2nicshasees,thFe1y2eff=ecFti2vy1e=int0e,raicrtrieosnpebcettivweeeonf
explained by the overlapof density coats aroundthe ob- the inter-inclusion distance.
stacles that leads to the onset of a dense region between
them acting like a repelling barrier, see Fig. 3a. Longitudinalalignment (Fig.5). Inthecaseoflowcon-
In contrast, at n >1/2, the overlap of the individual centration(n <1/2),thetypicalsituationforBrownian
0 0
density perturbation coats of the obstacles results in the systems is shown. An inclusion falling on the depleted
formationoftheextendeddensezoneaheadofthemthat wakeinducedbyanotherinclusioniseffectivelyattracted
6
Figure 5: Longitudinal configuration. The lattice gas parti-
Figure 6: (a) Concentration dependence of the dissipative
cleconcentrationdistributionsn(x,y)correspondingtothree forces F0x(n0), F1x(n0), and F2x(n0). (b) The forces F1x2(n0)
d—iffeeffreencttivreegaimttersacotfiotnhe(|dFi2sx1s|ip>ati|vFe1x2i|n)t,e(rbac)tnio0n:=(a0).5n—0 =an0t.i2- aconndfiFg2ux1r(ant0io)n,.acTtihnegvbaeltuweesenmathrkeeidncwluitshioncsiricnletshceolrornegspitounddintaol
Newtonian interaction (F2x1 = F1x2), (c) n0 = 0.8 — effective theprofiles presented on Fig. 5.
repulsion (|F2x1| < |F1x2|). The external field g (|g| = 0.5) is
directed along the x-axis; the impermeable (n0 =1) circular
inclusions are of the radius R=7 (in units of a), their posi-
tions being marked with black circles; the distance between V. CONCLUSIONS
inclusion centers d = 2x0 = 10R. The gray background cor-
responds to the equilibrium gas concentration n0 for every In this paper we have shown the switching of the non-
contour plot, in consistence with thecolorbar.
equilibrium depletion force between fixed impurities or
obstaclesimmersedinaflowingandinteractingBrownian
gas. To this end we have considered the simplest model
of a lattice gas with two obstacles whose sizes are much
toitsincethefrictionforceinthemoredepletedregionis
larger than the lattice constant. We also used both the
weaker[1,3]. Thistypeofeffectiveinteractionisreferred
mean-field and the long-wavelength approximations, ne-
to as the wake-mediated [35]. As Fig. 6b suggests, the
glecting short-range correlations and fluctuations in the
second obstacle does not practically affects the first one,
Fx ≈ 0. In contrast, at high concentrations n > 1/2, gas.
12 0
thesecondobstacledoesnotfeeltheinfluenceofthefirst Thenon-equilibriuminteractionbetweentheobstacles
one, Fx ≈ 0, whereas the first obstacle comes under is caused by the density gas perturbation or wakes in-
21
theexcesspressureofthe densegasregioncreatedahead duced aroundthem by the gas flow. The force switching
of the second one due to the blockade effect. As a re- isduetotheblockadeeffectofthegasparticlesandman-
sult, the effective interaction changes its sign switching ifestsitselfbychangingthedirectionofforcetoitsoppo-
from attraction to repulsion, Fig. 6b. In particular case site. With increasing the equilibrium gas concentration,
of n = 1/2, the effective interaction between the inclu- the blockade effect provokes the wake inversion (or the
0
sions becomes strictly anti-Newtonian, Fx = Fx 6= 0 wake profile turn-over) that, in turn, leads to switching
12 21
(see Fig. 6b), which holds for arbitrary inter-inclusion of non-equilibrium depletion interaction, e.g., from the
distance. effective repulsion to attraction.
In contrast to the equilibrium case, the non-
Note that for a dense gas in the blockade regime, the
equilibriumdepletion forceexhibits long-rangecharacter
second obstacle “pushes” the first one upstream, thus re-
due to extended wake tail behind the obstacles which
ducing the total friction force Fx exerted on the first
1 damps accordingto a power law which canbe estimated
obstacle, Fig. 6a.
in the linear approximation. However, non-linear effects
Thedissipativeinteractionbetweentheinclusionsnat- become significant near obstacle. The density pertur-
urally vanishes in the limit of strongly sparse (empty) bation near its surface exhibits a pronounced step-like
medium n → 0, due to wake depletion. The same is profilethatdescribestheformationofasparsecavity-like
0
true in the total jamming limit n →1. regionofthegasbehindoradensedrop-likeregionahead
0
7
the obstacle in the cases of high and low equilibrium gas take into account the fluctuation-induced (Casimir-like)
concentration, correspondingly. In turn, this can lead to forces, see, e.g., [29, 36–41].
the effect of dissipative pairing for two closely located
obstaclesthataccompaniedbythe creationofacommon
coat of gas density perturbation around them. However,
Acknowledgments
with using the mean-field approximation we lose infor-
mation onshort-rangecorrelationswhich may be signifi-
cant near the obstacle surfaces and in the case of closely We are grateful to A.A. Chumak, B.I. Lev, and
locatedobstacles,see [14, 21]. Moreover,neglecting fluc- V.V. Gozhenko for helpful discussions and comments on
tuationsinagas,e.g.,thetermδJα inEq.(1),wedonot the manuscript.
i
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