Non-local response in a lattice gas under a shear drive
NNon-local response in a lattice gas under a sheardrive
Tridib Sadhu
Institut de Physique Th´eorique, CEA/Saclay, F-91191 Gif-sur-Yvette Cedex, France.
Satya N. Majumdar
Univ. Paris-Sud, CNRS, LPTMS, UMR 8626, Orsay F-01405, France.
David Mukamel
Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel.
Abstract.
In equilibrium, the effect of a spatially localized perturbation is typicallyconfined around the perturbed region. Quite contrary to this, in a non-equilibriumstationary state often the entire system is affected. This appears to be a genericfeature of non-equilibrium. We study such non-local response in the stationary stateof a lattice gas with a shear drive at the boundary which keeps the system out ofequilibrium. We show that a perturbation in the form of a localized blockage at theboundary, induces algebraically decaying density and current profile. In two examples,non-interacting particles and particles with simple exclusion, we analytically derive thepower-law tail of the profiles.
Keywords : Non-equilibrium, Shear drive, Long-range correlation, Multi-lane exclusionprocess.
Submitted to:
J. Phys. A: Math. Gen. a r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug on-local response in a lattice gas under a shear drive
1. Introduction
Systems far from equilibrium often display some novel and unexpected features thatare in striking contrast to equilibrium [1, 2, 3]. One of the intriguing features ofnon-equilibrium stationary state is the presence of long-range correlation at genericparameter values [4]. It is well known that non-equilibrium stationary state of systemsevolving under a dynamics which conserves some variables (such as energy, momentumor density) display slow decay of correlations, even when the dynamics is local [5]. Thisis observed in several studies of driven diffusive systems [6, 7, 8, 9, 10, 11, 12, 13, 14].A consequence of the long-range correlation is that a local perturbation can lead toglobal changes in the one-point function, like density profile. Such non-local responsehas been demonstrated earlier in a system of diffusive particles [15] and also in a lattice-gas with hard-core repulsion (simple exclusion) [14, 16]. In this paper we investigate ascenario different from those studied earlier: the non-equilibrium stationary state of alattice-gas under a shear drive along the boundary. Our purpose is to analyze the effectof a localized perturbation on this stationary state.An alternate motivation comes from studies in fluid medium, where a shear flowlocalized at the boundary often leads to non-local changes in the liquid structure[17, 18, 19, 20]. Here the long-range effect is related to the slow modes of the capillarywave which originates essentially from momentum conservation [20]. A natural questionto ask is, what happens when there is no inertia, and hence no convective currentwhich can carry the information of the boundary flow to the bulk. Does the long-rangeeffect survive? Such momentum non-conserving dynamics are important in describingprocesses within biological cells or mobility of individual cells — the world of lowReynolds number.Our study is based on a two-dimensional lattice gas with cylindrical boundarycondition. The shear drive is at the bottom lane which biases the motion of particlesin one direction along the lane (see Figure 1). In the bulk, there is no drive andthe particles move symmetrically across any bond. For our theoretical analysis weconsider two cases: one, where the particles are independent of each other, and thesecond, where the particles interact with simple exclusion such that no two particlescan occupy the same site at the same time. In the latter case, the model is essentiallya symmetric simple exclusion process (SSEP) coupled to a totally asymmetric simpleexclusion process (TASEP) at the boundary. The particles don’t have momentum andthe only conserved quantity is the total particle number.In both examples considered, the boundary shear drive, in itself, does not induceany current in the bulk and the average density profile remains the same as in theabsence of drive. However, if a defect is introduced on a single bond at the boundarylane which hinders the motion across the bond, particles on the boundary lane goaround the blockage by hopping to neighboring lanes. This way the boundary currentis transmitted to the bulk. The defect at the boundary lane can be considered as alocal perturbation on the non-equilibrium stationary state with shear drive. What is on-local response in a lattice gas under a shear drive /r with the distance r from the blockage, whereas the current decays as 1 /r . Weargue from a hydrodynamic description that the power-law decay is the same even forother short-range interactions among particles.Our result has direct relevance to several other systems. For example, the modelis closely related to the directed motion of molecular motors along a lane made ofcytoskeletal filament [21]. The motors are actively driven in one direction when theyare attached to a cytoskeletal filament. Due to the thermal fluctuations, the motorsusually unbind after a certain time from the filament. While unbound, the motorsdiffuse freely in the surrounding fluid until they eventually re-attach to the filament.Lipowsky and co-workers have modelled this as a TASEP on a single lane coupled toa SSEP on the surrounding multiple lanes [22]. Our result shows that a defect onthe cytoskeletal filament generates current around the defect with a slowly decayingamplitude. This should be accessible to experiments in biological transport.Our model is also an example of coupled multi-lane exclusion process. In particular,the model is a multi-lane generalization of a well known model of one-dimensionalTASEP with a slow bond, first studied by Janowsky and Lebowitz in 1992 [23]. Anexact characterization of the stationary properties of the J-L model is still elusive[23, 24, 25, 26]. There have been several other studies on the multi-lane transport[27, 28, 29]. They were mostly introduced to model biological transport [21, 22],transport of spins in quantum systems [30], macroscopic clustering phenomena [31],and also in vehicular traffic [32, 33, 34, 35]. In the case of two lanes with opposite bias,competing currents on the coupled lanes often produce complex dynamical behavior andrich phase diagram [36, 37, 38, 30, 39, 40, 41, 42, 34, 43, 44, 45]. Particularly relevantto our model is the work in [44, 46] where a single lane asymmetric exclusion processis coupled to a single lane symmetric exclusion process. The idea of surface drive in amulti-lane lattice-gas has also been used in kinetic Ising model for simulating magneticfriction [47, 48, 49].We present the results of our work in the following order. In section 2, we introducethe model for the two cases: independent particles and particles with simple exclusion.In section 3, we present a detailed analysis of the independent particles case. In section 4, on-local response in a lattice gas under a shear drive
2. The model
We consider a lattice gas on a [0 , L − × [0 , M −
1] square lattice with cylindricalboundary condition, i.e. , with periodicity along the x direction and reflecting boundaryin the y direction (see Figure 1). Particles in the bulk diffuse symmetrically by jumpingto the neighboring sites. We consider two cases: (a) independent particles and (b)particles with hard core repulsion (simple exclusion). In the latter case, a jump isallowed only when the destination site is empty. At the boundary lane y = 0, there isa shear drive which forbids anti-clockwise jumps along the lane. In addition, there is adefect bond on this lane where the jump rate is slower than the rest of the lattice. Thetime scale is set by taking the jump rate across this slow bond as 1 − (cid:15) , and across therest of the bonds as 1.In summary, the jump rates are the following:(i) Across the bond between sites ( L − ,
0) and (0 , − (cid:15) , where 0 ≤ (cid:15) ≤ y = 0 lane, the clockwise jump rate is 1. Anti-clockwise hops are forbidden.(iii) For any other bond in the y > y = M − y = M −
2. All of them are with rate1.The total number of particles is conserved at any time.
3. Independent particles
In absence of the boundary drive, i.e. , when the jump rates everywhere are symmetric,the dynamics satisfy detailed balance, and the stationary state is in equilibrium. It iseasy to see that the average density of particles per site is uniform across the lattice,and equal to ρ = N/LM , where N is the total number of particles.When the boundary drive is switched on, the asymmetric jump rate drives a particlecurrent along the boundary lane. Because of the slow jump rate across the defect bond,there is a density gradient of particles around it. This induces a diffusive current in on-local response in a lattice gas under a shear drive − (cid:15) L − ,
0) (0 , xy Figure 1.
The model on a square lattice with periodic boundary condition along the x -axis, and reflecting boundary condition along the y -axis. The jump rate of particlesin the bottom lane is totally asymmetric, with anti-clockwise jumps forbidden. Theslow bond is between sites ( L − ,
0) and (0 , Figure 2.
A streamline plot of the stationary state current of non-interacting particlesgenerated in a Monte Carlo simulation on a 100 ×
100 square lattice. The figure iscentered around the broken bond placed between the sites ( − ,
0) and (0 , the neighboring lanes. A stream line plot of the currents generated using Monte Carlosimulation is shown in figure 2. We have not shown the current on the driven lane as itovershadows the diffusive current on the same scale.In the stationary state, the density is maximum at the site ( L − , , ρ .An example of the stationary density profile generated by Monte Carlo simulation isshown in figure 3. In the following, we shall show that the density per site, as well as on-local response in a lattice gas under a shear drive Figure 3.
The stationary state density profile of the independent particles diffusingon a 600 × − ,
0) and (0 , . the induced current decay algebraically with the distance from the defect bond.For simplicity we take M → ∞ , and consider a semi-infinite lattice. Let us definethe density φ t ( x, y ) as the ensemble averaged number of particles at site ( x, y ) at time t . Following the dynamics given in section 2, it is easy to write down the time evolutionof φ t ( x, y ). Using the rules ( i ) and ( ii ), for y = 0 we get ∂φ t ( x, ∂t = φ t ( x − , − φ t ( x, φ t ( x, − (cid:15)φ t ( L − ,
0) [ δ x, − δ x,L − ] . (1)The two Kronecker delta functions are due to the slow bond between the sites ( L − , , − , y ) ≡ ( L − , y )and ( L, y ) ≡ (0 , y ). Similarly, using the rule ( iii ) for y > ∂φ t ( x, y ) ∂t = φ t ( x − , y )+ φ t ( x +1 , y )+ φ t ( x, y +1)+ φ t ( x, y − − φ t ( x, y ) . (2)In the stationary state where the time derivative vanishes, the equations yield φ ( x − ,
0) + φ ( x, − φ ( x,
0) = (cid:15)φ ( L − ,
0) [ δ x, − δ x,L − ] for y = 0 (3)∆ φ ( x, y ) = 0 for y >
0, (4)where we dropped the time index, and also defined discrete Laplacian ∆ by,∆ φ ( x, y ) = φ ( x − , y ) + φ ( x + 1 , y ) + φ ( x, y + 1) + φ ( x, y − − φ ( x, y ) . (5)The solution of (3) and (4), with the boundary condition that far from the driven lanethe stationary density φ ( x, y ) approaches the global average ρ , determines the stationaryprofile. on-local response in a lattice gas under a shear drive L − (cid:80) x φ ( x, y ) is the same as that in absence of the drive. This can be verified easilyfrom the stationary state equations above. This property will be used later in thederivation.Equations (3)-(4) are a set of coupled linear equations, and their solution can bedetermined exactly. To begin with, we consider (cid:15)φ ( L − ,
0) = Q as a free parameter.Its value will be determined self-consistently at the end of the calculation.Due to the cylindrical boundary conditions, the stationary solution is periodic inthe x coordinate, so that, φ ( x + L, y ) = φ ( x, y ). Then, the normal modes of the densityprofile are the Fourier transform g ( n, y ) = 1 L L − (cid:88) x =0 e − iω n x φ ( x, y ) , (6)with ω n = 2 πn/L and n = 0 , , · · · , ( L − g ( n,
1) + (cid:2) e − iω n − (cid:3) g ( n,
0) = QL (cid:2) − e iω n (cid:3) for y = 0, (7) g ( n, y + 1) + g ( n, y −
1) = 2 [2 − cos ω n ] g ( n, y ) for y >
0. (8)As mentioned earlier, the average density per lane is ρ which yields g (0 , y ) = ρ , forall y . Then the stationary state profile in terms of these normal modes is φ ( x, y ) = ρ + L − (cid:88) n =1 g ( n, y ) e iω n x . (9)The Fourier amplitude g ( n, y ), for n ≥
1, can be determined iteratively in termsof g ( n, G ( n, z ) = ∞ (cid:88) y =1 g ( n, y ) z y , for all n > . (10)Note that g ( n,
0) has been excluded from the definition. Using (8) it is easy to showthat G ( n, z ) = zg ( n, − z g ( n, z − z − )( z + z + ) , (11)where z ± ( ω n ) = 2 − cos ω n ± (cid:113) (2 − cos ω n ) − . (12)Note that g ( n,
0) and g ( n,
1) are so far unknown variables related by (7). A secondindependent relation can be found following a pole-cancelling mechanism [50, 51] whichuses the convergence of the generating function. From equation (6) it is easy to seethat the amplitude of the normal modes are bounded | g ( n, y ) | ≤ L − (cid:80) x φ ( x, y ) = ρ .This implies that, the generating function G ( n, z ) for all n converges at any value of | z | <
1. On the other hand, it easy to show that z − < z + >
1. Then, to be on-local response in a lattice gas under a shear drive z = z − must not be a pole of the generating function G ( n, z ), implying thatthe numerator in equation (11) must vanish at z = z − . In other words, g ( n,
1) = z − g ( n,
0) for all n > . (13)This, together with (7) determines g ( n,
0) and g ( n,
1) in terms of Q . The generatingfunction can thus be expressed as G ( n, z ) = QL − e iω n (2 − e − iω n − z − ) z ( z − z + ) for all n >
0. (14)The amplitudes g ( n, y ) for all y , can be extracted from this expression by expanding(14) in a Taylor series around z = 0 and comparing it with the definition of G ( n, z ). G ( n, z ) = ∞ (cid:88) y =1 (cid:20) QL − e iω n z − − e − iω n z y + (cid:21) z y . (15)By definition, the term inside the parenthesis is g ( n, y ), for n >
0, which when combinedwith (9), yields the stationary density profile, φ ( x, y ) = ρ + QL L − (cid:88) n =1 γ ( ω n ) e iωx z y + , (16)where we defined γ ( ω n ) = e iω n − − z − − e − iω n . (17)The only remaining quantity to be determined is Q which appears as an overallnormalization constant of the density difference. This can be evaluated using the self-consistency condition Q = (cid:15)φ ( L − , Q = (cid:15)ρ + (cid:15) QL L − (cid:88) n =1 γ ( ω n ) e − iω n . (18)The analysis is simpler in the L → ∞ limit, where ω n ≡ ω can be considered as acontinuous variable, and we replace the summation by integration. Then, Q = (cid:15)ρ + (cid:15) Q π (cid:90) π dω γ ( ω ) e − iω n . (19)Performing the integration (see Appendix A) and simplifying, we get Q = 2 π(cid:15)ρ [2 π − (cid:15) ( π + 1)] . (20)The complete solution for the density profile is then given by φ ( x, y ) − ρ = ρ(cid:15) π − ( π + 1) (cid:90) π dω γ ( ω ) e iωx [ z + ( ω )] y . (21)This expression can be reduced to a compact form by a change of variable ω → q (fordetails see Appendix B) which yields φ ( x, y ) − ρ = 2 (cid:15)ρ [2 π − (cid:15) ( π + 1)] (cid:90) π/ cos [ q (2 x + 1)] cos q − sin [ q (2 x + 1)] (cid:112) q (cid:16) q + 2 sin q (cid:112) q (cid:17) y dq. (22) on-local response in a lattice gas under a shear drive . . φ ( x , ) − −
10 0 10 20 x MonteCarloTheory
Figure 4.
A comparison of the density profile in (22) with the Monte Carlo results atsites along the driven lane. The x is the site index on the driven lane. For conveniencewe denote the site ( L − i,
0) on the left of the broken bond as x = − i . The brokenbond is between x = − x = 0th site, while the drive is along the positive x direction. The difference in the density from the uniform profile ρ is proportional to the blockagestrength (cid:15) , and vanishes when (cid:15) = 0. Also higher the bulk density ρ , more pronouncedis the effect.To verify the relevance of the L → ∞ limit to finite L systems, the density profile iscompared with numerical data from Monte Carlo simulation in Figure 4. The simulationis performed on a 100 ×
100 lattice with a broken bond ( (cid:15) = 1) between (99 ,
0) site and(0 ,
0) site. In the starting configuration 5000 particles were distributed randomly onthe lattice leading to a global average density ρ = 1 /
2. We follow a random sequentialupdate rule: in every time step, all the particles are updated exactly once following thestochastic dynamics in section 2. In spite of the finite size of the lattice, the densityprofile matches very well with the theoretical result in (22) which corresponds to the L → ∞ limit. In Figure 4 we present only the results along the driven lane. The density has a maximum at the site ( L − ,
0) which is the left end of the slow bond,whereas it is has a minimum at the site (0 ,
0) on the right of the slow bond. Far fromthe slow bond the density approaches the global average value ρ . The convergence tothis value is slow, and most importantly the difference φ ( x, y ) − ρ decays algebraicallyas 1 / (cid:112) x + y , in all directions far from the slow bond, except along the diagonal y = x where it decays as 1 / ( x + y ) / . To show this power law tail, we analyze the solution(22) in three directions, namely, along the driven lane ( y = 0), along the x = 0 line, and on-local response in a lattice gas under a shear drive m . For convenience we denote the integral in (22) by I ( x, y ) = (cid:90) π/ cos [ q (2 x + 1)] cos q − sin [ q (2 x + 1)] (cid:112) q (cid:16) q + 2 sin q (cid:112) q (cid:17) y dq. (23) Along the driven lane:
For y = 0 the integral I ( x, y ) simplifies to I ( x,
0) = (cid:90) π/ dq cos [(2 x + 1) q ] cos q − (cid:90) π/ dq sin [(2 x + 1) q ] (cid:113) q. (24)It is easy to show that the first integral vanishes for all integer values of x except at x = − x = 0 where its value is π/
4. To evaluate the second integral, we make achange of variable with (2 x + 1) q = ξ . Then the density profile yields, φ ( x, − ρ = 2 (cid:15)ρ π − (cid:15) ( π + 1) (cid:104) π δ x, − + δ x, ) − x + 1 (cid:90) (2 x +1) π/ dξ sin ξ (cid:115) (cid:18) ξ x + 1 (cid:19)(cid:35) . (25)The term inside the square root in the integrand is slowly varying, and for large | x | it remains almost constant while sin ξ completes a cycle. Then the integral can beapproximated as (cid:90) π/ dξ sin ξ + x (cid:88) n =1 (cid:115) (cid:18) nπ x + 1 (cid:19) (cid:90) (2 n +1) π n − π dξ sin ξ. The first integral is equal to 1 whereas the one inside the summation is zero. This yields,for large | x | , the power law tail, φ ( x, − ρ (cid:39) − (cid:15)ρ π − (cid:15) ( π + 1) × x + 1 . (26)An interesting feature to note that, except for the two sites ( x = − x = 0)the profile is anti-symmetric with respect to the slow bond (see figure 4). Along the x = 0 line: Along this line, perpendicular to the driven lane, the integral in(23) yields, I (0 , y ) = (cid:90) π/ dq cos q − sin q (cid:112) q (cid:104) q + 2 sin q (cid:112) q (cid:105) y . (27)The denominator in the integrand has its minimum value at q = 0 and it monotonicallyincreases with increasing q within the interval of integration. Thus the contributions forlarge y comes predominantly from small q . Expanding around q = 0 the integral can beapproximated as I (0 , y ) (cid:39) (cid:90) π/ dq (cid:18) cos q − sin q (cid:113) q (cid:19) e − yq . (28) on-local response in a lattice gas under a shear drive q (cid:47) / y where e − yq (cid:39) I (0 , y ) (cid:39) (cid:90) / y dq (cid:18) cos q − sin q (cid:113) q (cid:19) (cid:39) y + O (1 /y ) . (29)Thus the density far from the driven lane, approaches the global average value ρ as φ (0 , y ) − ρ (cid:39) ρ(cid:15) π − ( π + 1) × y . (30) Along the line with slope m : Consider a line 2 y = m (2 x + 1) with m being the slope.As a start, let both m and x are positive. The integral in (23) yields I (cid:20) x, m (cid:18) x + 12 (cid:19)(cid:21) = (cid:90) π/ dq cos [ q (2 x + 1)] cos q − sin q (2 x + 1) (cid:112) q (cid:104) q + 2 sin q (cid:112) q (cid:105) m ( x +1 / . (31)Like the case discussed above, the integral can be approximated by considering thecontributions only from small q whereby it reduces to I (cid:20) x, m (cid:18) x + 12 (cid:19)(cid:21) (cid:39) (cid:90) π/ dq (cid:26) cos [ q (2 x + 1)] cos q − sin [ q (2 x + 1)] (cid:113) q (cid:27) e − m (2 x +1) q . (32)Applying a change of variable q (2 x + 1) = ξ and keeping only the leading orders in 1 /x the integral simplifies to I (cid:20) x, m (cid:18) x + 12 (cid:19)(cid:21) (cid:39) x + 1 (cid:90) ( x +1 / π dξ (cos ξ − sin ξ ) e − mξ . (33)The integral is now easy to compute, yielding the density difference φ (cid:20) x, m (cid:18) x + 12 (cid:19)(cid:21) − ρ (cid:39) ρ(cid:15) π − ( π + 1) × m − m + 1)(2 x + 1) . (34)Note that, the density difference changes sign as m crosses 1. In fact, at m = 1, i.e. , along the line 2 y = 2 x + 1, the leading order term in the integral I vanishes. Byconsidering the higher order contributions in 1 /x , it can be shown that the integral for m = 1 yields I (cid:20) x, (cid:18) x + 12 (cid:19)(cid:21) (cid:39) −
12 (2 x + 1) . (35)For more details see Appendix C, where the integral I can be extracted from (C.6) bysubstituting ρ = 0. Thus, along the diagonal, the density profile φ (cid:18) x, x + 12 (cid:19) − ρ (cid:39) − ρ(cid:15) π − ( π + 1) × x + 1) . (36)The analysis can be easily extended for negative x and m . The expression for theprofile (34) remains unchanged. on-local response in a lattice gas under a shear drive In polar coordinate:
It is instructive to express the density in terms of the polarcoordinates θ = arctan( y/x ) and r = (cid:112) x + y . At large distances, the density profile, φ ( r, θ ) − ρ (cid:39) √ ρ(cid:15) π − ( π + 1) sin( θ − π ) r . (37)For θ = π/ /r .The slow decay of the density profile is also reflected in the induced current. Asthe particles are independent of each other, the current is due to diffusion except alongthe driven lane. The particle current at any site ( x, y ) can be expressed in terms of thelocal density profile as J ( x, y ) = [1 − (cid:15)δ x, − ] φ ( x, y ) ˆx − ∂φ∂y ˆy for y = 0, (38) J ( x, y ) = −∇ φ ( x, y ) elsewhere, (39)where ˆx and ˆy are the unit vectors along the x and y directions, respectively. Fromthe density profile in (37), it is clear that far from the slow bond, the particle currentdecreases algebraically. Particularly, in terms of the polar coordinates the inducedcurrent in the bulk, J ( r, θ ) (cid:39) √ ρ(cid:15) π − ( π + 1) × sin( θ − π ) ˆr − cos( θ − π ) ˆ θ r , (40)where ˆr and ˆ θ are the unit vectors along the radial and the angular directions.
4. Exclusion interaction
Most of the qualitative features of the stationary profile remain unchanged from theindependent particle case. In the absence of the slow bond the surface drive does notaffect the density profile which remains uniform through out the lattice. However,introducing a slow bond on the driven lane makes the particles queue behind the bond,resulting in a density gradient that propagates far inside the bulk and induces diffusivecurrent. The difference in density from the uniform profile decays algebraically with thedistance r from the slow bond. The decay exponent is same as that in the independentparticles case.We now proceed to derive the density profile from the rate equations at thestationary state. Let n t ( x, y ) be the occupation variable of site ( x, y ) at time t whichtakes value 1 if the site is occupied and 0 if there is no particle. The density at time t isobtained by performing ensemble average, φ t ( x, y ) = (cid:104) n t ( x, y ) (cid:105) . Following the dynamicalrules in section 2, it is easy to show that the time evolution of φ t ( x, y ) is ∂φ t ( x, y ) ∂t = ∆ φ t ( x, y ) for y > φ t ( x − ,
0) + φ t ( x, − φ t ( x, −(cid:104) n t ( x, n t ( x − , (cid:105) + (cid:104) n t ( x, n t ( x + 1 , (cid:105)− (cid:15) (cid:104) n t ( L − , − n t (0 , (cid:105) [ δ x, − δ x,L − ] for y = 0, (41) on-local response in a lattice gas under a shear drive φ ( x, y ) = 0 for y > φ ( x − ,
0) + φ ( x, − φ ( x,
0) = (cid:104) n ( x, n ( x − , (cid:105) − (cid:104) n ( x, n ( x + 1 , (cid:105) + (cid:15) (cid:104) n ( L − , − n (0 , (cid:105) [ δ x, − δ x,L − ]for y = 0 , (42)where we dropped the time index in both φ and n .In the absence of the slow bond, it is easy to show that in the steady state allconfigurations are equally probable. This directly implies that the density profile isuniform. It is worth mentioning that, a very different behavior was observed in arelated work [45, 29] with nearest neighbor exclusion interaction, where a shear driverearranges the average population of the lanes.For non-zero (cid:15) , the slow bond breaks the translation invariance, and the uniformdensity profile is no longer a stationary state. It is important to note that, like inthe case of independent particles, the average population in each lane remains same, L − (cid:80) x φ ( x, y ) = ρ . This feature will be important in our derivation of the stationaryprofile. An exact analysis of the stationary profile is hard as the Equation (42) involves twopoint correlations of the occupation variables which in turn depends on higher ordercorrelations. It is almost impossible to circumvent this hierarchy. However, therate equations can be simplified within a mean field approximation, i.e , by imposingfactorization assumption (cid:104) n ( x, n ( x + 1 , (cid:105) = φ ( x, φ ( x + 1 , . (43)Such mean-field approximation have been successfully applied to a variety of drivendiffusive systems, see e.g. [52, 53]. Within mean field approximation, the stationaryEquation (42) becomes ∆ φ ( x, y ) = 0 for y >
0, (44) φ ( x − ,
0) + φ ( x, − φ ( x,
0) = φ ( x, φ ( x − , − φ ( x, φ ( x + 1 , (cid:15)φ ( L − , − φ (0 , δ x, − δ x,L − ]for y = 0 . (45)The boundary condition is that at large distances from the slow bond the densityconverges to a global average value ρ .The solution is still difficult because of the non-linearity. We proceed by using aperturbative expansion of the density φ ( x, y ) in powers of (cid:15) . We assume that φ ( x, y ) is on-local response in a lattice gas under a shear drive (cid:15) = 0 state where the density profile is uniform, ρ at all sites.Let φ ( x, y ) = ρ + ∞ (cid:88) p =1 , , (cid:15) p α p ( x, y ) . (46)The advantage is that to each order in the expansion the corresponding equationsbecome linear. In this section, we analyze only the linear order term which capturesthe power-law tail of the density profile. The higher order terms do not change thepower-law. A detailed analysis of the higher order terms is presented in the AppendixD. At large distances away from the slow bond, the density reaches a uniform profileequal to ρ . This implies that α ( x, y ) must vanish at large distances from the slow bond.Substituting the perturbative expansion into Equation (44)-(45) yields: ∇ α ( x, y ) = 0 for y > , (47)and, α ( x − ,
0) + α ( x, − α ( x,
0) + ρ ( α ( x + 1 , − α ( x − , ρ (1 − ρ ) ( δ x, − δ x, − ) for y = 0 . (48)This is a set of coupled linear equations and can be solved in a way similar to thesolution of Equation (3)-(4), in the independent particle case. Let, the Fourier modesare defined as g ( n, y ) = 1 L L − (cid:88) x =0 α ( x, y ) e − iω n x (49)where ω n = 2 πn/L with n = 0 , , · · · , L −
1. The subscript 1 in the Fourier amplitudesindicates that the calculations are for the order one term in (cid:15) . In terms of these Fourieramplitudes, the equations (47) and (48) yield g ( n, y + 1) = − g ( n, y −
1) + 2 [2 − cos( ω n )] g ( n, y ) for y > , (50)and g ( n,
1) = (cid:2) − iρ sin ω n ) − e − iω n (cid:3) g ( n,
0) + ρ (1 − ρ ) L (cid:0) − e iω n (cid:1) , (51)respectively.By definition in (49), g (0 , y ) = L − (cid:80) x α ( x, y ). Given that the average L − (cid:80) x φ ( x, y ) = ρ for all y , the amplitude of the zeroth Fourier mode must vanish, g (0 , y ) = 0, for all y . Taking this into account, the inverse Fourier transformation yieldsthe formal solution α ( x, y ) = L − (cid:88) n =1 g ( n, y ) e iω n x . (52) on-local response in a lattice gas under a shear drive g ( n, y ) of the Fourier modes can be calculated iteratively usingthe recurrence relation (50). A systematic approach of solving this recurrence relationis using the generating function, defined as G ( n, z ) = ∞ (cid:88) y =1 g ( n, y ) z y . (53)A similar method is used for solving the profile for the independent particles. Using(51) the generating function yields a similar expression as in (11), G ( n, z ) = g ( n, − zg ( n, z − z − ) ( z − z + ) × z, (54)for n > z ± as in (12).A pole cancelling argument used earlier for the independent particles case, simplifiesthe expression for the generating function and can be used to determine the Fouriermodes g ( n, y ). From (50), it is clear that | g ( n, y ) | ≤ L − (cid:80) x | α ( x, y ) | , and as at largedistances from the driven lane the density is close to the uniform profile, | g ( n, y ) | isfinite for y → ∞ . Then, the series in (54) has a radius of convergence | z | <
1. Then, thegenerating function should not have any pole within the unit circle around the originon the complex z -plane. This yields g ( n,
0) = Γ ρ ( ω n ) (cid:20) ρ (1 − ρ ) L (cid:0) e iω n − (cid:1)(cid:21) , (55)where Γ ρ ( ω n ) = 1 (cid:20)(cid:113) (2 − cos ω n ) − (cid:21) + i (1 − ρ ) sin ω n . (56)In addition, the Fourier modes for any y , yields g ( n, y ) = g ( n, z y + . (57)Incorporating these expressions of g ( n, y ) in (55) and (57) to the solution (52), α ( x, y )can be written as α ( x, y ) = ρ (1 − ρ ) L L − (cid:88) n =1 (cid:0) e iω n − (cid:1) Γ ρ ( ω n ) e iω n x [ z + ( ω n )] y . (58)Then, finally the solution for the density profile can be expressed as, φ ( x, y ) − ρ = (cid:15)ρ (1 − ρ ) 1 L L − (cid:88) n =1 γ ρ ( ω n ) × e iω n x [ z + ( ω n )] y + O (cid:0) (cid:15) (cid:1) , (59)where γ ρ ( ω n ) = ( e iω n − ρ ( ω n ) . (60)Notice that for ρ = 0, the γ ρ ( ω ) is same as γ ( ω ) defined in (17) for the independentparticle case. on-local response in a lattice gas under a shear drive L → ∞ limit, the profile can be expressed in terms of continuous variables ω ≡ ω n , and the summation over n can be approximated by an integration, yielding, φ ( x, y ) = ρ + (cid:15)ρ (1 − ρ ) (cid:90) π dω π γ ρ ( ω ) e iωx [ z + ( ω )] y + O (cid:0) (cid:15) (cid:1) , (61)It will be shown in the following section that this leading order term in (cid:15) alreadycaptures the algebraic profile at large distances. It turns out that, the higher ordercontributions do not alter the power-law tail of the profile; it only changes the overallamplitude. By calculating these higher order terms in the Appendix D we shall showthat at large distances the density profile, φ ( x, y ) = ρ + (cid:15)φ ( − ,
0) [1 − φ (0 , (cid:90) π dω π γ ρ ( ω ) e iωx [ z + ( ω )] y . (62)When compared with the profile (21) in the independent particles case, the integrandabove differs in the factor γ ρ ( ω ) which is a function of ρ . The density profile decays algebraically to the uniform value ρ , at large distances. Inmost directions away from the slow bond, the difference decays as 1 /r with r being thedistance from the slow bond. Only at a certain angle the decay is faster as 1 /r .To analyze the profile, we write the expression in (62) as φ ( x, y ) − ρ = (cid:18) (cid:15)φ ( − ,
0) [1 − φ (0 , π (cid:19) (cid:90) π/ dq − ρ (1 − ρ ) cos q ] × (1 − ρ ) cos [(2 x + 1) q ] cos q − sin [(2 x + 1) q ] (cid:112) q (cid:104) q + 2 sin q (cid:112) q (cid:105) y . (63)For details see Appendix B. Notice that the profile is composed of a symmetric (withcosine term) part and an anti-symmetric part (with sine term) under x → − ( x + 1). Itcan be shown that the symmetric part falls of exponentially with increasing x , and atlarge distances the profile effectively becomes anti-symmetric (see Figure 5).The analysis of the asymptotic profile in different directions is similar to that inthe independent particles case. A detailed calculation is deferred to the Appendix C. Along the driven lane: φ ( x, − ρ (cid:39) − (cid:18) (cid:15)φ ( − ,
0) [1 − φ (0 , π [1 − ρ (1 − ρ )] (cid:19) x + 1 (64) Along x = 0 line: For ρ (cid:54) = 1 / φ (0 , y ) − ρ (cid:39) (cid:18) (cid:15)φ ( − ,
0) [1 − φ (0 , π [1 − ρ (1 − ρ )] (cid:19) × − ρ y , (65)and for ρ = 1 / φ (0 , y ) − ρ (cid:39) − (cid:18) (cid:15)φ ( − ,
0) [1 − φ (0 , π [1 − ρ (1 − ρ )] (cid:19) × y , (66) on-local response in a lattice gas under a shear drive Along a line of slope m : For m (1 − ρ ) (cid:54) = 1, φ [ x, m ( x + 1 / − ρ = (cid:15)φ ( − ,
0) [1 − φ (0 , π [1 − ρ (1 − ρ )] × m (1 − ρ ) − m + 1 × x + 1 , (67)and for m (1 − ρ ) = 1, φ [ x, m ( x + 1 / − ρ = − (cid:15)φ ( − ,
0) [1 − φ (0 , π [1 − ρ (1 − ρ )] × [1 + 2 ρ (1 − ρ )] (1 − ρ ) (2 x + 1) . (68) In polar coordinate:
Like in the independent particles case, the above asymptoticresults can be put together in a simple expression in the polar coordinates r = (cid:112) x + y and θ = arctan( y/x ). For large r , and 0 ≤ θ ≤ π , φ ( r, θ ) − ρ (cid:39) (cid:15)φ ( − ,
0) [1 − φ (0 , π (cid:112) − ρ (1 − ρ ) × sin ( θ − Θ) r , (69)where Θ = π/ − arctan[(1 − ρ ) / (cid:112) − ρ ) ]. For θ = Θ where the leading termvanishes, the profile decays as 1 /r . The angle depends on the average density ρ , andvaries from π/ π/ ρ is increased from 0 to 1. We recall that, for the independentparticles case, this angle is at θ = π/ ρ → (1 − ρ ) the angle Θ → π − Θ. This results in a symmetry ofthe profile φ − ρ ( r, θ ) = 1 − φ ρ ( r, π − θ ) . (70)This is a direct consequence of the particle-hole (empty site) equivalence in the exclusionprocess: A particle jumping across a bond to an empty site can also be considered as ahole moving in the opposite direction.The long-range density profile, induces diffusive current in the bulk. Within themean-field approximation the stationary state current can be expressed as J ( x,
0) = φ ( x,
0) [1 − φ ( x + 1 , − (cid:15)δ x, − ] ˆx − ∂φ ( x, y ) ∂y ˆy for y = 0,(71) J ( x, y ) = − ∇ φ ( x, y ) elsewhere.(72)The ˆx and ˆy are the unit vectors in the x and y directions, respectively. It is clear fromthe algebraic decay of the density profile that the induced current in the bulk decays as1 /r in all directions, with the distance away from the slow bond. Particularly, in thepolar coordinates, using the density profile in (69), the induced current in the bulk canbe expressed as J ( r, θ ) (cid:39) (cid:32) (cid:15)φ ( − ,
0) [1 − φ (0 , π (cid:112) − ρ (1 − ρ ) (cid:33) × sin( θ − Θ) ˆr − cos( θ − Θ) ˆ θ r , (73)where ˆr and ˆ θ are the unit vectors along the polar coordinates r and θ , respectively. on-local response in a lattice gas under a shear drive . . . φ ( x , ) − −
10 0 10 20 x MonteCarloTheory
Figure 5.
A comparison of the computer simulation results of the stationary densityprofile φ ( x,
0) in the exclusion case with the solution (63) obtained using mean-fieldapproximation. The slow bond is between x = − x = 0 th sites. − − . . . ρ − φ ( x , ) x MonteCarloAsymptotic Result
Figure 6.
The density difference at sites on the right of the broken bond along thedriven lane. The density approaches the global average density ρ = 1 / x away from the broken bond. on-local response in a lattice gas under a shear drive − − . φ ( , y ) − ρ y MonteCarloAsymptotic ResultFinite Size Result
Figure 7.
The power law tail of the difference in density from its global average value ρ , with the distance away from the driven lane. The Asymptotic result is obtained fromthe mean-field solution (65), whereas the finite size result is generated by numericaliteration of the mean-field equation (44)-(45) on a finite system. Although the profile (63) is derived within a mean-field approximation, it describesthe numerical result quite well. We demonstrate this by comparing with the profilegenerated by a Monte Carlo simulation on a 600 × ρ = 1 /
9. We consider the case with (cid:15) = 1, i.e. , no jumps are allowed across the bondbetween ( L − ,
0) and (0 ,
0) sites. In the simulation the particles are evolved followinga random sequential update, where in every step of the iteration a site is chosen atrandom. If the site is occupied, the particle is transferred to a randomly chosen nearestneighbor site only when the latter is empty. One Monte Carlo time step consists of LM such moves. Starting with a random distribution of particles, the system is evolved fora long time to ensure that a stationary state distribution is reached. The density isaveraged over one million configurations at intervals of 100 Monte Carlo time steps.The profile along the driven lane around the broken bond is shown in figure 5.For brevity we denote the sites x = L − i on the left of the broken bond as x = − i .The theoretical results of the density are calculated from the mean-field solution bysubstituting y = 0 and (cid:15) = 1 in the expression in (63). The integration is evaluatednumerically. Although the expression (63) is applicable only at large distances, itdescribes the Monte Carlo result quite well, even near the broken bond.The power-law decay of the profile can be seen in figure 6 where the profile alongthe boundary lane on the right of the slow bond is plotted on a log-log scale. The x range extends from the neighbor of the slow bond to the half of the system length. Theresult is consistent with the 1 /x decay in the asymptotic solution in Equation (64). on-local response in a lattice gas under a shear drive ).The second direction where we compare the results is perpendicular to the drivenlane along the x = 0 line. The Monte Carlo simulation data and the results derived fromthe asymptotic expression in (65) are plotted in figure 7. The saturation at the tail ofthe Monte Carlo data is due to the finite sample size on which the data is averaged: thenumber fluctuation is comparable to the mean value. It is expected that when averagedover larger sample size the density difference would vanish at large distances. In fact,theoretically, on a finite system size the power-law decay of the density difference is validonly up to a distance ∼ L , and beyond this it decays exponentially. This is shown inFigure 7 by a continuous line which is obtained by numerical solution of the mean-fieldequation (44)-(45) on a 600 × φ ( x, y ) − ρ satisfies Laplace’s equation in the bulk, whose solutionfor finite L can be written as φ ( x, y ) − ρ = L − (cid:88) n =0 (cid:20) A n cos 2 nπxL + B n sin 2 nπxL (cid:21) exp (cid:18) − nπyL (cid:19) , (74)where A n and B n are the amplitudes of the Fourier modes. It is clear that beyond y ∼ L the right hand side decays exponentially.
5. Electrostatic correspondence
The algebraic decay of the density profile is simple to understand from an electrostaticanalogy. A similar correspondence proved useful in a related study of a lattice gas withlocalized bulk drive [16, 14].Let us first consider the case of independent particles. For simplicity, we considerthe slow bond between the sites ( − ,
0) and (0 , x ranging from −∞ to ∞ and y from 0 to ∞ . In the continuum limit, the stationary stateequation in (3)-(4) yields ∇ φ ( x, y ) = 0 for all y > , (75) − ∂ y φ ( x, y ) (cid:39) (cid:15)φ ( − ,
0) [ δ ( x + 1) − δ ( x )] − ∂ x φ ( x, y ) for y = 0 , (76)the δ ( x ) being the Dirac delta function. In analogy with electrostatic, φ ( x, y ) is thepotential on the upper half-plane due to a line charge density σ ( x ) = − ∂ x φ ( x,
0) at theboundary y = 0 and a dipole of moment (cid:15)φ ( − ,
0) at the origin.As the charge density itself a function of the potential, the solution has to bedetermined self-consistently. It is not difficult to see that a potential which has a dipoleprofile at large distances, is a consistent solution. It is consistent with the dipole at theorigin. On the other hand, as the potential φ ( x, y ) jumps discontinuously across the slowbond (see Figure 3), the σ ( x ) consists of a positive charge at the origin and a distributednegative charge elsewhere. It is easy to verify that the total amount of the charge σ ( x )summed over x is zero. Moreover, corresponding to the dipole solution, the charge σ ( x )decays as 1 /x and it generates a quadrupolar potential which is sub-dominant to the on-local response in a lattice gas under a shear drive (cid:15)φ ( − , − φ (0 , σ ( x ) = − [1 + 2 φ ( x, ∂ x φ ( x, y ) . Again, the total charge of σ ( x ) is zero, and it generates a quadrupolar potential atlarge distances. Then the potential due to the dipole at the origin determines the largedistance profile, as found in the solution of the mean-field equation in section 4.
6. Summary
One of the motivations for our work is to study how a shear flow at the boundary caninduce current deep inside the bulk. In a fluid medium, this non-local effect is notsurprising as the particles due to their momentum carry the directional informationas they move away from the sheared layer. One simple example is the Couette flowin viscous liquids, where a steady flow at the surface induces current in the bulkwhose amplitude decays linearly with the distance from the sheared layer [54]. Anotherexample is provided by experimental [55] and theoretical [56, 57] studies of the effectof shear drive on a fluctuating interface placed away from the boundary. In the presentpaper, we showed that even in absence of the momentum degree of freedom (fluid withlow Reynolds number), a similar non-local current can be induced, when there is ablockage at the sheared layer. Essentially, due to the particle conservation, the blockageat the boundary generates a non-local density gradient across the system which in turninduces non-vanishing diffusive current. We demonstrated this in a simple lattice-gasmodel with diffusive transport in the bulk and shear flow at the boundary, along witha slow bond. We showed that in presence of the slow bond the density profile decaysas 1 /y with the distance away from the shear layer. As a result, the diffusive currentdecays as 1 /y .We expect that the 1 /y decay of the current is quite general and holds for arbitrarylocal inter-particle interactions. Typically in a diffusive system, away from criticality,the large scale properties of the conserved density field is effectively described by adiffusion equation ∂φ t ( x, y ) ∂t (cid:39) − D ∇ φ t ( x, y ) , (77)with D being a diffusion constant. Then in the stationary state the φ ( x, y ) follows theLaplace’s equation. On the semi-infinite upper half-plane, as in our problem, a sheardrive at the boundary induces a profile φ ( x,
0) at y = 0. The solution of the Laplace’sequation with this boundary condition can in general be written as φ ( x, y ) − ρ = yπ (cid:90) ∞−∞ dx (cid:48) φ ( x (cid:48) , − ρy + ( x (cid:48) − x ) , (78) on-local response in a lattice gas under a shear drive φ ( x, y ) approaches ρ far from the boundary. This is known as the Poisson formula.It is evident that for a non-uniform profile φ ( x,
0) at the boundary, the difference φ ( x, y ) − ρ decays as 1 /y . As a result the diffusive current decays as 1 /y . Theseresults can be generalized in higher dimensions, d , as well, where the induced current isfound to decay as 1 /r d with the perpendicular distance r away from the shear drive. Acknowledgments
We thank A. Bar, O. Cohen, M.R. Evans, O. Hirschberg, and S. Prolhac for helpfuldiscussions. The support of the Israel Science Foundation (ISF) and the MinervaFoundation with funding from the Federal Ministry of Education and Research isgratefully acknowledged. S.N. Majumdar acknowledges support by ANR grant 2011-BS04-013-01 WALK-MAT and in part by the Indo-French Centre for the Promotion ofAdvanced Research under Project 4604-3.
Appendix A. Integration in (19)
To perform the integration I = 12 π (cid:90) π dωγ ( ω )e − iω = 12 π (cid:90) π dω − e − iω (cid:112) (2 − cos ω ) − i sin ω , (A.1)consider a change of variable ω → q . In terms of q , the integral reduces to, I = 1 π (cid:90) π dq e − iq cos q − i (cid:112) − sin q , (A.2)which in terms of trigonometric functions yields I = 12 π (cid:90) π dq (cid:20) cos q + sin q (cid:113) q (cid:21) − iπ (cid:90) π dq (cid:20) sin 2 q − q (cid:113) q (cid:21) . (A.3)The imaginary part of the integral vanishes, whereas the real part reduces to I = 14 + 12 π (cid:90) − √ − x = 12 + 12 π . (A.4) Appendix B. Integration in (21)
Let I = (cid:90) π dωγ ( ω ) × e iωx [ z + ( ω )] y = (cid:90) π dω e iω ( x +1) − e iωx (cid:20)(cid:113) (2 − cos ω ) − i sin ω (cid:21) (cid:20) − cos ω + (cid:113) (2 − cos ω ) − (cid:21) y . (B.1) on-local response in a lattice gas under a shear drive ω → q and simplifying, the integration reduces to, I = 2 (cid:90) π dq e iq (2 x +1) (cid:16) cos q − i (cid:112) q (cid:17) (cid:16) q + 2 sin q (cid:112) q (cid:17) y . (B.2)Further simplification reduces the integration to I = (cid:90) π dq cos q cos [ q (2 x + 1)] (cid:16) q + 2 sin q (cid:112) q (cid:17) y − (cid:90) π dq sin [ q (2 x + 1)] (cid:112) q (cid:16) q + 2 sin q (cid:112) q (cid:17) y + i (cid:90) π dq cos q sin [ q (2 x + 1)] (cid:16) q + 2 sin q (cid:112) q (cid:17) y + i (cid:90) π dq cos [ q (2 x + 1)] (cid:112) q (cid:16) q + 2 sin q (cid:112) q (cid:17) y (B.3)The last two integrands are asymmetric around q = π/
2, and the integrals vanish. Thisis also expected as the density difference in (21) is a real number. On the other handthe first two integrands are symmetric around q = π/
2. Then, finally, I = 2 (cid:90) π/ dq cos q cos [ q (2 x + 1)] − sin [ q (2 x + 1)] (cid:112) q (cid:16) q + 2 sin q (cid:112) q (cid:17) y (B.4) Appendix C. Asymptotic analysis of the profile in (63)
Appendix C.1. Along y = 0 . Let us first consider the profile along the driven lane at y = 0, where integral in theexpression (63) yields I = (1 − ρ ) (cid:90) π/ dq cos [(2 x + 1) q ] cos q − ρ (1 − ρ ) cos q − (cid:90) π/ dq (cid:112) q − ρ (1 − ρ ) cos q sin [(2 x + 1) q ] . (C.1)Using the trigonometric identity 2 cos [(2 x + 1) q ] cos q = cos 2 xq + cos [2( x + 1) q ] and achange of variables 2 xq = η , 2( x + 1) q = η (cid:48) and (2 x + 1) q = ξ the integral yields I = 1 − ρ x (cid:90) xπ dη cos η cos (cid:2) η x (cid:3) − ρ (1 − ρ ) cos (cid:2) η x (cid:3) + 1 − ρ x + 1) (cid:90) ( x +1) π dη (cid:48) cos η (cid:48) cos (cid:104) η (cid:48) x +) (cid:105) − ρ (1 − ρ ) cos (cid:104) η (cid:48) x +1) (cid:105) − (cid:90) ( x +1 / π dξ sin ξ (cid:113) ξ x +1 − ρ (1 − ρ ) cos ξ x +1 . (C.2) on-local response in a lattice gas under a shear drive x , the terms involving x in the integrands vary slowly compared to the rest.Then, in the first two integrals the range of integration can be divided in intervals oflength π where within each such integrals the slowly varying terms can be replaced bytheir approximate value within that window. For example, for η ∈ [ nπ, ( n + 1) π ] with n being integer, cos( η x )1 − ρ (1 − ρ ) cos ( η/ x ) (cid:39) cos( nπ x )1 − ρ (1 − ρ ) cos ( nπ/ x ) . Then, it is easy to show that under this approximation the first two integrals vanish. Inthe last integral, by dividing the range of ξ in 0 to π/ π , ityields I (cid:39) − − ρ (1 − ρ ) (cid:90) π/ dξ sin ξ − x (cid:88) n =1 (cid:114) (cid:16) (2 n +1) π x +1) (cid:17) − ρ (1 − ρ ) cos (cid:16) (2 n +1) π x +1) (cid:17) (cid:90) ( n +1 / π ( n − / π dξ sin ξ (C.3)The integral in the right most term vanishes for all n , wherein the first integral yields1. Then, finally, I (cid:39) − − ρ (1 − ρ ) × x + 1 . (C.4)Then the asymptotic density profile for large | x | along the driven lane φ ( x, − ρ (cid:39) − (cid:18) (cid:15)φ ( − ,
0) [1 − φ (0 , π [1 − ρ (1 − ρ )] (cid:19) × x + 1 . (C.5) Appendix C.2. Along y (cid:39) m x . Next, we consider the profile on sites along the straight line 2 y = m (2 x + 1). As y ispositive for all sites on the lattice, m ≥ x ≥ m and x . Along this line the integral in (63) yields I = (cid:90) π/ dq (1 − ρ ) cos q (2 x + 1) cos q − sin q (2 x + 1) (cid:112) q { − ρ (1 − ρ ) cos q } (cid:104) q + 2 sin q (cid:112) q (cid:105) m ( x +1 / . (C.6)The term inside the square bracket in the denominator achieves its maximum value at q = 0 and monotonically decreases as q increases within the range of integration. Then,for large x , the leading contribution in the integral comes from small q . In this range,although q is small, q (2 x + 1) could be large. By expanding in terms of q the integrandyields I = (cid:90) π/ dq (cid:34) (1 − ρ ) cos q (2 x + 1) × (cid:8) − q (cid:9) − sin q (2 x + 1) × (cid:8) q (cid:9) { − ρ (1 − ρ ) (1 − q ) } (cid:2) q − q (cid:3) m ( x +1 / + O ( q ) (cid:3) . (C.7) on-local response in a lattice gas under a shear drive x can be expanded as (1 + 2 q − q / − m ( x +1 / =exp[ − m (2 x + 1) q ] { m (2 x + 1) q / O ( q ) } . In addition consider an expansion1[1 − ρ (1 − ρ )(1 − q )] = 1[1 − ρ (1 − ρ )] (cid:26) − ρ (1 − ρ )1 − ρ (1 − ρ ) q + O ( q ) (cid:27) (C.8)With these, and keeping only the terms up to order q , the integral in terms of a quantity ξ = (2 x + 1) q yields I = 12 x + 1 × − ρ (1 − ρ ) (cid:90) (2 x +1) π dξ [(1 − ρ ) cos ξ − sin ξ ] e − mξ + 1(2 x + 1) × − ρ (1 − ρ ) (cid:34) m (cid:90) (2 x +1) π dξ [(1 − ρ ) cos ξ − sin ξ ] ξ e − mξ − ρ (1 − ρ )1 − ρ (1 − ρ ) (cid:90) (2 x +1) π dξ [(1 − ρ ) cos ξ − sin ξ ] ξ e − mξ − (cid:90) (2 x +1) π dξ [(1 − ρ ) cos ξ + sin ξ ] ξ e − mξ (cid:35) + O (cid:20) x + 1) (cid:21) . (C.9)The integrations on the right hand side are easy to perform using method of integrationby parts, whereby the result yields,[1 − ρ (1 − ρ )] I = 12 x + 1 (cid:20) m (1 − ρ ) − m + 1 (cid:21) + 1(2 x + 1) × (cid:20) m (cid:26) × (1 − ρ ) + 4 m − − ρ ) m − m + (1 − ρ ) m ( m + 1) (cid:27) − ρ (1 − ρ )1 − ρ (1 − ρ ) (cid:26) × − − ρ ) m − m + (1 − ρ ) m ( m + 1) (cid:27) − (cid:26) − × − ρ ) m − m − (1 − ρ ) m ( m + 1) (cid:27)(cid:21) (C.10)Then for m (1 − ρ ) (cid:54) = 1, the asymptotic dependence on x is determined by the leadingterm which decays as 1 / (2 x + 1).Along the line with slope m (1 − ρ ) = 1 the leading term vanishes, and the sub-leading term determines the profile. By replacing m in terms of ρ , the expressionsimplifies to I (cid:39) − [1 + 2 ρ (1 − ρ )] (1 − ρ ) × [1 − ρ (1 − ρ )] × x + 1) . (C.11)Then from (63), the asymptotic density profile for m (1 − ρ ) (cid:54) = 1, φ [ x, m ( x + 1 / − ρ (cid:39) (cid:15)φ ( − ,
0) [1 − φ (0 , π [1 − ρ (1 − ρ )] × m (1 − ρ ) − m + 1 × x + 1 (C.12)and for m (1 − ρ ) = 1, φ [ x, m ( x + 1 / − ρ (cid:39) − (cid:15)φ ( − ,
0) [1 − φ (0 , π [1 − ρ (1 − ρ )] × [1 + 2 ρ (1 − ρ )] (1 − ρ ) (2 x + 1) (C.13)The analysis can be easily extended to negative x and m . The expression for the profileremains the same. on-local response in a lattice gas under a shear drive Appendix C.3. Along x = 0 . The last direction we analyze is along the x = 0 line perpendicular to the driven lane,where the integral in (63) yields I = (cid:90) π/ dq (1 − ρ ) cos q − sin q (cid:112) q { − ρ (1 − ρ ) cos q } (cid:104) q + 2 sin q (cid:112) q (cid:105) m ( x +1 / . (C.14)Following the same argument as used for the analysis of (C.6) the leading order term inthe integral can be calculated as I (cid:39) − ρ (1 − ρ ) (cid:90) π/ dq (cid:2) (1 − ρ ) − q + O ( q ) (cid:3) exp ( − yq ) . (C.15)Due to the exponential term, the integrand is sharply damped beyond q (cid:39) / y andthe leading contribution comes from q within this range. As a result,[1 − ρ (1 − ρ )] I (cid:39) − ρ y − y + O (cid:0) y − (cid:1) . (C.16)Then the density profile (63) along this line yields φ [0 , y ] − ρ (cid:39) (cid:15)φ ( − ,
0) [1 − φ (0 , π [1 − ρ (1 − ρ )] × (cid:20) − ρ y − y (cid:21) (C.17) Appendix D. A perturbative solution of the exclusion case.
The mean-field equation in (44)-(45) is non-linear in φ , and in general difficult to solve.As mentioned earlier, a method of solving the equations is by perturbative expansion,where the solution φ ( x, y ) is expanded in a series of small parameter (cid:15) , as shown in(46). Perturbation expansions of this type have been studied in recent years to solvecoupled non-linear equations [58, 59]. The central idea is that, applying this expansionin the non-linear equation, decomposes it order by order in (cid:15) into an infinite sequence ofinhomogeneous linear problems which are all formally solvable. The inhomogeneity inthe equation for the p th order depends on the solution of all the previous p − (cid:15) and extract the large distance densityprofile from the solution.Applying the series from (46) into (44)-(45) and equating the terms of same powerin (cid:15) , yields ∆ α p ( x, y ) = 0 for y >
0, (D.1)and α p ( x − ,
0) + α p ( x, − α p ( x,
0) + ρ [ α p ( x + 1 , − α p ( x − , Q p ( δ x, − δ x, − ) − F p ( x ) . (D.2)We defined Q p = α p − ( − , − p − (cid:88) q =0 α q ( − , α p − q − (0 , , (D.3) on-local response in a lattice gas under a shear drive α ( x, y ) = ρ . The Q p depends only on the lower order solutions α k , with k < p , atthe two adjacent sites of the slow bond.Note that, using (46) yields, ∞ (cid:88) p =1 (cid:15) p Q p = (cid:15)φ ( − ,
0) [1 − φ (0 , . (D.4)This identity will be used later in the derivation.The last quantity F ( x ) in (D.2) also depends only on the lower order solutions F p ( x ) = p − (cid:88) q =1 α q ( x,
0) [ α p − q ( x + 1 , − α p − q ( x − , . (D.5)The infinite set of linear equations (D.1) and (D.2) can be solved recursively. Theanalysis is similar to the solution of (47)-(48). We only present the final solution here.The solution for the p th order in the expansion α p ( x, y ) = Q p L L − (cid:88) n =1 (cid:0) e iω n − (cid:1) Γ ρ ( ω n ) e iω n x [ z + ( ω n )] y + L − (cid:88) n =1 f p ( ω n )Γ ρ ( ω n ) e iω n x [ z + ( ω n )] y , (D.6)where ω n = 2 πn/L with n = 0 , , · · · , L −
1, and f p ( ω n ) is the Fourier transform of F p ( x ), defined as f p ( ω n ) = 1 L L − (cid:88) x =0 F p ( x ) e − iω n . (D.7)The function Γ ρ ( ω n ) is defined in Equation (56).The first term in the solution, comes from the term containing the Kronecker deltafunctions in (D.2), and the second from F p ( x ). For convenience of presentation, let usdefine a p ( ω n ) = Q p + L f p ( ω n ) e iω n − . (D.8)Notice ( e iω n −
1) does not vanish in the range of 1 ≤ n ≤ L − α p ( x, y ) = 1 L L − (cid:88) n =1 γ ρ ( ω n ) × e iω n x [ z + ( ω n )] y a p ( ω n ) , (D.9)where γ ρ ( ω n ) = ( e iω n − ρ ( ω n ). Notice that for ρ = 0 the γ ρ ( ω ) is same as γ ( ω ) definedin (17) for the independent particles case.Then, using (46), yields the expression for the density profile, φ ( x, y ) − ρ = 1 L L − (cid:88) n =1 γ ρ ( ω n ) e iω n x [ z + ( ω n )] y × ∞ (cid:88) p =1 (cid:15) p a p ( ω n . ) (D.10)Using (D.4) and (D.8), yields ∞ (cid:88) p =1 (cid:15) p a p ( ω n ) = (cid:15)φ ( − ,
0) [1 − φ (0 , Le iω n − ∞ (cid:88) p =1 (cid:15) p f p ( ω n ) . (D.11) on-local response in a lattice gas under a shear drive φ ( x, y ) − ρ = (cid:15)φ ( − ,
0) [1 − φ (0 , L L − (cid:88) n =1 γ ρ ( ω n ) e iω n x [ z + ( ω n )] y + 1 L L − (cid:88) n =1 γ ρ ( ω n ) e iω n x [ z + ( ω n )] y × Σ ( ω n ) . (D.12)where Σ ( ω n ) = Le iω n − ∞ (cid:88) p =1 (cid:15) p f p ( ω n ) . (D.13)This is the complete solution of profile, with the f p ( ω n ) defined in (D.7). Note, thedensity difference vanishes as (cid:15) →
0, as expected.
Appendix D.1. Asymptotic profile
To study the large distance profile it is simpler to analyze in the L → ∞ limit, where ω n ≡ ω becomes a continuous variable, and the summation over n can be approximatedby an integration. This yields, φ ( x, y ) − ρ = (cid:15)φ ( − ,
0) [1 − φ (0 , π (cid:90) π dω γ ρ ( ω ) e iωx [ z + ( ω )] y + 12 π (cid:90) π dω γ ρ ( ω ) e iωx [ z + ( ω )] y × Σ ( ω ) . (D.14)As shown later in the Appendix D.2, the Σ( ω ) is analytic around ω = 0 and alsovanishes as ω →
0. This implies, the Σ( ω ) is Taylor expandable with the lowest orderbeing ω . At large distances, the profile is essentially determined by the small ω modes.Then clearly, at large distances, the second integral involving Σ( ω ) in (D.14) is sub-dominant to the first and the profile is φ ( x, y ) − ρ (cid:39) (cid:15)φ ( − ,
0) [1 − φ (0 , π (cid:90) π dω γ ρ ( ω ) e iωx [ z + ( ω )] y . (D.15) Appendix D.2. Analyticity of Σ( ω ) . We shall show that the Σ( ω ) is analytic around ω = 0 and has a Taylor expansionin powers of ω . The expression in (D.13) involves f p ( ω n ) which can be determinedrecursively order by order. However, it is not straightforward to prove the analyticityof Σ( ω ) from this expression. We argue in the following steps by expressing Σ( ω ) in analternate form where the ω dependence is clearer.A straightforward algebra using (D.5) and (D.9) yields,Σ( ω ) = 12 π (cid:90) ω dω (cid:48) e − iω (cid:48) γ ρ ( ω − ω (cid:48) ) γ ρ ( ω (cid:48) ) ∞ (cid:88) p =1 (cid:15) p p − (cid:88) q =1 a q ( ω − ω (cid:48) ) a p − q ( ω (cid:48) ) , (D.16) on-local response in a lattice gas under a shear drive a p ( ω ) defined in (D.8) can be determinedrecursively using a ( ω n ) = Q (D.17)and a p ( ω n ) = Q p + 12 π (cid:90) ω dω (cid:48) e − iω (cid:48) γ ρ ( ω − ω (cid:48) ) γ ρ ( ω (cid:48) ) p − (cid:88) q =1 a q ( ω − ω (cid:48) ) a p − q ( ω (cid:48) ) . (D.18)It is more convenient to express Σ( ω ) in terms of the function β n ( ω ) defined as β ( ω ) = 1 , and (D.19) β p ( ω ) = 12 π (cid:90) ω e − iω γ ρ ( ω − ω (cid:48) ) γ ρ ( ω ) p − (cid:88) q =1 β q ( ω ) β p − q ( ω − ω (cid:48) ) , for p > a p ( ω ) in (D.18) can be expressed as a linear combination of β p ( ω ),as follows, a ( ω ) = Q , (D.21) a ( ω ) = Q + Q β ( ω ) , (D.22) a ( ω ) = Q + 2 Q Q β ( ω ) + Q β ( ω ) , (D.23)and so on. On the other hand, by definition in (D.8) and (D.13), ∞ (cid:88) p =1 (cid:15) p a p = ∞ (cid:88) p =1 (cid:15) p Q p + Σ( ω ) . (D.24)Then, clearly Σ( ω ) can be expressed in a linear combination of β p asΣ ( ω ) = A β ( ω ) + A β ( ω ) + · · · , (D.25)where A p are constants depending on Q p and (cid:15) . In this way the ω dependence of Σ( ω )is solely in terms of β p ( ω ).In the next step we show that the functions β p ( ω ) are analytic around ω = 0 andsmoothly vanishes as ω →
0, for all p >
1. This can be seen by explicitly writing theexpression of γ ρ ( ω ) from (60) in the definition of β p ( ω ) in (D.20). This yields, β p ( ω ) = e iω/ π (cid:90) ω/ dω (cid:48) e − iω (cid:48) (1 − ρ ) cos( ω (cid:48) ) − i (cid:113) ( ω (cid:48) ) × (cid:80) p − q β q ( ω ) β p − q ( ω − ω (cid:48) )(1 − ρ ) cos( ω − ω (cid:48) ) − i (cid:113) ( ω − ω (cid:48) ) , (D.26)which for small ω can be approximated as β p ( ω ) (cid:39) e iω/ π (cid:90) ω/ dω (cid:48) e − iω (cid:48) (cid:80) p − q β q ( ω ) β p − q ( ω − ω (cid:48) )(1 − ρ − i ) . (D.27) on-local response in a lattice gas under a shear drive β ( ω ) = 1, it is easy to show that β p ( ω ) ∼ w p − as ω →
0. (D.28)Using this result in the Equation (D.25) yields that Σ( ω ) is expandable in positiveinteger powers of ω around ω = 0, asΣ ( ω ) = C ω + C ω + · · · , (D.29)with C k depending on Q k and (cid:15) . References [1] T. Chou, K. Mallick, and R. K. P. Zia. Non-equilibrium statistical mechanics: from a paradigmaticmodel to biological transport.
Rep. Prog. Phys. , 74:116601, 2011.[2] M. R. Evans. Phase transitions in one-dimensional nonequilibrium systems.
Braz. J. Phys. , 30:42,2000.[3] M. Henkel and M. Pleimling.
Nonequilibrium Phase Transitions Ageing and Dynamical Scalingfar from Equilibrium , volume 2. Springer, Heidelberg, 2010.[4] G. Grinstein. Generic scale invariance in classical nonequilibrium systems (invited).
J. App.Phys. , 69:5441, 1991.[5] B. Schmittmann and R.K.P. Zia. Statistical mechanics of driven diffusive systems. In C. Domband J.L. Lebowitz, editors,
Statistical Mechanics of Driven Diffusive System , volume 17 of
PhaseTransitions and Critical Phenomena . Academic Press, 1995.[6] H. Spohn. Long range correlations for stochastic lattice gases in a non-equilibrium steady state.
J. Phys. A , 16:4275, 1983.[7] S. Katz, J. L. Lebowitz, and H. Spohn. Phase transitions in stationary nonequilibrium states ofmodel lattice systems.
Phys. Rev. B , 28:1655, 1983.[8] S. Katz, J. L. Lebowitz, and H. Spohn. Nonequilibrium steady states of stochastic lattice gasmodels of fast ionic conductors.
J. Stat. Phys. , 34:497, 1984.[9] M. Q. Zhang, J. S. Wang, J. L. Lebowitz, and J. L. Vall´es. Power law decay of correlations instationary nonequilibrium lattice gases with conservative dynamics.
J. Stat. Phys. , 52:1461,1988.[10] P. L. Garrido, J. L. Lebowitz, C. Maes, and H. Spohn. Long-range correlations for conservativedynamics.
Phys. Rev. A , 42:1954, 1990.[11] G. Grinstein, D. H. Lee, and S. Sachdev. Conservation laws, anisotropy, and “self-organizedcriticality” in noisy nonequilibrium systems.
Phys. Rev. Lett. , 64:1927, 1990.[12] I. Pagonabarraga and J. M. Rub´ı. Long-range correlations in diffusive systems away fromequilibrium.
Phys. Rev. E , 49:267, 1994.[13] Shin ichi Sasa. Long range spatial correlation between two brownian particles under externaldriving.
Physica D , 205:233, 2005.[14] T. Sadhu, S. N. Majumdar, and D. Mukamel. Long-range correlations in a locally driven exclusionprocess.
Phys. Rev. E , 90:012109, 2014.[15] C. Maes, K. Netoˇcn´y, and B. M. Shergelashvili. Nonequilibrium relation between potential andstationary distribution for driven diffusion.
Phys. Rev. E , 80:011121, 2009.[16] T. Sadhu, S.N. Majumdar, and D. Mukamel. Long-range steady-state density profiles induced bylocalized drive.
Phys. Rev. E , 84:051136, 2011.[17] A. Onuki and K. Kawasaki. Nonequilibrium steady state of critical fluids under shear flow: Arenormalization group approach.
Ann. Phys. , 121:456, 1979.[18] A. Onuki. Phase transitions of fluids in shear flow.
J. Phys , 9:6119, 1997.[19] D. Winter, P. Virnau, J. Horbach, and K. Binder. Finite-size scaling analysis of the anisotropiccritical behavior of the two-dimensional ising model under shear.
EPL , 91:60002, 2010. on-local response in a lattice gas under a shear drive [20] M. Thi´ebaud and T. Bickel. Nonequilibrium fluctuations of an interface under shear. Phys. Rev.E , 81:031602, 2010.[21] J. Howard.
Mechanics of Motor Proteins and the Cytoskeleton . Sunderland, MA, 1 edition, 2001.[22] S. Klumpp and R. Lipowsky. Traffic of molecular motors through tube like compartments.
J.Stat. Phys. , 113:233, 2003.[23] S. A. Janowsky and J. L. Lebowitz. Finite-size effects and shock fluctuations in the asymmetricsimple-exclusion process.
Phys. Rev. A , 45:618, 1992.[24] S. Janowsky and J. Lebowitz. Exact results for the asymmetric simple exclusion process with ablockage.
J. Stat. Phys. , 77:35, 1994.[25] G. Schtz. Generalized bethe ansatz solution of a one-dimensional asymmetric exclusion processon a ring with blockage.
J. Stat. Phys. , 71:471, 1993.[26] T. Sepplinen. Hydrodynamic profiles for the totally asymmetric exclusion process with a slowbond.
J. Stat. Phys. , 102:69, 2001.[27] V. Popkov and M. Salerno. Hydrodynamic limit of multichain driven diffusive models.
Phys. Rev.E , 69:046103, 2004.[28] T. Ezaki and K. Nishinari. Exact solution of a heterogeneous multilane asymmetric simpleexclusion process.
Phys. Rev. E , 84:061141, 2011.[29] F. Q. Potiguar and R. Dickman. Lattice gas with nearest-neighbor exclusion in a shear-like field.
Braz. J. Phys. , 36:736, 2006.[30] T. Reichenbach, E. Frey, and T. Franosch. Traffic jams induced by rare switching events in two-lane transport.
New J. Physics , 9:159, 2007.[31] G. Korniss, B. Schmittmann, and R. K. P. Zia. Long-range order in a quasi one-dimensionalnon-equilibrium three-state lattice gas.
EPL , 45:431, 1999.[32] D. Helbing. Traffic and related self-driven many-particle systems.
Rev. Mod. Phys. , 73:1067, 2001.[33] D. Chowdhury, L. Santen, and A. Schadschneider. Statistical physics of vehicular traffic and somerelated systems.
Phys. Rep. , 329:199, 2000.[34] H. W. Lee, V. Popkov, and D. Kim. Two-way traffic flow: Exactly solvable model of traffic jam.
J. Phys. A , 30:8497, 1997.[35] M. Kanai. Two-lane traffic-flow model with an exact steady-state solution.
Phys. Rev. E ,82:066107, 2010.[36] Ekaterina Pronina and Anatoly B Kolomeisky. Two-channel totally asymmetric simple exclusionprocesses.
J. Phys. A , 37:9907, 2004.[37] E. Pronina and A. B. Kolomeisky. Asymmetric coupling in two-channel simple exclusion processes.
Physica A , 372:12, 2006.[38] R. J. Harris and R. B. Stinchcombe. Ideal and disordered two-lane traffic models.
Physica A ,354:582, 2005.[39] R. Jiang, M. Hu, Y. Wu, and Q. Wu. Weak and strong coupling in a two-lane asymmetric exclusionprocess.
Phys. Rev. E , 77:041128, 2008.[40] Christoph Schiffmann, Ccile Appert-Rolland, and Ludger Santen. Shock dynamics of two-lanedriven lattice gases.
J. Stat. Mech. , 2010:P06002, 2010.[41] A. Melbinger, T. Reichenbach, T. Franosch, and E. Frey. Driven transport on parallel lanes withparticle exclusion and obstruction.
Phys. Rev. E , 83:031923, 2011.[42] V. Popkov and I. Peschel. Symmetry breaking and phase coexistence in a driven diffusive two-channel system.
Phys. Rev. E , 64:026126, 2001.[43] T. Mitsudo and H. Hayakawa. Synchronization of kinks in the two-lane totally asymmetric simpleexclusion process with open boundary conditions.
J. Phys. A , 38:3087, 2005.[44] K. Tsekouras and A. B. Kolomeisky. Parallel coupling of symmetric and asymmetric exclusionprocesses.
J. Phys. A , 41:465001, 2008.[45] R. Dickman and R. R. Vidigal. Particle redistribution and slow decay of correlations in hard-corefluids on a half-driven ladder.
J. Stat. Mech. , 2007:P05003, 2007.[46] V. Yadav, R. Singh, and S. Mukherji. Phase-plane analysis of driven multi-lane exclusion models. on-local response in a lattice gas under a shear drive J. Stat. Mech. , 2012:P04004, 2012.[47] D. Kadau, A. Hucht, and D. E. Wolf. Magnetic friction in ising spin systems.
Phys. Rev. Lett. ,101:137205, 2008.[48] A. Hucht. Nonequilibrium phase transition in an exactly solvable driven ising model with friction.
Phys. Rev. E , 80:061138, 2009.[49] H. J. Hilhorst. Two interacting ising chains in relative motion.
J. Stat. Mech. , 2011:P04009, 2011.[50] R. Rajesh and S. N. Majumdar. Conserved mass models and particle systems in one dimension.
J. Stat. Phys. , 99:943, 2000.[51] R. Rajesh and S. N. Majumdar. Exact calculation of the spatiotemporal correlations in thetakayasu model and in the q model of force fluctuations in bead packs.
Phys. Rev. E , 62:3186,2000.[52] D. Mukamel. Phase transitions in nonequilibrium systems. In M. E. Cates and R. Evans, editors,
Soft and Fragile Matter: Nonequilibrium Dynamics, Metastability and Flow , Proceedings ofScottish Universities Summer School in Physics. Taylor & Francis, 2000.[53] R. A. Blythe and M. R. Evans. Nonequilibrium steady states of matrix-product form: a solver’sguide.
J. Phys. A , 40:R333, 2007.[54] B.R. Munson, D.F. Young, and T.H. Okiishi.
Fund. Fluid Mech.
John Wiley, 2005.[55] D. Derks, D. G. A. L. Aarts, D. Bonn, H. N. W. Lekkerkerker, and A. Imhof. Suppression ofthermally excited capillary waves by shear flow.
Phys. Rev. Lett. , 97:038301, 2006.[56] T. H. R. Smith, O. Vasilyev, D. B. Abraham, A. Macio(cid:32)lek, and M. Schmidt. Interfaces in drivenising models: Shear enhances confinement.
Phys. Rev. Lett. , 101:067203, 2008.[57] T. H. R. Smith, O. Vasilyev, A. Macio(cid:32)lek, and M. Schmidt. Laterally driven interfaces in thethree-dimensional ising lattice gas.
Phys. Rev. E , 82:021126, 2010.[58] C. M. Bender, S. Boettcher, and K. A. Milton. A new perturbative approach to nonlinear partialdifferential equations.
J. Math. Phys. , 32:3031, 1991.[59] Bernard J. Laurenzi. An analytic solution to the thomas–fermi equation.