Vortex and translational currents due to broken time-space symmetries
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n Vortex and translational currents due to broken time-space symmetries
S. Denisov , Y. Zolotaryuk , S. Flach , and O. Yevtushenko Institut f¨ur Physik, Universit¨at Augsburg, Universit¨atsstr.1, D-86135 Augsburg, Germany Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, 03680 Kiev, Ukraine Max-Planck-Institute for the Physics of Complex Systems,N¨othnitzer Str. 38, D-01187 Dresden, Germany and Physics Department, Arnold Sommerfeld Center for Theoretical Physics,Ludwig-Maximilians-Universit¨at M¨unchen, D-80333 M¨unchen, Germany (Dated: October 31, 2018)We consider the classical dynamics of a particle in a d = 2 , PACS numbers: 05.45.-a, 05.60.Cd, 05.40.-a
The idea of directed motion under the action of an ex-ternal fluctuating field of zero mean goes back to Smolu-chowski and Feynman [1]. It has been intensively stud-ied in the past decades again [2]. It is believed to beconnected with the functioning of molecular motors, andcan be applied to transport phenomena which range frommechanical engines to an electron gas (see [3, 4] and ref-erences therein).The separation of the fluctuating fields into an uncorre-lated white noise term and a time-periodic field was usedto perform a symmetry analysis of the most simple case -a point-like particle moving in a one-dimensional periodicpotential [5]. It allowed to systematically choose spaceand time dependencies of potentials and ac fields such,that a nonzero dc current is generated. Various studiesof the dynamical mechanisms of rectification have beenreported (e.g. [6]). Among many experimental reports,we mention the successful testing of the above symme-try analysis using cold atoms in one-dimensional opticalpotentials [7]. By use of more laser beams, experimen-talists can already fabricate two- and three-dimensionaloptical potentials, with different symmetries and shapes[8], with the aim of even more controlled stirring of coldatoms in these setups.A particle which is moving in a d = 2 , d = 2 , µ ¨ r + γ ˙ r = g ( r , t ) + ξ ( t ) , g ( r , t ) = − ∇ U ( r , t ) . (1)Here r = { x, y, z } is the coordinate vector of the par-ticle, the parameter γ ≥ µ ≥ g ( r , t ) = { g α ( r , t ) } , α = x, y, z , istime- and space-periodic: g ( r , t ) = g ( r , t + T ) = g ( r + L , t ) . (2)The absence of a dc bias implies h g ( r , t ) i L ,T ≡ Z T Z L g ( r , t ) dt dxdydz = 0 (3)where the spatial integration extends over one unit cell.The fluctuating force is modeled by a δ -correlatedGaussian white noise, ξ ( t ) = { ξ x , ξ y , ξ x } , h ξ α ( t ) ξ β ( t ′ ) i =2 γDδ ( t − t ′ ) δ αβ ( α, β = x, y, z ). Here D is the noisestrength. The statistical description of the system (1) isprovided by the Fokker-Planck equations (FPE) [11]:˙ P ( r , v , t ) = {−∇ r · v + ∇ v · [ γ v − g ( r , t )] ++ γD △ v } P ( r , v , t ) , µ = , v = ˙r , (4) γ ˙ P ( r , t ) = − [ ∇ r · g ( r , t ) + D △ r ] P ( r , t ) , µ = . (5)Each of the linear equations (4-5) has a unique attractorsolution, ˆ P which is space and time periodic [11]. Directed transport.
Let us consider the dc componentof the directed current j ( t ) = v = ˙ r in terms of theattractor ˆ P : J = h v · ˆ P ( r , v , t ) i T, L , µ = 1 , (6) J = 1 γ h g ( r , t ) · ˆ P ( r , t ) i T, L , µ = 0 . (7)The strategy is now to identify symmetry operationswhich invert the sign of j , and, at the same time, leaveEq. (1) invariant. If such symmetries exist, the dc cur-rent J will strictly vanish. Sign changes of the current canbe obtained by either inverting the spatial coordinates,or time (simultaneously allowing for shifts in the othervariables). Below we list all operations together with therequirements the force g and the control parameters haveto fulfill: b S : r → − r + r ′ , t → t + τ ; b S ( g ) → − g , (8) b S : r → r + Λ , t → − t + t ′ ; b S ( g ) → g (if γ = 0); b S ( g ) → − g (if µ = 0) . (9)Here t ′ and r ′ depend on the particular shape of g ( r , t ).The system must be invariant under a spatial translationby the vector 2 Λ in space and 2 τ in time, respectively.The vector Λ is therefore given by Λ = P α n α L α / n α = 0 ,
1, while τ = 0 , T /
2. By a proper choice of g all relevant symmetries can be broken, and one can thenexpect the appearance of a non-zero dc current J [12].To be more precise, we consider the case of a particlemoving in a two-dimensional periodic potential and beingdriven by an external ac field: g ( r , t ) = − ∇ V ( r )+ E ( t ) ≡ f ( r ) + E ( t ). The symmetry b S holds if the potential forceis anti-symmetric , f ( − r + r ′ ) = − f ( r ), and the drivingfunction is shift-symmetric , E ( t + T /
2) = − E ( t ). Thesymmetry b S holds at the Hamiltonian limit, γ = 0, if thedriving force is symmetric , E ( t + t ′ ) = E ( − t ). Finally, thesymmetry b S holds at the overdamped limit, µ = 0 , if thepotential force is shift-symmetric, f ( r + Λ ) = − f ( r ) andthe driving force is anti-symmetric, E ( t + t ′ ) = − E ( − t ).In order to break the above symmetries, we choose V ( r ) = V ( x, y ) = cos( x )[1 + cos(2 y )] , (10) E x,y ( t ) = E (1) x,y sin t + E (2) x,y sin(2 t + θ ) . (11)The potential (10) is shift-symmetric, Λ = {± π, } . Thesymmetry b S is broken since E is not shift-symmetric.Therefore in general we expect J = 0.In Fig.1 we show the computational evaluation of equa-tions (4,5) [13]. We confirm the presence of a nonzero dccurrent. Applying operations b S and θ → θ + π we con-clude J ( θ + π ) = − J ( θ ), which allows for an easy inversionof the current direction, as also confirmed by the data inFig.1(a). In the overdamped limit µ = 0, b S is restoredfor θ = 0 , ± π , and therefore J ( − θ ) = − J ( θ ) (thick linesin Fig.1(a)). Upon approaching the Hamiltonian limit, θ -0.4-0.200.20.4 J x , y γ θ x (a) π/2π (c) x y −5 0 5 10 15−50510 (i) (ii) (b) FIG. 1: (color online) (a) Dependence of the current compo-nents, J x (solid line) and J y (dashed line), on θ for (1),(10)-(11), with D = 1, E (1) x = − E (2) x = 2, E (1) y = − E (2) y = 2 . µ = 0 , γ = 1, thick lines)and underdamped ( µ = 1 , γ = 0 .
1, thin lines) cases, respec-tively; (b) The time evolution of the mean particle position,¯ r ( t ) = R r P ( r , v , t ) d r d v , for θ = 0. The trajectories are su-perimposed on the contour plot of the potential (10). Curve( i ) corresponds to E (1) x = 3, E (2) x = E (1) y = 0, E (2) y = 3 . ii ) - to the parameters of panel (a). The otherparameters are: D = µ = 1 , γ = 0 .
1; (c) The phase lag θ (0) x as a function of the dissipation strength γ . γ →
0, the points where J = 0 shift from θ = 0 , π to θ = ± π/ b S is restored (thin linesin Fig.1(a)). In the underdamped regime, the dc-currentcan be approximated as J α ∝ J (0) α sin[ θ − θ (0) α ( γ )] , α = x, y . The phase lag is equal to θ (0) x,y = π/ θ (0) x,y = 0in the Hamiltonian and overdamped limits, respectively(Fig.1(c)) [14].Even more control over the current direction is possi-ble, by imposing the symmetry conditions (8-9) on eachcomponent g α ( r , t ) independently. For (10) and (11) with E (2) x = E (1) y = 0, θ = 0 the symmetry transformation b S c : x → − x, y → y, t → t + π implies that the cur-rent along the x -direction is absent, J x = 0, and directedtransport is happening along the y -axis [see Fig.1b, curve(i)]. We may conclude, that the symmetry analysis turnsout to be a powerful tool of predicting and controllingdirected currents of particles which move in d = 2 , Vorticity.
At variance to the one-dimensional case,particles in two and three dimensions can perform vor-tex motion, thereby generating ring currents, or nonzeroangular momentum. First of all we note that the particledynamics is not confined to one spatial unit cell of theperiodic potential U ( r , t ). Even in the case when a di-rected current is zero due to the above symmetries, J = 0,the particle can perform unbiased diffusion in coordinatespace. In order to distinguish between directed transportand spatial diffusion on one side, and rotational currentson the other side, we use the angular velocity [17] Ω ( t ) = [˙ r ( t ) × ¨ r ( t )] / ˙ r ( t ) , J Ω = h Ω ( t ) i t , (12)as a measure for the particle rotation, where h ... i t =lim t →∞ t R t ...dt ′ . Ω ( t ) is invariant under translations inspace and time. It describes the speed of rotation withwhich the velocity vector ˙r (the tangential vector to thetrajectory r ( t )) encompasses the origin.Using the above strategy, we search for symmetry op-erations which leave the equations of motion invariant,but do change the sign of the angular velocity. If suchsymmetries exist, rotational currents strictly vanish onaverage. The sign of Ω can be inverted by either (i) time inversion t → − t together with an optional spaceinversion r → ± r , or (ii) the permutation of any twovariables, e.g. b P xy : { x, y, z } → { y, x, z } . That leads tothe following possible symmetry transformations: b R : r → b P r + r ′ , t → t + τ ; b R ( g ) → g , (13) b R : r → ± r + Λ , t → − t + t ′ , b R ( g ) → g (if γ = 0); b R ( g ) → − g (if µ = 0) . (14)Here b P stands for any of the following operations: b P xy , b P yz or b P zx and t ′ and r ′ again depend on the particularshape of g ( r , t ).To be concrete, we will again consider a particle mov-ing in a two-dimensional periodic potential and beingdriven by an external ac field: g ( r , t ) = − ∇ V ( r )+ E ( t ) ≡ f ( r ) + E ( t ). For d = 2 there is an additional transforma-tion due to a mirror reflection at any axis, ˆΣ x : { x, y } →{ x, − y } or ˆΣ y : { x, y } → {− x, y } , b R : r → ˆΣ r , t → t + T / b R ( g ) → g (15)Symmetry b R can be satisfied for the Hamiltonian , un-derdamped and overdamped cases if b P f ( b P r + r ′ ) = f ( r )and b P E ( t + t ′ ) = E ( t ). The symmetry b R apply for thesame cases as their counterparts b S if there is no spaceinversion. In the presence of space inversion they can besatisfied both in the Hamiltonian [if f ( − r ) = − f ( r ) and E ( t + t ′ ) = − E ( − t )] and overdamped [if f ( − r ) = f ( r )and E ( t + t ′ ) = E ( − t )] limits. The symmetry b R isrelevant if ˆΣ x can be applied: f x ( x, − y ) = f x ( x, − y ), f y ( x, − y ) = − f y ( x, − y ), E x ( t + T /
2) = E x ( t ) and θ -0.2-0.100.10.20.3 J x -2002040 y -1 0 1 x -3-2-10 y γ θ Ω (a) (b) FIG. 2: (color online) (a) Dependence J Ω ( θ ), Eq.(12), for(1), (16)-(17), with µ = 1, D = 0 . E (1) x = 0 . E (1) y = 0 . γ = 0 . γ = 0 .
05 (dashed line), and γ = 2(dashed-dotted line). Insets: the trajectory (left insert) andthe corresponding attractor solution, ¯ r ( t ), (right inset) for thecase γ = 0 . θ = π/
2. We have used N = 10 independentstochastic realizations to perform the noise averaging; (b) Thephase lag θ as a function of the dissipation strength γ . E y ( t + T /
2) = − E y ( t ). Similar conditions can be foundfor ˆΣ y .We performed numerical integrations of the equationof motion (1) with the following potential and drivingforce: V ( x, y ) = [ − x + cos y ) + cos x cos y ] / , (16) E x ( t ) = E (1) x cos t , E y ( t ) = E (1) y cos( t + θ ) . (17)Averaging was performed over N = 10 different stochas-tic realizations [18]. Fig.2 shows the dependence of therotational current (12) on the relative phase θ . The sys-tem is invariant under the transformation b S (8), there-fore the directed current J = 0. However, for the under-damped case, γ = 0, all the relevant symmetries (13-15)are violated, and the resulting rotational current (12)is nonzero, and depends on the phase θ (Fig.2a). Notethat symmetry b R is restored when θ = 0 , ± π , thus thecurrent disappears in the Hamiltonian and overdampedlimits for these values of the phase. The left upper insetin Fig.2 shows the actual trajectory of a given realiza-tion, confirming that the particle is acquiring an averagenonzero angular momentum, while not leaving a small fi-nite volume due to slow diffusion and absence of directedcurrents.The exact overdamped limit, µ = 0, is singular for thedefinition (12) since the velocity of a particle, ˙r (t), is anowhere differentiable function. The overdamped limitcan be approached by increasing γ at a fixed µ = 1. Al-ternatively, one may remove the restriction on µ allowingfor an infinitesimal value 0 < µ ≪ γ = 1. Both parameter choices equally regular-ize (12). Numerical simulations for the former way of reg-ularization show that if γ/µ ≥ θ ( γ ) in Fig.2(b)).Let us discuss the relation of our results to the caseof multidimensional stochastic tilting ratchets under theinfluence of a colored noise studied previously [19]. Sinceequivalent (in a statistical sense) stochastic processes, ξ ( t ), have been used as driving forces, the symmetry b R (13) can be violated only by an asymmetric potential.But all potentials considered in Refs. [19] are invariantunder the permutation transformation b P . As a conse-quence, vortex structures for a local velocity field pre-sented in Refs. [19] are completely symmetric (clockwisevortices are mapped into counterclockwise ones by b P )and, therefore, the average rotation for any trajectoryequals zero.The phase space dimension is five for d = 2 and sevenfor d = 3. Therefore, in the Hamiltonian limit ( γ = 0),Arnold diffusion [20] takes place. The particle dynamicsis no longer confined within chaotic layers of finite width.That leads to unbounded, possibly extremely slow, diffu-sion in the momentum subspace via a stochastic web [20].Therefore a direct numerical integration of the equationsof motion may lead to incorrect conclusions.There is rich variety of physical systems, where multi-dimensional ratchets can be observed: cold atoms in two-and three-dimensional potentials (optical guiding) [21],colloidal particles on magnetic bubble lattices [22], fer-rofluids [23], and vortices in superconducting films withpinning sites [24]. Our methods of directed current con-trol can be used for an enhancement of the particle sep-aration in the laser beams of complex geometry [25].To conclude, we formulated space-time symmetries forthe absence of both directed currents, and rotational cur-rents, for particles moving in spatially periodic poten-tials, under the influence of external ac fields. Properchoices of these potentials and fields allow to break theabove symmetries, and therefore to generate and controldirected and rotational vortex currents in an indepen-dent way. Numerical studies supplement the symmetryanalysis and confirm the conclusions. Acknowledgments.
This work has been partially sup-ported by the DFG-grant HA1517/31-1 (S. F. and S.D.), National Academy of Sciences of Ukraine throughthe special program for young scientists (Y.Z.). O.Y.acknowledges support from SFB-TR-12. [1] M. von Smoluchowski, Phys. Zeitschrift
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