Stochastic ratcheting of two dimensional colloids : Directed current and dynamicaltransitions

Dipanjan Chakraborty1 and Debasish Chaudhuri2

1Indian Institute of Science Education and Research, Mohali, Punjab-140306, India. ∗2Indian Institute of Technology Hyderabad, Yeddumailaram 502205, Andhra Pradesh, India †

(Dated: July 11, 2014)

We present results of molecular dynamics simulations for two-dimensional repulsively interactingcolloids driven by an one dimensional asymmetric and commensurate ratchet potential, switching onand off stochastically. The resultant time- and space-averaged directed current exhibits resonancewith change in ratcheting frequency. The resonance frequency itself varies non-monotonically withdensity. We use scaling arguments to derive analytic expression for the directed current whichreproduces these features. Our simulations reveal re-entrant dynamical transitions between solidand modulated liquid phases as a function of ratcheting frequency, associated with the variation ofdirected current.

PACS numbers: 05.40.Jc, 50.60.-k, 64.60.Cn

A flashing ratchet refers to a time-averaged directedmotion of Brownian particles under the influence of anspatially periodic and asymmetric potential, switchingon and off stochastically [1–4]. Stochastic ratcheting hasbeen extensively studied, in the context of active dynam-ics of molecular motors [5–7], dynamics of colloidal dis-persion in electrical [8–10], magnetic [11, 12] or opticaldrive [13, 14], as a mechanism of particle segregation [15–17], transport of cold atoms in optical lattice [18], andin the motion of flux quanta [19, 20]. While a largebody of work has been concentrated on the ratchetingof individual particles, fewer studies focused on the ef-fects of interaction [21–25]. Recent studies on two di-mensional (2D) paramagnetic particles under one dimen-sional (1D) magnetic ratcheting found relation betweendynamics and local structure in terms of particle coor-dination number [12], and emergence of large transversediffusion due to random disorder [11].

In colloidal suspensions, ratchet-like directed motion ofparticles have been achieved using suitable laser poten-tials [13, 14]. Confinement and laser trapping in colloids,on the other hand, is known to give rise to interesting me-chanical properties and phase transitions [26–31]. Cou-pling 2D interacting colloids to a 1D time-independentspatially periodic potential with periodicity commensu-rate with the mean particle separation, leads to the phe-nomena of laser induced freezing (LIF) and re-entrantmelting with increase in the potential strength. This wasdemonstrated in experiments using standing wave pat-tern of interfering laser beams [30, 31], and was under-stood in terms of a dislocation unbinding theory [9, 32].

We consider transport of a 2D system of particles inter-acting via soft-core repulsion and driven by an 1D asym-metric flashing ratchet, using molecular dynamics (MD)simulations in the presence of a Langevin heat bath. Theratcheting potential breaks time-reversal symmetry andgenerates an average directed current along the direc-tion of ratcheting (Fig. 1(a)). We choose a periodicity of

FIG. 1. (Color online) (a) Schematic of 2d colloids in 1Dasymmetric ratchet potential periodic in y- and constant inx- direction. The arrow between the (green) corrugated andflat surfaces denote switching of external potential betweenon and off state with rate f . (b) Time-averaged directed cur-rent along y-direction 〈jy〉 as a function of frequency f , anddensity ρ. The dashed line indicates variation of maximal cur-rent with density. (c) – (e): Superimposed positions of 103

uncorrelated configurations, from center of mass coordinates,at a density ρ = 1.0 with ratcheting frequencies (c) f = 0.11,(d) f = 1.67 and (e) f = 10. The color code denotes local den-sity of points from red/light (high) to blue/dark (low). Thereciprocal lattice vectors G1,2 and the corresponding latticeplanes are indicated in (e).

the potential commensurate with the inter particle sep-aration. At switching frequencies much faster than the

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intrinsic relaxation times, the system experiences a time-averaged effective periodic potential, which in the limit ofweak asymmetry is expected to lead to a situation similarto that of LIF.

In the 2D ratchet system that we study, averaged di-rected current shows non-monotonic variation with den-sity and ratcheting frequency, with the maximal currentachieved at their intermediate values. The behavior dif-fers significantly in detailed functional dependence from1D ratchet. Using scaling arguments, we derive expres-sions for the directed current which fully capture the sim-ulation results. Our study on 2D ratchet reveals two fas-cinating properties which are unlike 1D ratchet: (i) dy-namical structural transitions between solid and modu-lated liquid phase associated with the variation of cur-rent, (ii) crossover from ballistic to diffusive transportwith density, captured by a non-monotonic density- de-pendence of resonance frequency. Our predictions areamenable to verification in experiments on, e.g., steri-cally stabilized colloids driven by suitably tunable opticalor magnetic ratchets [12, 13].

Model: As a model colloid, we consider a system ofpurely repulsive particles interacting via a shifted andtruncated soft-core potential βU(r) = (σ/r)12 − 2−12

with a cutoff distance rc = 2σ, so that βU(r) = 0for r > rc. Here kBT = 1/β and σ set the energyand length scales, respectively. The asymmetric ratchetpotential Uext(y, t) = V0(t) [sin (2πy/λ) + α sin (4πy/λ)],where V0(t) switches between U0 and 0 with a frequencyf , and we used α = 0.2 the asymmetry parameter (seeFig. 1(a)). In all our simulations we use βU0 = 1. Theexternal potential is kept commensurate to the densityof the particles, such that λ = ay, with the separationbetween consecutive lattice planes ay =

√3a/2 in a tri-

angular lattice at a density ρ = 2/√

3a2. MD simulationsare performed using the standard leap-frog algorithm [34]with a time step δt = 0.001τ , where τ = σ

√m/kBT is

the characteristic time scale. We choose the mass of theparticles m = 1, and set the temperature T = 1.0 ε/kBby using a Langevin thermostat [35] with an isotropicfriction γ = 1/τ . We used N = 4096 particles in oursimulations.

The soft-core particles, in the absence of external po-tential, freezes at a density ρ∗ ≈ 1.01 (see Fig. 6(a) inthe Supplemental Material [36]). The limit of α = 0and V0(t) = U0 corresponds to the equilibrium situa-tion of laser induced freezing [32]. At a density close tothe liquid-solid transition, the system freezes into a tri-angular lattice solid (LIF) which remelts into a densitymodulated liquid with increasing U0 [31, 32]. In soft-coreparticles, the LIF with βU0 = 1 occurs at ρ = 0.95 [9].Similar freezing transition at this density is observed for aweakly asymmetric ratchet (α = 0.2) of strength βU0 = 1in the limit of high switching frequency, much faster thanthe typical relaxation time, such that the colloids ex-perience an effective periodic potential (see Fig. 7(d) in

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FIG. 2. (Color Online) Average directed current as a functionof frequency, at particle densities, ρ = 0.1(◦), 0.5 (�) and1.0 (�). The solid lines show fit to Eq. 2.

the Supplemental Material [36]). In the other limit ofextremely slow switching, the system comes to quasi-equilibrium with the instantaneous strength of externalpotential, and one obtains a slow variation between amodulated liquid and a solid phase. The most inter-esting dynamics takes place at intermediate frequencies.The ratchet- driven averaged directed current shows reso-nance with frequency, and non-monotonic variation withdensity (Fig. 1(b)). At suitable densities, the systemshows dynamical re-entrant transition from a soft solidto modulated liquid to solid with increase in ratchetingfrequency (Fig. 1(c)–(e) and Fig. 5).Transport properties: The steady state dynamics is

characterized in terms of a space and time-averaged di-rected current of particles flowing along the direction ofratcheting

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Clearly, for switching frequency f = 0, the systemis in equilibrium and 〈jy〉 = 0. In the other limit ofrapid switching, f � ν the inverse of intrinsic relaxationtime, the system experiences a time-averaged effectivepotential profile, and thus again 〈jy〉 = 0. Calculationsfor ratcheting in one dimensional non-interacting parti-cles showed 〈jy〉 ∼ f at small switching frequencies, and〈jy〉 ∼ 1/f at high frequencies [37]. We observe similarfrequency dependence in these two limits of ratchetingfrequencies, from our MD simulations of 2D system ofsoft-disks (Fig. 2). This behavior may be described bythe interpolation formula g(ν, f) = νf/(ν2 + f2), whereν is an intrinsic relaxation frequency. We use a simpleansatz that 〈jy〉 = κg(ν, f)ρv0 where κ is a dimensionless

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FIG. 3. (Color Online) Resonance frequency f0 as a functionof the density ρ for non-interacting particles (squares), andsoft-core particles (circles). The solid line shows ballistic formf0 ∼

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ρ(1− ρ/ρc) with ρc = 1.07.

proportionality constant, and ρv0 has the dimension ofcurrent with v0 an intrinsic velocity. As we show below,the form of v0 and ν allows us to describe the whole den-sity and frequency dependence of directed current. Therelation

〈jy〉 = κνf

ν2 + f2ρv0, (2)

shows good agreement with simulation results (Fig. 2).The above frequency dependence is obeyed even if theratcheting wavelength λ is incommensurate with density(see Fig. 7(a)-(c) in the Supplemental Material [36]). Asimilar frequency dependence was recently found for astochastic pump model of one-dimensional system of in-teracting particles [38]. Fitting the MD simulation dataof Fig. 2 to Eq.(2) we find the resonance frequenciesf = f0 = ν which show a non-monotonic variation withthe mean density of colloids ρ (Fig. 3).

The intrinsic relaxation frequency ν, controlling thebehavior of time-averaged dynamics, may arise from aballistic or diffusive relaxation of the particles over thecharacteristic length scale λ. We use ratcheting poten-tial commensurate with the density such that λ2 ∼ 1/ρ(for treatment using incommensurate potential, see Sup-plemental Material [36]). For under-damped motion,the ballistic time-scale τb for a particle to traverse thepotential valley is obtained from the kinematic rela-tion λ ∼ (U0/λ)τ2b , that leads to τb ∼ (ρU0)−1/2. Onthe other hand, the relaxation time in the over-dampeddiffusive regime is given by τD = λ2/D ∼ (Dρ)−1.The self diffusion constant D decreases with densityfor two-dimensional repulsively interacting particles asD = D0(1− ρ/ρc) [2, 3] (see Fig. 6(b) in the Supplemen-tal Material [36]).

In the underdamped case, the velocity scale is set by

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FIG. 4. (Color Online) Average particle flux 〈jy〉 as a functionof density ρ, for soft-core particles at ratcheting frequenciesf = 1.43 (•), 0.71 (�), 0.36 ( �).The dot-dashed lines arefit to Eq.(3) in the regime ρ < 0.5 with fitting parameterκ = 0.04, 0.03, 0.02 for the three data sets respectively. Thesolid lines are fit to Eq.(4) in the regime ρ ≥ 0.5 with fit-ting parameters κ = 0.12, 0.05, 0.02 and ρc = 1.03, 1.05, 1.03.Inset: The same quantity for free particles at two differentratcheting frequencies f = 0.71 (◦) and 0.36 (�). The dashedlines are fit to Eq.(3).

vb0 = λ/τb = U1/20 . Using this and ν = 1/τb in this

regime, one finds

〈jy〉 ' κfU0

ρU0 + f2ρ3/2. (3)

The resonance frequency is then f0 = (ρU0)1/2. Onthe other hand, the velocity scale in the over-dampedregime may be obtained using the time-scale for freediffusion 1/(ρD0) over mean inter-particle separation λ,vD0 = D0ρ

1/2. Thus, using ν = 1/τD the averaged di-rected current becomes

〈jy〉 ' κfD2

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2(1− ρ/ρc)2 + f2ρ5/2(1− ρ/ρc). (4)

The corresponding resonance frequency is f0 = D0ρ(1−ρ/ρc).

Our simulations show that the resonance frequency,and therefore the intrinsic relaxation frequency, followsballistic behavior f0 ∼

√ρ at low densities (and for

non-interacting particles), and diffusive behavior f0 ∼ρ(1 − ρ/ρc) at high densities (Fig. 3). The dynamicalbehavior changes from ballistic to diffusive with increasein density. This may be understood in terms of whathappens to a directed current in the presence of direc-tion randomizing scattering events. At low densities, thetime- and space- averaged motion of a test particle withsmall number of scattering events remains ballistic onan average. However, at large densities mean free pathreduces, and consequently, a large number of scatteringevents randomizes the direction of motion leading to apredominantly diffusive dynamics.

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FIG. 5. (Color Online) Amplitude of steady state structurefactor, for the reciprocal lattice vectors G1 (a) and G2 (b), asa function of ratcheting frequency f for densities ρ = 0.98 (•,◦), 0.99 (�,�), 1.00 (H,O), 1.01 (N,4) and 1.02 ( �,�).

In Fig. 4 we show the density dependence of the di-rected current at various switching frequencies. The plotsshow non-monotonic variation, the low density limit ofwhich is fully captured by Eq.(3), and the high densitylimit by Eq.(4). Near the density ρc, the system getsinto a jammed state where the directed current vanishesas 〈jy〉 ∼ ρ5/2(1− ρ/ρc). Note that, the overall density-dependence that we find in 2D ratchet is quite unlikethe 〈j〉 ∼ ρ(1 − ρ) behavior of directed current foundin repulsively interacting 1D ratchet [25]. The collectivedynamics of the 2D ratchet can be further characterizedin terms of the density and ratcheting- frequency depen-dence of longitudinal and transverse diffusivity Dx,y(ρ, f)(see Fig. 8 and Fig. 9 in the Supplemental Informa-tion [36]).

Dynamical transitions: The reduction of directed cur-rent at high densities and subsequent jamming is asso-ciated with freezing of the system into a triangular lat-tice solid. Our MD simulations showed similar structuraltransitions are also associated with change in current asa function of ratcheting frequency (Fig. 1(b)), a fully dy-namical effect. In Fig. 1(c)–(e), we plot the superim-posed positions of 103 uncorrelated configurations fromthe center of mass frame, for a system at a mean densityρ = 1.0, and ratcheting frequencies f = 0.11, 1.67, 10.This suggests frequency dependent re-entrant transitionfrom a triangular lattice solid (f = 0.11), to density mod-ulated liquid (f = 1.67), to again a triangular lattice solid(f = 10) order. Note from Fig. 2 that the modulated liq-uid at ρ = 1.0 and f = 1.67 corresponds to the resonancefrequency in directed current.

The interplay of structure and dynamics is furtherquantified with the help of time-averaged steady statestructure factor S(G) = 〈 1

N2

∑i,j exp(−G.(ri − rj))〉

with reciprocal lattice vectors G1 = (0,±2π/ay) andG2 = (±2π/a,±2π/

√3a) (see Fig. 1(e)). In Fig. 5

we show the frequency dependence of |S(G1,2)| at var-ious densities. The presence of ratcheting potentialkeeps |S(G1)| > |S(G2)| corresponding to stronger den-sity modulation in the y-direction. The non-monotonic

variation of |S(G1)| with frequency quantifies a reduc-tion followed by an increase in this density- modula-tion. At very high frequencies, the solid- order parameter|S(G2)| > 0.31 for densities ρ & 0.96, signifying freez-ing into a triangular lattice structure (see Fig. 6(a) andFig. 7(d) in Supplemental Material [36]), reminiscent ofLIF transition [9]. The solid order parameter |S(G2)| atdensities ρ ≥ 1 shows significant non-monotonic varia-tion with frequency, pointing to a dynamical re-entranttransition from a solid to modulated liquid to solid phase.Thus ratcheting frequency provides a means to structuralcontrol during transport, and may be utilized in experi-ments.

Summary and outlook: Our study on a 2D system ofsoft-core particles under 1D ratchet drive, have shown in-teresting relation between transport properties and struc-tural phases. Using scaling arguments we obtained thedensity and ratcheting frequency dependence of averageddirected current 〈jy〉, which fully captured the simula-tion results. The resonance frequency of 〈jy〉 showeda curious cross-over from ballistic to diffusive behaviorwith increasing density, related to reduction of meanfree path. Within a range of densities, we found a dy-namical re-entrant transition from solid- to modulatedliquid- to solid- phase with increasing ratcheting fre-quency. The fact that ratcheting frequency providesa control over both the emergent directed current andstructural phases, may have useful applications. Ourpredictions may be verified in experiments on repulsivelyinteracting colloids, e.g., using magnetic ratcheting [12],or optical ratcheting [13] in a suitably modified 2D lasertrapping setup [31].

Acknowledgment: We thank Madan Rao for a valuablesuggestion. Debasish Chaudhuri thanks Surajit Sen-gupta, Sriram Ramaswamy, Narayanan Menon, SwarnaliBandopadhyay for useful discussions, Abhishek Chaud-huri, Bela M. Mulder for critical comments on themanuscript, and MPI-PKS Dresden for hosting him atvarious stages of this work.

∗ chakraborty@iisermohali.ac.in† debc@iith.ac.in

[1] F. Julicher, A. Ajdari, and J. Prost, Reviews of ModernPhysics 69, 1269 (1997).

[2] P. Reimann, Physics Reports 361, 57 (2002).[3] R. D. Astumian and P. Hanggi, Physics Today 55, 33

(2002).[4] P. Hanggi, Reviews of Modern Physics 81, 387 (2009).[5] F. Julicher and J. Prost, Physical Review Letters 75, 2618

(1995).[6] F. Julicher and J. Prost, Physical Review Letters 78, 4510

(1997).[7] R. D. Astumian, Science 276, 917 (1997).[8] J. Rousselet, L. Salome, A. Ajdari, and J. Prost, Nature

370, 446 (1994).

4

[9] S. Leibler, Nature 370, 412 (1994).[10] C. Marquet, A. Buguin, L. Talini, and P. Silberzan, Phys-

ical Review Letters 88, 168301 (2002).[11] P. Tierno, P. Reimann, T. H. Johansen, and F. Sagues,

Physical Review Letters 105, 230602 (2010).[12] P. Tierno, Physical Review Letters 109, 198304 (2012).[13] L. Faucheux, L. Bourdieu, P. Kaplan, and A. Libchaber,

Physical Review Letters 74, 1504 (1995).[14] B. Lopez, N. Kuwada, E. Craig, B. Long, and H. Linke,

Physical Review Letters 101, 220601 (2008).[15] C. F. Chou, O. Bakajin, S. W. Turner, T. a. Duke, S. S.

Chan, E. C. Cox, H. G. Craighead, and R. H. Austin,Proceedings of the National Academy of Sciences of theUnited States of America 96, 13762 (1999).

[16] C. Kettner, P. Reimann, P. Hanggi, and F. Muller, Phys-ical Review E 61, 312 (2000).

[17] S. Matthias and F. Muller, Nature 424, 53 (2003).[18] C. Mennerat-Robilliard, D. Lucas, S. Guibal, J. Tabosa,

C. Jurczak, J.-Y. Courtois, and G. Grynberg, PhysicalReview Letters 82, 851 (1999).

[19] C.-S. Lee, B. Janko, I. Derenyi, and A.-L. Barabasi, Na-ture 400, 337 (1999).

[20] C. Olson, C. Reichhardt, B. Janko, and F. Nori, PhysicalReview Letters 87, 177002 (2001).

[21] I. Derenyi and T. Vicsek, Physical review letters 75, 374(1995).

[22] I. Derenyi and A. Ajdari, Physical Review E 54, R5(1996).

[23] P. Reimann, R. Kawai, C. V. den Broeck, and P. Hanggi,Europhysics Letters (EPL) 45, 545 (1999).

[24] D. C. Rapaport, Comput. Phys. Commun. 147, 147, 141(2002).

[25] Y. Aghababaie, G. Menon, and M. Plischke, PhysicalReview E 59, 2578 (1999).

[26] M. Koppl, P. Henseler, A. Erbe, P. Nielaba, and P. Lei-

derer, Physical Review Letters 97, 208302 (2006).[27] K. Mangold, P. Leiderer, and C. Bechinger, Physical Re-

view Letters 90, 158302 (2003).[28] D. Chaudhuri and S. Sengupta, Physical review letters

93, 115702 (2004).[29] U.Siems, C.Kreuter, A.Erbe, N.Schwierz, S.Sengupta,

and P. P.Leiderer, Scientific Reports Nature PublishingGroup 2, 1015 (2012).

[30] A. Chowdhury, B. J. Ackerson, and N. A. Clark, Phys.Rev. Lett. 55, 833 (1985).

[31] Q.-H. Wei, C. Bechinger, D. Rudhardt, and P. Leiderer,Physical Review Letters 81, 2606 (1998).

[32] E. Frey, D. R. Nelson, and L. Radzihovsky, Phys. Rev.Lett. 83, 2977 (1999).

[33] D. Chaudhuri and S. Sengupta, Physical Review E 73,11507 (2006).

[34] D. Frenkel and B. Smit, Understanding molecular simu-lation: from algorithms to applications (Academic press,NY, 2002).

[35] G. S. Grest and K. Kremer, Phys. Rev. A 33, 3628(1986).

[36] See Supplemental Material for the details on equilib-rium liquid-solid transition and density- dependent diffu-sivity, further characterization of the ratcheting dynamicsin terms of frequency- and density- dependent effective dif-fusivity, and impact of incommensurate ratcheting on theaveraged directed current.

[37] A. Mielke, Annalen der Physik 4, 721 (1995).[38] D. Chaudhuri and A. Dhar, EPL (Europhysics Letters)

94, 30006 (2011).[39] J. Lahtinen, T. Hjelt, T. Ala-Nissila, and Z. Chvoj, Phys-

ical Review E 64, 021204 (2001).[40] E. Falck and J. Lahtinen, The European Physical Journal

E 13, 267 (2004).

5

Supplementary Information:

Soft-core particles – in the absence of external potential

The phase behavior and dynamics of repulsively interacting colloidal suspensions have been extensively studiedin literature [1–3]. The two-dimensional fluid is known to undergo a freezing transition into a triangular latticesolid, arguably via a hexatic phase, with increasing density [4–7]. Without going into the intricacies of identifyingthe hexatic [1, 7], to determine the fluid-solid transition point we calculate the solid- order parameter |S(G2)| ofthe soft-disk system, in the absence of any external driving, as a function of the density of the system, depicted infig. 6(a). At phase transition, ρ∗ = 1.01, |S(G2)| shows a discontinuous increase with density (Fig.6(a) shows MDsimulation result of 4096 particles). At this point the order-parameter jumps increase to a value |S(G2)| = 0.31. Inall our simulations, even in the presence of ratchet-driving, we identify a phase transition to a solid phase whenever|S(G2)| crosses the value 0.31. Note that this value is close to the phase transition criterion of |S(G2)| = 0.35 used inrecent literature [8]. As depicted in the inset of Fig.6(a), the pressure versus density behavior shows non-monotonicitysignifying a phase-coexistence of a solid with density ρs = 1.01 and liquid with ρl = 0.99 [1].

In the overdamped regime, the internal relaxation is due to diffusive motion of particles. The density dependenceof diffusivity D(ρ) in soft-disk particles has been previously studied by Lahtinen et. al [2, 3] and exhibits a lineardependence on ρ. Using our MD simulations, we studied the same to obtain the linear density dependence D =D0(1− ρ/ρc), where D0 is the diffusivity of a noninteracting system, and the fitted value ρc = 1.04 (Fig.6(b)).

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Soft-core particles – under stochastic ratcheting

In order to study the effect of stochastic ratcheting, the system was evolved under the potential βUext(y, t) =βV0(t) [sin (2πy/λ) + α sin (4πy/λ)], with βV0(t) switching between 0 and 1, and asymmetry parameter α = 0.2. Theswitching is done stochastically with a rate f . During our MD simulation, at each time step the value of the externalpotential V (t) is switched between 0 and 1 with a probability fδt, and left unaltered with a probability (1 − fδt),where δt is the size of integration time-step. We have always waited for the system to reach steady state beforecollecting data presented in all our analysis.

In this section, we first focus on systems driven by a ratchet of fixed periodicity λ = 1σ. The driving potential isnot commensurate with the density, unlike the system discussed in the main text. The impact on the non-interactingsystem itself is different, as λ is no more a function of density. The integrated directed current 〈jy〉 exhibits resonanceat a fixed value of the frequency of f0 = 3.5, independent of densities (see Fig. 7 (a)). The ballistic time-scale τb to

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FIG. 7. (Color Online) (a) Plot of the averaged directed current for a non-interacting system driven by a periodic potentialwith periodicity 1σ, for densities ρ = 0.1 (◦), 0.2 (�), 0.3 ( 2) and 0.6 (4). The resonance in the measured flux occurs at afixed frequency f0 = 3.5. (b) Plot of the measured flux for an interacting system driven by a periodic potential with peroidicity1σ, for densities ρ = 0.1 (◦), 0.2 (�), 0.3 ( 2) and 0.6 (4). The solid lines are fit to the data using the functional form ofEq. (2) in the main text. (c) Plot of the measured flux for an interacting system driven by a periodic potential with peroidicity1σ, for densities ρ = 0.8 (•), 0.92 (�) and 0.98 ( �). The solid lines are fit to the data using the functional form of Eq. (2) inthe main text. The inset depicts the dependence of the resonance frequency on the density of the suspension. The solid line isa linear fit of f0 ∼ (1 − ρ/ρc) to the data with ρc ≈ 1.13. (d) Plot of the measured order parameter for the reciprocal latticevectors G2 (empty symbols) and G1 (inset, filled symbols) for densities ρ = 0.94 (•, ◦) and 0.96 (�,�).

travel a potential-valley of length λ follows the kinematic relation λ ∼ (U0/λ)τ2b leading to τb ∼ λ/√U0, independent

of density as λ itself is a constant. As a result the resonance frequency is also constant.

If, instead, an interacting system is driven by the same ratcheting potential with λ = 1σ, the resonance frequencyremains approximately constant at lower densities (ρ . 0.5) indicating a free-particle like ballistic motion. However,at higher densities the resonance frequency drops approximately linearly with density (inset of Fig. 7 (c)). Thisindicates a cross-over from ballistic to diffusive transport with diffusive time-scale τD ∼ λ2/D0(1 − ρ/ρc), and thecorresponding resonance frequency ∼ (1 − ρ/ρc). This behavior should be contrasted against the f ∼ ρ(1 − ρ/ρc)behavior for commensurately ratcheting soft- disks at high densities (Fig.3 of main text). Note that the reasoningused here is similar to that we applied for interacting particles driven by a commensurate ratchet (See Eq.s (2)–(4)of main text).

It is known that, in case of laser induced freezing (LIF), the soft-core system of particles undergo a modulated liquidto solid transition near ρ = 0.95 and a periodic commensurate potential of strength βU0 = 1 [9]. As we argued in themain text, at very high frequencies of weakly asymmetric commensurate ratcheting, one expects to recover fluid-solidtransitions similar to LIF. In Fig. 7 (d) we present the dependence on the ratcheting frequency f of the solid order

7

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parameter S(G2) (defined in main text) at densities ρ = 0.94 and 0.96. This shows that at high enough frequenciesf � 10, although the effective time-integrated periodic potential strength βU0 < 1, the system at ρ = 0.96 alreadyshows freezing, a reminiscent of LIF.

Finally, we consider the asymmetric diffusivity of the soft-core particle system, driven by a ratchet with periodicitycommensurate with the density. We obtain the density and ratcheting-frequency dependence of the diffusivity. InFig. 8, we show the frequency dependence of diffusion coefficients Dy, Dx along and perpendicular to the direction ofratcheting drive, respectively. We observe that Dy < Dx, since to diffuse in the direction of driving, particles have toclimb potential barriers. As with the directed current, we also observe a resonance in the diffusion coefficient when thedensity is relatively small (see fig. 8(a) and (c)). The properties of Dx,y(f, ρ) is further presented as a surface plot inFig. 9, in the low density regime where the resonance structure is seen. The magnitude of both Dx and Dy increaseswith density, the effect being more pronounced for Dx and persists for densities lower than ≈ 0.3 (see Fig. 9), beyondwhich the magnitude of the diffusion constants steadily decreases. At higher densities, the resonance structure is lost,the diffusivities remain approximately constant at low frequencies, decreasing at frequencies f & 1 (Fig.s 8(b) and(d)).

∗ chakraborty@iisermohali.ac.in† debc@iith.ac.in

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[1] J. Broughton, G. Gilmer, and J. Weeks, Physical Review B 25, 4651 (1982).[2] E. Falck and J. Lahtinen, The European Physical Journal E 13, 267 (2004).[3] J. Lahtinen, T. Hjelt, T. Ala-Nissila, and Z. Chvoj, Physical Review E 64, 021204 (2001).[4] J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973).[5] D. R. Nelson and B. I. Halperin, Phys. Rev. B 19, 2457 (1979).[6] A. P. Young, Phys. Rev. B 19, 1855 (1979).[7] S. Sengupta, P. Nielaba, and K. Binder, Physical Review. E 61, 6294 (2000).[8] C. Mondal and S. Sengupta, Phys. Rev. E 85, 020402 (2012).[9] D. Chaudhuri and S. Sengupta, Physical Review E 73, 11507 (2006).

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FIG. 9. Plot of the measured diffusion coefficient,normalized by the diffusion coefficient of a free system D0(ρ), for an interactingsystem driven by a periodic and a commensurate potential, along (a) and perpendicular (b) to the direction of the externaldrive, as a function of frequency and density.

10