Spontaneous unidirectional rotation of a symmetric gear driven by spherical active particles

: Time reversal asymmetry and spatial anisotropy are considered two prerequisites for Brownian ratchet. An intriguing realization can be achieved by placing an asymmetric gear in the suspension of motile rod-like bacteria. Usually, alignment interactions caused by anisotropic collisions or hydrodynamics would boost the ratchet effect. Here, we are concerned with a perfectly isotropic system, i.e., symmetric gear immersed in a bath of spherical active Brownian particles. We find that, under certain conditions, kinetic symmetry-breaking arises spontaneously, i.e., the symmetric gear keeps rotating in one direction. Unexpectedly, such ratchet phenomenon does not rely on the direct many-particle interactions and moreover the introduction of alignment interaction would counterintuitively prevent it from happening! Further investigation reveals that such spontaneous symmetry-breaking phenomenon shares similarities with the equilibrium phase transition of the Ising model. Our results provide new insights and enhance our understanding of the fundamental aspects of active ratchet phenomena.


INTRODUCTION
The Brownian motor or ratchet, also known as the Smoluchowski-Feynman ratchet, is a fascinating topic in thermodynamics [1,2].The fundamental physics behind the operation of the Brownian ratchets is the second law of thermodynamics, which prohibits extraction of useful work from a single equilibrium heat bath.Generally, to achieve the rectification phenomena, or ratchet effect, both time and spatial symmetries must be broken [1,3].This requires driving the system out of thermal equilibrium.Numerous models of Brownian ratchets that could break both the symmetries have been developed and studied, including temperature ratchets [4], friction ratchets [5], on-off ratchets [6], travelling potential ratchets [1], fluctuating potential ratchets [7,8], drift ratchets [9], and more.Many of them can be classified as pulsating ratchets or tilting ratchets [1].In the past two decades, a new class called active Brownian ratchets has attracted significant research interest [3,10].Unlike conventional Brownian ratchets, the self-propelling motion of agents in active Brownian ratchets inherently breaks detailed balance or time-reversal symmetry.And the breaking of spatial symmetry is then often achieved by introducing asymmetric potentials/structures [2,3,11,12] or material properties [13].Other possibilities are related to asymmetric dynamics either intrinsic (e.g., chiral active particles) [14][15][16] or generated spontaneously by alignment-induced collective motion [17,18].
An intriguing realization of active Brownian ratchets is achieved by immersing an asymmetric gear in a bacterial bath, which was able to rotate unidirectionally [19][20][21][22].This demonstrates a simple design of autonomous micromachine powered by microorganisms.A consensus among these researches is that the asymmetric shape of the gear, which breaks spatial symmetry, is essential for the persistent directed rotation.However, in this paper, we focus on a perfectly isotropic system, i.e., we study by numerical simulations a symmetric gear immersed in a bath of spherical active Brownian particles (ABPs).In this simple dry model, there is no shape/property asymmetry, chiral dynamics, hydrodynamics or collision-induced alignment interactions.Overall, there does not seem to be any factor that could cause the breaking of spatial symmetry.Surprisingly, we find that without any additional nonequilibrium perturbations, the perfectly symmetric gear can persistently rotate in one direction under certain conditions.This is a spontaneous symmetry-breaking phenomenon.Simulations also show that such ratchet phenomenon does not rely on the direct many-particle interactions.Moreover, if isotropic spherical ABPs are replaced by rod-like ones (akin to elongated E. coli cells), the thus introduced alignment effect (by collisions) would counterintuitively prevent the ratchet motion from happening.Further investigations reveal similarities between this spontaneous kinetic symmetry-breaking phenomenon and the equilibrium phase transition of the Ising model of the ferromagnetic systems.Our results provide new insights into the roles of asymmetric shape, alignment effect, and manyparticle interactions in active-matter-powered rotors, and enhance our understanding of the fundamentals of active ratchet phenomena.

MODEL AND METHODS
In our two-dimensional model, a symmetric gear with eight teeth (Figure 1A-C) is mixed with round ABPs of mass m and size σ.The rim of the gear is composed of highly overlapped beads of the same size σ.The inner and outer radii of the gear are set as 8σ and 14σ, respectively.All particles and beads interact with the repulsive Weeks-Chandler-Andersen (WCA) potential, U r r r with a cutoff at r = 2 1/6 beyond which U = 0

WCA
. Each ABP undergoes Brownian dynamics according to the following translational and rotational equations of motion ( ) where U i is the total potential of particle i. and D r are the translational friction coefficient and rotational diffusion rate, respectively.The ABPs are non-chiral and propelled by a centric force of strength F a along the direction ( ) are unit-variance Gaussian white noises.We consider the gear as a rigid body.In order to focus on the rotational motion, we fix its center position while allowing it to rotate freely [23,24].The rotational motion of the gear is governed by .and R are large enough that both the motions of the ABPs and the gear are essentially overdamped [23,24].Keeping the inertial terms in Eqs. ( 1) and (3) allows us to use the velocity-verlet algorithm (by the home-modified LAMMPS software) and a large time step.

The phenomenon of unidirectional rotation
The propelling strength F a , the rotational diffusion rate D r and the density (area fraction) of ABPs are three key variables that characterize the activity and amount of active agents in the bath.We find that under some conditions such as F = 8 a , D = 0.001 r and = 0.15, the symmetric gear can persistently rotate along one Natl Sci Open, 2024, Vol.3, 20230066 direction (see Figure 1A-C and supplementary movie S1).In the beginning, the ABPs aggregate densely around the gear (Figure 1A) and the rotation then starts in a direction stochastically determined by the imbalanced push from the surrounding ABPs.It is worth noting that no shape asymmetry of any kind is introduced into the system unlike prior works where either intrinsic shape asymmetry (asymmetric gear) [19][20][21][22]25] or spontaneously generated shape asymmetry (collective deformation of chains grafted on the colloidal disk) [23,24] was found to play a crucial role in persistent unidirectional rotation.The central questions here are why and under what conditions could a symmetrical gear persistently rotate in one direction, and why has this phenomenon not been observed in previous works?
By tuning the rotational diffusion rate D r and/or the density of ABPs, we observe three kinds of typical time sequences of the rotating angular velocity of the gear, t ( ) (Figure 1D-F and supplementary movies S1- S3).The time interval for measuring the "instantaneous" angular velocity is .We denote the three types of time sequences as three rotating states: unidirectional rotating state (URS), transitional state (TS) and random rotating state (RRS).In the URS, t ( ) fluctuates around a non-zero mean value and the accumulative angle, t ( ), increases (counterclockwise) or decreases (clockwise) nearly monotonically (Figure 1D, G).ABPs continuously enter and leave the vicinity of the gear during rotation.Unlike the asymmetric gear powered by bacteria or rod-like active particles [19], there is no reversal of rotation at the beginning for the symmetric gear (Figure 1D).In the TS, the gear could rotate in one direction for an extended period (Figure 1E, G), but the direction would occasionally reverse due to sudden, large fluctuations.And, in the RRS, there is no apparent directional rotation at all (Figure 1F, G).

Phase diagram
Figure 2A depicts the "phase diagram" for the states of rotation in D r -space.We set the upper bound of as 0.2, below which ABPs do not spontaneously form large clusters in the bulk.This avoids strong irregular disturbance to the gear rotation from the ABP aggregates in the bulk.D 1 / r characterizes the persistence time of the active motion of ABPs.Smaller D r (stronger activity of the ABPs) and higher would cause more ABPs to gather around the gear, presumably leading to stronger push on the gear and less impact of fluctuations.Therefore, both small D r and large are conducive to persistent directional rotation as demonstrated in Figure 2A (URS appears at the upper left part of the phase diagram).As D r and vary, the state of rotation continuously changes among the three states.In Figure 2B, we show the typical probability distributions of in the three rotating states.One big peak around = 0 splits into two symmetric peaks at nonzero values when the gear changes from the RRS to the TS/URS.In the TS, the probability around zero angular velocity P( = 0), is notably above zero, indicating that the rotational direction reverses from time to time.However, it is almost zero in the URS, implying that the change of rotational direction hardly occurs.

Mechanism for the unidirectional rotation
The symmetric gear in our model is not expected to rotate consistently in one direction, since neither spatial asymmetry nor alignment rules that can cause collective directional motion are included.However, we do observe the URS at small D r and moderate .Figure 3A displays the long-time-averaged particle-flow field, Natl Sci Open, 2024, Vol.3, 20230066 around the gear, where ρ represents the local particle number density and v represents the local mean velocity.As anticipated, the ABPs form a circular flow field near the gear, following its rotation.At first glance, the situation seems to resemble a travelling potential ratchet [1], where the rotating gear serves as a traveling potential that rectifies the motion of ABPs.However, the essential difference lies in the fact that the traveling potential (the rotating gear) is not externally controlled but generated by the motion of ABPs themselves.Interestingly, when we choose the rotating gear as the reference frame, the ABPs form small vortices between the teeth with opposite orientations relative to the rotation of the gear (Figure 3B).The formation of these vortices and the maintenance of the unidirectional rotation are closely related.In the schematics of Figure 3C-F, we ignore the change in the propelling direction of an ABP during its interaction with the gear since the URS happens at low D r .The behaviors of ABPs after hitting the two sides of a rotating tooth are different (see supplementary movies S4 and S5).We refer to these two sides as "head-on side" (HOS) and "rear-end side" (RES), respectively (Figure 3C).When an ABP collides onto the HOS, it tends to slide inward in the next moment due to the rotation of the gear (Figure 3D) and is then trapped in the groove (Figure 3E) until the gear rotates by a large enough angle that the propelling direction on the ABP becomes outward.In contrast, when an ABP collides onto the RES, it tends to slide outward and leave the gear (Figure 3D and E).Such asymmetric kinetic trend or biased trapping causes the inverse local particle vortices relative to the rotating gear.In view of the resultant torque on the gear, ABPs hitting the HOS initially generate a torque that impedes the rotation of the gear (Figure 3D).But after the gear rotates for a certain amount of angle, their push on the gear becomes promoting the rotation (Figure 3E).The analysis of the line density of normal force on the rim of the gear (Figure 3F) demonstrates that ABPs trapped near the grooves contribute the major driving torque to the rotation.The rotation of the gear and its consequent asymmetric impacts or biased trapping on incident ABPs form a positive feedback loop that sustains the unidirectional rotation.Why has the unidirectional rotation of symmetrical gear not been observed in previous studies [19][20][21][22]25]?The probable reason, we believe, is the use of rod-like active particles/bacteria rather than spherical ones with the freedom of orientation.The propelling direction of rod-like active objects is related to their asymmetric shape.When they collide each other or a boundary, their bodies and hence propelling directions tend to align with each other or the boundary.When the rod density is low, the behavior of an active rod entering a rotating gear is illustrated in Figure 4A and B. Shortly after touching the rim of the gear, the active rod moves parallel to the rim with the direction dependent mostly on the incident angle rather than whether it hits the HOS or RES.Therefore, the aforementioned positive feedback loop between the rotation of the gear and the asymmetric kinetic behavior of particles no longer exists.When the rod density is high, rod-rod aligned packing also dominates over the impact of gear rotation (see supplemental movie 6).Our simulations on varied situations show that symmetric gear immersed in a bath of active rods does not exhibit unidirectional rotation (Figure 4C-E).These results indicate that alignment effects counterintuitively hinder the breaking of spatial symmetry and hence the emergence of unidirectional rotation in the case of symmetric gear.

Non-interacting ABPs
Collective motion is usually the key to spontaneous symmetry breaking in active-matter systems.In bulk, with steric repulsion, ABPs can spontaneously form clusters through self-trapping mechanism and [26][27][28][29][30] move collectively, including transient rotation [31].A question here is whether steric packing of ABPs is essential to the phenomenon of unidirectional rotation.To address it, we perform simulations with noninteracting ABPs by ignoring the ABP-ABP pair interactions in the model.The potential term U i in Eq. ( 1) now only contains the ABP-gear interaction.The gear is thus immersed in an "idea gas" of ABPs.Our results show that the URS still occurs at small D r and moderate (Figure 5A and B).The mechanism illustrated in Without the particle-particle steric interaction, trapped ABPs can overlap and accumulate around the cusp of the groove (see supplementary Figure S1).A very good linear relation between the mean angular speed ( represents the absolute value and averages over time and independent runs) and the propelling strength F a has been observed in the URS (Figure 5C).This linear relation can be explained by the following analysis [23,24].The rotation of the gear together with the trapped ABPs depends on the balance of torque.The driving torque arises from the propulsions on the trapped ABPs, while the damping torque results from frictions on both the rotating gear and the trapped moving ABPs.The driving torque where n trap is the mean number of trapped ABPs and R = 8 in is the inner radius of the gear.The factor α accounts for the mean ratio of the projection of a propelling force normal to the radial direction.The damping torque comprises two parts: , arising from the frictions on the gear and the trapped ABPs, respectively.In the overdamped limit, Equation ( 4) describes the F a relation very well (Figure 5C). is independent of F a and in all situations it varies in a narrow range ~0.31-0.34.n trap depends on F a (see supplementary Figure S2), but it appears in both the numerator and the denominator in Eq. ( 4).As a consequence, the factor in front of F a on the right hand side of Eq. ( 4) is nearly a constant, implying a linear relation between and F a .and the critical exponent 0.34.Even though our model is two-dimensional, the exponent is close to that of the three-dimensional Ising model.Note that the transition here differs significantly from the Ising model.First, the transition in our system is kinetic and nonequilibrium, while the transition in the Ising model is thermodynamic and equilibrium.Second, the transition in our system is intrinsically a phenomenon of finite size, rather than of thermodynamic limit like the Ising model (the order parameter in our system is the state of rotation of one gear, rather than the magnetization averaged over individual spins in the Ising model).Third, once the spontaneous magnetization occurs below T c in the Ising model, the broken symmetry is then definite (up or down).In contrast, the gear in our system enters the TS (red hollow circles in Figure 2C) right below the critical point D r c , in which the gear occasionally reverses its rotational direction.Hence, spatial symmetry does not break down on average over long times in the TS.

Comparison with the Ising model and the phenomenon of hysteresis
Only when entering the URS (black hollow squares in Figure 2C), the direction of rotation and the symmetry breaking are stable and definite (at least within the time window of our simulations).Last, we cannot make a comprehensive analogy between the transitions in our system and in the Ising model.For example, we cannot find quantities in our system that corresponds to the heat capacity and the correlation length associated with the transition.Hence, we cannot obtain all the critical exponents and compare them with the Ising model.
To mimic the effect of external magnetic field in the Ising model, we add an extra term, representing external torque, M e to the equation of motion of the gear R R e The gear can rotate with a stable angular velocity M = / 0 e R in the absence of ABPs.We choose a URS (D = 0.001 r and = 0.15) in the phase diagram as an example and add external torque M e of different amplitudes.As shown in Figure 6A, the of becomes more and more asymmetric as M e increases.Positive M e not only increases the probability of the gear turning in the positive direction (higher peak for > 0), but also increases (for > 0)/decreases ( < 0) the angular velocity (shift of the peaks to the right in Figure 6A).It is worth noting that the probability curves at = 0 remain near zero, indicating that the gear is still in the URS under external torque.In other words, the gear can maintain unidirectional rotation even if the rotational direction is opposite to that of M e .However, when M e is sufficiently large, the rotational direction of the gear becomes completely determined by it and the stochasticity of the rotational direction Natl Sci Open, 2024, Vol.3, 20230066 ceases to exist (see M e = 1000 in Figure 6A).
Since the gear remains in the URS in the presence of external torque, we use, for simplicity, the mean angular velocity as the new order parameter instead of the peak value peak ., within which the rotational direction of the gear is indefinite, exhibiting a certain degree of stochasticity. Figure 6B is reminiscent of the magnetic hysteresis loop.In Figure 7, we calculate the mean angular velocity averaged over time and independent runs of both positive and negative rotations, taking into account their signs.Starting from random initial conditions, as in all the previous calculations, we obtain the blue curve in Figure 7.The hollow symbols signify that both positive and negative rotations are observed in the 100 independent runs for each case, while the solid symbols indicate that only positive or negative rotations occur.As expected, the symbols on the blue curve become solid after M e reaches M e t .The blue curve is analogous to the initial magnetization curve.Then, starting from an initial state of M M = 1500 >  Figures 1D and 5A) or 's of retentivity in Figures 7 and 8).Therefore, it is anticipated that the rotation of gear driven by non-interacting ABPs will be more heavily influenced by the initial state and the threshold (M e t ) and coercivity (M e c ) external torques will be greater, i.e., larger hysteresis loop.In Figure 8, M 2500   The blue and green lines in Figure 8 are predictions of Eq. ( 6), which are in good agreement with the simulation data.

SUMMARY
In summary, complementary to current understandings, we find that perfectly symmetric gear can also rotate unidirectionally when powered by simple ABPs.Conventionally, alignment interactions and steric packing between active objects promote collective motions, but we find that they have no or even negative effects on the ratchet phenomena of a symmetric gear.In this paper, we only consider a "dry" model that could be realized and tested through macroscopic granular/robotic experiments [23].While the impact of shapeinduced alignment effects has been discussed, the impact of hydrodynamic interactions remains open for future research.In some situations, the impact is believed to be quantitative rather than qualitative [19,23,24].However, hydrodynamic effects are complicated and may have significant impacts in our model.Additionally, the translational motion of the gear is turned off in this study, leaving the cooperation between the translational and rotational motion in the ABP-driven kinetics of a symmetric gear as another open topic for future research.Our findings provide new insights into the fundamental aspects of active ratchet phenomena.

/ 2 . 1 B,
R RNatl Sci Open, 2024, Vol.3, 20230066where I is the rotational inertia, R the rotational friction coefficient, and M R the total torque on the gear due to the gear-ABP interactions.The simulations are performed in a box of 150 × 150 with periodic boundary conditions.m, and are used as scaling units and the corresponding time unit m = We set k T =

Figure 1
Figure 1 Rotation of the symmetric gear in the ABP bath.(A)-(C) Snapshots at sequential times of the simulation.The solid cyan circles are ABPs.The symmetric gear is marked with a blue arrow and the black curved arrow indicates the direction of rotation.(A) The aggregation of ABPs around the gear just before the start of rotation.Typical time sequences of t ( ) for the three rotating states: URS (D), TS (E) and RRS (F).(G) The corresponding accumulative angles t ( ). (A)-(D) D r = 0.01, (E) D r = 0.007, (F) D r = 0.02, and F = 8 a , = 0.15 for all.

Figure 2 (.
Figure 2 (A) Phase diagram in the D r -space.Black squares, red circles and blue triangles represent the URS, TS and RRS, respectively.(B) Typical probability distributions of in the three states corresponding to Fig. 1(D)-(F).(C) Peak value of ( peak ) as a function of D r (F = 8 a

Figure
Figure Example long-time averaged flow field of ABPs in the lab (A) and the gear (B) frames (F = 8 a , D = 0.001 r and = 0.15).(C)-(E) Schematics of collision of an ABP with the "head-on side" and the "rear-end side" of the gear in the URS.(F) Long-time averaged distribution of normal force density along the boundary of the gear exerted by surrounding ABPs.

Figure 4 (Figure 3
Figure 4 (A) and (B) Schematics of collision of a rod-like active object with the gear.Long-time averaged flow field of rod-like active particles (C), time sequence of (D) and the corresponding accumulative angle (E) all indicate the disappearance of the unidirectional rotation (F = 8 a , = 0.15 and effective D r of a free rod-like active particle ~0.001).

Figure 5
Figure 5 Unidirectional rotation of the gear in the bath of non-interacting ABPs.(A) Time sequence of (F = 8 a , D = 0.001 r and = 0.15).(B) Time sequences of accumulative angles for F = a 4, 6, 8, 10, 12, from bottom to top (D = 0.001 r and = 0.15).(C) Averaged angular speed as a function of propelling strength F a .Colored symbols are simulation data and lines are fittings by Eq. (4).Colors represent = 0.10 (black), = 0.15 (red), and = 0.20 (blue); symbols represent D = 0.001 r

Figure
Figure 2B shows the example probability distributions of in the three typical states.To quantify the transition between the RRS and the TS/URS, we choose peak (the peak values of the distribution curves) as the order parameter.Its variation with ( ) D log r at F = 8 a angular velocity averaged over time and independent runs of positive (negative) rotations.The relation between p ( ) n and M e is shown in Figure 6B.The dashed lines indicate the threshold values of external torque M ± 600 e t rotation), we decrease M e to obtain the black curve.Due to the biased initial condition, definite positive rotation extends to around M = 0 e (far below M e t ).Below M = 0 e , the probability of negative rotation gradually increases.When the external torque reaches the negative "coercivity" of positive and negative rotations become equal.Further decreasing M eto below M e t , the rotation becomes definite negative.A similar red curve is obtained, starting from an initial state of M = 1500 e (define negative rotation).The black and red curves form a hysteresis loop between M ± e t .In reality, the lines of solid symbols in Figure7align with those in Figure6B.The of retentivity is around 0.023, which is just the p or n at zero external torque.The hysteresis phenomenon is also observable for non-interacting ABPs as shown in Figure8.Since the

Figure 6
Figure 6 Rotation of the gear under external torque.F a = 8, D r = 0.001 and = 0.15, i.e., the gear is in the URS in the absence of external torque.(A) Distribution of angular velocity of the gear under different positive external torques.Over 100 independent runs are performed from random initial conditions.(B) Mean angular velocities as functions of external torque, which are averaged over time and independent runs of positive ( ) p and negative ( ) n rotations, respectively.
than four (five) times that in Figure 7.It is worth noting that, for example, the gear maintains definite positive rotation until approximately M = 1000 e , when decreasing from the initial state of M = 2700 e .The bare angular speed driven by external torque of M = 1000 e is = 0.1 0 , much larger than p or n at zero external torque, which again manifests the strong dependence of the gear rotation on the initial state.Near M = 0 e , the number and distribution of trapped ABPs around the gear are not greatly

Figure 7
Figure 7 Hysteresis loop.Mean angular velocity under external torque, which is averaged over time and independent runs of both positive and negative rotations, considering the positive and negative signs.Simulations of the blue curve start from random initial conditions, while those of the black and red curves start from the URS at M = 1500 e

Figure 8
Figure 8 Hysteresis loop for non-interacting ABPs.(A) Mean angular velocity under external torque as in Figure 7. Simulations of the black and red curves start from the URS at M = 2700 e