Collective motion of pulsating active particles in confined structures

The collective motion of pulsating active particles with periodic size contraction is investigated in a two-dimensional asymmetric channel. Our findings reveal that changes in particle size can act as a non-equilibrium driving force, disrupting the system’s thermodynamic equilibrium and leading to the transformation of self-contraction motion into directional motion in the asymmetric channel. The specific direction of motion is dictated by the symmetrical properties of the channel. Furthermore, our study identifies an optimal degree of channel opening (or self-pulsation frequency) at which the average velocity reaches its peak value. At lower frequencies, the average velocity demonstrates a peak function in relation to the self-pulsation amplitude (or particle number density). Conversely, at higher frequencies, the average velocity increases with the self-pulsation amplitude (or particle number density). The system exhibits three distinct states: the arrested ordered state, disordered state, and cycling ordered state. Notably, particle rectification reaches its optimum in the disordered state.


Introduction
Theoretical and experimental research has extensively explored active matter in both biological and physical systems.Distinguishing themselves from passive particles, active particles, commonly known as self-propelled particles, microswimmers, or nanoswimmers, possess the unique capability to absorb energy from their surroundings and convert it into directed motion.A defining characteristic of active matter systems is the injection of energy into each individual unit, a disruption that fundamentally alters the system's equilibrium [1][2][3][4][5][6].Understanding the collective behavioral patterns in systems involving active matter can provide insights into out-of-equilibrium phenomena observed in various biological systems, such as fish schools, bacterial suspensions, and artificial microswimmers.
The challenge of inducing directed motion in stochastic environments poses long-standing theoretical and pragmatic implications [7,8].Active matter, however, can be rectified on asymmetric substrates without external catalysts, offering a plethora of potential applications, including cargo transportation, sorting, and micro-machine construction .Experimental investigations [10][11][12][13][14][15] have illuminated the critical nature of self-propulsion in driving a nano-sized ratchet-shaped wheel (demonstrating that active particles can drive asymmetric ratchets, rotors, and gears), or in rectifying cell motion within an array of asymmetric funnels (exposed to external vibration to stimulate macroscopic particles).For instance, a bacterial collection can negotiate past funnel-shaped barriers by establishing and sustaining a chemoattractant gradient [16].Researches by Ghosh and colleagues [18,26,28] on the ratchet transport of Janus particles revealed that rectification can be significantly stronger when compared to typical thermal potential ratchets.Similarly, theoretical observation of the rectification phenomenon in overdamped swimming bacteria was conducted, employing a system with an array of asymmetric barriers [21].Pietzonka and colleagues [22] investigated the dynamics and energetics of asymmetrically shaped passive obstacles immersed in active baths.Research on the transport of run-and-tumble particles in a periodic potential conducted by Angelani et al [25] revealed that the asymmetric potential contributed to a net drift speed.Furthermore, Li et al [24] successfully manipulated the transport of overdamped point-like Janus particles in constricted, two-dimensional corrugated channels.In other research, Pototsky et al [33] discovered that spatially modulated self-propelled velocity could incite directed transport.Moreover, rectification and sorting of chiral active particles were possible in complex and congested environments [29].Consequently, active matter, due to its inherent non-equilibrium driving forces or speeds, can transform random motion into directed motion under conditions marked by spatial or temporal asymmetry.
A growing subset within the realm of active matter, known as pulsating active matter, has recently garnered attention [42].Diverging from traditional counterparts, these particles operate without non-equilibrium driving forces or specified speeds.Instead, the particles' activity stems from cyclical size changes in each individual.The interplay between repulsion and synchronization triggers an instability that gives rise to a myriad of dynamic patterns, ranging from spiral waves to defect turbulence.Conventionally, the rectification of active matter predominantly relied on explicit non-equilibrium driving.However, this prompts an intriguing question: Can periodic alterations in particle size alone disrupt the thermodynamic equilibrium of the system and transform size fluctuations into directed motion?
In this study, we explore this question within the context of a dense system comprising pulsating active particles in a two-dimensional asymmetric channel.Our findings reveal that fluctuations in particle size can function as an implicit non-equilibrium driving force, disrupting the thermodynamic equilibrium of the system.Consequently, these size variations can be translated into directed motion within asymmetrical structures.The investigation delineates three distinct states: the arrested ordered state, the disordered state, and the cycling ordered state.In the arrested ordered state, particles lack directional motion, while the cycling ordered state exhibits moderate directional movement.Conversely, in the disordered state, the presence of directional motion is most pronounced.

Model and methods
We consider the collective motion of N self-pulsating active particles in a two-dimensional asymmetric channel.The channel has periodic boundaries along the x-direction and hard-wall boundaries along the y-direction.The shape of the channel, shown in figure 1(a), is determined by its half-width h(x), where L x = L 1 + L 2 represents the periodicity of the channel, and W denotes the bottleneck width.For simplicity, we define L x /L y = 3/2.The asymmetry of the channel is described by ∆ = (L 1 − L 2 )/L x , where ∆ = 0 indicates symmetry.The opening of the channel is defined as α = W/L y , ranging from α = 0 (closed channel) to α = 1 (straight channel).
In various biological systems, such as cell membrane [43][44][45], actin [46,47], and epithelium [48][49][50][51], oscillations in area or perimeter have been observed.We focus on the idealized self-pulsating active particle, whose radius σ is dependent on its phase angle θ i [42], Here, σ 0 represents the largest reachable radius, λ < 1 is the self-pulsation amplitude.The phase angle θ i cyclically varies within the range [0, 2π] at a frequency ω, leading to periodic radius contraction, as shown in figure 1(b).When particles overlap, synchronization Γ and repulsion U occur.The dynamic characteristics of particle i are determined by its position r i and phase angle θ i [42], where {µ, D} are respectively the mobility and diffusion coefficients of positions, µ θ represents the diffusion coefficient of phases, and ξ i (t) denotes Gaussian white noise with a unit variance and a zero mean.The range for interactions is given by a , where U 0 and ϵ determine the strengths of repulsion and synchronization, respectively.Otherwise U(a ij ) = Γ(a ij , θ i − θ j ) = 0. To highlight the influence of phase periodicity (ω) and inter-particle interactions (repulsion and synchronization) on the directed transport of pulsating active particles, we have removed the noise term in equation ( 4).In the presence of high-intensity noise, the pulsating effects of particles will be significantly diminished.For repulsion, its distancing effect (∂ r i U) prevents particle overlap, while its contractile effect (∂ θ i U) causes a reduction in particle size.Synchronization facilitates the alignment of particle sizes.Interestingly, when a larger particle overlaps with a smaller particle, there exists a size competition for the smaller particle between repulsive contraction (resulting in size reduction) and synchronization (resulting in size increase) (figure 1(c)).This competition significantly influences the order and disorder of particle sizes within the system.
In discussing non-equilibrium driving, we introduce the effective entropy potential, assuming equilibrium in the y direction for convenience.In this system, the entropy is directly related to the number of microscopic states.The number of microscopic states is determined by the available spatial width for particle diffusion, denoted as where 2h(x) represents the width of channel and h p (x, t) = i =1 σ i [x, θ i (t)] denotes the total width occupied by particles already positioned at position x.By utilizing the equations S(x, t) = k B ln Ω(x, t) and U eff (x, t) = −TS(x, t), we obtain the expression for the effective entropy potential as Here, k B represents the Boltzmann constant, and T represents temperature.Moreover, variations in particle phase are transformed into oscillations of the entropy potential.
To quantify the rectification effect along the x-direction, we measure the average velocity V x of particles in the asymptotic long-time state where is the average displacement of particles at time t along x-direction.x i (t) is the center of mass of particle i.The symbol ⟨. ..⟩ denotes an average over the random initial conditions.
To characterize the diffusion behavior of particles, we employ the time-dependent mean square displacement (MSD) Here, the calculation is performed in the reference frame of the center of mass, denoted by r cm = 1 N i r i , where r ′ i = r i − r cm .The ensemble average, denoted by the brackets, is calculated under steady-state conditions.
To analyze the overlap and average size of particles, we define the packing fraction φ as the ratio of the occupied area by particles to the total available area, The packing fraction φ varies with time as particle radius undergoes periodic contraction.In this study, we focus on the time-averaged packing fraction ⟨φ ⟩.In addition, we define the particle number density as To describe the size variations of particles, we introduce the synchronization parameter A value of r = 1 indicates complete order, where all particles exhibit the same cyclic rhythm and size simultaneously.As r decreases, the system becomes more disordered, with particles cycling with phase differences and exhibiting larger variations in size at the same time.Similarly, we are interested in the time-averaged order parameter ⟨r⟩.Equations ( 3) and ( 4) are integrated using the forward Euler algorithm.The integration time step is chosen to be 10 −4 , and the total integration time is 10 5 , which ensures that the system can reach a nonequilibrium steady state.For each simulation run, particle positions are initialized using a uniform random distribution within the channel, and the initial angles are randomly chosen over the interval of [0, 2π].Unless otherwise specified, our simulations utilize the following parameter sets: N = 100, µ = µ θ = 1.0,D = 0.1, σ 0 = 0.5, U 0 = 1.0, and ϵ = 10.The presented results have shown robustness when these parameters are reasonably altered.

Results and discussion
We begin by discussing the implementation of rectification conditions for Brownian particles in our model.The ratchet mechanism requires two key components: (a) a non-equilibrium driving force that disrupts thermal equilibrium and prevents directed transport in accordance with the Second Law of Thermodynamics, and (b) asymmetry, either temporal or spatial, which breaks the left-right symmetry of the response.In the current system, by examining the effective entropic potential U eff (x, t) defined in equation ( 6), we observe that the periodic variation in particle size results in a time-dependent effective entropic potential, thus generating a periodic force over time.This can be interpreted as a non-equilibrium driving force.Therefore, this non-equilibrium driving disrupts thermodynamic equilibrium, resulting in directed motion of the particles in the asymmetric channel.In the subsequent section, we will study the collective motion of pulsating active particles by varying the asymmetry parameter ∆, the opening of the channel α, the self-pulsation frequency ω, the self-pulsation amplitude λ, and the particle number density ρ 0 .

The impact of channel properties on collective motion
Figure 2 illustrates the relationship between the average displacement, ∆X, and time, t, for different values of ∆.It is evident that the direction of particle transport is determined by the sign of ∆.When ∆ < 0 (e.g.∆ = −0.8), the particles exhibit an average rightward movement.Conversely, when ∆ > 0 (e.g.∆ = 0.8), the particles demonstrate an average leftward movement.If ∆ = 0, directed transport is not observed.Consequently, further discussion will be provided to elucidate the rectification mechanism of particles.
We discuss the rectification mechanism using the effective entropy potential as shown in figures 3(a) and (b).In the ordered state, due to the synchronous cycling of particles, the entropy potential at each position x increases or decreases synchronously as shown in figure 3(a).Specifically, in the disordered state, particles at different x-positions undergo cyclic oscillations with phase differences, resulting in unsynchronized changes in the entropy potential accompanied by small peaks and valleys as shown in figure 3(b).We define the effective entropy force as f(x, t) = −∂U eff (x, t)/∂x.A larger entropy force implies that particles face greater resistance in surmounting the entropy barrier.Notably, larger particles experience a higher entropy force (f ′ ) at a given position compared to smaller particles (f ).Taking the ordered state for ∆ < 0 as an example (figure 3(a)).When a small particle is in state (i), the entropy force from the left (f L )  outweighs that from the right (f R ).Consequently, the small particle tends to overcome the rightward entropy force and migrate towards position (ii).(This process is more challenging for larger particles due to the stronger entropy force (f ′ R ).)As the small particle moves rightwards, its radius continuously increases.In state (iii), the prominence of entropy potential (greater entropy force (f ′ L ) relative to small particles) aids the smooth descent of large particles into the subsequent potential well.Therefore, the oscillation of the entropy potential (forces), acting as non-equilibrium driving, effectively facilitates each sub-process of rectification towards the right: lower entropy (force) enables particles to escape from the previous potential well, while higher entropy (force) accelerates the smooth entry into the next potential well.To clarify, in a disordered state, the peaks and valleys do not impact the overall terrain of the entropy potential (figure 3(b)), thereby not affecting the rectification direction.
Similarly, particles move towards the left when ∆ > 0. When ∆ = 0, the channel is entirely symmetric, resulting in equal probabilities of particle movement towards either side and the absence of directed motion.Due to the fact that particles, on average, move in the opposite direction for ∆ and −∆, we will focus on the case where ∆ = −0.8 in the subsequent analysis.
Figures 4(a) and (b) show the average velocity V x as a function of the opening of the channel α in various conditions.It is found that the average velocity V x is a peaked function of the opening of the channel α.This phenomenon can be explained as follows.When the channel is too narrow (α → 0), it is almost completely closed, inhibiting passage and causing V x to approach zero.Conversely, when α → 1, the channel becomes nearly straight, disregarding the channel's spatial asymmetry and eliminating the ratchet effect, leading to V x converging to zero.Therefore, an optimal value of approximately α ≈ 0.25 exists, maximizing V x , and this value remains almost consistent despite changes in parameters of system.In the subsequent analysis, we will focus on the case of α = 0.3.

The impact of self-pulsation of particle size on collective motion
Figure 5 displays the average velocity V x as a function of the self-pulsation frequency ω for different λ.Notably, V x exhibits a distinct peak in response to changes in ω.To gain a deeper understanding, we consider figure 3(a) as an illustrative example.If ω → 0, the size oscillation occurs at a slower rate, leading to two effects that contribute to a reduction in V x .On one hand, the long duration of the larger size increases the likelihood of particles being trapped at position (i) due to higher potential barriers.On the other hand, particles spend more time in the smaller size, which diminishes the impact of the entropy potential and increases the likelihood of failure when entering the next potential well.Conversely, when ω is too large, as in state (iv), particle size increases prior to reaching the right barrier.Under a stronger entropy force (f ′ R ), larger particles easily slide back to position (i), interrupting rectification.If ω → ∞, the system fails to perceive oscillation and the particles remain stationary, as if their radii have never undergone any change.Therefore, there exists an optimal value of ω at which V x takes its maximal value.Some scholars have proposed an explanation based on the resonating effect between particles' size oscillations and interactions [52].It is worth noting that excessively large or small ω hinders particles' interactions.For more comprehensive discussions on this topic, please refer to below.
Figure 6(a) illustrates the relationship between the average velocity V x and the self-pulsation amplitude λ for various moderate or larger values of ω.When λ = 0, signifying rigid particles with the largest radius, the system loses its non-equilibrium characteristics, resulting in V x = 0 without diffusion (figure 6(d)).As λ steadily increases from zero, the average velocity V x consistently rises.This phenomenon can be elucidated from two perspectives.Firstly, as λ increases, the average size of particles in the system becomes smaller (confirmed by a decrease in the average packing fraction ⟨φ ⟩ in figure 6(b)), which differs from other models involving both expansion and contraction [52][53][54][55].The smaller size makes it easier for particles to pass through the channel, thus increasing V x .Secondly, as particles become smaller on average, the competition between repulsive contraction and synchronization becomes apparent.When the delicate balance is reached, the system transforms from a cyclic ordered state to a disordered state (figure 6(c)).Disorder means that particles at different x positions oscillate with phase differences, leading to the appearance of entropy potential in small peaks and valleys (shown in figure 3(b)).As time evolves, the valley (cyan line) rises while the peak (orange line) falls, acting as a booster that propels particle from position (i) to position (ii), thus enhancing rectification.Therefore, the reduction of the average particle size and the presence of competition-induced disorder lead to an increase in V x in response to λ.
Figure 6(e) shows the average velocity V x as a function of the self-pulsation amplitude λ for different smaller values of ω.It is found that the average velocity V x is a peaked function of the self-pulsation amplitude λ.V x initially rises with increasing λ, but ultimately drops to zero.The attenuation results from a system blockage caused by the prolonged maintenance of larger particle sizes at low frequencies (for λ = 0.40 as evidenced by MSD in figure 6(h) and ⟨φ ⟩ ≈ 0.85 in figure 6(f)).Despite slight fluctuations, the system consistently is located in an arrested ordered state (proved by ⟨r⟩ ≈ 1 in figure 6(g)).In addition, the position of the peak in the curves shifts to large λ with the increase of ω.
In order to thoroughly investigate the interdependence of the average velocity V x on ω and λ, a phase diagram was constructed.This diagram, illustrated in figure 7, emphasizes the average velocity in the ω − λ representation at ρ 0 = 1.2, ∆ = −0.8, and α = 0.3.Optimal rectification performance is observed at moderate ω values and larger λ values.
To explore the relationship between system states (cycling ordered, arrested ordered, and disordered) and rectification efficiency, different state regions were delineated on figure 7 using dashed lines.Dash line (i) separates disordered and ordered states based on the value of the mean synchronization parameter ⟨r⟩.States with ⟨r⟩ below 0.9 are classified as disordered, while those above 0.9 are classified as ordered.Within the ordered states, dashed line (ii) distinguishes between arrested ordered and cycling ordered states, based on the behavior of the packing fraction φ with respect to time t.
To visually illustrate the three distinct states, figure 8 provides snapshots of temporal evolution of points A, B, and C in figure 7. The blue arrows represent the velocity vectors of the particles at a given time.In the arrested ordered state (figures 8(a)-(d)), particles remain at their maximum size, densely aggregating and resulting in a high packing fraction φ that remains constant over time.Minimal particle motion, indicated by the blue velocity vectors, leads to zero average rectification.In the cycling ordered state (figures 8(e)-(h)), particles synchronously change size, causing variations in the packing fraction φ over time.Initially, the particles are sparsely distributed (figure 8(e)), gradually transiting to a tightly packed configuration (figure 8(g)).At the same time, although the particles are moving in different directions, overall, a greater  number of particles are moving towards the right side.In the disordered state (figures 8(i)-(l)), particles evolve with a consistent phase difference in size, resulting in non-uniform local density resembling wave propagation.For example, at the beginning, the left side exhibits a lower local density compared to the higher density on the right side (figure 8(i)).Over time, the local density increases on the left side while decreasing on the right side (figure 8(l)).It's important to note that the direction of density wave propagation may not align with the average particle motion direction.The density wave propagation acts as the local oscillations in the entropy potential, as shown in figure 3(b), promoting to overall particle rectification.
By comparing system states (cycling ordered, arrested ordered, and disordered) regions and rectification effect (figure 7), we have observed the following phenomena.In arrested ordered state, particles exhibit no directional motion.In the cycling ordered state, they show weaker directional movement, while in disordered state, directional motion is most pronounced.Therefore, the ordered states are not conducive to the directional motion of particles, whereas the disordered states promote the directional motion of particles.

The impact of the particle number density on collective motion
Figure 9(a) depicts the relationship between the average velocity V x and particle number density ρ 0 for different moderate or larger values of ω.As ρ 0 increases, V x initially rises, then slightly declines, and eventually increases again.To comprehensively explain these patterns, we analyze two separate sections: the general increase and the local decrease.In the range of 0.8 < ρ 0 < 1.7, an increase in density leads to a decrease in the average spacing between particles (|r i − r j |), as confirmed by the rise in the average packing fraction ⟨φ ⟩ shown in figure 9(b).This reduced spacing enhances repulsive distancing force (dependent on a ij ), promoting the passage of particles through channels and resulting in the general increase in V x .For the specific density range of 1.3 < ρ 0 < 1.5, we focus on figure 9(c) to explore the slight decrease in average velocity V x .Here, ρ 0 increases, repulsive contraction (rather than synchronization) also increases.As density increases (1.0 < ρ 0 < 1.3), the contractile effect of repulsion reaches a comparable level with synchronization, causing the transition from cycling ordered state to disordered state (indicated by the decrease in ⟨r⟩).As mentioned earlier, the appearance of disorder significantly increases V x .However, at higher densities (1.3 < ρ 0 < 1.5), repulsive contraction gradually surpasses synchronization, breaking delicate relationship.Some particles reduce in size due to the stronger contractile torque, and the system gradually shifts from a disordered state to a cycling ordered state again (evident by the increasing ⟨r⟩ and the actual cyclic frequency being less than ω).This results in a decrease in V x .Therefore, the local drop in V x arises from the breakdown of competition-induced disorder.It's worth noting that unlike rigid particles, pulsating active particles do not experience blockage at high ρ 0 (under the appropriate ω as a premise condition), and even exhibit superdiffusive behavior (for ρ 0 = 1.7 in figure 9(d)).This is due to their cyclic nature, allowing them to pass through the channel when they reach the smaller sizes.Note that for very large ω values (e.g.ω = 50), V x monotonically increases without declining (figure 9(a)).The dominance of periodic diving over interactions leads to continuous cycling ordered states (figure 9(c)), which rely solely on repulsive distancing forces to promote rectification.
For small ω values (e.g.ω = 2.7), slow circulation causes congestion at high density (as observed for ρ 0 = 1.5 in figure 9(h) and V x returning to zero in figure 9(e)).Increasing ω and relaxing constraints slightly shift the critical density for blockage higher (figure 9(e)).When jamming occurs, the average packing fraction ⟨φ ⟩ remains relatively constant despite increasing ρ 0 (figure 9(f)).Notably, a highly disordered state with contraction waves induces a superdiffusive tendency among particles (shown in figure 9(g) for ρ 0 = 1.4 with ⟨r⟩ ≈ 0.35, and supported by the MSD in figure 9(h)).
Figure 10 depicts the phase diagram that illustrates the relationship between the average velocity V x , the self-pulsation frequency ω, and the particle number density ρ 0 .This diagram resembles figure 7 as it also delineates three distinct states with dashed lines based on ⟨r⟩ values and whether φ changes with t.In the arrested ordered state, there is an absence of particle directional motion, whereas the disordered state showcases the most pronounced particle directional motion.The subtle distinction lies in the fact that, under high-density conditions, the cycling ordered state also exhibits improved directional movement.

Conclusion and outlook
In this study, we studied the directed transport of pulsating active particles in a two-dimensional asymmetric channel.The fluctuations in particle size lead to time-dependent changes in the effective entropy potential, creating a non-equilibrium driving force.This force disrupts the thermodynamic equilibrium of the system, causing the conversion of size periodic motion into directional motion within the asymmetrical periodic channel.The direction of transport depends on the symmetrical properties of the channel, resulting in positive average velocity when ∆ < 0, zero average velocity at ∆ = 0, and negative average velocity when ∆ > 0. Particle rectification reaches its optimal state when the opening of the channel, α, is approximately 0.25.The existence of an optimal ω value yields the maximum average velocity.The relationships between average velocity and both the self-pulsation amplitude and particle number density are complex.At lower frequencies, the average velocity exhibits a peak function in relation to self-pulsation amplitude (or particle number density), whereas at higher frequencies, the average velocity increases with the amplification of self-pulsation (or particle number density).The system exhibits three distinct states: the arrested ordered state, disordered state, and cycling ordered state.In the arrested ordered state, particles exhibit no directional motion, whereas directional motion is prominent in the disordered state.Additionally, the cycling ordered state displays advantageous directional movement facilitated by a high particle number density.Note that the waves propagate without a preferential direction.The probability of the wave propagating to the left is equal to the likelihood of it propagating to the right.There is no inherent correlation between the direction of wave propagation and the average motion direction of particles.
In contrast to traditional particle rectification, where random external forces induce non-equilibrium motion converted into directed motion, our study introduces a novel mechanism.Here, fluctuations in particle size act as the driving force for directed particle motion, expanding the current paradigms of particle rectification.Additionally, size pulsations can occur naturally in living systems or can be controlled through external stimuli.Previous studies have demonstrated reversible or significant changes in the size of nanomaterials [56][57][58][59][60].The outcomes of our research exhibit potential applications in these systems.

Figure 1 .
Figure 1.(a) Mechanism of pulsating active particles in a two-dimensional asymmetric channel.The asymmetric channel structure is described by the half-width function h(x) (equation (1)).The dynamics of particles is governed by equations (3) and (4).(b) The radius of a pulsating active particle undergoes periodic contraction at a frequency of ω when the phase angle varies (equation (2)).(c) Competitive mechanisms of repulsive contraction and synchronization.The color represents the radius of particle, with larger sizes depicted in yellow and smaller sizes in red.

Figure 2 .
Figure 2. The correlation between the average displacement in the x-direction and time t for different ∆.The sign of ∆ controls the direction of transport.The other parameters are α = 0.3, ρ0 = 1.0, ω = 5.0, and λ = 0.5.

Figure 3 .
Figure 3.The profile of the effective entropic potential U eff (x, t) for large particles with a blue line and for small particles with a red line.(a) Rectification mechanisms for larger and smaller particle in a cycling ordered state.(b) The effect of promoting rectification in a disordered state.The above is based on ∆ = −0.8.

Figure 7 .
Figure 7. Phase diagram of the average velocity Vx in the ω − λ representation at ρ0 = 1.2, ∆ = −0.8, and α = 0.3.The background represents the value of Vx according to the color bar on the right.The dashed lines divide the three states according to ⟨r⟩ and the behavior of φ with respect to time t.The disordered state greatly promotes directional motion.

Figure 8 .
Figure 8. Snapshots of the dynamic evolution of particles in three states over time.(a)-(d) In the arrested ordered state (ω = 0.5 and λ = 0.4), particles are confined within the larger sizes, while the packing fraction φ remains consistently constant throughout the process.(e)-(h) In the cycling ordered state (ω = 3.5 and λ = 0.3), particle sizes demonstrate synchronized cyclic changes, while the packing fraction φ fluctuates throughout the cycle.(i)-(l) In the disordered state (ω = 2.0 and λ = 0.5), particles exhibit cyclic variations in size, resulting in an uneven local density that propagates in a wave-like manner.The blue arrows indicate the velocity vectors of the particles at a specific moment.The particle size is visually highlighted through the red-to-yellow color gradient, while the gray shapes represent the channels.

Figure 10 .
Figure 10.Phase diagram of the average velocity in the ω-ρ0 representation at λ = 0.5, ∆ = −0.8 and α = 0.3.The background represents the value of Vx according to the color bar on the right.The dashed lines divide the three states according to ⟨r⟩ and the behavior of φ with respect to time t.