Perspective on Reversible to Irreversible Transitions in Periodic Driven Many Body Systems and Future Directions For Classical and Quantum Systems

Reversible to irreversible (R-IR) transitions arise in numerous periodically driven collectively interacting systems that, after a certain number of driving cycles, organize into a reversible state where the particle trajectories repeat, or remain irreversible with chaotic motion. R-IR transitions were first systematically studied for periodically sheared dilute colloids, and appear in a wide variety of both soft and hard matter systems, including amorphous solids, crystals, vortices in type-II superconductors, and magnetic textures. In some cases, the reversible transition is an absorbing phase transition with a critical divergence in the organization time scale. R-IR systems can store multiple memories and exhibit return point memory. We give an overview of R-IR transitions including recent advances in the field, and discuss how the general framework of R-IR transitions could be applied to a much broader class of periodically driven nonequilibrium systems, including soft and hard condensed matter systems, astrophysics, biological systems, and social systems. Some likely candidate systems are commensurate-incommensurate states, systems exhibiting hysteresis or avalanches, and nonequilibrium pattern forming states. Periodic driving could be applied to hard condensed matter systems to see if R-IR transitions occur in metal-insulator transitions, semiconductors, electron glasses, electron nematics, cold atom systems, or Bose-Einstein condensates. R-IR transitions could also be examined in dynamical systems where synchronization or phase locking occurs. We discuss the use of complex periodic driving such as changing drive directions or multiple frequencies as a method to retain complex multiple memories. Finally, we describe features of classical and quantum time crystals that could suggest the occurrence of R-IR transitions in these systems.


I. INTRODUCTION
Driven many-body deterministic nonlinear systems generally exhibit disordered or chaotic dynamics, as found in turbulence 1 , particle flow over disordered media 2 , plasmas 3 , sheared materials 4,5 , granular matter 6 , earthquakes 7 , gravitational systems 8 , and biological systems 9 .Chaotic dynamics can arise even in systems with only three degrees of freedom 10,11 so it can be expected that driven many body disordered systems with hundreds or thousands of degrees of freedom will generally exhibit fluctuating or chaotic dynamics.Recently, a growing number of many body systems have been shown to exhibit a transition from time periodic or reversible motion to chaotic irreversible motion under oscillatory driving [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] .In these studies, the particle positions are compared from one driving cycle to the next.In the chaotic phase, the particles do not return to the same positions and undergo diffusive motion away from the initial positions over many driving cycles.For certain driving amplitudes or system parameters, however, the particles can organize over many cycles into a reversible state in which they return to the same positions after every or every few cycles, and the long time diffusive behavior is lost.
Reversible behavior in viscosity-dominated flows was famously demonstrated by G. I. Taylor 27 using a two cylinder setup in which the inner cylinder is rotated multiple times and then rotated back.Pine et al. used the same shearing geometry and viscous fluid as Taylor but considered the case where there are additional colloidal particles in the fluid that can collide with each other, so that any irreversible behavior would be due to the particle collisions rather than the fluid itself 12 .The simple periodically sheared dilute colloidal particles experiment allows for the systematic study of transitions from irreversible to reversible motion in many-body systems.Pine et al. showed that there is a critical strain amplitude below which the steady state behavior is reversible and above which the behavior becomes diffusive or irreversible.The colloids are in a viscous fluid and are large enough that thermal effects are negligible; since they are also electrically neutral, the only interactions capable of producing an irreversible state are contact forces between the colloids during collisions.During the initial shear cycles, particles move by different amounts and some par-ticles collide with one another.In a steady irreversible state, collisions occur during each cycle and cause the particles to wander away from their initial positions in a Brownian-like diffusion, where the distances traveled along the shear direction, x 2 , and perpendicular to the shear direction, z 2 , increase linearly with time.
In further work, Corte et al. 13 studied the number of drive cycles required for the system to reach a reversible state from a randomized initial state.In particular, during the first cycles the motion is chaotic, but after many cycles the system may organize to a state where collisions are absent.The number of transient cycles n τ spent reaching the reversible state diverges at a critical density or at a critical shear amplitude γ c according to a power law n τ = |γ − γ c | ν 13 .This critical behavior suggests that the transition to the reversible state is a nonequilibrium phase transition, and the observed exponents β = 0.45 (2)  and ν = 1.33 (2) are similar to the critical exponents of the two-dimensional directed percolation (DP) universality class, namely β = 0.584(4) and ν = 1.295(6) 28 .Although extensive theoretical studies of DP transitions have been performed, clear observations of these transitions in experimental systems have only been obtained relatively recently [29][30][31] .
Since R-IR transitions were first observed in dilute systems, it might seem reasonable to imagine that these transitions only arise under specialized circumstances where the interactions are of sufficiently short range or the system is sufficiently dilute that it is possible to reach a state where collisions never occur.This would imply that it would be difficult to find reversible states in disordered strongly coupled systems where the particles are always in contact or where long range interactions are relevant; however, R-IR transitions have in fact been observed for periodically driven granular systems 17 where the particles are always in contact as well as for vortices in type-II superconductors 14,24 at densities for which the vortices are strongly interacting.One of the most extensively studied strongly interacting systems exhibiting an IR-R transition is periodically sheared amorphous solids 15,16,18,19,21,23,25,[32][33][34] .Here the particles or atoms are always in contact and long range strain fields are present.Reversible behavior is expected to occur in solids when the strain amplitude is small so that the system behaves completely elastically; however, the R-IR transitions in the amorphous solids appear well into regimes where strong plastic deformations occur in which particles are exchanging neighbors.In general, in strongly coupled systems the particle trajectories during reversible states show more complex loops or return to the same position not after every drive cycle but after multiple drive cycles, such as in examples where the motion repeats every six cycles 16,21,34,35 .Plasticity is thought to be an inherently irreversible process; however, these works indicate that it is possible for plastically deforming reversible regimes to emerge, suggesting that similar transitions to reversible states could arise in many other strongly coupled systems.There have now been several other studies of many body interacting systems undergoing IR-R transitions including crystals 36 , systems with quenched disorder 14,24,37 , chiral active matter 38,39 , skyrmions 40 , and magnetic materials 41 .Additionally, it has been shown that in the reversible phase, the system can be trained to store multiple memories 20,33 .
In this paper we give an overview of work on reversibleirreversible transitions in both dilute and strongly coupled systems, and discuss how ideas about R-IR transitions could be more broadly applied to other areas including polymeric soft matter, pattern forming systems, and frictional systems.The R-IR transition could be induced via global driving such as a shear or external field, or via local driving such as the periodic motion of a local probe.We also discuss classes of active matter systems that could also exhibit R-IR transitions in the limit where thermal effects are negligible but the activity can be treated as periodic.Most existing studies involved a periodic drive applied in a single direction at one frequency; however, much more complicated periodic drives could be applied with multiple frequencies or changing directions, opening a new area of investigation.Such systems might organize to more complex reversible states and could retain a memory of past drive protocols as the driving is changed from more to less complex.We also discuss possible applications of R-IR transitions to a wider number of hard condensed matter systems under periodic driving such as metal-insulator systems, charge ordering systems, Bose-Einstein condensates, superfluids, sliding charge density waves, Wigner crystals, systems showing nonlinear transitions, classical time crystals, and numerous magnetic textures including magnetic domain walls and magnetic bubbles.These types of systems can be subjected to numerous forms of periodic driving, such as electric currents, magnetic fields, or optical excitations.The most promising quantum systems for observing R-IR transitions are the time crystals 42 , which already show a number of features consistent with a transition from a chaotic state to a time periodic state.There is also a broader class of nonlinear coupled systems such as coupled oscillators or networks that have been shown to exhibit synchronization and phase locking effects 43,44 and nonequilibrium pattern formation 45 .Synchronization effects can occur in many body coupled systems when the many degrees of freedom become coupled so that the response looks like that of a few body system, and we argue that such transitions could be viewed as R-IR transitions, suggesting that these systems could also be candidates for exhibiting transitions into absorbing states.

II. SHEARED SYSTEMS
The dilute colloid experiments mentioned above that were performed by Pine et al. 12 were the first studies of transitions from irreversible to reversible states.In Fig. 1, representative configurations show the situation in FIG. 1.A schematic of the system studied by Pine et al. 12 , consisting of a dilute suspension of colloids (black disks) subjected to periodic shearing.Arrows indicate the shearing direction.In (a) and (b), the system is in the dilute limit, and the comparison of the starting (black) and ending (red) positions of the particles shown in (c) indicates reversible behavior.In (d) and (e), the system is in the dense limit, the particles collide repeatedly under shear, and the image of starting and ending positions in (f) indicates the occurrence of irreversible motion.
the reversible versus irreversible steady states for systems with low [Fig.1(a)] and high [Fig.1(b)] colloid densities subject to the same amplitude of shearing.Comparison of the starting (black dots) and ending (red dots) positions of the particles, shown in Fig. 1(c,f) for the two densities, reveals that the motion is reversible in the low density system where collisions do not occur and irreversible in the higher density system where collisions are ongoing.Figure 2(a) shows the experimentally measured particle positions from Ref. 12 at the end of each cycle over multiple cycles in the irreversible regime.Here the dynamics is Brownian-like, and the particles gradually diffuse away from their initial positions.In Fig. 2(b), the corresponding measured mean square displacements x 2 in the direction of drive versus accumulated stain increases linearly as expected for Brownian motion.The displacements z 2 in the direction perpendicular to the drive show the same behavior but have a lower value.
In general, for any random initial condition in sheared colloidal systems, during the first few cycles the particles do not return to their original positions after each cycle.Instead, over time the particles either organize to a reversible state or remain in an irreversible state.Corte et al. 13 used a combination of simulation and experiments to further explore the system studied by Pine et al. 12 , and applied different strain amplitudes to identify the manner in which the system reaches a steady irreversible or reversible state.Figure 3(a,b) shows the positions of the active particles, or particles that did not return to their original positions after each cycle, for a simulation of two-dimensional (2D) sheared hard disks at two different strain amplitudes of γ 0 = 3.0 [Fig.3(a)] and γ 0 = 2.0 [Fig.3(b)] 13 .At the end of the first shear cycle, there are numerous active particles in each case, but over time the system reaches a steady state that is either irreversible with a finite number of active particles, as shown for γ 0 = 3.0 in Fig. 3(a), or reversible with no active particles, as shown for γ 0 = 2.0 in Fig. 3(b).A time series of the fraction of active particles as a function of shear cycle number appears in Fig. 3(c) for the same two strain amplitudes.After 2500 cycles, there are no active particles remaining in the γ 0 = 2.0 sample, but the activity in the γ 0 = 3.0 sample plateaus at a finite steady state value where about 39% of the particles remain active.The inset in Fig. 3(c) shows the steady state active particle fraction versus strain amplitude, indicating that there is a transition from irreversible (active) to reversible (nonactive) behavior near a critical strain of γ c 0 = 2.66.By fitting the curves in Fig. 3(c) to a stretched exponential, Corte et al. 13 obtained the mean time τ required to reach a steady reversible or irreversible state, plotted in Fig. 4 as a function of strain amplitude γ 0 .At the critical amplitude γ c 0 , τ diverges as a power law according to In the 2D simulations, it was found that ν ≈ 1.33 on both sides of the transition, while the three-dimensional (3D) experiments produced a similar divergence with ν = 1.1.This transition has the hallmarks of what is known as an absorbing phase transition, which appears in nonequilibrium systems and often falls in the directed percolation (DP) universality class 28,46 .The irreversible state can be viewed as a dynamically fluctuating state in which the particles continue to exchange positions and long time diffusion is occurring.In contrast, the reversible state corresponds to the absorbed state where the fluctuations are lost and the behavior becomes completely time repeatable, indicating that the system is dynami- cally frozen or trapped in a limit cycle.Corte et al. 13 called the transition into the reversible state "random organization" since the particles form a random configuration in which collisions do not occur.Since that time there have been further studies of random organization in these sheared dilute colloidal systems 47 , as well as a number of studies indicating that randomly organized states near the critical threshold are hyperuniform 22,38,[48][49][50][51][52][53] .In hyperuniform states, there are no large fluctuations at long length scales.Periodic crystals are hyperuniform by definition; however, certain random structures also have hyperuniform properties 54 , and there is ongoing work to understand the conditions under which random hyperuniform states can occur.In a random organized state, the particles form configurations where collisions are absent.This increases the average distance between particles and reduces large local variations in the particle positions, thereby diminishing large density fluctuations and giving a more uniform density that extends out to long length scales.An open question is whether all systems near a R-IR transition exhibit hyperuniformity, or whether this is a property found only in systems that are dilute or that have short range interactions.
There has also been some work on ordered pattern formation in dilute systems, where the R-IR transition overlaps with a disorder (irreversible) to order (reversible) transition.One example is hard disk colloidal particles interacting with a periodic array of obstacles 55 .Figure 5(a) shows the fraction R n of active particles at the end of each cycle for periodically driven disks moving through an ordered array of posts at a fixed drive amplitude for different disk densities φ.When φ > 0.3716, the system remains in an irreversible state, while for φ ≤ 0.3716, the disks organize into a reversible state.Figure 5(b,c) illustrates the particle positions in the steady state, which is irreversible and disordered in Fig. 5(b) at φ = 0.3962, and reversible and ordered at φ = 0.3716 in Fig. 5(c).In Ref. 55 , the authors also found a power law divergence of the time required to reach the reversible state.The power law exponents are similar to those observed in 2D random organization systems, suggesting that the periodically driven disks fall in the same universality class as those systems.Another interesting feature of the reversible state in Fig. 5(c) is that collisions are not absent but instead the particles collide with the obstacles in a repeating pattern.Pattern formation in a reversible state was also studied for bidisperse systems in which half of the particles move in circles and the other half do not, where it was shown that there is a transition from a mixed fluid to a pattern forming phase separated reversible state 39 .

III. CONDENSED SYSTEMS
We next consider R-IR transitions in systems close to or just at jamming, as well as for systems deep in the jamming phase and amorphous solids.In dilute systems, the irreversible state generally shows a liquid structure and the particles do not form a solid.At the transition into the absorbing state, the particles either experience a small number of repeating collisions or have no collisions at all.As a result, it might be assumed that R-IR transitions are limited only to systems that are dilute or have contact interactions, providing the particles with enough space to rearrange and organize into a reversible state.
FIG. 6. R-IR transition in simulations of a 2D bidisperse assembly of hard disks from Ref. 56 .The dynamic phases are plotted as a function of the strain amplitude γmax versus the disk density φ.Black symbols indicate reversible states where the disks return to their original locations after a single cycle.For the green symbols, the states are loop reversible with complicated disk trajectories that repeat after one or more periods.In the region with red symbols, the steady state is irreversible.Reprinted with permission from C. F. Below we show that this is not the case.

A. Reversibility near Jamming and Packing
Schreck et al. 56 studied a granular matter version of R-IR transitions for a periodically driven 2D bidisperse disk assembly, and focused on the formation of a reversible state below the jamming transition, φ < φ J = 0.84.Well below jamming, the system organizes into a reversible state where there are no collisions.Schreck et al. term this a "point reversible" state, and it is the same as a random organization state.For higher densities, the system can still organize to a reversible state, but the disk trajectories become much more complex and involve some collisions.Schreck et al. named these "loop reversible" states since the reversible orbits form loop structures instead of straight lines.At densities near jamming, an irreversible steady state emerges.Figure 6 shows the phase diagram as a function of strain amplitude γ max and disk density φ from Ref. 56 , where the black region is point reversible, the green region is loop reversible, and the irreversible states are colored in red.Schreck et al. also obtained similar results in 3D systems.Recent studies by Gosh et al. on disk packings showed that the transition to a reversible state coincides with a transition to a crystalline state, which is interesting because it was not expected a priori that these transitions would occur at the same point 57 .
Both experiments and simulations have been performed for R-IR transitions in granular matter as a function of varied shear amplitude 58 and friction 17 .Other works address transitions to reversible states or random organization at the approach to jamming or random close packing 59 , as well as possible ways to connect jamming and yielding in a unified framework [60][61][62] .Open questions for systems transitioning between jammed and unjammed states include how the nature of the trajectories changes across jamming and whether there could be different types of absorbing transitions.Other effects to consider would be adding quenched disorder sites to a jamming system 63,64 in order to determine how the R-IR transition is affected, or to study whether R-IR transitions change in the presence of Griffiths 65 or Gardner transitions 66 .

B. Amorphous Solids
For systems such as solids or glasses well above the jamming density subjected to shear, it is known that at small strains, the response is elastic and plastic rearrangements do not occur, while for intermediate strains, plastic events start to appear, and for even higher strains the system exhibits plastic yielding.Currently there is a considerable amount of work on elucidating the nature of the yielding transition in sheared solids and glasses, understanding the shape of the stress-strain curve, and determining the way in which shear response and the yielding transition depend on how the system is prepared.Figure 7(a) shows the potential energy of an amorphous solid subject to an athermal strain increase every simulation step.The energy increases parabolically, as is expected from a rigid elastic material, and decreases discontinuously when a plastic rearrangement of the particles occurs.The most basic plastic rearrangement involves a change of nearest neighbors called a "soft-spot" [67][68][69][70] that is illustrated in Fig. 7(b).Most plastic events involve several soft-spots, as will be discussed below.Amorphous and crystalline systems can also be subjected to periodic shearing, and for strains where the response is plastic under unidirectional shear, it could be assumed that the system would only exhibit irreversible states; however, Fiocco et al. 15 , Regev et al. 16 and Priezjev 71 studied 3D and 2D model glasses subject to cyclic shear and showed the existence of a transition from reversible to irreversible dynamics at a critical strain amplitude.Priezjev 71 demonstrated that, at a finite temperature, there is a transition from an almost periodic, subdiffusive regime, to a diffusive regime.Fiocco et al. 15 and Regev et al. 16 used athermal quasi-static simulations to show that below the critical point the dynamics is exactly periodic and particles repeat the same positions after each cycle.Regev et al. 16 found that the number of cycles needed to reach a limit-cycle diverges at this point.In their study, Fiocco et al. 15 also showed that the post-yield dynamics involves a loss of memory of the initial configuration.
Figure 7(c) shows the potential energy as a function of strain cycles for increasing strain amplitudes.At the lowest amplitude, on the top portion of the panel, the system is initially in an irreversible state and settles after a short transient into a reversible state, as indicated by the transitions from a fluctuating non-repeating signal to a periodic signal.As the strain amplitude increases, it takes longer for the system to reach a reversible state.Remarkably, even in the reversible states the system shows large scale reversible plastic deformations.Figure 8 shows an example of a reversible plastic avalanche event for the system from Fig. 7 32 .Figure 7(d) illustrates the number of cycles required to reach a reversible state as a function of strain amplitude along with the stress-strain curve.At yielding, marked by the red vertical line, there is a divergence in the time needed to reach the reversible state.This work also showed that there is a power-law divergence in the time scale to reach the reversible state as a function of strain amplitude; however, the observed critical exponent ν ≈ 2.6 differs from the value ν ≈ 1.33 obtained by Corte et al. 13 , suggesting that the transition is in a different universality class.There are several possible reasons why this might be the case.For a reversible state in dilute systems, there are no collective effects since the particle-particle contacts van- ish, whereas in the amorphous system, the reversible plastic events indicate that there is strong coupling among the particles, implying that there may be a dynamical length scale present and that the microscopics of the reversible state are different in the amorphous and dilute systems.Another feature in the work of Regev et al. 16 is that the divergence in the reorganization time scale correlates well with the point at which a yielding transition appears, so that below yielding, the oscillating drive creates a reversible plastic state, while above yielding, the system can never reach a reversible state.
As shown in Fig. 9 and Ref. 16 , the reversible state does not have to recur during each drive cycle, but may instead recur after multiple driving cycles.For example, the pattern might repeat after two cycles, and in fact limit cycles of up to seven or more cycles have been observed 16,34,35 .Multiple other studies found that periodically sheared amorphous systems can organize to reversible states with a varied number of limit cycles.This behavior is reminiscent of the routes into chaotic states that arise in low dimensional systems 10,11,72 .Keim et al. 34 showed that the multi-periodicity can be explained as resulting from interactions between soft-spots such as the ones shown in Fig. 10, while Szulc et al. 73 explained how oscillations in the activation thresholds of the soft-spots cause multiperiodicity.
There have also been a number of experiments on transitions into reversible plastic states in amorphous jammed systems.Keim et al. found that jammed systems can organize into reversible plastic states 18 .An example of this appears in Fig. 11, which highlights a detail of how the system organizes to a state with reversible plastic events.Nagamanasa et al. 74 also found a power law diverging time scale for the transition to an irreversible state in a binary colloidal glass.
In several works [75][76][77][78][79] , the effect of sample preparation on the potential energy in the steady-state was considered.For sub-critical amplitudes, the steady-state potential energy of samples with different initial mean energies depends on the initial conditions, but for post-critical initial conditions, the steady-state potential energy is independent of the preparation protocol.These studies also showed that for post-critical amplitudes, the potential energy increases up to saturation.These observations were explained using models of dynamics on random energy landscapes 78,80,81 .Studies on "ultra-stable" glasses, or amorphous samples prepared using special quenching protocols, have shown that in these samples the transition from reversible to irreversible dynamics are abrupt and occur at critical amplitudes that depend on the preparation protocol 79,82 .This is contrary to samples prepared using "standard" quenching protocols where the critical amplitude does not depend on sample preparation.Other studies have shown that within the reversible state, although the system remains solid it becomes a softer solid, particularly when the reversible trajectories form loops 83 .Efforts to model the R-IR transition in amorphous solids subject to cyclic shear have so far focused on using integer automata models that represent an amorphous solid as a lattice of soft-spots interacting by elastic interactions.Using such a model, Khirallah et al. 23 found that such systems indeed exhibit a transition at a critical amplitude from asymptotically periodic, where the system repeats after n forcing cycles, to asymptotically diffusive dynamics.They also found that, similar to what is observed in particle simulations, the transition between the reversible and irreversible phases is marked by a power law divergence in the number of cycles required to reach a reversible state.The exponent observed in this case was ν ≈ 2.7, consistent with the exponents found by Regev et al. 16 and lending further support to the idea that the universality class of R-IR transitions in amorphous solids differs from that of the dilute systems.Khirallah et al. also observed that similar to the observations from particle simulations 15,84 , the diffusion coefficient on the irreversible side of the transition increases as a power law in the magnitude of the strain amplitude, and showed that there are still a large number of reversible plastic events that occur within the irreversible state.This work suggests that cellular automata models, which are computationally faster than particle simulations and are easier to model theoretically, can capture many of the relevant behaviors at R-IR transitions, and that similar reversible plastic to irreversible diffusive phases could arise in other types of cellular automata models.
There are several possible avenues for continued research in the study of amorphous solids under cyclic shear.First, there are a variety of different amorphous systems in which R-IR transitions were not yet studied such as polymer glasses, metallic glasses, gels, and emulsions.It is not clear if such systems will show the same transition and, if so, whether the transition would be of the same character.Second, the theoretical under- standing of the transition in amorphous solids remains undeveloped.Although it is clear that in dilute systems, reversible dynamics arises due to the emergence of states where the particles are spaced in such a way that they are not interacting, it is not clear why in some amorphous configurations the interactions between soft-spots lead to irreversible dynamics whereas in others the same interactions allow for periodic states.Third, it would be interesting to study how the R-IR transition varies for different kinds of particles.For example, particles in densely packed emulsions can undergo a variety of shape changes under compression or have very different types of elastic properties compared to hard particles 85 .It may be that the ability of individual particles to distort would introduce another form of dissipation that could increase the range of reversible behavior; however, such distortions could also inject additional degrees of freedom, giving more possible ways for the packing to change and promoting irreversible motion.
Beyond disordered solids, R-IR transitions have also been studied in polycrystals 19,36 and point dislocation models 83 .There have been only a few studies of R-IR transitions in crystalline systems 36 , but there are a variety of effects that could be studied in ordered states, such as the motion of grain boundaries or of isolated defects such as dislocation lines and disclinations.In crystalline systems, application of periodic driving could generate defects, leading over time to work hardening and eventual failure; however, there could be regimes in which the system reaches a steady reversible state under cyclic driving.This could be tested for crystalline systems found in soft matter, materials science, and certain hard condensed matter systems such as superconducting vortices or Wigner crystals.In Fig. 12, dislocations in a 2D colloidal assembly are manipulated using local stresses, shears, and dilatations 86 .One could consider applying local or global periodic driving to such a system in order to determine whether the motion of the individual defects illustrated in Fig. 12(a,b,c,d) or of the grain boundary illustrated in Fig. 12(e) is reversible.

C. Magnetic Systems
Hysteresis is frequently observed in condensed matter and materials science, and the best known example is in magnetism where cycling an applied field generates a hysteretic magnetic response in the material 87 .Hysteresis in magnetic systems is a result of disorder and exchange interaction between the spins of different atoms.In the case of ferromagnetic interactions, disorder can take the form of a local random field, as in the random field Ising model (RFIM) which has the Hamiltonian 88 : where s i = {−1, 1} is the direction of the ith spin, J > 0 is a constant ferromagnetic coupling constant, h i is the random field and h is an externally applied field.Alternatively, there can be randomness in the effective spin-spin interactions, as captured by the Edwards-Anderson (EA) spin-glass model 89 : In this case the coupling constant J ij is a random variable that can be both positive and negative, leading to geometric frustration.Models with ferromagnetic interactions where J > 0 always reach a limit cycle after a transient of two cycles or less due to the "no-passing" property first proved by Middleton 88,90 .For this reason, such systems cannot have a R-IR transition.Models with couplings that can be both positive or negative, such as a spin-glass, can have long transients and thus, in principle, can have both reversible and irreversible behavior.In the case of the EA spin-glass and related systems, each spin has only two states, which may hinder the emergence of completely irreversible dynamics.At the thermodynamic limit, however, the transients may become infinitely long, and the system can then become effectively irreversible.Basak et al. 41 considered cyclic driving of uniaxial random field XY models with disorder.They found that for increasing field amplitude, the system goes from an Ising ferromagnetic state to a paramagnetic state that does not repeat, as shown in Fig. 13.In the reversible case, the spin patterns repeat after n cycles, and the number of cycles to reach a reversible state increases as the critical point is approached.The plot of the y and x magnetizations m y versus m x in Fig. 13(c) indicates that there is an initial transient before the system settles into a period-2 limit cycle.This implies that the magnetic system can organize into a repeatable pattern spanning one or multiple periods, similar to what was found in the amorphous solids discussed previously.
Uniaxial random field XY models can be applied to many systems, including Josephson junctions 91 , superfluids in a uniaxially stressed aerogel 92 , uniaxially stressed 2D Wigner crystals 93 , the half-integer quantum Hall effect, and the graphene quantum Hall ferromagnet 94 .Electron nematics 95,96 are also promising.Each of these are candidate systems in which to look for R-IR transitions under periodic driving.

IV. FUTURE DIRECTIONS IN CONDENSED SYSTEMS A. Commensurate-Incommensurate Systems
Another class of systems that are good candidates for examining R-IR transitions is that in which commensurate-incommensurate transitions can occur [97][98][99][100] .These systems can be described in terms of interacting particles on a periodic substrate, where the ratio of the number of particles N to the number of substrate minima N s is given by the filling ratio f = N/N s .For example, Fig. 14(a) shows a schematic of charged colloidal particles interacting with a 2D periodic egg-carton substrate at a filling close to f = 1 101 .Commensurate conditions can arise for fractional matching when f = n/m with integer n and   cific systems include atomic ordering on surfaces 97,98 , cold atoms in optical traps 102 , vortices in nanostructured superconductors 103,104 , vortices in Bose-Einstein condensates 105 , and colloidal systems 101,[106][107][108] .In quantum systems, there can be transitions from commensurate Mott phases to incommensurate superfluids 109,110 .For most of the systems listed above, the particles are not strictly elastically coupled but can undergo exchanges with one another, permitting plastic deformations, defect generation, and phase slips to occur.
In typical commensurate-incommensurate systems, the particles are largely localized under commensurate conditions, while for slightly incommensurate states there are kinks or anti-kinks that are more mobile than the particles and that depin first under an external drive 100 .At more strongly incommensurate states, the system becomes increasingly disordered and can exhibit glass-like behavior.Since the amount of order can be tuned by changing the filling, it would be possible to examine R-IR transitions for varied fillings.For example, under commensurate conditions the system may be strongly reversible for a range of drives, whereas incommensurate fillings might either organize to a reversible state similar to what is found in the sheared systems or remain in an irreversible state.There could be a critical drive amplitude marking the R-IR transition, or there could be fillings for which the system becomes fluid-like and is always irreversible.Whether the response of a commensurateincommensurate system is reversible or irreversible may also depend on the type of drive applied.In addition to their relevance to a wide class of systems, another advantage of studying commensurate-incommensurate systems is that a variety of distinct types of defect structures can be realized, ranging from individual solitons or kinks and domain walls to strongly amorphous phases.To illus-trate why commensurate-incommensurate systems could show different reversible or irreversible behaviors, Fig. 15 shows that a wide range of defect morphologies emerge for colloids on periodic substrates at different incommensurate fillings 111 .It would be interesting to understand whether the grain boundaries and isolated defects that emerge in a system such as this one would exhibit reversible or irreversible dynamics under periodic driving, and whether a possible R-IR transition would resemble that found in dilute or amorphous systems, or whether it would fall into an entirely new universality class that has not yet been observed for R-IR transitions.Such a study would relate to the broader question of the nature of R-IR transitions in interacting soliton systems.

B. Local Driving
In the systems discussed up to this point, the driving is applied globally via shearing, a current, or a field; however, it is also possible to subject a system to local driving.In soft matter, such driving is referred to as active rheology, where a single particle is dragged through a background medium in order to create measurable perturbations [112][113][114][115][116] .Studies of this type have been used to examine changes in the drag 114,116 and fluctuations 112,115,117 , as well as to determine whether there is a threshold force needed for motion of the probe particle 112,113,118 as the system passes through different types of glassy or jamming transitions.Figure 16(a) shows an example of a single probe particle driven through a glassy background of bidisperse colloidal particles, and the corresponding velocity time series contains a series of jumps that indicate the occurrence of plastic events in the medium surrounding the probe particle 112 .Similar active rheology techniques have also been used to study plastic rearrangements in crystalline systems 119,120 .
In hard condensed matter systems, local driving can be achieved using a magnetic force tip or scanning tunneling microscope for superconducting vortices 121,122 or an optical probe for both superconducting vortices 123 and magnetic skyrmions 124 .Other possible methods include injecting a current at a single localized spot or applying time dependent inhomogeneous fields.Most studies of active rheology have used dc driving, but ac driving could also be applied.Local driving has been applied to plastic to elastic transitions as well as situations where the probe particle particle creates local plastic deformations in the system.Similar techniques could be used to explore local R-IR.For example, if the probe particle is periodically driven at different driving amplitudes, as illustrated schematically in Fig. 17, the system could organize to an elastic reversible state with no plastic deformation, to a state with reversible plastic events or loop reversibility, or to a state that is continuously fluid-like or irreversible.It would be possible to measure whether a reversible or irreversible interface extends out from the FIG.18. Example of clump (left), stripe (center), and void (right) patterns that form for particles with competing long range repulsion and long range attraction from a system such as that described in Ref. 126 .region containing the driven particle, or if there is a fluctuating or continuous border surrounding the local probe region.To test for memory effects, the ac drive could be applied in one direction until the system reaches a reversible state, and could then be rotated into a new direction to see whether the system still remains reversible.Such an approach would also be useful for understanding how the dissipation or drag is affected when the system transitions from an irreversible to a reversible state.In work examining sedimenting colloids under a periodic shear 125 , the presence of a density gradient made it possible to determine that the denser regions are irreversible while the less dense regions are reversible, and that the reversible and irreversible regions are separated by a well defined coexistence front.

C. Complex Interactions
Up until this point, R-IR transitions have been studied only for systems with relatively simple short range interactions.There are, however, numerous systems in which more complex interactions appear, such as long range repulsion and short range attraction, frustrated interactions, and interactions with multiple length scales.When these types of complex interactions are present, pattern formation or phase separation often occurs, as found in many soft matter systems [126][127][128][129] as well as in hard condensed matter systems such as electron liquid crystals [130][131][132] , doped Mott insulators 133 , charge ordering in Jahn-Teller systems 134 and magnetic domains 135 .In Fig. 18 we show some examples of pattern formation in a system with multiple length scale interactions, which cause both short range crystalline order and larger scale ordering to occur.It would be interesting to test whether systems of this type could exhibit R-IR transitions under periodic driving.For soft matter systems, this could be achieved through shearing, while for hard condensed matter systems such as electron liquid crystals, the driving could take the form of an oscillating current 131,132 .Other systems with long range interactions where some form of periodic driving can be imposed include dusty plasmas or charged dust particles that form microscale crystals 136,137 .Such systems could be periodically sheared or driven with other methods 138 .
Due to the multiple length scales that are present in the interaction, it is possible that individual particles might undergo irreversible deformations even as the larger scale pattern remains reversible, producing a situation with microscopic irreversibility but macroscopic reversibility.If this were the case, there might be multiple R-IR transitions as well as the possibility for complex memory formation.

D. Quasi-Reversibility and Universality Classes
The systems described so far are in either a reversible or an irreversible state during one or multiple drive cycles.Another possibility is that even in the absence of long time diffusion, the system could follow one of several available loop paths, where on any given cycle the particular path that is chosen is selected randomly.This could occur if the system shows only a local ergodicity or if there are regions where the particles can locally explore many different possible states but local fluctuation paths remain confined, so that the particles are unable to diffuse over long times.There could also be glassy reversibility.In many of the systems described so far, the irreversible states were characterized by observing diffusive behavior of the particles; however, there could be regimes of very long time diffusion in which the system is trapped in one cyclic orbit for long times before making a rare jump to a new orbit.Glassy behaviors of this type could arise when thermal fluctuations become important.For example, the system could be in a reversible regime for T = 0, but become irreversible at finite T when it can occasionally thermally hop to different periodic orbits.Finally, there could be orbits that are quasi-periodic in time, in analogy to orbits that are quasi-periodic in space.
Another possibility is smectic-like irreversibility in which there is long time diffusion in one direction but not in the perpendicular direction.In driven systems with quenched disorder, moving smectic states have been observed in which the diffusion is finite in one direction but not in the other 100 .For colloids or amorphous solids under a periodic drive, 1D reversibility of this type could occur for anisotropic particles or layered systems.There could then be two irreversible transitions that occur separately for each direction.It would be interesting to understand whether there are only a few types of distinct universality classes for R-IR transitions, or whether multiple universality classes could be observed by changing the nature of the particle-particle interactions or the driving.

V. VORTICES, SKYRMIONS, AND OTHER HARD CONDENSED MATTER SYSTEMS
There are many other systems containing assemblies of particles that can be periodically driven.These in- clude cases in which the particles are coupled to random or periodic substrates 2,100 , as found for sliding friction 139 , vortices in type-II superconductors 140,141 , sliding charge density waves 142,143 , colloids 144 , pattern forming systems 129 , active matter 145 , election liquid crystals 146 , and Wigner solids 147 .The driving can be applied uniformly using an applied current, an electric field, or a magnetic field.If there is no quenched disorder present and the drive is uniform, the particles simply translate back and forth; however, when disorder is present, certain particles can become trapped while other particles move past them, leading to plastic events.Under an increasing dc drive, these systems typically exhibit a pinned phase, a plastically deforming phase, and a moving crystal or moving smectic state.If the disorder is weak, the system depins elastically without the creation of topological defects.
Mangan et al. 14 performed numerical simulations of superconducting vortices in a 2D system with random disorder and ac driving where the driving period, vortex density, and pinning strength are varied.Figure 19(a) shows the vortex positions at the end of each cycle after 25 cycles under a drive for which each vortex moves a distance of 45λ during a single cycle, where λ is the London penetration depth.Here the system is in a reversible regime.In Fig. 19(b), when the driving amplitude is increased so that each vortex moves a distance of 160λ during every cycle, the vortices do not return to the same positions and the system is irreversible.For the reversible state, the diffusion is zero and the time series of the average vortex velocity repeats the same pattern during every drive cycle, which would produce a periodic voltage signal if measured experimentally.In the irreversible regime, the vortices exhibit a finite diffusion and the voltage signal is chaotic.Okuma et al. 37 studied R-IR transitions experimentally by shearing supercon- ducting vortices in a Corbino geometry, and mapped out the phase diagram for the transition from reversible to irreversible flow as a function of vortex displacement per cycle.
Additional work has since been performed 24 to more clearly show that superconducting vortices can also exhibit point reversible states similar to the random organization found for dilute colloids, loop reversible states similar to those observed in amorphous systems, and irreversible states, as illustrated in Fig. 20.At low densities where the vortices are far apart, the behavior resembles that of a dilute colloidal system, and point reversible states emerge, while at higher density where interactions between the vortices become important, the system shows loop reversibility.For large densities and large driving amplitudes, the system is irreversible.Ref. 24 also showed that there is a diverging time scale for the system to settle into a reversible or irreversible state.The power law divergence reported in Fig. 4 of Ref. 24 is similar to that found in the random organization system and has an exponent ν = 1.33 that is close to the value expected for directed percolation 29 .Even though the vortices can exhibit collective effects due to their longer range interactions, the exponents observed are consistent with the dilute colloidal system rather than with amorphous solids, where exponents closer to ν = 2.6 are found.This could be due to the nature of the plastic events that occur in the different systems.There has also been work showing evidence for R-IR transitions in ac driven superconducting vortices in linear geometries 148 .
Magnetic skyrmions are another particle-like magnetic texture that have many similarities to superconducting vortices in that they can be driven by an applied current 149,150 .An important distinctive feature is that skyrmions have a strong gyrotropic or Magnus force.Simulations of skyrmions under ac drives with quenched disorder 40 showed an R-IR associated with a diverging time scale where the exponent is close to ν = 1.29.In this system, the Magnus term enhances the irreversible behavior by increasing the number of dynamically accessible orbits.This is in contrast to the behavior of an overdamped system, such as strongly damped skyrmions and vortices in type-II superconductors.It has been shown that in the overdamped limit, there appear to be two different R-IR transitions, since the diffusion constant first drops to zero in the direction perpendicular to the drive, followed by a regime in which the behavior is reversible both parallel and perpendicular to the drive.This suggests that the number of possible R-IR transitions may depend on the effective dimensionality of the system.Experimental results that seem consistent with the predictions of Ref. 40 were found in the vortex system 151 .

VI. FUTURE DIRECTIONS FOR SYSTEMS WITH QUENCHED DISORDER
R-IR transitions under periodic driving could be explored in charge density wave systems 142,143 , which often exhibit narrow band noise or temporal ordering that may be the hallmarks of reversible behavior.It would be interesting to understand if there is also an irreversible regime and if there is a diverging time scale for reaching the reversible regime.This is of particular relevance since recent work on classical time crystals has suggested that sliding charge density waves under ac driving are examples of discrete time crystals 152 , which we discuss further in a later section.Other elastic systems that can be driven periodically include magnetic domain walls 153 , moving contact lines 154 , depinning of interfaces 155 , and slider block models 2 .Examples of possible materials science systems are ac driven grain boundaries 156 , twin planes 157 , and dislocation patterns 158,159 , as illustrated in the schematic of Fig. 21.It is possible that systems of this type would always organize to a reversible state; however, there could be different kinds of reversible states separated by transitions with diverging time scales.For example, there could be a critical point separating point reversible states and loop reversible states.

A. Systems With Avalanches and Noise
Further directions include studying systems with hysteresis that exhibit repeating avalanches to seek diverging scales under repeating hysteretic cycles.Figure 22 illustrates experimentally measured avalanches across the metal-insulator transition of VO 2 160 .In systems of this type, it would be possible to perform repeated cycles to determine whether the same resistance values recur, and whether there is a finite number of cycles that must be performed before the system reaches a repeating pattern.
In many condensed matter systems, noise is generated under the application of a current or field.This noise can take the form of avalanches or crackling noise 161,162 , telegraph noise 163 , switching 164 , narrow band 165,166 , or broad band noise 166,167 , and can be characterized by using the power spectrum or second spectrum 162 .Under applied periodic driving, it would be interesting to explore whether the system exhibits the exact same noise pattern, and if so, whether a finite number of cycles must be performed before the system reaches a reversible state.

B. Astrophysical Systems
Another set of systems that can be considered to have periodic driving and that also exhibits avalanches or bursts are pulsar glitches in neutron stars [168][169][170][171] and brightness variability for certain stars 172 .Here the periodic driving arises from the rotation of the stars.When a pulsar glitch occurs, there is a shift δν in the rotation frequency ν of the pulsar.In general, glitching pulsars fall into two classes: those with Poisson-like waiting times [168][169][170] , and those with unimodal or quasiperiodic waiting times 170,171 .Figure 23 shows some example time series of pulsar glitches exhibiting different levels of periodicity.It is possible that the glitches could be a sign that over time the star is organizing to a more periodic state, and if so, a reversible state could emerge in which there is a power law distribution of periodic waiting times.It would be interesting to understand whether there could be a transition from chaotic glitch intervals to periodic or quasiperiodic glitches, and whether a diverging time scale appears in these systems.Sheikh et al. 172 measured avalanches in the brightness variability of stars and found scaling exponents that suggest the system may be near a nonequilibrium critical point.A system of this type might be irreversible on one side of the nonequilibrium point and reversible on the other side.

VII. MEMORY EFFECTS
Sheared colloidal systems can organize to a reversible state when the applied strain γ i satisfies γ i < γ c , where γ c is the critical stain below which the system is always reversible.If the strain amplitude is changed so that γ i > γ c , reversible behavior cannot appear.Paulsen et al. 173 considered a colloidal random organization model in which a memory effect emerges for strains smaller than γ c .In Figure 24(a), a numerical study 174 of a system with a critical strain level of γ c = 4.0.showed that when a training shear pulse amplitude of γ 1 = 3.0 is applied, a "reading" of the system with a trial strain level of γ produces a signature or memory of γ 1 in the form of a cusp.It is also possible to store the values of two distinct strain levels, as shown in Fig. 24(b) for γ 2 = 2 and γ 2 = 3.0, where there are now two kinks that appear as the trial strain level is varied.This implies that multiple memories can be storied in these systems as long as the strain amplitude remains below the critical stain where the system becomes irreversible and loses all memory.Fiocco et al. 33 studied a protocol similar to that used by Keim et al. 174 but employed an amorphous solid in which the particles always remain in contact.They find that the same memory effect shown in Fig. 24(a) can be achieved by using a training shear amplitude.
Mungan et al. 175 showed that the configurations of an amorphous solid sheared along a fixed plane can be represented by a directed graph where nodes represent stable particle configurations and arrows represent transitions between them due to plastic events.Figure 25 shows an example of such a network where all the transitions are reversible.In each configuration a soft spot is switched on or off.Regev et al. 176 further showed that the strongly connected components of the graphs, representing clusters of configurations where every two configurations (nodes) are reachable by a path of plastic deformation, include all the possible limit-cycles of the system.Since transitions between different strongly connected components are irreversible, as the strain magnitude increases to a level close to the R-IR transition, irreversible transitions become more frequent and the strongly connected components become smaller and can contain only small limit-cycles, such as the yellow arrows in Fig. 25.Some open questions include whether intro- ducing a change in the direction of shearing or driving in this regime could reduce the persistence of the memory.Although memory has been studied in sheared colloids and sheared amorphous solids, less is known about whether similar memory effects can occur in other systems.For example, Dobroka et al. 177 found that kinks similar to those found by Paulsen et al. 173 can occur for periodically driven superconducting vortices.
Memory can also be associated with the hysteresis that appears in magnets, semiconductors 178 , and metalinsulator systems 179 under oscillating voltage, and at dynamic transitions in magnetic systems subject to fast oscillations where the hysteresis is with respect to dynamic variables 180 .For example, in the phenomenon of return point memory 181,182 , spin configurations are examined at the end of each minor loop cycle to see if the configurations fully overlap from cycle to cycle according to an overall function.In periodically driven colloidal spin ice systems 183 , the spin overlap function q was examined for repeated minor loops.A value q = 1.0 means that exactly the same effective spin configuration appears during each cycle, indicating reversible behavior.In Ref. 183 , the system did not adopt the same spin configurations for every cycle, but as shown in Fig. 26, q increases with time until saturating at a reversible state after a fixed number of cycles, similar to the random organization observed for dilute colloids.In addition, including quenched disorder increased the memory of the system.

VIII. FUTURE DIRECTIONS FOR TIME CRYSTALS AND QUANTUM SYSTEMS
Another promising system in which to seek R-IR transitions is time crystals, which have been proposed for both quantum 184 and classical systems 185 .In a time crystal, the lowest energy state of the system is periodic not only in space but also in time.Shortly after the initial proposals, it was recognized that true time crystals as originally envisioned cannot occur under conditions of strict equilibrium 186,187 ; however, the time crystal concept has now generated a wealth of ideas for creating time periodic systems that could arise under nonequilibrium conditions 42,188 , such as periodic driving.A key feature of a time crystal is that the system can exhibit subharmonics of the oscillatory driving [188][189][190] (multi-periodic response).Yao et al. 152 have noted that there are a number of classical systems that also show subharmonic entrainment, including Faraday waves 45 and predator-prey models 191 , phase locking in driven charge density wave systems 192 or Josephson junctions 193 , and superconducting vortex 194 or magnetic skyrmion 195 motion on periodic pinning arrays, meaning that these systems could be examples of Classical Discrete Time Crystals (CDTCs).In quantum systems, time crystals have been studied by measuring subharmonic entrainment in unitary manybody systems or Floquet systems 190 .In some cases, time crystals can arise in quantum systems where many-body localization can prevent thermalization 42,188 .Up until now, work on time crystals has focused on identifying systems that support time crystals; however, the dynamics of how a system can organize into a time crystal state or the general phase diagram near the boundary from chaotic to time crystal behavior has many similarities to the R-IR transitions discussed above.
An example of a classical time crystal that appears close to systems in which R-IR transitions can happen appears in the work of Yao et al. 152 , where a onedimensional array of coupled oscillators was studied at finite temperature.Figure 27 shows a stroboscopic view of the position coordinate q j of each oscillator under periodic driving as a function of time.In the initial state, the system is disordered or fluctuating, but over time it organizes into an antiferromagnetic repeating state.This is reminiscent of an initially irreversible state organizing over time into a reversible state.The reversibility does not have to appear after a single driving cycle; instead, the system can repeat after two or more driving cycles.It would be interesting to see whether there is a time scale for the organization into the time periodic or CDTC state, and if so, whether this time diverges as a function of the driving parameters, similar to what is found for the R-IR transition in colloidal systems.The work of Yao et al. 152 can also be applied to the much broader class of systems in higher dimensions and at zero temperature, such as systems that exhibit phase locking effects where nonlinearities give rise to higher order harmonic phase locking.There could be a variety of other systems on periodic flashing substrates that could organize to reversible states over time, such as choreographic colloidal time crystals that have a liquid-like state and a time ordered state 196 .
As discussed in a previous section, the reversible states can exhibit memory and training effects, so similar phenomena along with memory encoding studies could be explored in time crystal systems, where much more complicated periodic driving protocols can be employed.For example, the system might organize to a CDTC for one set of driving frequencies, but if additional frequencies were added, the system could organize into a new time crystal state.The question would be whether, if the system is first trained with the initial set of frequencies, these frequencies could be retained in the time crystal state formed with additional driving frequencies.Although we have focused on classical time crystals, similar R-IR transitions could arise in periodically driven quantum time crystals, such as periodically driven trapped atomic ions 197 , driven spins in diamond 198 , and arrays of superconducting qubits 199 .

IX. COMPLEX NETWORKS AND DYNAMICAL SYSTEMS
In general, there are many other coupled systems that can organize into synchronized states.Systems that exhibit ergodicity breaking would also be candidates for study in the framework of R-IR transitions.For example, dynamical many-body coupled oscillators are known to show transitions to synchronized states 200 .Other systems include particles coupled to or flowing on a complex network 201 under periodic driving.In this case, the particles could form repeating loop paths on the network in the reversible state, such as those illustrated in Fig. 10(c).Such systems could include those for which particles or carriers of information cannot pass though each other and the network is rigid.Other systems such as those with nonreciprocal interactions 202 or odd viscosity 203 show transitions from chaotic to ordered edge states, and could organize to patterns containing ordered loop currents or flocking.
Complex networks and dynamical systems exhibiting synchronization arise across many biological 204,205 , robotic 206 , social 207 , and economic systems 208 .It is known that these systems can all enter chaotic or strongly fluctuating states; however, there could be situations in which, when some type of periodic driving is applied, time repeating states could emerge similar to the reversible states we have discussed above.Here, similar types of memory formation could arise, or there could be some form of power law divergence in a time scale near a critical point.

X. SUMMARY
We have given an overview of the recent work examining transitions from chaotic irreversible states to time periodic or reversible states for systems under periodic driving.In these systems, the locations of the particles are compared to their positions on the previous driving cycle.For irreversible states, the particles do not return to the same positions, and over multiple cycles they exhibit diffusive motion away from their initial positions.In a reversible state, the particles return to the same positions after one or an integer number of driving cycles, and there is no long time diffusion.Reversibleirreversible (R-IR) transitions were initially studied for periodically sheared dilute colloids, in which a process termed random organization produces reversible states where particle-particle collisions no longer occur, while the irreversible states have continuous collisions.R-IR transitions have also been studied in strongly interacting systems such as amorphous solids and jammed systems.Another hallmark of the R-IR transition is that there is a critical drive or density above which the system remains in an irreversible state, and there is a power law divergence on either side of the transition for the time required for the system to settle into a steady irreversible state or organize into reversible motion.In dilute systems, the R-IR transition is consistent with directed percolation.For amorphous solids, similar R-IR behavior is found but the transition appears to fall into a different universality class, and for dense systems, the critical amplitude coincides with the yielding transition.On the reversible side of the R-IR transition it is possible to store a series of memories by applying training pulses.Such memory effects have been observed in both dilute and dense systems.R-IR transitions have also been studied in solid state systems such as periodically driven superconducting vortices and magnetic skyrmions, both of which show similar behavior to that found in dilute sheared colloidal systems.
We highlight how the general features of the irreversible to reversible dynamics transition could be applied to much broader classes of soft matter, hard matter and dynamical systems.For example, many solid state systems show hysteretic behavior, and systems of this type would be fertile ground for studying transitions to reversible states under repeated cycling.The impact of repeatable noise, avalanches, or return point memory could be studied, as well as the number of cycles required to reach the reversible state, which could show critical behavior similar to that found in colloidal systems.Such studies could be performed for magnetic systems, metalinsulator transitions, charge ordering systems, and semiconductors.Other classes of system in which to look for R-IR transitions include commensurate-incommensurate systems, frustrated systems, crumpled and corrugated sheets 209,210 , and even active matter systems.These systems may also be able to store more complex memo-ries.For example, amorphous solids or colloids could be sheared with a periodic but increasingly complex protocol to see if the system can still reach a reversible state.We also discuss some other possible states that are not time periodic but for which there is no long time diffusion, so that quasiperiodic dynamics can be explored, as well as systems that do not follow the same path on each cycle but can still only trace out a limited number of cycles, which could arise for frustrated states.
We discuss how the R-IR framework could be applied to classical and quantum discrete time crystals where the transitions to time periodic or harmonically entrapped states could be an example of an R-IR transition.Power law divergences in time could appear for the formation of time crystal states.Other systems to consider include more general cyclic systems such as pulsars, coupled oscillators, social systems, economics, biological systems, and particle flow dynamics on complex networks where the reversible state could arise through the formation of loop currents.

FIG. 2 .
FIG. 2. (a) Particle trajectories measured experimentally for the sheared colloid system in Ref. 12 in the irreversible or chaotic regime.(b) The corresponding mean square particle displacements in the direction parallel, x 2 (filled squares), and perpendicular, z 2 (open squares), to the drive, showing diffusive motion.Reprinted by permission from: Springer Nature, D. J. Pine et al., "Chaos and threshold for irreversibility in sheared suspensions", Nature (London) 438, 997 (2005).

FIG. 3 .
FIG. 3. (a,b) Snapshots from simulations in Ref. 13 of a hard disk model of the experiments in Fig. 2 as a function of time for different strain amplitudes γ0.Black disks are moving irreversibly between cycles and open disks are returning to the same position between cycles.(a) γ0 = 3 in the irreversible regime.(b) γ0 = 2 in the reversible regime.(c) For the same system, the fraction of particles that are active (moving irreversibly) in each cycle as a function of the number of shear cycles applied at γ0 = 3 (red) and γ0 = 2 (blue).The inset shows the steady state fraction of active particles as a function of γ0.Reprinted by permission from: Springer Nature, L. Corté et al., "Random organization in periodically driven systems", Nature Phys.4, 420 (2008).

FIG. 4 .
FIG. 4. For the simulation of sheared 2D disks from Fig. 3, a plot of the time τ to reach a steady state versus shear amplitude γ0 for the reversible (blue diamonds) and irreversible (red squares) regimes.The inset shows the same data as a power law plot of τ versus |γ0 − γ c 0 |, where the critical shear amplitude is γ c 0 = 2.66 ± 0.05.The lines indicate fits of the data to τ ∝ |γ0 − γ c 0 | ν with ν = 1.33 ± 0.02.Reprinted by permission from: Springer Nature, L. Corté et al., "Random organization in periodically driven systems", Nature Phys.4, 420 (2008).

FIG. 5 .
FIG. 5. Simulations of cyclically driven disks in a periodic array of obstacles from Ref. 55 .(a) The number Rn of active or irreversibly moving disks versus cycle number n for different disk densities ranging from φ = 0.335 (bottom) to φ = 0.3962 (top) under shearing with an amplitude of A = 0.031623 at an angle of θ = 18.435 • from the x-axis symmetry direction of the obstacle array.(b,c) Images of the disk locations (blue) and obstacles (red) in a portion of the sample for the same driving amplitude and direction as in panel (a).(b) An irreversible state at φ = 0.3962.(c) A reversible state at φ = 0.3716.Reprinted from C. Reichhardt and C.J.O.Reichhardt, J. Chem.Phys.156, 124901 (2022) with the permission of AIP Publishing.

FIG. 7 .
FIG. 7. (a) The energy as a function of simulation steps for an athermal quasi-static simulation of an amorphous solid.The discontinuous drops in the energy occur due to plastic rearrangements.Reprinted under CC license from I. Regev et al., Nature Commun.6, 8805 (2015).(b) An illustration of particle motion in the most fundamental plastic rearrangement event, the soft-spot.(c) Three different potential energy time series for three different maximal strain amplitudes, which increase from top to bottom.The red lines mark the onset of repetitive behavior or the formation of a limit cycle.(d) Stress-strain curve under linear shear (green line).The red vertical line marks the R-IR transition, while the red data points indicate the number of cycles required to reach a limit cycle under oscillatory shear.The inset shows a similar stress-strain curve obtained for different initial conditions.Reprinted with permission from I. Regev et al., Phys.Rev. E 88, 062401 (2013).Copyright by the American Physical Society.
FIG. 8. (a) A repeating avalanche in the reversible state of the system from Fig. 7. Arrows and colors indicate the direction and magnitude, respectively, of the displacements during the avalanche motion.Reprinted under CC license from I. Regev et al., Nature Commun.6, 8805 (2015).(b) The potential energy E per particle in the steady state under zero strain (γ = 0) for different values of maximum shear strain γmax increasing from bottom to top under temperatures T = 1.0 (open symbols) and T = 0.466 (closed symbols).Reprinted with permission from D. Fiocco et al., Phys.Rev. E 88, 020301 (2013).Copyright by the American Physical Society.

FIG. 9 .
FIG. 9. Particle trajectories in an amorphous solid following a multi-periodic limit-cycle.(a) A system of 4096 particles subject to periodic shear.(b) A blow-up showing individual particles and the trajectories performed by their centers marked in blue and green, where blue represents the first cycle and green represents the second cycle.(c) A blow-up showing the trajectory of a single particle.During the first cycle the particle performs the blue trajectory, followed by the green trajectory during the second cycle.(d) The strain as a function of simulation steps (quasistatic equivalent of time) in the cycle.Reprinted under CC license from I. Regev et al., Nature Commun.6, 8805 (2015).

FIG. 11 .
FIG. 11.Experimental measurements of local plastic deformation in a 2D jammed system.(a,b) Reversible (blue) and irreversible (red) motion after (a) 8 shear cycles and (b) 20 shear cycles showing that the system settles into a reversible state.(c) Detailed view of the residual displacement for the reversible cluster outlined with a red box in panel (b).(d) Relative maximum displacements of the particles in this cluster at the point of minimum shear (red) and at maximum shear (open blue circles).(e) The corresponding flow streamlines.(f, g, h) Images showing the details of a plastic rearrangement event in the same region.Reprinted with permission from N. C. Keim and P. E. Arratia, Phys.Rev. Lett.112, 028302 (2014).Copyright by the American Physical Society.

FIG. 12 .
FIG. 12. Motion of dislocations and grain boundaries being controlled with optically induced "topological" tweezers in experiments on a colloidal assembly.(a) Inducing glide with localized shearing.(b) Inducing climb by dilatation.(c) Dislocation fissioning through applied shear.(d) Glide of a dislocation that has been trapped by opposing shear stresses.(e) Moving a grain boundary by applying a potential that is commensurate with the lattice on one side of the boundary.From W.T.M. Irvine et al., Proc.Natl.Acad.Sci.(USA) 110, 15544 (2013).

FIG. 13 .
FIG. 13.Cyclic driving of uniaxial random field XY models with disorder.(a) The driving field angle φ versus the magnetic response in the x direction mx for increasing driving strengths, where purple is the lowest drive and red is the highest drive.(b) A plot of φ versus φ indicating that the applied field is rotated counterclockwise.(c) The y direction magnetic response my plotted versus mx.(d) my versus φ.Reprinted under CC license from S. Basak et al., Nature Commun.11, 4665 (2020).

m 100 .
FIG. 15.Voronoi tessellations of colloidal particles interacting with a periodic substrate for different nearly commensurate fillings ranging from f = 3.075 to f = 8.80.Dark blue polygons are fourfold coordinated, light blue polygons are fivefold coordinated, gray polygons are sixfold coordinated, and red polygons are sevenfold coordinated.A variety of grain boundary and dislocation structures appear.Reproduced from Ref. 111 with permission from the Royal Society of Chemistry.

FIG. 16 .
FIG. 16.(a) Particle locations (small black dots) and trajectories (lines) in a simulation of a single probe particle (large black dot) being dragged through a glassy background of bidisperse colloidal particles.(b) Time series of the velocity of the probe particle in panel (a) shows jumps corresponding to plastic rearrangement events.Reprinted with permission from M. B. Hastings et al., Phys.Rev. Lett.90, 098302 (2003).Copyright by the American Physical Society.

FIG. 19 .
FIG. 19.Stroboscopic snapshots of the positions during 25 cycles of superconducting vortices cyclically driven over a random landscape.(a) The reversible regime for small ac drive amplitudes.(a) The irreversible regime for large ac drive amplitudes.Reprinted with permission from N. Mangan et al., Phys.Rev. Lett.100, 187002 (2008).Copyright by the American Physical Society.

FIG. 20 .
FIG. 20.Schematic illustration of R-IR phases in a superconducting vortex system as a function of shear amplitude d versus vortex or particle density n.Point reversible states appear for low densities or small shears, while a region of loop reversibility emerges prior to the transition to irreversible behavior at high densities and large shears.Reprinted under CC license from S. Maegochi et al., Sci.Rep. 11, 19280 (2021).

FIG. 21 .
FIG. 21.Schematic of other elastic systems that could exhibit R-IR transitions under ac driving (arrows), such as a moving contact line (left) or a sliding charge density wave (right).

FIG. 22 .
FIG. 22. Experimentally measured hysteresis loops in a VO2 sample at the metal-insulator transition.(a) Illustration of the sample geometry showing that eight devices are present on a single sample; the widths of two of the devices are marked with arrows.(b) Full hysteresis loop as a function of resistance versus temperature.Main panel: Consecutive resistance versus temperature cycles shown zoomed in on the portion of the full hysteresis loop circled in panel (b).Jumps occur that are not repeatable from cycle to cycle.Reprinted with permission from A. Sharoni et al., Phys.Rev. Lett.101, 026404 (2008).Copyright by the American Physical Society.

13 FIG. 23 .
FIG. 23.Observations of glitches δν (in µHz) of the rotation frequencies ν for different pulsars as a function of time in units of the modified Julian date (MJD).The total number of glitches observed for each pulsar is listed as Ng, and windows of time in which no observations were made for at least 3 months are shaded in gray.Reprinted under CC license from J.R. Fuentes et al., Astron.Astrophys.630, A115 (2019).

FIG. 24 .
FIG. 24.The fraction of moving or irreversible particles fmov versus the trial strain level γ for different numbers of cycles in a numerical model of sheared colloidal particles, demonstrating the emergence of memory.(a) Application of a single shear level γ1 gives a kink in the trial strain γ at γ = γ1; reversible behavior eventually emerges for γ < γ1 but the system remains irreversible for γ > γ1.(b) Application of a smaller shear level γ2 for five cycles followed by application of a larger shear level γ3 for one cycle gives the trial strain level γ a memory of the value of both γ2 and γ3.When the system organizes to a reversible state, only memory of the larger strain level γ3 is retained.Reprinted with permission from N. C. Keim and S. R. Nagel, Phys.Rev. Lett.107, 010603 (2011).Copyright by the American Physical Society.

FIG. 25
FIG. 25.A network representation of the transitions between different states of a model of an amorphous solid subjected to cyclic shear deformations.The inset shows the soft-spot locations and numbers.Each state is represented by a binary number where 1 or 0 in the second digit represents a state where soft spot number 2 is "switched on" or "switched off," respectively.Arrows represent transitions due to an increase (black) or decrease (red) in the strain causing soft-spot switching.The green and purple shaded regions are limit-cycles with soft-spot number 6 switched off or on, respectively.Reprinted with permission from M. Mungan et al., Phys.Rev. Lett.123, 178002 (2019).Copyright by the American Physical Society.

FIG. 26 .
FIG.26.Return point memory for minor loops in an artificial spin ice system constructed from colloidal particles.When q = 1, the system reaches a reversible state.For both the (a) square ice and (b) kagome ice geometries, q increases as the number of minor loop cycles performed increases (numbers), indicating the emergence over time of reversible behavior.Reprinted with permission from A. Libál et al., Phys.Rev. E 86, 021406 (2012).Copyright by the American Physical Society.

FIG. 27
FIG. 27.A stroboscopic image as a function of time versus position j of a one-dimensional system of coupled oscillators.Oscillators that have a position coordinate qi less than zero are red, those with qi > 0 are blue, and those with qi ≈ 0 are white.Period-doubled oscillations emerge from the uniform state over time, forming a Classical Discrete Time Crystal (CDTC).The behavior is similar to the organization of a fluctuating system into a reversible state as shown earlier.Reprinted by permission from: Springer Nature, N. Y. Yao et al., "Classical discrete time crystals," Nature Phys.16, 438 (2020).