The article is structured as follows. In the first section, we discuss results such as diffusion dynamics of the tracer particle in terms of MSD, , diffusion constant, and a more detailed elucidation in terms of van-Hove function, and TAMSD. This is followed by a summary and conclusions drawn from the results section. Finally, we elaborate on the model used for our study.
The physical properties of a self-propelled probe particle immersed in a complex, ordered, non-inert system are primarily dictated by the interplay between the particle's activity and the strength of attractive interactions within the crowded environment, which are influenced by the system's packing fraction. This study focuses on the emergence of exotic anomalous, non-Gaussian behavior arising from the interactions between the probe particle and the crowded medium. Our analysis encompasses both the time-dependent diffusion dynamics and the long-term diffusion coefficient, shedding light on the intricate mechanisms driving such behavior.
To elucidate the dynamics of the tracer particle, we first compute the mean squared displacement (MSD), which is defined as , where is the position of the particle at time t. The MSD of a Brownian particle takes the power-law form: where is the anomalous exponent. At very short time scales in free space, the tracer particle exhibits ballistic motion with . Over longer time scales, its motion transitions to normal diffusion, characterized by . However, deviations from normal diffusion occur in crowded media, resulting in anomalous diffusive behavior where . During the ballistic regime at very short times, the tracer particle's motion remains largely unaffected by the crowded environment. At intermediate times, the influence of the surrounding crowders slows down the particle, leading to anomalous diffusion. Eventually, the particle exhibits normal diffusion with . Introducing attractive sites within the crowded environment further alters MSD. The presence of these sites traps the tracer particle, thereby reducing its MSD. As the stickiness () of the environment increases, the motion of the particle becomes increasingly restricted due to stronger attractive interactions, resulting in a more pronounced reduction in MSD.
To validate our simulation methodologies, we first investigate the motion of a passive tracer particle in free space, for both underdamped and overdamped limits. We validate the underdamped dynamics by comparing our results with the solution of the Ornstein-Uhlenbeck process,
In this case, the particle exhibits characteristic ballistic motion at short times, where the mean squared displacement scales as , transitioning to diffusive behavior () at longer times due to the cumulative effect of damping and stochastic forces. Our simulation accurately captures this two-regime behavior, further demonstrating the reliability of our numerical approach in modeling tracer dynamics under various physical conditions.
In the overdamped regime, where inertial effects are negligible due to high friction, the simulation outcomes are evaluated against the analytical expression , which was shown in Fig. S1 The simulation data show excellent agreement with the theoretical prediction, confirming the expected purely diffusive behavior () on all time scales in this regime.
Subsequently, we extended our investigation to study the dynamics of a self-propelled tracer particle under both overdamped and underdamped conditions. Once again, we compared our simulation results with the analytical expression
The results show good agreement with the expected behavior, demonstrating a diffusive regime at short times, followed by an intermediate super-diffusive phase, and eventually returning to diffusive motion at long times, as shown in Fig. S1. In the absence of the active force term () in the governing equation, the system exhibits purely passive dynamics, and the particle's motion corresponds to that of a passive tracer. In the underdamped limit, we validate our simulation as shown in Fig. S2, which is initial ballistic and long time diffusive, with a constant shift in the MSD profile, which characterizes the propulsion strength. The underdamped passive tracer dynamics validated with the solution of the Ornstein-Uhlenbeck process. Representative trajectories of both passive and self-propelled particles are presented in Fig. S3 to highlight the distinct nature of their motion.
First, we studied the dynamics of a passive tracer particle in a non-inert, crowded environment. Thermal fluctuations and attractive interactions among the crowders govern the motion of these particles. The behavior of the tracer particle in this system is influenced by two key parameters: the packing fraction of the medium (), which determines the density of crowders, and the stickiness () of the crowders, which quantifies the strength of the attractive interactions associated with the fixed crowders.
Figure 2a,b illustrates the variation in the mean squared displacement (MSD) for two different stickiness values ( and ) at various packing fraction values (). At lower stickiness (), the tracer particle diffuses freely, as shown in Fig.1a, resulting in negligible variation in the MSD (Fig.2a). This behavior can be attributed to the larger available space between the crowders, which allows the tracer particle to have more freedom of movement. However, when the stickiness increases to , as depicted in Fig.2b, the crowder size significantly influences the diffusion dynamics. As the size of the crowder increases, the MSD value decreases, likely due to dynamic caging caused by the larger attractive surface area of the fixed crowders. Tracer particles cannot overcome this caging because they cannot have sufficient energy from the thermal fluctuations of the medium. At large crowd sizes (), the attractive potential surfaces overlap, forming a valley of equipotential surfaces. This geometrical frustration limits the tracer particle's movement, allowing it to navigate along the semi-equipotential surfaces connecting the crowders.
At low and intermediate packing fractions or crowder sizes (), the tracer particle becomes trapped due to the sticky surfaces of the crowders, exhibiting subdiffusive behavior (). As the packing fraction increases, the attractive surfaces of the crowders approach each other, forming an overlapping potential valley that behaves like a single potential surface. This overlap allows the tracer particle to move more freely, resulting in higher MSD values. The observed behavior suggests an optimal crowder size, where overlapping potential valleys are just beginning to form, creating geometric frustration and slowing diffusion, particularly at intermediate values of packing fraction (), as shown in Fig. 1c.
The active tracer particle's behavior in non-inert crowding system is influenced by three key parameters: the packing fraction of the medium (), which determines the density of crowders, the stickiness () of the crowders, which quantifies the strength of the attractive interactions associated with the fixed crowders, and the activity of the tracer particle (Pe), which signifies the self-propelling forces acting on it. In this system, we observe an interplay between the tracer particle's activity and the crowder's attractive interaction. At a fixed attractive strength, at lower activity, the tracer particle's MSD exhibits two distinct regimes: ballistic at short times and diffusive at long times. At short times, due to the active force, the tracer particle gains mobility, enhancing its MSD, and the tracer exhibits ballistic motion. As time proceeds, the existence of crowders begins to impact the motion of the tracer particle, resulting in a slowing down of its motion.
In the case of a self-propelled particle, the dynamics is given by the activity (Pe) along with medium properties, including packing fraction() and stickiness (). In Fig. 2e, we have shown the variation of MSD at a fixed value of stickiness (), a fixed value of activity () at different packing fraction values.
At a higher packing fraction, as the size of the crowders increases, the crowding effect becomes more pronounced. At higher stickiness and larger crowder size, the diffusion of active tracer particles is like a "tug-of-war" situation between the activity and the geometrical trapping by crowders. This results in slowing the diffusion of the tracer particle. On increasing the tracer particle's activity to shown in Fig. S4, it recovers from the trapping, which in turn results in higher diffusion.
As shown in Fig. 2f, increasing the activity to a higher value (), enhances the tracer's mobility, enabling it to hop between lattice sites. This activity induced mobility effectively overcomes the attractive interactions (stickiness) of the crowders, diminishing the trapping behavior typically observed in a crowded environment. At high propulsion strengths, the role of interaction is negligible, allowing the particle to move through dense surroundings with reduced obstruction and exhibit free diffusion.
For a smaller crowder size or packing fraction, the active tracer particle can easily move through the system. However, an optimum crowder size exists, where the tracer particle starts to feel the existence of the overlapping potential valleys created by the crowders. This leads to geometric frustration and slows the diffusion of tracer particles, specifically at the intermediate packing fraction (). Subsequently, at the larger crowder size, the tracer particle escapes from the crowded environment, and eventually, the dynamics become diffusive. Furthermore, rotational inertia has a significant role in the particle dynamics to create an inertial delay between the velocity and orientation of the particle. This effect has been experimentally observed for single active particles and theoretically verified by explicitly accounting for rotational inertia. The persistence of single-particle trajectories is maintained by rotational inertia until interrupted by either a crowder interaction or by reorientation due to rotational diffusion. Consequently, rotational inertia effectively increases persistence time. Although we did not explicitly analyze the rotational mean-square displacement, our results indicate that rotational inertia strongly influences the diffusion-to-subdiffusion transition. By prolonging the persistence of propulsion direction, rotational inertia enhances the likelihood of particles remaining trapped at crowded sites, thereby amplifying subdiffusive behavior. At long times, however, the rotational mean-squared displacement becomes independent of rotational inertia.
The diffusive behavior of the tracer can be further characterized by evaluating the anomalous local exponent of time , which describes the time dependence of the mean squared displacement (MSD) or the mean squared displacement averaged over time (TAMSD). This exponent is defined via the scaling relation , and is typically computed as the local slope of the MSD on a log-log plot:
To obtain well-resolved and statistically reliable profiles for determining the local time exponent, the ensemble-averaged MSD and TAMSD are often computed over a large number of trajectories.
The time evolution of the anomalous exponent , shown in Fig. 2c, corresponds to a passive tracer particle diffusing in a medium of inert crowders at varying packing fractions . This analysis provides insight into how tracer dynamics evolve over time in the absence of significant interactions. Consistent with the MSD behavior, exhibits a characteristic two-regime pattern in the non-interacting case (): an initial ballistic regime with , followed by a crossover to normal diffusion with at longer times. No intermediate subdiffusive () or superdiffusive () anomalous behavior is observed during this transition. The absence of anomalous scaling indicates that, without attractive interactions, the tracer undergoes a conventional ballistic-to-diffusive crossover, unimpeded by trapping or caging effects that typically arise in more complex or interacting environments.
At higher attraction strength (), the tracer dynamics exhibit a three-stage behavior, as shown in Fig. 2d. Initially, at short times, the tracer displays ballistic motion with , as it is not yet significantly influenced by the crowded environment. At intermediate times, drops below unity, indicating pronounced subdiffusive behavior. This regime corresponds to transient trapping of the tracer in overlapping attractive potential wells formed by neighboring crowders. At long times, the tracer escapes these traps and transitions to normal diffusion with .
The degree of subdiffusion is most significant at intermediate packing fraction (), where the separation of attractive regions is optimal for trapping as the packing fraction increases further (e.g., ), the overlap becomes more extensive but less effective at confining the tracer, resulting in a weaker subdiffusive regime. While similar subdiffusive behavior has been reported in polymer-crowding environments, our results highlight the critical role of attractive interactions ("stickiness") for passive tracers. A more detailed analysis of the impact of activity (Pe) will be presented in a later section.
Now, examine how activity influences the anomalous exponent . In Fig. 2g,h, we show how varies with the packing fraction of crowders at a fixed stickiness strength (), for two distinctly different activity characterized by low and high Péclet numbers , and , respectively. At low activity (), the tracer's self-propulsion is weak and insufficient to overcome the hindering effects of the crowded environment. Consequently, the packing fraction continues to exert a strong influence, and subdiffusive behavior () persists over an extended time window. However, compared to the passive case, a modest enhancement in is observed (see Fig. 2g), indicating that even weak activity introduces persistent motion that facilitates tracer escape from local traps. As activity increases (), the tracer can overcome adhesive interactions with the crowders. This increases across intermediate and long time scales. Eventually, the tracer reaches a normal diffusion limit with , as shown in Fig. 2h. These results demonstrate that self-propulsion plays a key role in enhancing tracer mobility in sticky, crowded environments. Activity helps the tracer escape from transient potential wells formed by overlapping attractive regions between crowders. This leads to a systematic increase in effective diffusivity and a transition from subdiffusive to diffusive dynamics at long times, a trend robust across parameter regimes.
To characterize the diffusive behavior of the tracer particle, we compute the normalized long-time diffusion coefficient, defined as , where is the mean squared displacement MSD(t). We analyze the dimensionless normalized diffusion coefficient , of the passive tracer particle as a function of packing fraction for different attractive interaction strengths and propulsion strength . Here, denotes the diffusion coefficient of the tracer particle in the absence of crowders (i.e., in free space). The results reveal a non-monotonic dependence of the diffusion coefficient on the packing fraction, indicating a complex interplay between crowding geometry and attractive interactions.
The variation of the normalized diffusion coefficient of the passive tracer particle as a function of the packing fraction , for different values of the attractive strength (stickiness), is illustrated in Fig. 3a. The results show that the diffusion coefficient decreases monotonically with increasing packing fraction, indicating increased hindrance to tracer mobility at high packing fraction () created by the larger obstacles. This behavior reflects the growing spatial confinement and longer interaction times between the tracer and the attractive crowders as increases. At a particular intermediate packing fraction, the influence of crodwer size becomes most effective and leads to the lowest diffusion constant. Beyond a critical packing fraction (), the diffusion constant increases consistently across all attraction strengths. This enhancement is attributed to the formation of quasi-equipotential, valley-like pathways between lattice sites, which facilitate less obstructed tracer motion. At larger values of packing fraction (), the potential well overlap becomes ineffective for trapping, allowing unrestricted diffusion along these valleys. In contrast, at intermediate crowder sizes, the spatial separation between crowders hinders hopping and enhances trapping, leading to a minimum in the diffusion constant. This dip can be attributed to anomalous dynamics in the crowded environment, where the tracer particle is more confined and experiences transient trapping effects. The turnover points where this decrease-increase transition occurs are shown in Fig. 3a. For both small and large packing fractions, even moderate attractions (e.g., ) can induce partial trapping and reduce mobility. At intermediate attraction strengths (e.g., ), hopping remains possible but is increasingly hindered. For strong attractions (e.g., ), we observed pronounced trapping and significantly suppressed diffusion.
Now, we study as a function of for different packing fraction in Fig. 3b. The results show a general monotonic decrease in the diffusion coefficient with increasing stickiness, indicating that stronger attractive interactions between the tracer and crowders hinder particle mobility. This suppression arises due to the increased likelihood of the tracer being temporarily trapped at the crowder lattice sites. However, an interesting deviation was observed from this trend for optimal packing fraction (), where the diffusion coefficient exhibits a relative enhancement regardless of the stickiness value. This suggests that at this particular crowd size, the crowd's spatial organization or dynamic rearrangements may facilitate more efficient tracer motion, partially offsetting the effects of increased attraction.
To better elucidate the dependence of the diffusion constant on system parameters, we present a heat map in Fig. 3c, where both attraction strength () and packing fraction () are varied. The color scale denotes the diffusivity, with brighter regions indicating higher diffusion. As expected, diffusivity is highest at low attraction strengths. Even with moderate stickiness (), enhanced diffusion persists at both low and high packing fractions. In contrast, diffusivity exhibits a pronounced minimum at intermediate crowder densities, highlighting the geometry-driven competing effects of crowding and attraction-induced trapping.
Next, we examine the variation of the normalized diffusion constant of a self-propelled tracer particle, as shown in Fig. 4. Figure 4a-c shows the dependence of on system parameters at a fixed intermediate propulsion force corresponding to a Péclet number . The diffusion constant exhibits a non-monotonic dependence on packing fraction , indicating that tracer mobility is sensitive to the geometric constraints imposed by the crowders. At low attraction strengths, crowders have only a weak effect on particle mobility. As the attraction strength increases, the tracer becomes increasingly restricted, with the strongest suppression occurring at intermediate packing fractions where crowders are large but still relatively far apart. Further increasing the packing fraction brings crowders closer together, creating overlapping potential valleys that enable the tracer to hop between neighboring crowders more easily, resulting in enhanced diffusion. Figure 4b shows the variation of with attraction strength for different values of packing fraction. For intermediate crowder sizes (), diffusion decreases sharply as increases, indicating that strong attractive interactions significantly hinder tracer mobility. In contrast, for small and large values of packing fraction (), diffusion exhibits a slow, monotonic decay with increasing . For small crowders, the sparse distribution results in weaker confinement and relatively high mobility regardless of attraction strength. For large crowders, overlapping excluded volumes create interconnected voids, facilitating tracer motion even under strong adhesion. Figure 4c presents a phase plot of in the versus plane. At low , where attractive interactions are negligible, decreases monotonically with increasing packing fraction. However, at higher , a non-monotonic behavior emerges. This indicates that crowding-induced enhancement of diffusion arises only in the sticky situation.
Now we observe the variation of presents as a function of propulsion strengths at a fixed packing fraction for different attraction strengths in Fig. 4d. At low , diffusion remains nearly constant across all propulsion strengths. However, with increasing , diffusion becomes highly sensitive to propulsion: tracers with low propulsion are strongly hindered, whereas highly propelled tracers overcome adhesive interactions and maintain higher mobility. In Fig. 4e, we show as a function of at a fixed packing fraction for different Pe. The diffusion coefficient consistently decreases with across all propulsion strengths, highlighting the dominant role of attraction strength in minimizing transport even inherent tracer propulsion. But higher propulsion shows a significant role in the enhancement of transport. In the phase plot of in vs plane (see Fig.4f) D is monotonically increasing as we increases Pe.
Figure 4g shows as a function of for different Pe at a fixed . For relatively high attraction strengths, at any propulsion strengths, the D exhibits nonmonotonic dependency on . Figure 4h shows as a function of for different at a fixed . The diffusion increases with activity across all crowder sizes, with larger crowders exhibiting a stronger enhancement, suggesting that large crowders facilitate active tracer transport by reducing spatial hindrance.
Figure 4i demonstrates the combined effects of crowder size and propulsion strength. The color scale represents the magnitude of normalized diffusion, providing a comprehensive view of its dependence on crowder size, stickiness, and propulsion. Both parameters enhance tracer diffusion, with larger crowders amplifying the activity-induced mobility.
For a more profound understanding of the underlying trapping mechanism of the tracer particle in the ordered environment, we analyze the self-part of the van-Hove correlation function , which describes the probability distribution of the tracer particle's displacement, and is defined as
where and x(t) represent the tracer particle positions along the x-direction at times and t, respectively. The van-Hove function is independent of direction, reflecting the system's symmetry and the isotropic nature of tracer particle motion.
For reference, we first calculated the van-Hove function for a free Brownian particle in free space. The probability distribution function is expected to give a Gaussian distribution,
where measures the width of the distribution, is the tracer particle displacement at any given lag time.
Now, we calculate the van-Hove function of a passive tracer particle within the crowded environment at two short and long lag times, at respectively, for various packing fractions. Figure 5a,b show the probability distribution function of the passive tracer at these lag times in an inert environment (). At shorter times, Fig.5a, for all values of packing fraction (), the distributions collapse onto a single curve, indicating that the crowder size has no significant effect for the inert situation. Even at longer timescales, Fig.5b, there remains no noticeable dependence of the probability distribution on . As increases, the van-Hove function broadens, reflecting the tracer particle's exploration of larger spatial regions. The packing fraction does not significantly affect the tracer's dynamics because the inert crowders exert minimal interaction forces, allowing the tracer to behave like a free Brownian particle.
As we introduce the attractive interaction between the crowders and the tracer (), the deviation from Gaussian behavior remains minimal at shorter lag times () , as shown in Fig. 5c. However, at longer lag times (), this deviation becomes more pronounced, leading to prominent non-Gaussian dynamics. The resulting non-Gaussian behavior can be characterized by the fitting function defined as
Here, w represents the width of the Gaussian distribution, denotes the gamma function, and quantifies the extent of Gaussianity. When , the distribution is purely Gaussian, while any deviation from this value indicates the degree of non-Gaussianity.
In a crowded environment with attractive interactions, as and increase, the particle gets attached to the crowder's surface. This confined motion causes the distribution to become narrower. For a passive tracer, when reaches , the Gaussian nature of the distribution breaks down, and at a lag time of , the van-Hove curve exhibits a double hump. For , a minimum is observed in the MSD, resulting in an inner Gaussian distribution, indicating the slowest diffusion. This behavior can be interpreted in terms of a potential well, where both its size and depth are key parameters influencing the emergence of the double-hump feature. Two distinct Gaussian distributions are observed in the van-Hove function: a broader one associated with hopping dynamics and a narrower, stiffer one corresponding to the tracer particle stickiness to the surrounding crowders, see Fig. 5d. The double Gaussian profile can be approximated by the sum of two non-Gaussian distributions, defined as
The parameter w represents the resultant width of the curve, derived from the widths of two Gaussian distributions, and . Similarly, the mean is determined based on the individual means of the two distributions.
For better visualization, we fit individual curves in the supplementary information. Figure S5a shows the fitting of curves for model parameters for attraction strength . Figure S5b shows the fitting of a double hump for the packing fraction value(), where we observe the sum of inner Gaussian and outer non-Gaussian profiles.
For self-propelled particles (SPPs), the van-Hove correlation function exhibits broader distributions than passive tracers, reflecting enhanced spatial exploration over a given time interval. Figure 6 presents the self-part of the van-Hove correlation function for SPPs at two lag times: low () and high (). It highlights how activity (Pe) and packing fraction () influence displacement statistics.
At short lag times (), Fig. 6a,b show the van-Hove function for two propulsion strengths (, and 10), fixed attraction strength (), and varying packing fraction . For (Fig. 6a), the profiles for different nearly merge onto a single curve, indicating negligible influence of crowder size. In this regime, the tracer rarely reaches crowder sites due to weak propulsion, and trapping by attractive crowders is minimal. Consequently, the van-Hove remains close to Gaussian for both small and large , with only slight deviations at intermediate sizes captured by the non-Gaussian fit in eq. 5.
In contrast, at higher activity () (Fig. 6b), the van-Hove function broadens and displays noticeable deviation from Gaussianity due to increased tracer-crowder interactions. This deviation is well captured by a stretched exponential (non-Gaussian) form with in eq. 5. The enhanced propulsion enables the tracer to explore attractive regions more frequently, causing intermittent trapping and leading to a broader, non-Gaussian displacement distribution. However, with further increases in propulsion, the activity overcomes trapping, and the distribution gradually restores to Gaussian behavior, see Fig. 6c.
At longer lag times (), Fig. 6d,e shows that the influence of crowder size becomes more prominent. For , Fig. 6d, Gaussian behavior persists at small , but at intermediate sizes (), a double-hump structure emerges. This structure reflects two populations: one trapped near crowders (inner Gaussian) and the other corresponding to hopping events (outer Gaussian). This behavior is accurately modeled using the double Gaussian expression in eq. 7.
As increases, the geometric frustration diminishes and the double-hump structure gradually vanishes. For higher propulsion () [Fig. 6e], the van Hove function becomes broader and non-Gaussian across all , consistent with active exploration overcoming confinement. In particular, soft multi-step pattern emerge in the distribution at intermediate values of , attributed to repeated hopping between attractive lattice sites. These features arise from competition between activity and attraction, and are best described by a sum of two non-Gaussian profiles, as given by Eq. 7:
where and denote the widths, and are the means of the two components, and , quantify deviations from Gaussianity. When , the distribution is Gaussian; otherwise, it exhibits non-Gaussian characteristics.
At high propulsion strengths, activity overcomes attractive interactions, diminishing non-Gaussian features, and restoring Gaussian-like dynamics as shown in Fig. 6f. The soft multi-step pattern observed in the van-Hove correlation function arising from the interplay between activity and attraction is well captured by the double non-Gaussian fitting (Eq.7), as demonstrated in Fig. S5d. Additional fitting details are provided in the Supplementary Information.
To investigate potential non-ergodic behavior arising from the presence of attractive crowders, the individual time averaged trajectories (TAMSD) are evaluated using the following expression:
Here, is the total simulation time, denotes the lag time, specifying the duration of the sliding time window applied to the trajectory r(t). Further, we averaged the individual trajectories over the available ensembles N, defined as We compare the individual TAMSDs, , with the ensemble-averaged mean squared displacement (MSD), , and the ensemble average of the individual TAMSDs, , to assess their consistency. Prior studies have established that if the MSD and the ensemble-averaged TAMSD exhibit the same scaling behavior over a given time scale, the dynamics can be considered ergodic in that regime. In contrast, significant deviations between these quantities indicate a breakdown of ergodicity, suggesting non-ergodic dynamics.
We begin by examining the dynamics of a passive tracer particle (), in a non-inert crowded environment with low attraction strength (), and packing fraction () (Fig. S6). Under these conditions, the system exhibits ergodic behavior, as indicated by the close agreement between the ensemble-averaged MSD and the ensemble average of individual TAMSDs.
As the attraction strength increases, deviations from ergodicity emerge. At the critical packing fraction , where diffusion is notably suppressed, a transient breaking of ergodicity is observed. This deviation is not persistent and occurs primarily at intermediate time scales, suggesting a dynamic crossover between ergodic and non-ergodic regimes.
Upon further increases in attraction strength to and packing fraction as shown in Fig. 7a, the system remains ergodic at short time scales, but shows a clear divergence between the MSD and ensemble-averaged TAMSD at intermediate time scales. At long times, diffusive behavior resumes, and ergodicity is restored. This intermittent ergodicity breaking arises from the transient trapping of the tracer particle by attractive crowders. At high attraction strengths and optimal crowder sizes, the tracer becomes temporarily localized, unable to escape the attractive domains for extended periods.
Figure 7b presents a comparative analysis of the above three properties for a self-propelled tracer particle at a propulsion strength of . Similar to the passive tracer particle, the self-propelled tracer also exhibits ergodic behavior at low attraction strength and packing fraction , as shown in Fig. S6, indicating that the addition of self-propulsion does not alter this behavior under these conditions. For higher attraction strength () and the optimal packing fraction (), unlike the passive tracer, it shows less deviation of MSD from the TAMSD at the intermediate times. An increase in propulsion strength enables the tracer particle to overcome the confinement induced by sticky crowders; as a result, the dynamics is ergodic for the complete time scale as shown in Fig. 7c. Hence, we can conclude that self-propulsion helps to revive ergodicity even in complex media.
The tracer particle can overcome the geometric frustration created by the crowders at the intermediate crowder size. Anomalous diffusion is frequently associated with non-ergodic dynamics. But in this situation, the anomalous behavior slowly diminishes due to propulsion. Overall, propulsion helps restore ergodic behavior.