Activity controls fragility: A Random First Order Transition Theory for an active glass

How does nonequilibrium activity modify the approach to a glass? This is an important question, since many experiments reveal the near-glassy nature of the cell interior, remodelled by activity. However, different simulations of dense assemblies of active particles, parametrised by a self-propulsion force, $f_0$, and persistence time, $\tau_p$, appear to make contradictory predictions about the influence of activity on characteristic features of glass, such as fragility. This calls for a broad conceptual framework to understand active glasses; here we extend the Random First-Order Transition (RFOT) theory to a dense assembly of self-propelled particles. We compute the active contribution to the configurational entropy using an effective medium approach - that of a single particle in a caging-potential. This simple active extension of RFOT provides excellent quantitative fits to existing simulation results. We find that whereas $f_0$ always inhibits glassiness, the effect of $\tau_p$ is more subtle and depends on the microscopic details of activity. In doing so, the theory automatically resolves the apparent contradiction between the simulation models. The theory also makes several testable predictions, which we verify by both existing and new simulation data, and should be viewed as a step towards a more rigorous analytical treatment of active glass.

A ctive systems, consisting of particles that convert energy supplied to it into mechanical work, are a fascinating class of driven nonequilibrium systems (1,2). The range of systems that fall under this ambit include living systems, from cells and their motor-cytoskeletal extracts to collections of cells constituting tissues, which utilize chemical energy to perform biological function (3), and synthetic systems, such as magnetic colloidal beads (4), light-activated colloidal swimmers (5), and vertically vibrated grains (6,7). So far, theoretical and experimental studies have focused on dilute collections of active particles, however the situation in cells and tissues lie at the other extreme-for instance, the material within a cell and a collection of confluent cells within a tissue are both at high densities and require activity to facilitate either molecular rearrangements (8) or collective cell movement (9)(10)(11)(12)(13)(14); thus, it is of interest to study the effect of activity in systems where the dynamics are slow and multiparticle correlations are significant and that approach a collective jammed or glassy state.
Motivated by this, there have been a number of simulation models (15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28) of dense assemblies of stochastically driven particles, parametrized by a self-propulsion force of magnitude f0 and an orientational persistence time τp. Each of these models characterizes the statistics of activity in a slightly different way, however their predictions on the variation of signature glassy behavior, such as fragility, on the extent of activity have been divergent. Unlike in the case of dilute collections of active particles, there is no analytically calculable framework to understand the dynamics of active particles in the dense limit. There have been attempts to extend the usual mode-coupling theory (MCT) to include active self-propulsion (15,16,25,29), however such theories are not applicable to the glassy dynamics at high densities where relaxation typically occurs via activated processes (30,31).
The random first-order transition (RFOT) theory (32)(33)(34)(35)(36) has been remarkably successful (37)(38)(39) in describing a glassy system both above and below the regime where activated processes dominate the dynamics. RFOT theory generalizes the theory of first-order crystallization transition to that of freezing to a disordered structure (34,36), describing the system in terms of a mosaic of local aperiodic domains (SI Appendix, Fig. S1) and a mismatch energy at the interface between domains (37)(38)(39). Including activity within this basic RFOT picture presents a major challenge (SI Appendix). Here we propose an extension of the RFOT theory to an active system, which should be viewed as a first step toward a more rigorous analytical treatment of active glass. Even so, the theory presented here makes several testable predictions, regarding how the active parameters affect glassy behavior: Whereas f0 always inhibits glassiness, the effect of τp is more subtle and depends on the microscopic details of activity. This not only helps reconcile the apparently divergent results of the simulation models discussed above but also makes predictions that we verify with numerical simulations (Materials and Methods).

Significance
Understanding how activity affects the physics of dense glassy suspensions is of fundamental interest in a variety of cellular and tissue contexts. However, current simulations of dense active systems make qualitatively different predictions about the influence of activity on characteristic features of glass, such as fragility. Acknowledging the need for a broad theoretical framework, we extend random first-order transition (RFOT) theory to active glasses, based on an effective independent-particle treatment. We find that the analytic model predicts that the behavior of the active glass is strongly influenced by the microscopic details of activity. This not only resolves the apparent contradictions posed by previous studies but provides a number of testable predictions, some of which we verify using numerical simulations.
Active RFOT: General Formalism The RFOT theory for passive glasses describes the activated reconfigurations of such mosaics through a mechanism similar to nucleation (32-34, 37, 40). Consider a domain of volume v and length scale R ∼ v 1/d , in d dimension (SI Appendix, Fig. S1). The RFOT theory posits that the driving force for reconfiguration of the mosaics is the configurational entropy density, sc(Φ, T ), which is a function of the potential energy density Φ at temperature T (36,41). We can write the free-energy gain of the system due to such reconfiguration as where Ω d is the volume and S d the surface of a region with unit radius in dimension d andγ is the surface mismatch energy density. The driving force responsible for the reconfiguration of a domain of size R is obtained by differentiating Eq. 1 with respect to R. The critical nucleation size * is determined by force balance. If the cavity size R > * , then the volume contribution drives the reconfiguration of the mosaic. On the other hand, if R < * , the surface contribution dominates and the state within the cavity remains the same. Thus, the system consists of mosaics of correlated domains with a typical length-scale * . Note that this length-scale may be different from dynamic length-scales associated with domains of correlated motion (11,42,43).
In extending the conventional RFOT theory to active systems that are manifestly out of equilibrium, we must first find a physically reasonable definition of the configurational entropy density sc for such systems (SI Appendix). The configurational entropy of passive thermal systems is usually defined in terms of the multiplicity of the local minima of the potential energy ("inherent structures") whose basins of interaction are visited by the system at a particular temperature T . We continue to use the same definition of the configurational entropy for active systems. Particle configurations in a steady state of active systems can be used as starting points of a minimization procedure to locate a set of inherent structures characteristic to the steady state being considered. The multiplicity of inherent structures obtained this way would be different from that of the inherent structures obtained for an identical system without activity at the same temperature. Therefore, the temperature dependence of the configurational entropy (defined as the fragility) of an active system would be different from that for its passive counterpart.
To proceed further, we need to evaluate the active corrections to the configurational entropy, which we do within a Langevin dynamics of active particles (see SI Appendix for details). Since the typical persistence time of the active force is larger than the time scale of thermal fluctuations, the effects of activity, to leading order, can be accommodated in a renormalized potential energy density,Φ = Φ + δΦ, whose form δΦ = −N −1 i f a i · xi depends on the precise model of activity (44,45). Here, xi is the position vector of particle i, f a i is the active force parametrized by a self-propulsion force of magnitude f0 and an orientational persistence time τp, and · · · denotes an average over a steadystate distribution of the particle configurations (SI Appendix). Expanding the active configurational entropy about its value for the passive system, we get where the leading correction involves the derivative of the configurational entropy evaluated at zero active force f a i . We write this correction term as κa δΦ, where we have introduced an active fragility parameter, κa , that quantifies the sensitivity of the configurational entropy to changes in active force. We assume, for simplicity, that κα does not depend on the temperature. In addition, to arrive at an active extension of RFOT, we must also evaluate the active surface contribution. Here we simply take it to beγ(f0, τp, T ) (Eq. 1).
Since the Kauzmann temperature TK is defined as the temperature where the configurational entropy sc(Φ, T ) of the passive system vanishes (46), we take, following the prescription in refs. 34 and 36, sc(Φ, T ) = ∆Cp(T − TK )/TK close to but above TK , where ∆Cp is the jump in specific heat from the liquid to the crystalline state (46). Further, we assume, as in the passive glass (47), that the temperature dependence ofγ is linear:γ = Ξ(f0, τp) T , where Ξ is a function of the active parameters alone. As in the passive glass, the detailed T dependence ofγ does not change the qualitative results because the change in configurational entropy plays the major role in governing the dynamics (47). . [

3]
This expression is similar in spirit to what Wang and Wolynes proposed for active network materials (48). Note that while in principle, D depends on the active parameters, f0, τp, for simplicity we will consider it to be independent of the activity. The relaxation dynamics of the system is characterized by the reconfiguration of a domain of length-scale * . The typical potential energy barrier height for the relaxation of the system is ∆( * ) ∼ * ψ , where ψ is a scaling exponent. Then the relax- where ∆0 is an energy scale and τ0 a microscopic time scale that depend on T , interparticle interactions, and the active parameters. If, following refs. 33, 36, we take ψ = θ = d /2, and assuming ∆0 = κT , we obtain, close to TK , where E = κD = c d κΞTK /∆Cp. The alternative suggestion (42), θ = d − 1, ψ = 1, would change the expression for * but lead to the same expression for τ . Therefore, within this active extension of the RFOT, all one needs is the activity correction to the configurational entropy κa δΦ. The sign of this correction is important because it tells us whether the presence of activity increases or decreases the configurational entropy. We have carried out test simulations (see SI Appendix for details) to check how the average energy of inherent structures in the model of active glass considered in ref. 19 depends on the strength of activity f0 (SI Appendix, Fig. S2). We find that the average energy of inherent structures increases as f0 is increased at a fixed temperature. This observation implies that the configurational entropy also increases with f0-that is, the sign of the activity correction is positive. This conclusion is consistent with the results of recent work by Preisler and Dijkstra (49). A positive entropy correction implies that the temperature at which τ diverges for an active system is lower than TK -that is, increasing the active force decreases the glass transition temperature.
Active RFOT: Application to Self-Propelled Particle Models We will now explicitly evaluate the active correction in Eq. 2, for a system of self-propelled particles (1,2,17,19), described by the following overdamped dynamics: where xi is the position of the ith particle, µ is the particle mobility, φi ≡ j =i φ(xi , xj ) is the instantaneous many-body potential experienced by particle i, and f a i (t) and fi (t) are the active and thermal noises, respectively. To calculate the active correction to the configurational entropy (SI Appendix), we use an approximate theory where we replace the many-particle dynamics (Eq. 5) by the overdamped dynamics of a single self-propelled particle with friction γ, caged by the other particles, whose effect is represented as an effective confining harmonic potential of strength k (29,50,51). We then calculate the correction to the configuration entropy κa f a · x , where the angular bracket is an average over the steady-state probability distribution of the caged active particle. Details of this "mean-field" approximation are provided in SI Appendix. We consider two different models of active forces that have been widely used in the simulation literature and show that they give fundamentally different behavior as a function of the orientational persistence time τp, although the dependence on the magnitude of the active force f0 is the same for both models. These opposite effects of the propulsion force and the persistence time on the glass transition in active glasses, which were reported in simulations in refs. 19, 26, were a puzzle and find a natural resolution in our active RFOT theory. We further discuss the distinction between these two models and their phenomenological relevance in SI Appendix. Model 1. The active noise f a i with fixed amplitude f0 has zero mean and a shot-noise temporal correlation: This is the realization of the active noise considered in the simulation studies of ref. 19. For this model of active forces, the active RFOT calculation detailed in SI Appendix gives for the correction to the configurational entropy appearing in Eqs. 3 and 4, thus leading to * =   D (T − TK ) + [9] where H = T K κa γ ∆Cp , G = k /γ, and as before, θ = ψ = d /2. Eq. 8 shows that both f0 and τp decrease the effective Kauzmann temperature T eff K = TK − Hf 2 0 τp/(1 + Gτp), defined as the temperature when * and τ diverge. In other words, as we increase either f0 or τp, the system shows glassy behavior at lower T compared with the corresponding passive system-that is, both f0 and τp promote fluidization (or suppress glassiness) for small τp. This prediction with respect to f0 is consistent with many simulation studies (16,17,19,20,43). . [13] with the same expressions for H and G as above.
While the effect of Tsp is same as that of f 2 0 in Model 1, the effect of changing τp is entirely different. At a fixed Tsp, an increase τp now makes the system show glassy behavior at higher T -that is, τp promotes glassiness. This prediction with respect to τp is consistent with another set of simulations (26).
Thus, what was perceived as contradictory results reported in simulations, namely that increasing τp may either promote fluidization or glassiness, finds a natural resolution within our active RFOT framework. As demonstrated, the trends should indeed depend on the microscopic details of activity.
We now discuss other predictions that emerge from our active RFOT theory of models 1 and 2 and verify their consistency with our simulations of an active glass (Materials and Methods) and ref. 26, respectively.

Active RFOT Confronts Simulations-Model 1
We now perform molecular dynamics (MD) simulations of a glass former driven by active propulsion forces in three dimensions (Materials and Methods) and make a quantitative comparison with our theoretical predictions of model 1. We first tune f0 at fixed τp and plot * and τ as a function of T in Fig. 1 A  and B, respectively. For fixed T , we see that * and τ decrease as we increase f0. Next, we calculate the modification to the glass transition temperature Tg by the activity. We define Tg as the temperature where the relaxation time of the system increases beyond the threshold value of τ /τ0 = 10 6 . From Eq. 9 we see that Tg gets modified due to activity as Tg = E /(6 ln 10) + TK − Hf 2 0 τp/(1 + Gτp), similar to T eff K . Using this definition, we compare our theory with the MD simulation data (see Materials and Methods and ref. 19), by rewriting Eq. 9 as where Λ = H τp/(1 + Gτp) is a constant, since τp is held fixed. Analysis of our simulation data gives τ0 = 0.135 and TK = 0.3. We obtain the values of the other two constant parameters by fitting our analytical expression to one particular dataset, at f0 = 1.50 (fitting to the data for other values of f0 works equally well), and obtain E = 1.55 and Λ = 0.1. The Angell plot (53) shown in Fig. 1C demonstrates the excellent agreement between active RFOT theory and simulation data. We emphasize here that the theoretical lines are not individual fits to the simulation data, since all of the plots of Eq. 14 use the same constant parameter values obtained from one initial fit. Comparing Eq. 14 to the Vogel-Fulcher-Tammann-form ln(τ /τ0) = 1 m (T /T eff K − 1) −1 , we obtain for the fragility parameter We find that this differs from the simulation data by a scale factor of ∼ 1.25 (this is true even in the passive limit); however, it is remarkable that our simple theory captures the functional dependence on f0 correctly (Fig. 1D). This leads to an important result, namely when τp is fixed and the active force f0 increases, the behavior of τ becomes closer to Arrhenius law: The system becomes a stronger glass former (53) with increasing activity.  19, and lines are from active RFOT theory for model 1, Eq. 14. As activity increases, the behavior of τ becomes closer to Arrhenius law, making the system a stronger glass former. (D) Behavior of fragility as a function of activity. Data taken from MD simulations (19) and lines from the active RFOT theory for model 1, Eq. 15. As mentioned in the main text, the theory curve has been scaled by a factor of 1.25.
We next fix f0 and tune activity by changing τp. The active RFOT theory of model 1 gives explicit formulae for * (Eq. 8), τ (Eq. 9), and the fragility parameter m = TK /E − Hf 2 0 τp/E (1 + Gτp). This predicts that the relaxation time decreases for increasing τp, while the dynamics become independent of τp, when τp 1/G. We have tested these predictions by obtaining τ as a function of τp at different values of f0 from our simulations (Materials and Methods). The agreement is good for low f0 and τp ( Fig. 2A) and become systematically worse at larger activity-for example, at large τp for f0 = 1.0 and 2.0.
We next compare the theoretical predictions for the fragility parameter with our simulations (Fig. 2B). We again see fair agreement for small f0 and τp and a clear deviation between theory and simulation data for larger f0 and τp. Within model 1, both τ and m decrease as τp increases: Increasing τp fluidizes the system and makes it a stronger glass former.
Many experimental and numerical studies of glass have shown that the relaxation of density fluctuations in supercooled liquids is well approximated by a stretched exponential exp(−t/τ ) β , where β is the stretch exponent [Kohlrausch law (54)]. Using our simulations, we can measure the change in the β exponent due to activity. A fit to the computed density correlation function Q(t) (SI Appendix, Eq. S32) shows that β increases with both f0 and τp (Fig. 2C), with the change ∆β scaling as f 2 0 τp (Fig. 2D). In Discussion, we provide a rationalization for this.

Active RFOT Confronts Simulations-Model 2
We first tune Tsp at fixed τp; The active RFOT theory for model 2 shows that both τ (Eq. 13) and fragility parameter m = T K E − HTsp E (1+Gτp ) decrease with increasing Tsp. For this case, model 2 shows the same qualitative behavior as model 1. We then fix Tsp and change activity by tuning τp. We find that at fixed T , * increases monotonically with τp (Fig. 3A). Thus, we predict that τp promotes glassy behavior in the sense that larger τp drives the system more toward the glassy state. Using the active RFOT theory for model 2, we obtain the effective Kauzmann temperature, T eff K , and the glass transition temperature, Tg . The Angell plot (53) (Fig. 3B), τ versus Tg /T at different values of τp, shows a systematic departure from Arrhenius behavior as τp increases: Here, fragility increases with τp.
Since the simulations of ref. 26 with which we would like to compare our results were done in the athermal limit, we have to take T → 0 in Eq. 13. In this athermal limit, the fit parameters are E , H , G, and τ0 (note that TK ∼ 0.3 is obtained from simulations of the passive system and hence is not a fit parameter). We find that with E = 1.255, G = 3.801, and H = 1.0, we obtain a good fit as long as we choose τ0 to depend on τp (Fig. 3C). Indeed, this dependence of τ0 on τp was reported in the simulations of ref. 26; this is replotted in SI Appendix, Fig. S3 (40) for completeness. We emphasize that the parameters E , G, and H are obtained from one set of data in Fig. 3C and use the same values for the rest of the plots.
Our active RFOT theory makes a prediction for the dependence of the fragility parameter on τp: m = T K E − HTsp E (1+Gτp ) , at fixed Tsp > 0, the fragility is at first insensitive to τp and then increases before saturating to the passive value TK /E (Fig. 3D). Irrespective of this detailed behavior, the point is that this system becomes more fragile as τp increases. This is opposite to the behavior of model 1 (at small τp). This is Tg when τ /τ 0 becomes 10 6 and plot log(τ /τ 0 ) as a function of Tg/T for Tsp = 0.64, using Eq. 13. As τp increases, the curves depart further from Arrhenius behavior; the system becomes more fragile with increasing τp.
(C) Relaxation time of the system as a function of 1/Tsp at different τp when T = 0 (athermal limit), allowing us to make comparisons with simulation data (26) (see text for discussion of fit parameters). (D) Our active RFOT theory for model 2 predicts that, for fixed Tsp > 0, the fragility parameter is at first insensitive to τp and then rises before saturating to the passive value T K /E. also different from the trend shown on increasing Tsp within this model.

Discussion
We have developed an RFOT theory for active glasses by proposing an analytic expression for the active contribution to the configurational entropy, using an effective theory that involves the dynamics of a typical active particle within a caging potential. It will be interesting to explore, in future simulations, the extent of validity of this single-particle approach and the relation between the effective parameters of this single-particle theory and the many-body properties of the system. While our effective single-particle model captures the dependence of the mean relaxation time on the activity parameters rather well, it shows deviations at large f0 and τp where nonequilibrium effects are stronger, and the assumption that the effective cage parameters (γ and k ) are independent of activity breaks down. Our theory should be viewed as a first step toward a more rigorous analytical treatment of active glasses. Within our approximate single-particle picture, valid for small f0 and τp, we see that the effects of activity can be described by an effective temperature T eff . This is related to the fact that within this approximate theory, the steady-state probability distribution of the typical particle in a caging potential driven by the exponentially correlated noise has a Boltzmann form (SI Appendix). Thus, for example, Eq. 9 can be interpreted as changing the temperature T to T eff = T + (Hf 2 0 τp)/(1 + Gτp). This quadratic dependence of the effective temperature on f0 is consistent with the results reported in ref. 49. Both here (Fig. 2) and in ref. 49, one observes deviations from the model at large active forces.
This observation that at low f0 and τp one may interpret our results as arising from a change in T → T eff > T provides a rationalization for the observed increase in the stretch exponent β due to activity reported in Fig. 2 C and D. As discussed in ref. 55, within the RFOT framework, the origin of the stretch exponent β is due to fluctuations in the configurational entropy barrier associated with the critical nucleus * . This gives rise to a formula for the stretch exponent β = (1 + α 2 ) −1/2 , where α is the ratio of the fluctuations in the configurational entropy barrier to temperature. Replacing the temperature with T eff > T leads to the prediction that the β-exponent for the active system is larger than the equivalent passive system (Fig. 2C). Indeed, as shown in ref. 19, the form of the decay of the overlap function (SI Appendix) depends on f 2 0 τp, for a limited range of values of f0 and τp, implying that β should scale as f 2 0 τp, as seen in Fig. 3D. While consistent with our simulations (Fig.  2 C and D), we would like to revisit this in a more rigorous calculation.
What are the implications of our results for cells and tissues that show glassy behavior? Recent studies have shown that active processes facilitate molecular rearrangements in the dense interior of cells (8,13). In the context of tissues, it has been observed that active processes induced relaxation via collective cell movement in a confluent monolayer of cells (9)(10)(11)(12)(13)(14). In both these situations, there exist long-lived structures that rearrange by activation, thus making it appropriate to analyze their slow dynamics using an active extension of the RFOT theory. To ascertain which model of activity is more suitable, we will need to know the statistical nature of the active fluctuations in these systems. In cells, activity is often realized in the form of active stresses generated by ATP-driven motor proteins that bind/unbind onto an isotropic cytoskeleton (chemical energy transduced per ATP molecule under physiological conditions is ≈ 20kB T). The effective temperature estimated from fluctuations (56,57) in the active stress arising from independent motors is proportional to f 2 0 τm , where τm is the activity correlation time of the force generators (58). This is precisely of the form Eq. 9 obtained from model 1, following which we would expect that higher ATP concentration decreases the fragility, driving the system toward a strong glass former, and that activity, controlled through either f0 or τp, would always fluidize the system. This prediction agrees with early experiments (9,12) and is confirmed in a recent microrheology study (59) of the cytoplasm of a variety of cells, which shows strong glassy behavior in wild type, which becomes more fragile upon ATP depletion.
Beyond biology, active glasses present a challenge to physics theory, with the main advances in this field relying so far on numerical simulations. In this manuscript, we provide a theoretical framework that should be viewed as a step toward a more rigorous analytical treatment of active glass. Our analytic expressions give an excellent description of available numerical simulation results and resolve the apparent contradiction between the effects of f0 (or Tsp) and τp that have been reported in the literature. Since our theory is based on effective singleparticle active dynamics in a caging potential, we believe our qualitative results are likely to be independent of the precise form of the interparticle interactions. Our theory could then be made to confront precision experiments in synthetic systems (27), where the active forces and persistence time can be controlled independently.

Materials and Methods
We perform an MD simulation of a Kob-Anderson binary mixture (60) of Aand B-type particles at a 80:20 ratio interacting via a Lennard-Jones pair potential (see SI Appendix for details) following the simulation strategy of ref. 19. The B particles are driven with a self-propulsion force f 0 and a persistence time τp.
acknowledges the Council of Sceintific and Industrial Research for financial help and Indian Institute of Science and Simons Center at the National Centre for Biological Sciences for computational resources. N.S.G. is the incumbent of the Lee and William Abramowitz Professorial Chair of Biophysics and this research was made possible in part by the generosity of the Harold Perlman family.