Trajectory of a particle, i.e. monomer, in a MD simulation of glassy polymer in 3D. It vibrates and then hops from one metastable state to another [3].  Probability distribution of particle displacement during time interval τ at temperature T, i.e. Van Hove correlation function. The main peak is mainly contributed by particles which only vibrate. A secondary peak at a displacement comparable to the particle diameters emerges at low temperature, signifying activated hopping at deep supercooling [3]. 
(Above) Three particles in a 3D polymer simulation which hop together four times, i.e. two roundtrips, shown as animation and timecoarsegrained particle trajectories. (Right) Movie of trajectories of hopped particles in a 3D polymer simulation with 1500 chains each of 10 particles. Random colors denote the time at which a hop occurs. A trajectory with a changing color thus indicates backandforth hopping events. Numerous repetitive stringlike particle hopping motions are observed [3]. 
We first recognized the importance of repetitive particle hops in the study of glassy polymer films, which is a topic of interest because one may find insights not available in the bulk. Our experiments on glassy polystyrene films show the existence of a nonglassy liquid mobile layer on its surface [10]. Surface mobility is well understood for crystals, e.g. silicon, where the surface atomic layer is weakly bonded and dominate the dynamics. In our polymers with chains each having 22 monomers, mobility enhancement must penetrate much deeper than merely the surface atoms since a whole chain needs to move together. The mobile layer found to be about 2 nanometers thick thus consists of many atomic layers and is indeed surprisingly thick. The properties in the surface layer converge gradually to those of the bulk when going deeper into the film. By examining how deep surface effects penetrate into a film using MD simulations, we identified a peculiar inner surface layer, which is the deepest layer away from the free surface still enjoying surface effects. In this layer, particle hopping rate measured from shorttime displacements is bulklike, as expected from bulklike structural properties at this depth into the film. However, hopping rates inferred from longertime displacements are clearly surface enhanced. Therefore, the free surface provides a unique situation in which longtime dynamical behaviors can be perturbed, while structural and shorttime dynamical properties are kept invariant. The enhanced longtime dynamics was found to be due to a break down of backandforth hopping motions. Since the surface layer is found nonglassy experimentally [10], our results support that backandforth particle hopping motion is of central importance and its presence or absence and make or break the glass [4].  Particle hopping rate R(τ) as a function of perpendicular coordinate z in the film from 3D glassy polymer film simulations, with the free surface at z ≅ 10. Hops are considered to have occurred if particle displacements during a time interval τ is comparable to or larger than the average particle diameter. Dependence of R(τ) on τ indicates temporal correlations, i.e. backandforth hops. In the inner surface layer (green region), R(τ) at small τ is bulklike. However, R(τ) at large τ is larger than its bulk value, signifying surface enhanced mobility [4]. 
Glass formers in general consist of rather hardcore particles, the hops of which necessarily require empty spaces, i.e. free volumes. A free volume of size comparable to a particle is called a void. At deep supercooling, we found that sequences of particle hops are easily rationalized as motions induced by voids [3,11]. In an earlier discussion with David Weitz, he told us that he looked very carefully at his colloidal samples, but could not see any void. After viewing the repetitive particle motions in our simulations, he remarked that "If voids cannot exist in real space, they must be quasiparticles in the phase space." Now, our view is that contiguous voids of sizes comparable to the particles are rare. To increase the entropy, surrounding particles displace inward, so that a void is split into a few free volume fragments in a small neighborhood. We define a quasivoid as a quasiparticle consisting of a few free volume fragments in a small neighborhood with a combined size comparable to the particle size. A direct evidence of quasivoid came from the experimental observation of its reversible transformation into a vacancy, a wellestablished quasiparticle, across a glasscrystal interface. This result was obtained quite accidentally. When producing crystalline colloidal systems, there are inevitably some "bad" samples which have regions of glass coexisting with crystalline domains. We observe that vacancies can diffuse into the disordered regions and induce very typical stringlike hopping motions there. A quasivoid is also able to diffuse back to the crystalline region to regenerate a vacancy. Besides demonstrating the relevance of quasivoid, this is also the first time that the precise structural feature, i.e. a vacancy, causing a stringlike motion is unambiguously identified [11]. ^{4}  Eight particles in a 3D glassy polymer simulation which hop repeatedly. Since particles cannot overlap, the white particle on the left, for example, must move away before one of the red particles can fill its position. The motions can be easily rationalized by considering that all hops are induced by void moving along the chain, which naturally exclude any unphysical particle overlapping event. 
Assuming that particle hops are induced by quasivoids, backandforth hops at deep supercooling then signify the trapping of a quasivoid by the potential energy landscape (PEL) to motions within a few metastable positions. How can quasivoid be untrapped? We observe that isolated stringlike motions repeat itself for much longer time. In contrast, multiple nearby stringlike motions reconfigure much more readily [3,11]. Opposite to PEL dictating quasivoid motions, movements of a quasivoid perturb the PEL of other quasivoids. ^{5} This is because quasivoid motions rearrange particles along its path. As particles are in general distinct in glass, this alters the PEL as experienced by another quasivoid in the immediate neighborhood. This can open up new pathways of motion of the latter. In terms of stringlike motions of particles, this leads to a form of dynamic string interactions [3]. As a result, multiple quasivoids in close proximity mutually open up new hopping pathways for each other and facilitate their motions. ^{6} 

In the DPLM, each particle on a square lattice is of its own type. Any two nearest neighboring particles k and l interact with an energy V_{kl} randomly sampled at the beginning of the simulation from a probability distribution g(V), which can be, e.g. a uniform distribution. ^{7} The random interactions effectively account for the random pair interactions due to random particle separations in a disordered system. A particle can hop to an empty nearest neighboring site, i.e. a void, with a standard kinetic Monte Carlo rate following detailed balance. As particle moves around, the interaction between any two neighboring sites also changes. The model thus has randomness quenched in the configuration space but not quenched in the real space. As a nice surprise, exact equilibrium statistics of the DPLM is known: every interaction simply follows the Boltzmann distribution independently [12]. This greatly assists both simulations and analytic calculations. ^{8} The DPLM demonstrates a wide range of glassy phenomena as well as facilitated dynamics. As temperature decreases, isolated voids are increasingly trapped by the rugged PEL formed by the random pair interactions. In contrast, coupled voids are more mobile due to facilitated dynamics analogous to those observed in MD simulations [12]. The DPLM also successfully reproduces an aging phenomenon called Kovacs paradox, which have not been previously reproduced by any other particle simulations. When the temperature of a sample is abruptly increased (upjumped), the sample seems to persistently 'remember' its initial temperature. In the DPLM, we find strong spatial heterogeneity in the rate of warming, so that some unrelaxed cold domains persist and generate the 'memory'. This solves the paradox. A related and much better known phenomenon called Kovacs effect is also reproduced and is found to be caused by deviation of particle interactions from Boltzmann statistics under nonequilibrium conditions [9]. Different glasses classified as strong or fragile can both be modeled by choosing different particle pairinteraction distribution g(V). (Fragile glasses are those with highly superArrhenius dynamic slowdown, in a more dramatic manner than in strong glasses.) Fragile glass is obtained when g(V) has a bimodal distribution, with a small but finite chance that any two particles when neighboring each other have lowenergy interactions. As temperature decreases, these rare stable states become statistically important. Dynamics is thus highly constrained and slowed down as the system prefers to maintain a significant population of them [7]. Fragile glass has large heat capacity and exhibits strong heatcapacity hysteresis during coolingheating cycles. In the DPLM, heat capacity is dominated by particle pair interactions ^{9} . During cooling of fragile glass, as highenergy interactions are replaced by lowenergy ones in the bimodal distribution, large amount of energy is released, leading to a large heat capacity. In addition, as the particle pairings and the associated interactions relax slowly at low temperature, heat capacity hysteresis results [6]. Most glasses show heat capacity proportional to temperature when cooled to a few Kelvins, widely believed to be due to twolevel systems. As temperature decreases in the DPLM, voids are more and more tightly trapped by the PEL until most of them are completely immobile, but the rest of them mostly hop between two sites, forming twolevel systems. The linear relation between heat capacity and temperature is also explicitly reproduced ^{10} [2]. 

We construct a microscopic theory of glass in finite dimensions based on motions of quasivoids affected by and at the same time affecting the PEL. To account for facilitation, we analyze local regions of size V with m ≥ 1 coupled quasivoids. ^{11} A fit to 2D DPLM results in the end gives an area of V=12 lattice points, indicating the range of facilitation. Due to the rugged PEL, only a fraction q of nearest neighboring hops of each void is considered energetically favorable. Unfavorable hops are neglected. For the DPLM, q as a function of temperature T can be straightforwardly estimated, assuming that an energy excitation of C k_{B}T for a single hop is energetically acceptable. A fit to 2D DPLM results gives C=1.7 which is of order 1 as expected. We apply a fully interacting strings approximation in which every hop of a void completely changes the PEL of any other void in the region. All energetically favorable, i.e. reachable, particle configurations of a region can well be arranged in a random tree, with each edge corresponding to a hopping event of a void. We then solve the percolation problem of the diffusion of configuration in the random tree for various values of m. ^{12} Results are mapped to diffusion of void and further to the diffusion of particles in the real space. Applying the tree theory, we have obtained analytic expressions for the diffusion coefficient [5] and more generally for the particle mean square displacement at arbitrary time [1]. They are expected to be applicable to both molecular and lattice systems, but we have only applied them to the DPLM at this point. With only two tunnable parameters V and C, DPLM results for a wide range of temperature and void density can be reasonably accounted for.  Sketch of the local configuration tree energy landscape for a local region with a single void. A node denotes an arrangement of all particles in the region, i.e. a particle configuration. The energy for a node is represented schematically by the height of the cylinder at the node. Red (grey) cylinders indicate the configuration with lower (higher) energies, which are considered to be more (less) energetically favorable. Blue edges connect the nodes with red cylinders and represent the only possible transitions. The red nodes with the blue edges form the random configurationtree. 
Particle diffusion coefficient D against reciprocal temperature 1/T for various void density ϕ_{v} from DPLM simulations (points) and treetheoretical calculations (lines). The tree theory uses two adjustable parameters for all the curves.  Mean squared displacement (MSD) versus time t for decreasing temperatures (from top to bottom) from DPLM simulations (points) and treetheoretical calculations. We apply the same values of the two parameters used in the fit of D and there is no additional adjustable parameter. Despite the shallow plateaus, the system is deeply supercooled at the lower temperatures studied because particle vibrations are not considered. ^{13} 