^{1,a)}, Juan P. Garrahan

^{2}and David Chandler

^{3}

### Abstract

In a recent article [M. Merolle *et al.*, Proc. Natl. Acad. Sci. U.S.A.102, 10837 (2005)], it was argued that dynamic heterogeneity in -dimensional glass formers is a manifestation of an order-disorder phenomenon in the dimensions of space time. By considering a dynamical analog of the free energy, evidence was found for phase coexistence between active and inactive regions of space time, and it was suggested that this phenomenon underlies the glass transition. Here we develop these ideas further by investigating in detail the one-dimensional Fredrickson-Andersen (FA) model, in which the active and inactive phases originate in the reducibility of the dynamics. We illustrate the phase coexistence by considering the distributions of mesoscopic space-time observables. We show how the analogy with phase coexistence can be strengthened by breaking microscopic reversibility in the FA model, leading to a nonequilibrium theory in the directed percolation universality class.

The authors thank Fred van Wijland for important comments on links with the Ruelle formalism. They also benefited from discussions with Mauro Merolle and Tommy Miller. (R.L.J) was supported in part by NSF Grant No. CHE-0543158; (J.P.G.) by EPSRC Grants No. GR/R83712/01 and No. GR/S54074/01, and by University of Nottingham Grant No. FEF 3024 and (D.C.) by the U.S. Department of Energy Grant No. DE-FG03-87ER13793.

I. INTRODUCTION

II. MODELS, TRAJECTORIES, AND OBSERVABLES

A. Distribution of the event densities

B. Comparison with a model of appearing and annihilating excitations (AA model)

III. DYNAMICAL ACTION AND THERMODYNAMIC ANALOGY

A. Thermodynamic analogy

B. Discussion of magnetization distributions

C. Distribution of the dynamical action

IV. GENERALIZED MODEL

### Key Topics

- Trajectory models
- 13.0
- Statistical mechanics models
- 12.0
- Free energy
- 11.0
- Phase transitions
- 6.0
- Temperature inversion
- 6.0

## Figures

Illustration of a trajectory in a facilitated model with space-time “bubbles” of the inactive state. The boxes illustrate finite observation space-time windows: the top one corresponds to a typical region; the bottom one is a rare collective fluctuation of size much larger than those typical of the active state. On the right are trajectories from the one-spin facilitated one-dimensional Fredrickson-Andersen model (Ref. 10), at for observation windows of and (smaller observation windows of are also outlined).

Illustration of a trajectory in a facilitated model with space-time “bubbles” of the inactive state. The boxes illustrate finite observation space-time windows: the top one corresponds to a typical region; the bottom one is a rare collective fluctuation of size much larger than those typical of the active state. On the right are trajectories from the one-spin facilitated one-dimensional Fredrickson-Andersen model (Ref. 10), at for observation windows of and (smaller observation windows of are also outlined).

Sketch illustrating the effect of the choice of initial condition in a model of diffusing excitations that branch and coalesce. An initial state with no excitations (left) persists throughout the observation time. All other initial conditions result in the system exploring the active steady state (right).

Sketch illustrating the effect of the choice of initial condition in a model of diffusing excitations that branch and coalesce. An initial state with no excitations (left) persists throughout the observation time. All other initial conditions result in the system exploring the active steady state (right).

Distribution of trajectory magnetization at , , and various observation times. We use which is large enough so that does not depend on . The exponential tails of all have similar gradients: the dotted lines are with .

Distribution of trajectory magnetization at , , and various observation times. We use which is large enough so that does not depend on . The exponential tails of all have similar gradients: the dotted lines are with .

We show at for varying and . We use which is large enough that the results do not depend on . (Top) Increasing observation time at fixed . (Middle) Increasing box size at . As or is increased, we move from a regime in which the tail gradient is independent of the increasing parameter to a regime in which the gradient is proportional to that parameter. (Bottom) We show a typical trajectory for large and where the observation box is outlined: the size of the total spatial region shown is . For large the trajectories are of the form shown in Fig. 1.

We show at for varying and . We use which is large enough that the results do not depend on . (Top) Increasing observation time at fixed . (Middle) Increasing box size at . As or is increased, we move from a regime in which the tail gradient is independent of the increasing parameter to a regime in which the gradient is proportional to that parameter. (Bottom) We show a typical trajectory for large and where the observation box is outlined: the size of the total spatial region shown is . For large the trajectories are of the form shown in Fig. 1.

Data showing (approximate) scaling of in the FA model at various temperatures, scaled according to (9). We plot : the box sizes are ; the observation times are ; and we use . These temperatures are not very small, so there are subleading corrections to scaling, but there is no qualitative change to the scaled distribution on lowering the temperature. Further, the computational time required at is quite significant, so we cannot rule out small systematic errors arising from nonconvergence of our TPS procedure (see Appendix).

Data showing (approximate) scaling of in the FA model at various temperatures, scaled according to (9). We plot : the box sizes are ; the observation times are ; and we use . These temperatures are not very small, so there are subleading corrections to scaling, but there is no qualitative change to the scaled distribution on lowering the temperature. Further, the computational time required at is quite significant, so we cannot rule out small systematic errors arising from nonconvergence of our TPS procedure (see Appendix).

Plot of at , , and , showing secondary maximum at small . (Inset) Enlargement of the secondary peak, shown in a linear scale for .

Plot of at , , and , showing secondary maximum at small . (Inset) Enlargement of the secondary peak, shown in a linear scale for .

Distribution of (reduced) box magnetization in the FA and AA models. The reduced variable , where is the variance of the instantaneous magnetization. Parameters are , , and ; in the FA model ; in the AA model is given by (11) with . For the AA model, is close to Gaussian. The standard deviation is *not* trivially related to the variance of the box magnetization, so the fact that the Gaussian parts of the two distributions are very similar is a nontrivial consequence of the exact mapping between the two models.

Distribution of (reduced) box magnetization in the FA and AA models. The reduced variable , where is the variance of the instantaneous magnetization. Parameters are , , and ; in the FA model ; in the AA model is given by (11) with . For the AA model, is close to Gaussian. The standard deviation is *not* trivially related to the variance of the box magnetization, so the fact that the Gaussian parts of the two distributions are very similar is a nontrivial consequence of the exact mapping between the two models.

Distribution of the action in the FA model for , , obtained with . (Top) Contour plot of the joint probability distribution for action density and magnetization (obtained from independent trajectories). The contours are at . The dotted line is the prediction (34). (Bottom) Distribution of the action density [where ].

Distribution of the action in the FA model for , , obtained with . (Top) Contour plot of the joint probability distribution for action density and magnetization (obtained from independent trajectories). The contours are at . The dotted line is the prediction (34). (Bottom) Distribution of the action density [where ].

(Left) Action distribution in the ensemble with finite . The distribution at is that of Fig. 8 and is shown with symbols. To get data at we simply use (36) and rescale by a constant for convenience (these data are shown as simple lines). (Right) Action distribution with varying at , , and . For we use to ensure that data are independent of . The behavior at small is qualitatively similar to the behavior at small in that the gradient of the exponential tail decreases; at larger a secondary minimum appears. The inset shows an expanded view of the secondary minimum that is present at . Samples with the action exactly equal to zero are omitted from the plot: the probability of this happening is of the order of 1% at .

(Left) Action distribution in the ensemble with finite . The distribution at is that of Fig. 8 and is shown with symbols. To get data at we simply use (36) and rescale by a constant for convenience (these data are shown as simple lines). (Right) Action distribution with varying at , , and . For we use to ensure that data are independent of . The behavior at small is qualitatively similar to the behavior at small in that the gradient of the exponential tail decreases; at larger a secondary minimum appears. The inset shows an expanded view of the secondary minimum that is present at . Samples with the action exactly equal to zero are omitted from the plot: the probability of this happening is of the order of 1% at .

Sketch of the steady state density in the generalized model, as a function of , for different values of with . The axis is the FA model and the axis is a line of critical points. The dotted line separates the region in which the scaling of directed percolation (DP) will apply from those in which can be treated perturbatively [so the scaling will be that of the coagulation-diffusion (CD) fixed point]. The FA model is the unique case for which the critical scaling is coagulation diffusion; for finite the relevant critical point is DP.

Sketch of the steady state density in the generalized model, as a function of , for different values of with . The axis is the FA model and the axis is a line of critical points. The dotted line separates the region in which the scaling of directed percolation (DP) will apply from those in which can be treated perturbatively [so the scaling will be that of the coagulation-diffusion (CD) fixed point]. The FA model is the unique case for which the critical scaling is coagulation diffusion; for finite the relevant critical point is DP.

Sample trajectories at with conditions otherwise similar to Fig. 3 (, , and ). (Left) Sample from center of distribution. (Right) Sample with . Clearly there are more large inactive regions in these trajectories than in those of Fig. 1; increasing from zero leads to proliferation of large “bubbles.”

Sample trajectories at with conditions otherwise similar to Fig. 3 (, , and ). (Left) Sample from center of distribution. (Right) Sample with . Clearly there are more large inactive regions in these trajectories than in those of Fig. 1; increasing from zero leads to proliferation of large “bubbles.”

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