Varied phenomenology of models displaying dynamical large-deviation singularities

Singularities of dynamical large-deviation functions are often interpreted as the signal of a dynamical phase transition and the coexistence of distinct dynamical phases, by analogy with the correspondence between singularities of free energies and equilibrium phase behavior. Here we study models of driven random walkers on a lattice. These models display large-deviation singularities in the limit of large lattice size, but the extent to which each model's phenomenology resembles a phase transition depends on the details of the driving. We also compare the behavior of ergodic and nonergodic models that present large-deviation singularities. We argue that dynamical large-deviation singularities indicate the divergence of a model timescale, but not necessarily one associated with cooperative behavior or the existence of distinct phases.

Figure 1

(a–c) Large-deviation functions for the time-averaged position $a$ of a driven discrete random walker on a closed lattice of $L$ sites. (d) Walker trajectories showing the instantaneous position $x/L$ under the biased dynamics corresponding to the point $k=k^{★}$ of greatest curvature of $\lambda \left(k\right)$. These trajectories show intermittency, with the walker switching between two locations on either side of the lattice. The timescale for residence in the distinct lattice locations increases with increasing $L$. (e) Histograms of the instantaneous position $x/L$ for the trajectories in (d).

Figure 2

Analog of Fig. 1, now for an undriven walker. (a–c) As for the driven walker, large-deviation functions show increasingly sharp behavior as $L$ increases. (d) Trajectories showing the instantaneous position $x/L$ of the walker at the point of greatest curvature of $\lambda \left(k\right), k^{★}=0$. These trajectories do not exhibit intermittency. (e) As a result, histograms of the instantaneous position $x/L$ for the trajectories in (d) are unimodal.

Figure 3

(a–c) The large-deviation functions of Fig. 2–2, rescaled by $L^2$ to account for the timescale associated with diffusion. The resulting collapse indicates that these systems behave similarly when viewed on the natural timescale $T/L^2$. The large-deviation singularity in this case results from divergence of the diffusive timescale.

Figure 4

Negative log-probability per unit time for the driven walker to achieve a time-averaged position $a=x/L$ in a homogeneous way (blue lines) or a two-state intermittent way (cyan and red lines). In panel (a) the walker's driving is constant, $p\left(x\right)=1/4$, while in panel (b) the tendency of the walker to move left increases with rightward distance, $p\left(x\right)=[1-{(x/L)}^2]/4$. The lattice size $L=20$. When the straight lines lie below the blue lines it is more likely for the system to achieve the time-averaged position $a$ corresponding to $x/L$ in an intermittent way. This construction is consistent with conditioned trajectories of these models (Fig. 5), as long as the trajectory time comfortably exceeds the time on which switching occurs.

Figure 5

Large-deviation rate function $I\left(a\right)$ for the time-averaged position of the driven lattice random walker from Sec. 2 (red), together with that for a super-driven walker (cyan) for which the probability to move left increases with distance from the left-hand edge. Shown below are histograms of the instantaneous walker position from trajectories of the two models conditioned to produce various atypical values of $a$. Consistent with the simple arguments developed in this section, the driven walker displays intermittency, for most values of $a$, involving sites near the edges of the lattice. By contrast, the super-driven walker shows intermittency for $a$ not too far from the typical value $a_0$, involving sites near the middle of the lattice. For $a$ far from $a_0$, its conditioned trajectories are homogeneous. For both models, $L=20$.

Figure 6

Summary of the biased trajectory ensembles associated with three model systems, each generated at the point $k^{★}$ at which the SCGF associated with the time-integrated observable has largest curvature. For the undriven walker (left) and the growth model (right) we have $k^{★}=0$, while for the driven walker (center) we have $k^{★}\ne 0$. The top panels show the instantaneous dynamical observable associated with a single trajectory (two trajectories in the case of the growth model). The lower panels show $a$, the time-integrated version of $x$, for an ensemble of 100 trajectories. The trajectories of the top panels are indicated in the bottom panels by the dark blue or red lines.

Figure 7

Time-integrated observable $a$ for ensembles of trajectories of the irreversible growth model of Refs. [15, 16]. From left to right we show the one-phase region, the critical point, and the two-phase region. Panels are labeled by the values of the model parameter $J$. The black lines are derived from (B1).

Figure 8

Empirical rate functions $-T^{-1}ln{\rho }_T\left(A\right)$ for different values of trajectory time $T$ (denoted by symbols) for the irreversible growth model, compared with the steady-state rate function (B1) (black lines). Panels are labeled by values of the parameter $J$. (a) Far from the critical point $J_{\mathrm{c}}=1$ the empirical rate functions are convex in the one-phase region $(J<1)$ and concave in the two-phase region $(J>1)$. Close to the critical point the behavior of the model, on the times simulated, is influenced by a population of transient trajectories. (b) Enlargement of the right-hand portion of the plot for $J=3$ shows the empirical rate function to be consistent with the form (B1), indicating steady-state growth. (c) We again consider the case $J=1.3$, but now initiate simulations from a pre-made structure of size $T_0=1000$ and composition $a$ consistent with the position of the right-hand minimum of the steady-state rate function (B1). Doing so allows us to effectively access longer timescales than we could from “unassembled” initial conditions. The empirical rate function derived from the resulting ensemble of trajectories is again consistent with the form (B1).

Figure 9

In the two-phase regime of the growth model, trajectory ensembles begun from structures of increasing size $T_0$, having composition $a_0=0$ (top row), become increasingly strongly confined by the unstable dynamical attractor at $a=0$, corresponding to the central minimum of the function (B1). By contrast, large structures of initial composition $a_0\ne 0$ not consistent with any of the minima are driven toward the stable dynamical attractors (in this figure only, the symbol $a_0$ denotes initial composition and not a typical value of the trajectory ensemble.)