Finite-Size Scaling of a First-Order Dynamical Phase Transition: Adaptive Population Dynamics and an Effective Model

We analyze large deviations of the time-averaged activity in the one-dimensional Fredrickson-Andersen model, both numerically and analytically. The model exhibits a dynamical phase transition, which appears as a singularity in the large deviation function. We analyze the finite-size scaling of this phase transition numerically, by generalizing an existing cloning algorithm to include a multicanonical feedback control: this significantly improves the computational efficiency. Motivated by these numerical results, we formulate an effective theory for the model in the vicinity of the phase transition, which accounts quantitatively for the observed behavior. We discuss potential applications of the numerical method and the effective theory in a range of more general contexts.

Figure 1

The average activity ${⟨K⟩}_s$ for $L=36$ as a function of $sL$, estimated using the feedback method described in the text. As the number of copies $N_c$ increases, the estimators of ${⟨K⟩}_s$ converge to the correct result. The solid black line is the analytical form (11). The parameters $A$, $B$, $\kappa$, $s_c^L$in Eq. (11) are determined by fitting the data outside of the coexistence region ($sL=0.06–0.08$, 0.12–0.15) for $N_c=400$, where the method converges rapidly. The inset shows a comparison between the results obtained from the feedback method (blue dashed line) and from the standard method (blue dotted line) for $N_c=100$.

Figure 2

Exponential divergence of the dynamical susceptibility $\kappa$, defined in Eq. (8). We plot $\log \kappa$ as a function of $L$, as obtained from the feedback method (points) together with the theoretical prediction (10) (straight lines).

Figure 3

Schematic picture of the domain wall dynamics at dynamical phase coexistence, which is analogous to equilibrium phase coexistence on the (2D) surface of a long cylinder [44, 45] (in this latter case the horizontal axis is a spatial coordinate). In the effective model, the fluctuating variable $x\in \{1,2,\dots ,L\}$ represents the width of the active phase.