(a) Phase diagram for the 1d soft East model in the (s, ε) plane at J/T = 0.75 (so γ = 0.47). The solid line is the phase boundary between the active and inactive phase. The dashed line is the continuation of the symmetry line (4) into the 1 phase region. The red circles indicate a state point with ε = 5 × 10−4 < εc, on the coexistence line, in the two phase region. The blue triangles indicates a simulation point where ε = 5 × 10−3 > εc. The black × indicates a state near the critical point: ε = 1.5 × 10−3 ≈ εc. (The precise location of the critical point is not known for this model). The inset to (a) shows histograms of the intensive activity k for the three state points in (a) where N = 120 and t obs = 1280. (b) Plots of average intensive activity ⟨k(s)⟩ as a function of field s, for the values of ε, N, and t obs given in (a).
Sample trajectories from the three state points identified in Fig. 1 , taken from near the centers of the distributions P(k) (k ≈ 0.1). Thus, for ε ⩽ εc these trajectories are rare, coming from the trough in the histogram that lies between the two stable basins. Active sites are colored (n i = 1) and inactive sites are white (n i = 0). (a) Trajectory with ε < εc. showing space-time phase separation. (b) Trajectory at ε ≈ εc where the phases are still identifiable but the clusters no longer have a sharp interface. (c) Trajectory at ε > εc showing a single homogeneous phase.
Finite size scaling of the intensive activity ⟨k(s)⟩. In (a) we show ε = 5 × 10−4 < εc and in (b) ε = 1.5 × 10−3 ≈ εc. Inset to (b) shows scaling of the susceptibility for various system sizes t obs · N, with symbols and colors corresponding to the ε values shown in Fig. 1 . The dashed lines indicates linear scaling associated with a first order phase transition (for ε ⩽ εc) and the constant value of susceptibility expected in a single phase (ε > εc).
Schematic phase diagrams in the (s, T) plane. (a) The “hard” East model (U → ∞, varying T/J and s). The system is in the inactive phase for all s > 0 and in the active phase for s ⩽ 0. The equilibrium dynamics are characterized by diverging length and time scales as T → 0. (b) The softened East model with fixed U/J > 1, varying T/J and s. For an intermediate range of temperature, there is a first order phase transition at s > 0. This is also the temperature range in which facilitation effects are strong. Facilitation effects are weak both for large T (above the onset temperature) and for small T (below T x): there is no phase transition in these weak-facilitation regimes.
Article metrics loading...
Full text loading...