(a) Scatter plot of the two activity measurements k and k alt, in three different s-ensembles. The ensembles are characteristic of the active phase (s = 0.000), the coexistence region (s = 0.015), and the inactive phase (s = 0.025). The two activity measurements k and k alt are anti-correlated. The trajectory length is t obs = 400Δt. (b) and (c) Marginal distributions of k and k alt from the s-ensemble with s = 0.015. This bimodal behaviour is characteristic of the dynamical phase transition found in Ref. 20 . (d) Scatter plot of k and k alt for three values of s alt and t obs = 200Δt. The data for s alt = −3.0 × 10−5 are similar to the inactive data for s = 0.025. The dashed and dotted lines in (a) and (d) are the same in both panels and are obtained by linear regression analyses on data from (a) for the dots and (d) for the dashes.
(a) and (b) Averaged activities in biased ensembles. Note that panel (b) shows the negatives of the field and the activity, −s alt and ⟨ − k alt⟩. All data are for N = 150 and T = 0.6, except for the red-dashed lines, where N = 300 and we show the linear response behaviour about equilibrium: ⟨K⟩ s = ⟨K⟩0 + s⟨δK 2⟩0 + O(s 2), and similarly for s alt. These linear response results do not capture the non-trivial crossovers, but they do show that the mean and variance of K and K alt are approximately extensive in N, for s = 0 (there is a weak finite-size correction to ⟨k⟩0: particle motion in smaller systems is known to be slightly slower for this system, compared to the bulk 28 ).
(a) Distribution of eigenvalues of the Hessian for both phases. [(a), inset] The difference ΔD(ω2) = [D(ω2) s = 0.04 − D(ω2) s = 0.00] between the phases. The distribution for the active phase is slightly broader, and it associated mean value of ω2 is larger. (b) Distribution of ω where ω2 > 0 for both phases.
(a) Distribution of eigenvalues of the Hessian for inherent structures of both phases. (b) Distribution of ω for inherent structures of both phases. [(b), inset] Dividing D 1(ω) by ω2 emphasises the lack of low frequency modes associated with the inactive phase.
A schematic representation of the differences in the energy landscape between the active and inactive phases. In the inactive phase, the barriers between basins (inherent structures) are smaller making rearrangements on large length scales less likely. These correspond to small values of ω2. The strongly curving directions around basins are less steep in the inactive phase, allowing more motion on short length scales. These correspond to large values of ω2.
(a) Comparison of the partial pair correlation function for large particles between the active and inactive phases. Although there are some differences (the height of the first peak and the depth of the first trough) they are small. (b) The function 4πr 2 G AA(r) which can be integrated to give . The interesting part of this function occurs around the position of the first peak in the pair correlation function. The inset panel shows the difference in this function between the phases, ΔG AA(r) = [G AA(r)] s = 0.04 − [G AA(r)] s = 0.00. This serves to illustrate that the changes in K alt come from structural changes on short length scales.
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