Cartoon of the sandwich geometry. In the ‘parallel” (or αα) setup (top) both frozen walls are taken from the same equilibrium configuration, while in the “anti-parallel” (αβ) case, they come from different configurations (bottom).
Overlap at the center of the sandwich vs. sandwich half-width d in the parallel setup for (from left to right) T = 0.482, 0.350, 0.246, 0.202. Lines are exponential or compressed-exponential fits (see text). (Inset) Same data in semilog plot.
Overlap in the parallel setup as a function of z for a sandwich of half-width d = 4 at several temperatures. The free region is wide enough that the overlap can reach its bulk value q 0 near the center.
Overlap vs. d − z (distance from wall) at fixed d (same data as Fig. 3 ) with pure exponential fits. (Inset) Same data in semilog scale.
Overlap at center vs. sandwich half-width d at T = 0.203 and two values of L.
Excess energy per mobile particle at T = 0.246 and several values of d, together with the bulk (PBC) average value.
Excess energy vs. d at several temperatures and L = 16. Lines are exponential fits. (Inset) Same data in semilog scale.
Penetration length λ vs. point-to-set correlation length ξ.
Energy decay length l vs. point-to-set correlation length ξ.
Energy decay length vs. two-spin correlation length for the Ising model in the square lattice. Data are from Monte Carlo simulations on a 100 × 100 lattice with single-flip Metropolis dynamics performed above the critical point, at temperatures T = 2.5J, T = 2.4J, T = 2.35J and T = 2.32J, where J is the Ising coupling constant (the critical point is T c ≈ 2.269J). The correlation length was obtained from a fit of the spin-spin space correlation function C(r) = ⟨S(0)S(r)⟩. To determine the length l, sandwich configurations were prepared as explained for the liquid case, measuring the excess energy ΔE(d) = E αβ − E αα for d = 1, 2, 3, 4, 5, 7.5, 10, 15, 20, 25 and 45 lattice spacings, and fitting to an exponential decay.
Point-to-set correlation length ξ, and anomaly exponent ζ, from a fit of Eq. (19) ; penetration length λ, from a fit of Eq. (20) ; and excess energy decay lengthscale l, from a fit of Eq. (21) . At the highest temperature the value of l has large uncertainty as we have very few nonzero values of ΔE.
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