3  0.001  0.01  1000  10000  100000  1e+06  1e+07 E(t) t free constrained t -1/3  0  0.2  0.4  0.6  0.8  1 10 3 10 4 10 5 10 6 -2 t L -2  0  0.5  1  1.5  2  2.5  3 10 2 10 3 10 4 10 5 t L -2 Q=0.25 Q=0 FIG. 4: Energy difference Δ E ( t ) = E ( t ) − E 0 vs t at T = T$_{MC}$ with constrained dynamics, ˆ Q = 0 . 25. E 0 is a parameter of the fit E ( t ) = E 0 + γt − 1 / $^{3}$. The line is 1 /t 1 / $^{3}$, corresponding to the surface tension exponent θ = 2. FIG. 3: Left: A/B = ξ − $^{2}$( t ) (see Eq. 2) at T = T$_{MC}$ in the constrained (circles) and unconstrained (triangle) cases. Right: the same at T = 1 . 55 T$_{MC}$ . L = 16. parameter is conserved and constrained to take a value in the non-convex interval, phase separation occurs. We have clearly observed phase-separation. Can we define a thermodynamic potential displaying Maxwell’s construc- tion? unconstrained case ξ − 2 keeps well clear of L − $^{2}$, while in the constrained case ξ − $^{2}$( t ) unmistakably goes below L − $^{2}$. This is exactly what we expect in a system with nonzero surface tension undergoing phase separation. We studied two other sizes, L = 8 and L = 25, and in both cases ξ − $^{2}$( t ) drops below L − $^{2}$, indicating phase separa- tion. At higher temperatures, however, though the cor- relation is enhanced by the constraint, the latter is in- effective to make ξ − $^{2}$( t ) drop below L − 2 (Fig. 3, right). These results are consistent with the idea that the sur- face tension decays at high temperature, thus preventing phase separation [16]. Let us proceed minimalistically. Our phase-separating order parameter is the overlap Q , so it is a potential W ( Q ) we are after. Besides, the potential must deter- mine the observed probability distribution of Q through the relation, P ( Q ) = exp [ − NW ( Q )] θ ( Q − ˆ Q ), where the θ -function enforces the constraint [18]. If we com- pute the average linear fluctuation of Q and expand the exponential, we obtain, In systems with conserved order parameter undergo- ing phase separation the domains size ξ ( t ) grows as t 1 / 3 and the dynamics proceeds by reducing the total amount of interfaces, and therefore of energy, in the system [15]. The interface energy per domain scales like ξ $^{θ}$, where θ is the surface tension exponent. The total number of do- mains is L $^{d}$/ξ$^{d}$, so that the total interface energy density is Δ E ( t ) ∼ 1 /ξ ( t ) d − θ ∼ 1 /t ( d − θ ) / $^{3}$. In the standard case θ = d − 1, so that Δ E ( t ) ∼ 1 /t 1 / 3 [15]. Fig.4 shows that something remarkably similar happens in our case. After the constraint kicks in, Δ E ( t ) decays compatibly with an exponent 1 / 3. Hence, even though fitting coars- ening exponents is notoriously difficult, and one must be careful in drawing any conclusion, our data seems to be compatible with the ‘naive’ exponent θ = 2 [16, 17]. In general, phase-separation is the landmark of first order phase transitions and metastability. At the mean- field level one can normally define a thermodynamic po- tential as a function of the order parameter that, below some spinodal point, exhibits a stable and a metastable minimum, corresponding to the two phases. In finite di- mension Maxwell’s construction makes the potential con- vex, so that the derivative of the potential is constant (zero second derivative) in a finite interval (Fig. 5, in- set). Maxwell’s construction implies that when the order W ( Q ) is a finite-dimensional variant of the two-replica potential originally introduced in mean-field spin-glasses [20], and later generalized to structural glasses [21]. This potential is the free energy cost to keep a configuration (the running one in the present work) at fixed overlap Q with a generic equilibrium configuration (the initial reference one). Below a dynamic transition (roughly, the Mode Coupling temperature), the mean-field potential develops a metastable minimum at a finite value of Q . In this framework relaxation at low temperatures can be interpreted as a barrier crossing process, bringing the system from the metastable minimum (short times, finite Q ) to the stable minimum (long times, zero Q ) [22]. The This quantity is easy to compute: we let the system evolve until the constraint is hit, and then we measure the (very small) average fluctuation of the overlap Q over ˆ Q (see [19] for a different definition of the poten- tial). We report W $^{′}$( ˆ Q ) in Fig.(5). The second deriva- tive of the potential is clearly nonzero at high T , whereas around the Mode Coupling temperature a finite region with W $^{′′}$( ˆ Q ) ∼ 0 develops. This is evidence of Maxwell’s construction and it supports the link between phase sep- aration and metastability in our system. W $^{′}$( ˆ Q ) ∼ N − 1 〈 Q − ˆ Q 〉 − $^{1}$. (3)