 0  1  2  3  4  5  6  7  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7 w’(Q) Q T/T$_{mc}$ 2.13 1.55 1.09 1.00 0.95 0.89 w(Q) w’(Q) (2006). 4 FIG. 5: The derivative W $^{′}$( ˆ Q ) of the thermodynamic po- tential at different temperatures, ranging from 2 . 13 T$_{MC}$ to 0 . 89 T$_{MC}$ . Inset: a cartoon of Maxwell’s construction for the potential and its derivative. constraint is just a stratagem to keep the overlap within the nonconvex region of the potential, as to interrupt relaxation and therefore force phase separation. We have studied dynamic heterogeneities in a glass- forming liquid with constrained global overlap. At low temperature both the dynamic correlation function and the thermodynamic potential indicate that the system phase separates into regions of high and low overlap. On the contrary, at high temperature no phase separation occurs, supporting the view of a surface tension that de- creases at high T . The co-existence of regions belonging to different amorphous ‘states’ (here the high/low overlap patches) is reminiscent of the random first-order theory of thermodynamic relaxation [9]. In the dynamical case the evolution of these regions is driven by a classic coars- ening mechanism, which is stable with the constraint, but ephemeral in absence of the constraint. In the thermo- dynamic case, on the other hand, the evolution of these regions is presumably driven by an entropic mechanism [9]. Our results show that surface tension and metasta- bility stand as key links between the two frameworks. We thank G. Biroli, J.-P. Bouchaud, L. Cugliandolo, S. Franz, W. Kob and F. Zamponi for several important remarks, and ECT* and CINECA for computer time. The work of TSG was supported in part by grants from ANPCyT, CONICET, and UNLP (Argentina). [31] Due to the constraint, the space integral of the correla- tion function is zero, hence the single point S − $^{1}$(0 , t ) = [ ∫ d r G ( r , t )] − 1 must be excluded from the analysis. This also implies that the dynamical susceptibility, χ ( t ) = S (0 , t ) = V [ 〈 Q $^{2}$( t ) 〉 − 〈 Q ( t ) 〉 $^{2}$], a standard marker of het- erogeneous dynamics, is trivially zero and therefore use- less with the constraint. [5] A. Montanari and G. Semerjian, J. Stat. Phys. 125 , 23 [4] J.-P. Bouchaud and G. Biroli, J. Chem. Phys. 121 , 7347 (2004). [3] C. Donati, S. Franz, G. Parisi, and S. C. Glotzer, J. Non- Crys.Sol. 307 , 215 (2002). [2] C. Donati, S. C. Glotzer, and P. Poole, Phys. Rev. Lett. 82 , 5064 (1999). [1] M. D. Ediger, Annu. Rev. Phys. Chem. 51 , 99 (2000). [30] To enforce the constraint we modify the Metropolis algorithm: the probability to accept a move is p = min { 1 , exp − Δ $^{E/T}$} for Q ( t ) ≥ ˆ Q , and p = 0 for Q ( t ) < ˆ Q . [29] Actually, normalization is such that Q = ℓ 3 = 0 . 062876 for completely uncorrelated configurations, while Q = 1 for two identical configurations. [28] The so-called second-moment correlation length is ob- tained by computing A and B using only the first two points, ξ 2 = [ S − $^{1}$( k 1 ) /S − $^{1}$(0) − 1] /k 2 1 , where k 1 = 2 π/L [12]. However, we find that a linear fit to a few small- k points gives equivalent results for ξ 2 yet lowering statis- tical errors. Of course, the free-field form (2) of S ( k, t ) does not hold at generic values of k . [27] The side ℓ of the cells is such that the probability of finding more than one particle in a single box is negligible. [26] We simulate the 3- d soft-sphere binary mixture [23] with parameters as in ref. 7. Simulations were done with Metropolis Monte Carlo with particle swaps [24]. The mode-coupling temperature for this system is T$_{MC}$ =0.226 [25]. Our largest system has N = 16384 particles in a box of length L = 25 . 4. [25] J.-N. Roux, J.-L. Barrat, and J.-P. Hansen, J. Phys.: Condens. Matt. 1 , 7171 (1989). [24] T. S. Grigera and G. Parisi, Phys. Rev. E 63 , 045102 (2001). [23] B. Bernu, J. P. Hansen, Y. Hiwatari, and G. Pastore, Phys. Rev. A 36 , 4891 (1987). [22] S. Franz, J. Stat. Mech. p. P04001 (2005). [21] S. Franz and G. Parisi, Physica A 261 , 317 (1998). [20] S. Franz and G. Parisi, J. Phys. I (France) 5 , 1401 (1995). [19] L. Fernandez, V. Martin-Mayor, and D. Yllanes, Nuclear Physics B 807 , 424 (2009). [18] G. Parisi, arXiv:0911.2265v1 [cond-mat.soft] (2009). [17] C. Cammarota, A. Cavagna, G. Gradenigo, T. S. Grigera, and P. Verrocchio, J. Chem. Phys. 131 , 194901 (2009). [16] C. Cammarota, A. Cavagna, G. Gradenigo, T. S. Grigera, and P. Verrocchio, J. Stat. Mech. p. L12002 (2009). [15] A. Bray, Adv. Phys. 43 , 357 (1994). [14] C. Toninelli, M. Wyart, L. Berthier, G. Biroli, and J. Bouchaud, Phys. Rev. E 71 , 041505 (2005). [13] N. Lacevic, F. W. Starr, T. B. Schroder, and S. C. Glotzer, J. Chem. Phys. 119 , 7372 (2003). [12] S. Caracciolo, R. Edwards, A. Pelissetto, and A. Sokal, Nucl.Phys. B 403 , 475 (1993). [11] A. Montanari and S. Franz, J. Phys. A: Math Theor. 40 , F251 (2007). [10] T. Castellani and A. Cavagna, J. Stat. Mech. p. P05012 (2005). [9] T. Kirkpatrick, D. Thirumalai, and P. Wolynes, Phys. Rev. A 40 , 1045 (1989). [8] J. H. Gibbs and E. A. DiMarzio, J. Chem. Phys. 28 , 373 (1958). [7] G. Biroli, J.-P. Bouchaud, A. Cavagna, T. S. Grigera, and P. Verrocchio, Nature Phys. 4 , 771 (2008). [6] A. Cavagna, T. S. Grigera, and P. Verrocchio, Phys. Rev. Lett. 98 , 187801 (2007).