We further use the relation in Eq.(26) to compute the mean-field expression for F$_{0}$ ( g ) (by numerically minimiz- ing the free energy from Eq.(24)). The result is shown with the dash-dotted line in Fig.5. To compute the full free energy of the solid F ( g ), from Eq.(3), we must make a specific assumption on the penalty F$_{ex}$ ( g ) for dilating the sample. For the sake of argument, we have used a quadratic function F$_{ex}$ ( g ) /Na 3 = cg $^{2}$. A specific value c = 0 . 05 was chosen, so that the resulting barrier be- tween the two phases is about k$_{B}$T , since it is known that the one-particle barrier for surface melting is about k$_{B}$T . 7 The mean-field solution supports the asymptotic analy- sis from Section II that the fully mechanically stable solid is indeed the lowest free energy state; whereas the ape- riodic state, though significantly higher in free energy, is metastable owing to the surface tension between the two phases. 8 IV. SUMMARY AND DISCUSSION We have mapped the liquid activated-transport regime onto the dynamics of a spin model on a fixed lattice. The long-lived aperiodic arrangements, which are char- acteristic of the activated regime, correspond to aperi- odic spin configurations below the Curie point of the spin system. The presence of non-zero shear resistance in the long-lived structures is reflected in the spins hav- ing six components, which correspond to the six inde- pendent entries of the local deformation tensor. Each spin corresponds to a rigid, weakly interacting molecu- lar units (or “bead”) of the ROFT theory. The size of the bead is set unambiguously as the length scale of the inhomogeneity of a frozen-in stress distribution in the ac- tivated regime. The length of the vectors is determined self-consistently as the value of the order parameter for the periodic-to-aperiodic crystal transition. The mutual angles between spins change little during the cooperative reconfigurations, implying individual bonds are deformed nearly harmonically most of the time. The anharmonic- ity of the reconfigurations is reflected in a high nucle- ation barrier in the free energy profile of the transition, the progress coordinate being the number of beads that have already switched their positions to their value in the new arrangement. The present approach exhibits clear parallels with earlier, self-consistent phonon theories of aperiodic crystals, 1,19 which considered the uniform liq- uid state as the reference state and have demonstrated that the uniform liquid can cross-over to an aperiodic- crystal state, accompanied by a discontinuous change of an order parameter corresponding to the bead localiza- tion length. In contrast, here we construct the aperi- odic crystal using a fully mechanically-stable state as the starting point, thus allowing for a straightforward treat- ment of local, high-frequency shear resistance. We have also explicitly demonstrated that aperiodic crystals ex- hibit multiply degenerate states whose mutual intercon- versions can be mapped onto the dynamics of long-range, 6-spin model with anisotropic interactions. The present microscopic picture has implications for the temperature dependence of the activated barriers in supercooled melts that we can begin to discuss al- ready at the mean-field level. In the allowed part of the phase diagram in Fig.4, consider a transition from the J$_{compr}$ < J$_{shear}$ sector to the J$_{compr}$ > J$_{shear}$ sec- tor. As already mentioned, this transition would corre- spond to ordering of an Ising-like assembly of spins and and vice versa for the 5-component Heisenberg spins. To avoid confusion, we emphasize that this Ising-like order- ing takes place along one component of randomly ori- ented, six -component vectors. Now, according to accu- rate calculations, 38 the heat capacity jump upon order- 0 5 10 15 g ( a $^{3}$k$_{B}$ T  ) 1/2 5 10 15 20 25 30 F ( g )  / k B TN 0 5 10 g ( a $^{3}$/k$_{B}$ T ) 1/2 -2 -1 0 1 F  /  g 2 F$_{0}$ ( g ) FIG. 5: The dashed-dotted line shows the mean-field expres- sion for the F 0 ( g ) function. The thick solid line shows the full free energy from Eq.(3), with a specific choice of the func- tion F$_{ex}$ ( g ) /N = c ( g $^{2}$a$^{3}$), where the coefficient c = 0 . 05. The thin solid line is the common tangent of the two portions on the F ( g ) that correspond to the mechanically stable refer- ence state and the aperiodic metastable state. The dashed line shows the curve with the largest value of this coefficient ( c = 5 / 3), at which the F ( g ) curve still exhibits a non-concave portion. In the inset, we plot the second derivative of the mean-field F 0 ( g ). The discontinuity is at g $^{2}$a$^{3}$/k$_{B}$ T = 6, c.f. Eq.(27). when the vector length is large. According to Eqs.(14) and (24), the quadratic term in g from Eqs.(14) will can- cel out at large g , if As a result, After comparing this equation with Eq.(17), we conclude that the mean-field Curie temperature of our magnet is significantly higher than the temperature at which the aperiodic stress pattern sets in, since the actual value of g in the metastable phase is significantly greater than the critical value from Eq.(17), see Fig.5. This notion confirms that, at least in the mean-field limit, our map- ping activated transport onto an ordered spin model is internally consistent. max( J$_{compr}$, J$_{shear}$ ) = g $^{2}$a$^{3}$. (26) g 2 = 6 k$_{B}$T MF Curie a 3 . (27)