6 and the spin-spin coupling is the Fourier transform: of the following tensor: flexible functional form for the non-linear oscillators, in- cluding higher order multipole terms in the spin-spin in- teraction etc., are possible, however will result in largely quantitative corrections. where ˆ k ≡ k /k and the tensor Λ$_{ijkl}$ is from Eq.(11). The Fourier image ˜ J$_{αβ}$ ( k ) of the interaction J$_{αβ}$ ( r ) from Eq.(18) is a dimensionless 6 × 6 matrix, which means here that the interaction between the spins scales as 1 /r 3 at large distance, similarly to the usual dipole-dipole inter- action. We note that Hamiltonians similar to that in Eq.(18) have been employed in the context of configurational glasses. 32,34 Those treatments were restricted to defects of the type Q$_{ij}$ ∝ [ n$_{i}$n$_{j}$ − (1 / 3) δ$_{ij}$ ], where n is a three - component unit vector corresponding to the orientation of a polar group. Grannan at el. 32 point out that mag- nets of the type in Eq.(18) are frustrated even on periodic lattices. The classical Heisenberg-like ferromagnet from Eq.(18) exhibits many distinct states separated by barriers, below its Curie temperature. (See next Section for the mean- field phase-diagram of the model.) These distinct states correspond to distinct configurations in the correspond- ing quenched liquid. The bead motions during transitions between these distinct states correspond to rotations of the six-component, classical “spins” ⃗ g ( m $^{)}$. The relative displacement of two beads, say m and n , during a tran- sition is related to the transition-induced change in the vector ( ⃗ g ( m ) − ⃗ g ( n $^{)}$). The relation, however, is not obvi- ous because of the tensorial nature of the frozen-in stress resulting in vectors being six-component. As in any mag- net below its ordering transition, typical transitions be- tween distinct metastable states of the spin system from Eq.(18) will involve the more spins the lower the tem- perature. Such bigger cooperative regions are necessary because the density of states for a region of fixed size decreases dramatically for lower energies. Specifically, according to the RFOT theory, the cooperativity size in the present case should increase as 1 /s 2 / 3 c , 8 where s$_{c}$ is the entropy per bead (spin). Consistent with the coop- erativity, the individual vectors rotate little relatively to each other so as to minimize local strain, implying the beads will move nearly harmonically most of the time. The transition is, nevertheless, strongly anharmonic, as reflected in a high free energy barrier for the reconfigura- tion. An appropriate progress coordinate is the number of beads that have already moved. 10 We conclude this Section by pointing out that the ob- tained mapping between activated liquid transport and the dynamics of the 6-spin Hamiltonian from Eq.(18) is not exact, of course, but nonetheless, does capture the es- sential features of the activated-transport regime in liq- uids. Systematic improvements, such as using a more It is straightforward to show that the Fourier image ˜ J$_{αβ}$ ( k ) of the interaction J$_{αβ}$ ( r ) from Eq.(18) has the following simple property: ˜ J 2 = ˜ J (i.e. ∑ β ˜ J$_{αβ}$ ˜ J$_{βγ}$ = ˜ J$_{αγ}$ ), and that the trace of the ˜ J matrix equals three for any value of the wave-vector k . These two notions imply the matrix ˜ J has two eigenvalues: 1 and 0, both triply degenerate. For comparison, the analogous matrix for the regular elastic dipole-dipole interaction between 3- component spins, ˆ k$_{i}$ ˆ k$_{j}$ (see Appendix B), has eigenvalues 1, 0, 0. For the electric dipole-dipole interaction, the eigenvalues are − 1 , 0 , 0. We thus conclude that in the sense that electric dipoles form an anti -ferromagnet (for not too oblong samples) 35 the Hamiltonian from Eq.(18) corresponds to a ferromagnet. (This difference can be understood as stemming from the notion that in phonon- mediated interactions, it is like -charges that attract, in contrast with the Coulomb law.) In a mean-field limit, when the interaction J$_{αβ}$ ( r ) from Eq.(18) does not depend on the mutual separation r between spins, the Hamiltonian (18) indeed becomes a simple ferromagnet. This limit may be formally imple- mented by taking the k$_{micro}$ → 0 limit in Eq.(19) while renormalizing the integrand so that the total energy of the system is finite and scales with the total number of the 6-spins. Under these circumstances, the k -integration in Eq.(19) reduces to angular averaging of the ˜ K ten- sor. Upon the averaging, as outlined in Appendix B, the ˜ K tensor becomes a diagonal matrix with eigenval- ues that turn out to be dimensionless combinations of the microscopic elastic constants: one eigenvalue corre- sponding to uniform compression: K/ ( λ +2 µ ), and a five- fold degenerate eigenvalue corresponding to pure shear: (2 / 5)( K +2 µ ) / ( λ +2 µ ). In other words, in the mean field limit, the the dilatational components of local stress on one site do not interact with the isochoric components of local stress on another site. As a result, the mean-field limit of the Hamiltonian (18) is: where ⃗ s ( m ) ≡ ⃗ g ( m $^{)}$/g are unit vectors and denominators 2 N were introduced for convenience. Note that the cou- pling constants J above are proportional to g $^{2}$, according to Eq.(18). The numerical value of the J$_{compr}$/J$_{shear}$ ra- tio is equal to the ratio of the two distinct eigenvalues of H MF = − J$_{compr}$ 2 N [ N ∑ m s ( m ) 1 ] 2 − J$_{shear}$ 2 N 6 ∑ α $_{=2}$[ N ∑ m s ( m ) α ] 2 , (21) ˜ K$_{ijkl}$ = ˆ k$_{p}$ ˆ k$_{q}$ µ ( Λ$_{ijmp}$Λ$_{klmq}$ − λ + µ λ + 2 µ Λ$_{ijmp}$Λ$_{klnq}$ ˆ k$_{m}$ ˆ k$_{n}$ ) , (20) K$_{ijkl}$ ( r ) ≡ ∫ k micro 0 d $^{3}$( k a ) (2 π ) 3 cos( kr ) ˜ K$_{ijkl}$, (19) III. MEAN-FIELD ANALYSIS OF THE SPIN MODEL