Elementary flip

An elementary flip of a torus front is either a replacement of a northwestern sharp corner at (a, h) by a southeastern sharp corner at (a + 1,6 — 1) or vice versa. It may help to think of a torus front as a sort of flexible snake, where the sharp corners (meaning the vertex in the sharp corner, together with the two adjacent edges) can be flipped about (as if using ball-and-socket joints) the diagonal line connecting the neighbors of the sharp corner vertex. 

It is also clear that moving along edges in d>m,n,g corresponds to elementary flips in TFm,n-m,g- This is completely obvious as long as the flip does not concern the vertex (0,0) (or equivalently, the vertex (m, n — m)). If, on the... 

As mentioned above, each sharp corner defines an elementary flip of T, during which it remains a sharp corner but changes orientation. If this sharp corner were extreme, then the new sharp corner would be extreme (on the opposite side) if and only if w(T) < /2. Clearly, for any torus front, the elementary flips of northwestern extreme corners are mutually noninterfering, and therefore form one flip the same is true for the southeastern extreme corners. Furthermore, if w T) > V2/2, then these two sets of elementary flips do not interfere with each other either. Indeed, if they did, it would mean that... 

Assume now that w(T) > and consider the total flip of all extreme corners. It is geometrically clear that performing any combination of these elementary flips will not increase the width of the torus front, and that performing all of the northwestern flips, or all of the southeastern flips, simultaneously will for sure decrease it. 

Let us fix this width and denote it by w. To every cell a we can associate cells and as follows is obtained from a by adding elementary flips... 

On the other hand, if F is a maximal cell, we have w F) = /2. If we now choose a torus front T belonging to this cell, we will also have w(T) = /2, except for two cases if in each elementary flip we either always choose the northwestern path, or if we always choose the southeastern path. We see that each maximal cell contains exactly two torus fronts of width strictly smaller than a/2, and that these are opposite corners of the cube. Since ThinTO,n,s is connected, the conclusion follows. ... 

It should be realized that unlike the study of equilibrium thermodynamics for which a model is often mapped onto Ising system, elementary mechanism of atomic motion plays a deterministic role in the kinetic study. In an actual alloy system, diffusion of an atomic species is mainly driven by vacancy mechanism. The incorporation of the vacancy mechanism into PPM formalism, however, is not readily achieved, since the abundant freedom of microscopic path of atomic movement demands intractable number of variational parameters. The present study is, therefore, limited to a simple spin kinetics, known as Glauber dynamics [14] for which flipping events at fixed lattice points drive the phase transition. Hence, the present study for a spin system is regarded as a precursor to an alloy kinetics. The limitation of the model is critically examined and pointed out in the subsequent sections. 

It is recalled that the elementary atomic migration by breaking bondings with surrounding atoms is also driven by thermal activation process. This is modeled through the incorporation of the activation barrier, AG, in the spin flipping event via the following equation. 

In the PPF, the first factor P% describes the statistical average of non-correlated spin flip events over entire lattice points, and the second factor P2 is the conventional thermal activation factor. Hence, the product of P and P2 corresponds to the Boltzmann factor in the free energy and gives the probability that on of the paths specified by a set of path variables occurs. The third factor P3 characterizes the PPM. One may see the similarity with the configurational entropy term of the CVM (see eq.(5)), which gives the multiplicity, i.e. the number of equivalent states. In a similar sense, P3 can be viewed as the number of equivalent paths, i.e. the degrees of freedom of the microscopic evolution from one state to another. As was pointed out in the Introduction section, mathematical representation of P3 depends on the mechanism of elementary kinetics. It is noted that eqs.(8)-(10) are valid only for a spin kinetics. 

---