Memory of incomplete phase transitions from a random squares model

We present a simple two-dimensional model for a phase transition, then study its predictions, in particular the memory properties. The direct transformation is modeled by randomly placing small squares, "nuclei", on an initially empty surface. Then, the nuclei expand ("grow") up to finite final sizes which are randomly chosen in a given range, while keeping their square shape. An important issue is the "interaction" which forces some squares to remain at smaller sizes if the surrounding squares get in the way of their growth. Interestingly, this naturally leads to quasiequal total area covered by the squares of each size after a complete direct transformation. Next, it is shown that the system "remembers" incomplete ("arrested") reverse transformations taking place in reversed order of the squares sizes. The memory is "encrypted" in the distribution of the squares sizes after a next direct transformation and manifests as a significant imbalance between the areas covered by the "big" and "small" (relative to the arrest size) squares. We are able to also reproduce the so-called "hammer effect" and the memorizing of multiple arrest points. Our model is particularly relevant for the thermal memory effect in shape memory alloys, and we actually borrowed many features from existing thermodynamic models addressing this effect. However, here we eliminate the explicit thermodynamics and end up with a statistical geometry model, presumably easier to reproduce.