Multiple Temperature Model

The multiple temperature model is a natural generalization of the aTAM, where the temperature of a tile system is dynamically adjusted by the experimenter as self-assembly proceeds. This allows sections of assemblies that are bound with less strength to break off of the assembly whenever the temperature is raised. This adds an additional measure of complexity to the assembly system in the encoding of a sequence of temperature changes. This model works off the assumption that everything that can possibly happen in the system at the current temperature will before the temperature is changed to the next value in the given sequence.

Aggarwal, Cheng, Goldwasser, Kao, and Schweller proved that the number of tile types required to assemble “thin” \(k \times N\) rectangles can be reduced from \(\Omega (\frac{N^{1/k}}{k})\) (in the aTAM) to \(\Omega (\frac{log N}{log log N})\) if the temperature is allowed to change but once. Subsequently, Kao and Schweller discovered a clever “bit-flipping” scheme capable of assembling any \(N \times N\) square using \(O(1)\) tile types and \(\Theta(log N)\) temperature changes. Note that the multiple temperature model has a similar flavor to that of the staged self-assembly model in the sense that the input to a tile system in both models can be encoded into a sequence of laboratory operations.