A hybrid Monte Carlo algorithm for simulating phase transitions in dense polymer systems

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Lattice models of polymers such as the model of self-avoiding walks (SAW) on a regular lattice have been at the forefront of research in statistical mechanics for more than half a century; they have been of interest to chemists, physicists, and mathematicians alike. Indeed, the Encyclopædia Britannica gives SAW as one of two examples of classical combinatorial problems. Recent work on SAW includes a rigorous mathematical study by Fields medallist Stanislav Smirnov. Polymer models incorporating more realistic effects, such as internal structure and interactions, are considerably harder to analyse. Normally not even the existence of the thermodynamic limit is proven, and one focusses either on computer simulations or on the study of somewhat simplified models that allow explicit solutions for the partition function, while still generally showing very complicated structures. I propose to combine cutting-edge algorithms in the areas of Monte-Carlo simulations, thereby developing a novel hybrid algorithm suitable for the simulation of compact polymers in spatially restricted environments, with the aim of investigating behaviour of polymers in dense environments or at low temperatures. Prominent examples are given by polymer crystallisation or the folding of proteins. This project crosslinks many different disciplines, ranging across Mathematics, Chemistry, and Biology, and the techniques developed within the context of lattice polymers will have an impact reaching far beyond this setting: the proposed algorithm development will be of benefit for general rare-event simulations in the context of complexity theory.