This grant is funded jointly by the Divisions of Materials Research and Mathematical Sciences. The research focus is on the statistical mechanics of strongly interacting many-body systems and their connections with pure mathematics. Two topics of particular interest are the eight-vertex model and the hard-sphere fluid in D-dimensions. The results of these investigations will be used in a new textbook on statistical mechanics being written by the principal investigator (PI).
The PI has discovered an unexpected new infinite dimensional symmetry of the six-vertex model at rational values of the coupling constant that explains what had previously been mysterious degeneracies in the spectrum of the transfer matrix. In the course of extending this work to the eight-vertex model the PI has discovered significant problems with the solutions of this famous thirty-year old problem. It is proposed here to solve these problems and thus provide the first complete solution for the spectrum of the model.
The PI has also found that it is possible to greatly extend the computations of the virial coefficients of the hard sphere fluid. For suitable low orders numerical computations will be replaced by exact analytic computations and for higher orders significant new results will be obtained numerically. In particular the question of negative signs in the virial coefficients will be studied and the new data will be used to critically assess the existing theories of freezing.
Finally, it is proposed to use new developments in the Ising model susceptibility to investigate the possibility that there are deviations from the scaling theory of critical phenomena.
This research provides connections between pure mathematics and condensed matter physics. It also trains students in this interdisciplinary area of research. Finally, the proposed textbook on statistical mechanics will make this research available to a wider audience. %%% This grant is funded jointly by the Divisions of Materials Research and Mathematical Sciences. The research focus is on the statistical mechanics of strongly interacting many-body systems and their connections with pure mathematics. Two topics of particular interest are the eight-vertex model and the hard-sphere fluid in D-dimensions. The results of these investigations will be used in a new textbook on statistical mechanics being written by the principal investigator (PI).
This research provides connections between pure mathematics and condensed matter physics. It also trains students in this interdisciplinary area of research. Finally, the proposed textbook on statistical mechanics will make this research available to a wider audience. ***
