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This work is a study of the application of a theory proposed by J. S. Langer (J.S. Langer, Statistical Theory of the Decay of Metastable States, Annals of Physics 54, 258-275 (1969)) for the calculation of the decay rate (relaxation rate) of a metastable state. The theory is set in the context of statistical mechanics, where the dynamics of a system with a large number of degrees of freedom (order 1023) are reduced to N degrees of freedom, where N is small, when a steady state or equilibrium position is maintained by the entire system. In this thesis N equals 1 or 2 degrees of freedom for a single particle and N equals 4 degrees of freedom for two particles with interaction. In particular, we are interested in the mathematical details of the theory when applied to a Single Domain Ferromagnetic Particle in the Intermediate to High Damping limit, which represents one type of statistical mechanical system for which Langer's theory is applicable. We show by example, that the choice of coordinate system in which the ferromagnetic particle's energy function is written, i.e. with 2N conjugate spherical polar coordinates, has a direct bearing on the successful application of the theory. We elucidate the mathematical details of the application of Langer's theory to systems of non-interacting particles in the presence of zero and non-zero eigenvalues, with examples therein. Finally, we rigorously apply the theory to a system of interacting ferromagnetic particles.
Duff, Gerard . Langer's Method for the Calculation of Escape Rates and Its Application to Systems of Ferromagnets: Doctoral Thesis. Dublin :Dublin Institute of Technology, 2008.