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HOMOGENIZATION ON LATTICES: SMALL PARAMETER LIMITS, HMEASURES, AND DISCRETE WIGNER MEASURES
"... Abstract. We fully characterize the smallparameter limit for a class of lattice models with twoparticle long or short range interactions with no \exchange energy. " One of the problems we consider is that of characterizing the continuum limit of the classical magnetostatic energy of a sequence of m ..."
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Abstract. We fully characterize the smallparameter limit for a class of lattice models with twoparticle long or short range interactions with no \exchange energy. " One of the problems we consider is that of characterizing the continuum limit of the classical magnetostatic energy of a sequence of magnetic dipoles on a Bravais lattice, (letting the lattice parameter tend to zero). In order to describe the smallparameter limit, we use discrete Wigner transforms to transform the storedenergy which is given by the double convolution of a sequence of (dipole) functions on a Bravais lattice with a kernel, homogeneous of degree with N with the cancellation property, as the lattice parameter tends to zero. By rescaling and using Fourier methods, discrete Wigner transforms in particular, to transform the problem to one on the torus, we are able to characterize the smallparameter limit of the energy depending on whether the dipoles oscillate on the scale of the lattice, oscillate on a much longer lengthscale, or converge strongly. In the case where> N, the result is simple and can be characterized by anintegral with respect to the Wigner measure limit on the torus. In the case where = N, oscillations essentially on the scale of the lattice must be separated from oscillations essentially onamuch longer lengthscale in order to characterize the energy in terms of the Wigner measure limit on the torus, an Hmeasure limit, and the limiting magnetization. We show that the classical
Acknowledgements
"... to take me on as a student and for finding an interesting and novel problem on which to work. Thanks to Michael Fowler, we were lucky enough to have Tom Blum work with us during the summer of 1998. We made a great deal of progress and submitted our first publication on this work that summer, in larg ..."
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to take me on as a student and for finding an interesting and novel problem on which to work. Thanks to Michael Fowler, we were lucky enough to have Tom Blum work with us during the summer of 1998. We made a great deal of progress and submitted our first publication on this work that summer, in large part because of Tom’s diligent efforts. He also kept me from becoming lazy while my advisor was away. Tom Gallagher first brought the results of his experiments to our attention, and he was always willing to discuss this work with us. He also provided me with the opportunity to travel to the University of Paris and work with Vladimir Akulin during the spring of