Geometric Theory of Lattice Vibrations and Specific Heat
Mikhail Aleksandrovich Shubin, Toshikazu Sunada · Pure and Applied Mathematics Quarterly · 2006
We discuss, from a geometric standpoint, the specific heat of a solid.This is a classical subject in solid state physics which dates back to a pioneering work by Einstein (1907) and its refinement by Debye (1912).Using a special quantization of crystal lattices and calculating the asymptotic of the integrated density of states at the bottom of the spectrum, we obtain a rigorous derivation of the classical Debye T 3 law on the specific heat at low temperatures.The idea and method are taken from discrete geometric analysis which has been recently developed for the spectral geometry of crystal lattices.