The Ising model in three dimensions is fermionized by using integrals over anticommuting variables. The result is generalized to the Ising model in arbitrary dimensions and in a magnetic field. Approximation methods are developed to attack unsolved statistical mechanics models. Perturbation theory and the Hartree approximation are applied to the unsolved monomer-dimer problems. The result is a numerical solution to this unsolved class of problems. Anticommuting variables appear to be a powerful approach to unsolved problems.
Dynamics and statistical mechanics of kinks in one-dimensional classical Ising model in a transverse field is studied in a dilute kink-gas approximation along the line with the theory of Krumhansl and Schrieffer. The energy of a single kink is written in a simple analytic form by approximating an exact one-kink solution. Kinks in the system are then treated as gas of quasi-particles. It is shown that the partition function so obtained is coincident with that of the result obtained previously by using the transfermatrix method. The dynamical structure of the system is also calculated to show the existence of central peak due to kink excitations.