Quantum Methods for Interacting Particle Systems. II. Glauber Dynamics for Ising Spin Systems

#### M. Gianfelice, M. Isopi

1998, v.4, Issue 3, 411-428

ABSTRACT

Using the formalism and the results described in [M. Isopi, Quantum methods for interacting particle systems, I] and [M. Gianfelice, Quantum methods for interacting particle systems. III. Statistical mechanics of Ising models. J. Stat. Phys.], we discuss the approach to termodynamic equilibrium for discrete spin systems in a framework that generalizes the one originally proposed by R. Glauber. We prove a lower bound estimate for their exponetial rate of convergence to equilibrium in the high temperature regime which is better than those previously known (the case of $d=1$ is amenable to a more detailed analysis, see [R.A. Minlos and A. G. Trishch, Complete spectral decomposition of a generator of Glauber dynamics for the one-dimensional Ising model. Commun. Moscow Math. Soc., 1994, 49, 210-211]). We also give application to some (not necessarily ferromagnetic) Ising-spin models. These results provide an upper bound for the critical temperature of the $d$-dimensional Ising model.

Keywords: Glauber dynamics,spectral gap,interacting particle systems