  We consider random generalizations of a quantum model of infinite range
introduced by Emch and Radin. The generalization allows a neat extension from
the class $l_1$ of absolutely summable lattice potentials to the optimal class
$l_2$ of square summable potentials first considered by Khanin and Sinai and
generalised by van Enter and van Hemmen. The approach to equilibrium in the
case of a Gaussian distribution is proved to be faster than for a Bernoulli
distribution for both short-range and long-range lattice potentials. While
exponential decay to equilibrium is excluded in the nonrandom $l_1$ case, it is
proved to occur for both short and long range potentials for Gaussian
distributions, and for potentials of class $l_2$ in the Bernoulli case. Open
problems are discussed.
