  Nonequilibrium, quasi-stationary states of a one-dimensional "hard" $\phi^4$
deterministic lattice, initially thermalized to a particular temperature, are
investigated when brought into contact with a stochastic thermal bath at lower
temperature. For lattice initial temperatures sufficiently higher than those of
the bath, energy localization through the formation of nonlinear excitations of
the breather type during the cooling process occurs. These breathers keep the
nonlinear lattice away from thermal equilibrium for relatively long times. In
the course of time some breathers are destroyed by fluctuations, allowing thus
the lattice to reach another nonequilibrium state of lower energy. The number
of breathers thus reduces in time; the last remaining breather, however,
exhibits amazingly long life-time demonstrated by extensive numerical
simulations using a quasi-symplectic integration algorithm. For the
single-breather states we have calculated the lattice velocity distribution
unveiling non-gaussian features describable in a closed functional form.
Moreover, the influence of the coupling constant on the life-time of a single
breather has been explored. The latter exhibits power-law behaviour as the
coupling constant approaches the anticontinuous limit.
