  We analyze the spontaneous magnetization reversal of supported monoatomic
chains of finite length due to thermal fluctuations via atomistic spin-dynamics
simulations. Our approach is based on the integration of the Landau-Lifshitz
equation of motion of a classical spin Hamiltonian at the presence of
stochastic forces. The associated magnetization lifetime is found to obey an
Arrhenius law with an activation barrier equal to the domain wall energy in the
chain. For chains longer than one domain-wall width, the reversal is initiated
by nucleation of a reversed magnetization domain primarily at the chain edge
followed by a subsequent propagation of the domain wall to the other edge in a
random-walk fashion. This results in a linear dependence of the lifetime on the
chain length, if the magnetization correlation length is not exceeded. We
studied chains of uniaxial and tri-axial anisotropy and found that a tri-axial
anisotropy leads to a reduction of the magnetization lifetime due to a higher
reversal attempt rate, even though the activation barrier is not changed.
