  We present numerical results demonstrating the possibility of thermalization
of single-particle observables in a one-dimensional integrable system (a
quasicondensate of ultra-cold, weakly-interacting bosonic atoms being studied
as a definite example). These results may seem counterintuitive because the
physical system is integrable in both the quantum and classical (mean-field)
descriptions. However, we find a class of initial conditions that admits the
relaxation of distributions of single-particle observables to the equilibrium
state very close to the Bose-Einstein thermal distribution of Bogoliubov
quasiparticles. Our numerical results allow us to explain experimentally
observed thermalization in one-dimensional ultracold atomic gases on a short
(~1 ms) time scale.
