  The time irreversibility problem is the dichotomy of the reversible
microscopic dynamics and the irreversible macroscopic physics. This problem was
considered by Boltzmann, Poincar\'e, Bogolyubov and many other authors and
though some researchers claim that the problem is solved, it deserves a further
study. In this paper an attempt is performed of the following solution of the
irreversibility problem: a formulation of microscopic dynamics is suggested
which is irreversible in time.
  A widely used notion of microscopic state of the system at a given moment of
time as a point in the phase space and also a notion of trajectory does not
have an immediate physical meaning since arbitrary real numbers are non
observable. In the approach presented in this paper the physical meaning is
attributed not to an individual trajectory but only to a bunch of trajectories
or to the distribution function on the phase space.
  The fundamental equation of the microscopic dynamics in the proposed
"functional" approach is not the Newton equation but the Liouville equation for
the distribution function of a single particle. Solutions of the Liouville
equation have the property of delocalization which accounts for
irreversibility. It is shown that the Newton equation in this approach appears
as an approximate equation describing the dynamics of the average values of the
position and momenta. Corrections to the Newton equation are computed.
