  The Boltzmann kinetic equation is obtained from an integro-differential
master equation that describes a stochastic dynamics in phase space of an
isolated thermodynamic system. The stochastic evolution yields a generation of
entropy, leading to an increase of Gibbs entropy, in contrast to a Hamiltonian
dynamics, described by the Liouville equation, for which the entropy is
constant in time. By considering transition rates corresponding to collisions
of two particles, the Boltzmann equation is attained. When the angle of the
scattering produced by collisions is small, the master equation is shown to be
reduced to a differential equation of the Fokker-Planck type. When the dynamics
is of the Hamiltonian type, the master equation reduces to the Liouville
equation. The present approach is understood as a stochastic interpretation of
the reasonings employed by Maxwell and Boltzmann in the kinetic theory of gases
regarding the microscopic time evolution.
