  A path information is defined in connection with different possible paths of
irregular dynamic systems moving in its phase space between two points. On the
basis of the assumption that the paths are physically differentiated by their
actions, we show that the maximum path information leads to a path probability
distribution in exponentials of action. This means that the most probable paths
are just the paths of least action. This distribution naturally leads to
important laws of normal diffusion. A conclusion of this work is that, for
probabilistic mechanics or irregular dynamics, the principle of maximization of
path information is equivalent to the least action principle for regular
dynamics.
  We also show that an average path information between the initial phase
volume and the final phase volume can be related to the entropy change defined
with natural invariant measure of dynamic system. Hence the principles of least
action and maximum path information suggest the maximum entropy change. This
result is used for some chaotic systems evolving in fractal phase space in
order to derive their invariant measures.
