  We propose a new approach concerning the introduction of time-irreversibility
in statistical mechanics. It is based on a transition function defined in terms
of path integral and verifying a time-irreversible equation. We show first how
dynamic processes may enter in the description of equilibrium states. In order
to do that a characteristic time is associated with closed paths. For large
isolated systems at equilibrium or for systems in contact with a thermostat our
results are identical with those obtained with the Gibbs ensemble methods. For
a model used in the microscopic approaches of the brownian motion no new basic
assumption is required to predict a transition from a quantum state to a
classical one exhibiting a time-irreversible behavior. This demonstration is
sufficient to show that very well accepted approximations can lead to
time-irreversible behaviors for a large class of systems. The difference
between our work and the system+reservoir approaches is underlined. Here
equilibrium states and irreversible processes are described on the same footing
representing a progress in the question of time-irreversibility in statistical
physics. The transition function is also used for describing a small system for
which there is no thermodynamics. By adding to the transition function a second
one characterizing the reverse motion we may describe time-reversible systems.
In a simple case we replace two real valued transition functions by a complex
function verifying a Schrodinger like equation. From this we see how to break
the time-reversibility of this equation and how to investigate the connection
quantum mechanics-thermodynamics from a very fundamental point of view.
