  Nonequilibrium equalities have attracted considerable interest in the context
of statistical mechanics and information thermodynamics. What is remarkable
about nonequilibrium equalities is that they apply to rather general
nonequilibrium situations beyond the linear response regime. However,
nonequilibrium equalities are known to be inapplicable to some important
situations. In this thesis, we introduce a concept of absolute irreversibility
as a new class of irreversibility that encompasses the entire range of those
irreversible situations to which the conventional nonequilibrium equalities are
inapplicable. In mathematical terms, absolute irreversibility corresponds to
the singular part of probability measure and can be separated from the ordinary
irreversible part by Lebesgue's decomposition theorem in measure theory. This
theorem guarantees the uniqueness of the decomposition of probability measure
into singular and nonsingular parts, which enables us to give a well-defined
mathematical and physical meaning to absolute irreversibility. Consequently, we
derive a new type of nonequilibrium equalities in the presence of absolute
irreversibility. Inequalities derived from our nonequilibrium equalities give
stronger restrictions on the entropy production during nonequilibrium processes
than the conventional second-law like inequalities. Moreover, we present a new
resolution of Gibbs' paradox from the viewpoint of absolute irreversibility.
This resolution applies to a classical mesoscopic regime, where two prevailing
resolutions of Gibbs' paradox break down.
