  Macroscopic many-body systems always exhibit irreversible behaviors together
with the entropy increase. However, the underlying microscopic dynamics of the
many-body system, either the (quantum) von Neumann or (classical) Liouville
equation, guarantees the entropy of an isolated system does not change with
time. Notice that, in practical measurements, usually it is the partial
information (e.g., marginal distribution, few-body observable expectation) that
is directly accessible to our observations, rather than the full ensemble
state. But indeed such partial information is sufficient to give most
macroscopic thermodynamic quantities, and they exhibits irreversible behaviors.
At the same time, there is some correlation entropy hiding in the full
ensemble, i.e., the mutual information between different marginal
distributions, but difficult to be sensed in practice. We notice that such
correlation production is closely related to the macroscopic entropy increase
in the standard thermodynamics. In open systems, the irreversible entropy
production of the open system can be proved to be equivalent with the
correlation production between the open system and its environment. During the
free diffusion of an isolated ideal gas, the correlation between the spatial
and momentum distributions is increasing monotonically, and it could well
reproduce the entropy increase result in the standard thermodynamics. In the
presence of particle collisions, the single-particle distribution always
approaches the Maxwell-Boltzmann distribution as its steady state, and its
entropy increase indeed indicates the correlation production between the
particles. In all these examples, the total entropy of the whole isolated
system keeps constant. In this sense, the macroscopic irreversibility and the
reversible microscopic dynamics coincide with each other.
