According to Boltzmann, irreversibility emerges as an objective macroscopic property of a large collection of particles, all moving under time-reversible equations of motion. He also expressed his definition of the theoretical entropy in the form
where W is the phase volume of microscopic states available to the system. These views have withstood the test of a century of contemplation, and we see no reason to discard them now. But continued debate leads us to believe that they are in need of further elaboration and interpretation.
Recall the canonical description (5) of the equilibrium system, obtained
from Eq.(4) through the PME. Other properties of the equilibrium system
can now be predicted as expectation values using the set 
,
and the entropy in this scenario can be identified with the physical entropy
of Eq.(6), as well as with the experimental entropy. In addition, we see
that the phase volume of available states plays exactly the same role as
the multiplicity factor did above, in Eqs.(3) and (4). That is, it provides
a measure of the number of possible ways the equilibrium thermodynamic
state can be realized, and this is a large number for a system with 
degrees
of freedom.
To appreciate further the role of W, let the system be taken
from an equilibrium state 
to
an equilibrium state of higher entropy 
.
If the difference 
is
as small as a microcalorie at room temperature, we find that the ratio
of phase volumes in the two states is
The number of ways the state of higher entropy can be realized is incredibly
greater than that for the initial state, and it is just this leverage that
is at the core of irreversibility and the second law. So, when a system
is prepared in some arbitrary nonequilibrium state and then left to itself,
microscopic dynamics will cause the system to evolve to the state of maximum
entropy, subject to whatever constraints are present, because it can do
that in many more ways than any other possibility -- not simply a greater
number of ways, but overwhelmingly many more ways. Put another way,
entropy increase involves degradation of energy and a loss of organization,
and there are always many, many more ways for that to occur than any other
in a system with a large number of degrees of freedom. Thermodynamic variables
in the final state are then calculated as expectation values in the distribution
obtained
by maximizing the entropy subject to the constraints characterizing the
final state.
The weak form of the second law now follows immediately, allowing a
full appreciation of the leverage provided by W. As an example,
let a system be in thermal equilibrium at t=0 and be described by
a statistical operator 
,
experimental entropy 
,
and a theoretical entropy 
.
Then,
where W(0) describes the full range of initial microstates, as
well as measuring our ignorance regarding the actual initial microstate.
Apply a time-dependent dynamical perturbation that induces an adiabatic
change of state, and then allow the system to return to a new equilibrium
state at time t. Because the associated mapping of the Hilbert space
is unitary, we have W(t)=W(0) -- a purported defect
in the definition of 
that,
in fact, is just what yields the second law. At the conclusion of the experiment
a new equilibrium state is again defined by measuring the original macroscopic
variables, and these data provide constraints for a new maximization of
the entropy, denoted 
.
In this state this is also the experimental entropy, so that 
.
The experiment can only be reproducible if 
,
which then leads to a statement of the weak form of the second law:
The strong form simply states that the system will, indeed, evolve to a state of maximum entropy.
It is certainly possible to keep a system from thermal equilibrium with the aid of external forces, and some of these processes are important and useful: steady-state hydrodynamics governed by the macroscopic Navier-Stokes equations, and the highly-organized non-thermal-equilibrium states of living matter are just two examples. These are not situations described fundamentally by the thermodynamic entropy, however, but by rather complicated macroscopic dynamical laws. (This is not meant to imply that life is not a consequence of complex microscopic organization -- only that it is much more than a simple thermodynamic system.)
Irreversibility indeed originates from the microscopic dynamics of the many-body system, but not in the direct way envisioned or desired by many contemporary writers. First of all, it is the complete lack of control over those microscopic variables that leads to the thermodynamic description. If we had such control, it would certainly be reversible and we would have no need for probability distributions and their entropies. And secondly, it is just that large number of microscopic degrees of freedom that makes the state of maximum entropy so overwhelmingly probable as to be almost a certainty -- that it is not a certainty gives meaning to the possibility of fluctuations. Under given constraints, it is the macroscopic state that can be realized in the greatest number of microscopic ways.
Let us conclude with a curious observation. Many people express a strong desire, and often demand, to follow a nonequilibrium process in microscopic detail, but cannot -- for reasons we understand completely, as presented above. But the same people are not bothered in the least by the quantum-mechanical prohibition against following the electron in its transition from one atomic state to another -- for reasons we do not understand at all. This seems to be exactly opposite to the demands of logic -- but, as noted earlier, thermodynamics often elicits emotional, rather than logical, responses.