Boltzmann conceived the H-theorem to explain how a many-body system would approach equilibrium from an arbitrary nonequilibrium initial state. He derived this from what is now called the Boltzmann equation for the single-particle phase-space distribution function, based on the Stosszahlansatz, or collision-number hypothesis. The latter was eventually posited to be true only on average, owing to earlier objections raised by Loschmidt and Zermelo. Rigorous derivations of the Boltzmann equation are strictly valid only in the limit of zero number density, and thus at the very best it can be valid only for very dilute systems. But in 1973 Jaynes showed there exist nontrivial sets of H-theorem-violating thermodynamic states in ordinary systems, so that independently of Stosszahlansatz the H-theorem and Boltzmann equation cannot be universally valid even for dilute systems [15].
In a similar way, Gibbs attempted to construct an analogous scenario within the ensemble theory. Owing to Liousville's theorem, this proved impossible in terms of the actual distribution function, but he introduced the notion of coarse graining and demonstrated that one could exhibit a kind of H-theorem for the coarse-grained distribution. Unfortunately, there are no criteria by which to render this procedure unique, and one has never understood the need for it on a fundamental level. Hopf later attempted to generalize the idea and put it on a more rigorous basis with the concept of mixing. The idea is that, if we take it as a matter of principle that correlations present in the initial nonequilibrium state will decay rapidly, the initial distribution describing that state will decay to the canonical equilibrium distribution if the system is mixing. To get the latter, one now postulates that thermodynamic systems may be Bernoulli, as typified by the Baker's transformation, say. These matters are reviewed clearly and concisely by Lebowitz and Penrose [14].
The difficulty with all this is that it cannot be true, in the following
sense. On the one hand, if we prepare a closed system in some nonequilibrium
initial state described by a statistical operator 
,
the tenets of quantum mechanics require this operator to evolve unitarily.
On the other hand, for well over a hundred years it has been known that
the equilibrium state is described by the canonical distribution 
,
with Hamiltonian H. Under unitary transformation, however, not only
is the theoretical entropy of Eq.(1) invariant, but also every eigenvalue
of .
Hence, unless the eigenvalues of and
coincide,
there is no way that the former can evolve into the latter. Rather than
try to connect the distributions, which depend on the specific constraints
particular to their construction, a more reasonable goal might be to attempt
a demonstration that the expectation values computed with the one evolve
into those computed with the other. Indeed, at least a tacit adaptation
of this strategy is at the heart of attempts to establish kinetic equations
in this field, though we shall have more to say about these efforts below.
It is often stated that the thermodynamic system must be placed into contact with a heat bath, or external forces introduced in some other way, for the system to come to thermal equilibrium. Lamb has been a particularly strong advocate of this view with respect to isolated systems [17]. But these are clearly unnecessary complications, even though real systems are constantly in some kind of interaction with their surroundings. As Jaynes has pointed out [18], if external influences were important we would expect relaxation times to reflect this fact; but we calculate those without such considerations, and they agree quite well with experiment. To understand that an isolated system can come to equilibrium requires nothing more than a contemplation of particle interactions and energy-momentum transfer, along with the basic conservation laws.
We shall provide an explanation for the approach to equilibrium below, after a necessary discussion of irreversibility and ideas about the origin of the second law.