The formulas for the population fractions directly result from the normalization of the pure states
|ψβ;a〉, and|ψα;b〉. Notice that states|ψβ;a〉for different values ofβneed not be orthogonal to
one another. The above result may be applied to two different physical situations.
16.8.1 Second law of thermodynamics
First, suppose we view the entire universe as divided into a certain systema, in which we are
interested, and the environmentbwhich we do not study directly. The Hilbert space of the entire
universeH=H ⊗Hb. Supposeaandbinteract with one another, for example by exchanging
energy. Now, at an initial timet 0 , we prepare the systemain such a way that it has no statistical
correlations with the environment. This is clearly an idealized set-up, which is certainly hard
to realize experimentally. But if we assume absence of correlations at timet 0 , then the density
operator of the universeρ(t 0 ) is a tensor product,
ρ(t 0 ) =ρa⊗ρb (16.79)From the property of additivity of the statistical entropy,we then have
S(ρ(t 0 )) =S(ρa) +S(ρb) (16.80)Unitary time evolution of the entire universe implies that at a later timet > t 0 , we will have
S(ρ(t)) =S(ρ(t 0 )) (16.81)But the density matricesρa(t) andρb(t) at timetwill not be the same as the density matrices
ρa=ρa(t 0 ) andρb=ρb(t 0 ) respectively. In fact, they will not even be unitary evolutions of these
because the Hamiltonian will mix the time evolutions of states inHaandHb, since the systema
interacts with the environment. All we can do is use the definition of these density operators,
ρa(t) = TrHb(ρ(t))
ρb(t) = TrHa(ρ(t)) (16.82)But now, in general, the system at timetwill have statistical correlations with its environmentb,
so that the density matrixρ(t) will no longer be the tensor product ofρa(t) andρb(t). As a result,
by the subadditivity property of the statistical entropy, we have,
S(ρ(t))≤S(ρa(t)) +S(ρb(t)) (16.83)putting this together with (16.80) and (16.81), we obtain,
S(ρa(t 0 )) +S(ρb(t 0 ))≤S(ρa(t)) +S(ρb(t)) (16.84)for timest > t 0. We have just derived the second law of thermodynamics, under the assumptions
we have advocated: the sum of the entropy of the system and of its environment cannot decrease
with time.