  The classical Hertz entropy is the logarithm of the volume of phase space
bounded by the constant energy surface; its quantum counterpart, the quantum
Hertz entropy, is $\hat S = k_B \ln \hat N$, where the quantum operator $\hat
N$ specifies the number of states with energy below a given energy eigenstate.
It has been recently proved that, when an isolated quantum mechanical system is
driven out of equilibrium by an external driving, the change in the expectation
of its quantum Hertz entropy cannot be negative, and is null for adiabatic
driving. This is in full agreement with the Clausius principle. Here we test
the behavior of the expectation of the quantum Hertz entropy in the case when
two identical XY spin chains initially at different temperatures are quenched
into a single XY chain. We observed no quantum Hertz entropy decrease. This
finding further supports the statement that the quantum Hertz entropy is a
proper entropy for isolated quantum systems. We further quantify how far the
quenched chain is from thermal equilibrium and the temperature of the closest
equilibrium.
