Second Law of Thermodynamics and Particle Creation/Annihilation I have a question regarding the second law of thermodynamics:
In most proofs of the second law of thermodynamics (like Jaynes' proof) the phase space is considered to be of constant dimension.
However in quantum field theory the dimension need not be constant, as particles can be created and annihilated.
In Sir Roger Penrose's lectures on Conformal Cyclic Cosmology he notes that when black holes evaporate the phase space dimension reduces and as a result the entropy can reduce as well, which can explain how in a cyclic universe the entropy gets reset. He notes that the second law is not violated but transcended.
So I was wondering whether there is a formulation of the second law which takes into account this fact? Or is it really true, that the second law can be transcended.
 A: IMO when one is talking of "laws" one should keep in mind that they have a specific variable phase space for validity. Laws are distillations of experimental data and observations that are used in order to relate the mathematical variables of a theory to specific physical quantities that can be measured and observed.
When one goes from the mathematics of classical mechanics and statistical mechanics to field theory, one goes to a completely different mathematical phase space and different laws, which for quantum physics are called "postulates" in order to correlate measurements and observations to the mathematics used to model them. The quantum mechanics postulates have no obvious one to one correlation with the laws of classical mechanics and statistical mechanics. After all, quantum mechanics had to be invented because there were violations in the predictions of the classical theories which obeyed its laws.
There is continuity in the physics theories used , but that does not mean that one can take predictions from one region of validity and impose them by fiat on the other one.

he notes that when black holes evaporate the phase space dimension reduces and as a result the entropy can reduce as well,

Note that this statement is within the mathematical model of classical general relativity and mechanics.

which can explain how in a cyclic universe the entropy gets reset. He notes that the second law is not violated but transcended.

I think the expression "transcended" is used to cover the fact that gravity is not quantized definitively yet, and the model of an evaporating black hole needs quantization of gravity since  the way the black hole evaporates uses a phenomenology of an effective quantized gravity. IMO it is a hand waving term.
This is under research , see for example here .
