# Is there really the need for an entropy postulate or does it arise from our lack of understanding?

I'm just reading "Equilibrium and Non-Equilibrium Statistical Thermodynamics" from Michel Le Bellac, Fabrice Mortessagne and G. George Batrouni. I encountered in this book a postulate I've heared many times before in various forms:

Postulate of maximum statistical entropy Among all the density operators consistent with the macroscopic constraints, we must choose the density operator D that gives the maximum statistical entropy S[D]. At equilibrium, a macrostate will therefore be represented by this density operator. In more intuitive terms, we choose the most disordered macrostate consistent with the available information. The density operator D is the one which contains no information beyond what is necessary to satisfy the macroscopic constraints.

Why is there a need for this postulate? I know it leads to all of the thermodynamics relations we want. But shouldn't the axioms of quantum mechanics be all we need to arrive at all results of quantum statistical physics/thermodynamics? Is it just that we don't know (yet) how to do it without this assumption or why is it that this postulate is necessary besides the QM axioms?

• cannot say I understand it much, but this might be relevant: en.wikipedia.org/wiki/Ergodic_theory Commented Apr 28, 2020 at 11:00
• If I understand this correctly, the idea is, that thermodynamics should be derivable from fundamental physics just by using statistics and no additional postulates. The problem is how to translate time average into average over possible states Thermodynamics measures time averaged quantities, while statistical mechanics talks about state averages. The above mentioned postulate is linking these two together. Commented Apr 28, 2020 at 11:09
• @Umaxo yes it is very well possible that these things are related, but I can't see this realtion. It doesn't seem to be trivial...
– user224659
Commented Apr 29, 2020 at 11:42

I don't think this is due to lack of knowledge. The laws of thermodynamics are postulates for the study of thermodynamics, but they are explained in statistical mechanics, classically by Gibbs, and quantum mechanically by von Neumann. The explanations require rather more abstract argument than is needed for thermodynamics, so one simply takes the laws as a starting point for applications in that field.

• I think you misunderstood me. The book is about statistical mechanics. This postulate is still necessary to justify the microcanonical or canoncial form of the von neumann density operator. The author makes it seem like there is no way to get this density operators otherwise. And my question is if this is due to a lack of understanding. I feel like statistics and the QM axioms should be enough.
– user224659
Commented Apr 29, 2020 at 11:40
• Using the postulate and assuming the energy average is fixed leads directly to the canonical density operator. But I feel like using this postulate is like cheating.
– user224659
Commented Apr 29, 2020 at 11:43