All modern derivations of statistical quantum mechanics I've found in the literature, have relied on the axiom, that the physical density operator is the one which maximizes the Von-Neumann entropy $$ S=-k\cdot\textrm{tr}(\rho\log\rho)$$ under certain constraints. These constraints define different ensembles, e.g.

  • Micro canonical ensemble: $N=\textrm{const} \quad\wedge\quad E=\textrm{const.}$
  • Canonical ensemble: $N=\textrm{const} \quad\wedge \quad\langle E\rangle=\textrm{const.}$
  • Grand canonical ensemble: $\langle N \rangle=\textrm{const} \quad\wedge \quad\langle E\rangle=\textrm{const.}$

From an information theoretic perspective this can be stated as

Axiom: The density operator containing the least information is physical.

as the entropy is a measure for the information contained in the density distribution.

This Occam's razor approach is missing any microscopic/physical argument. Are people working on finding such argument or does it already exist?

  • $\begingroup$ What specifically are you looking for when you refer to a "microscopic/physical argument"? $\endgroup$ – probably_someone Jun 22 '20 at 22:24
  • $\begingroup$ @probably_someone something which only uses the quantum mechanical axioms. $\endgroup$ – user224659 Jun 22 '20 at 22:27
  • $\begingroup$ Which axioms are you labeling as "the quantum mechanical axioms"? $\endgroup$ – probably_someone Jun 22 '20 at 22:27
  • $\begingroup$ @probably_someone The Von-Neumann axioms or some axioms similar to these: cds.cern.ch/record/598319/files/0212126.pdf $\endgroup$ – user224659 Jun 22 '20 at 22:30

If by "a microscopic/physical argument" you're looking for an argument that only relies on the axioms usually assumed for ordinary quantum mechanics, then that's not possible. To see why, let's look at classical statistical mechanics.

We know, in principle, how to exactly compute the microscopic time-evolution of a classical system with fixed initial conditions. We just have to apply the axioms of classical mechanics. Unfortunately, we have a finite amount of computation power, and so it becomes tremendously impractical or impossible to even measure the microscopic initial conditions for a macroscopic system in the first place, let alone compute the actual time-evolution. This is an important point: it is impossible because we don't know the microscopic initial conditions of the system, since they're too complex to measure.

Classical statistical mechanics fixes this by saying that we don't have to know what the microscopic initial conditions are, because it turns out that the vast majority of initial conditions lead to basically the same macroscopic behavior. It does this by assuming something about the initial conditions, in place of our lack of knowledge. Namely, classical mechanics assumes the ergodic hypothesis, which essentially states that every microstate is equally probable. This is what makes the concept of entropy work: if we had to keep track of the probability of every single microstate, then that's just as much information that we don't have the ability to measure, which would render statistical mechanics useless. But if every microstate is equally probable, then we don't have to measure the initial microscopic properties, and instead we just have to keep track of how many microstates are associated with a particular macrostate (this is precisely what entropy is).

The same is true in quantum statistical mechanics: you need to assume something else, in addition to the axioms of quantum mechanics, in order to eliminate your reliance on microscopic information that you can't practically measure. The assumption of maximal von Neumann entropy is equivalent to assuming maximum ignorance about how the state was prepared, which is very similar to assuming the ergodic hypothesis (since, with no information whatsoever about the initial conditions, you have no reason to favor any particular microstate over another one).


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