# microcanonical distribution

We know that in an isolated system, the density matrix is the microcanonical distribution matrix. That this the possibility for all the states with energy in a certain interval is a constant? But how can I deduce this from the postulate of equal probability?

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Josh is saying it in a softer, vague way, but I will say it more clearly: it is not true that for an isolated system, the density matrix must be a microcanonical one. It may be but it is very unlikely. All other density matrices may describe an isolated system, too. That's why they exist. The words "microcanonical" and "equal probability" (among all states allowed in the microcanonical subset) are really synonyma, so any "explanation" why they're the same thing will be at most an an uninteresting tautology. – Luboš Motl Mar 26 '13 at 15:42

The assertion that the density matrix for an isolated system is that of the microcanonical ensemble implies the postulate of equal a priori probabilities since, as you indicate, it assigns equal probabilities to each of the energy eigenstates of the system.

I would then ask you the following question: On what grounds do you assert that the density matrix of an isolated system is microcanonical?

One answer you could give is that the microcanonical density matrix is precisely the one that maximizes the (von-Neumann) entropy of the system.

So then the question becomes: Why is the density matrix that which maximizes the entropy?

The best answers to this come from understanding the von-Neumann entropy in the context of information theory. More on that here. You might also find the following related Physics.SE question interesting:

Why is (von Neumann) entropy maximized for an ensemble in thermal equilibrium?

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