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Ignorance in statistical mechanics
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Consider this penny on my desc. It is a particular piece of metal, 
well described by statistical mechanics, which assigns to it a state, 
namely the density matrix $\rho_0=\frac{1}{Z}e^{-\beta H}$ (in the 
simplest model). This is an operator in a space of functions depending 
on the coordinates of a huge number $N$ of particles. 

The ignorance interpretation of statistical mechanics, the orthodoxy to 
which most introductions to statistical mechanics pay lipservice, claims
that the density matrix is a description of ignorance, and that the 
true description should be one in terms of a wave function; any pure 
state consistent with the density matrix should produce the same 
macroscopic result. 

However, it would be very surprising if Nature would change its 
behavior depending on how much we ignore. Thus the talk about 
ignorance must have an objective formalizable basis independent of 
anyones particular ignorant behavior.

On the other hand, statistical mechanics _always_ works _exclusively_ 
with the density matrix (except in the very beginning where it is 
motivated). 
Nowhere (except there) one makes any use of the assumption that the 
density matrix expresses ignorance. Thus it seems to me that the whole 
concept of ignorance is spurious, a relic of the early days of 
statistical mechanics.

In my opinion, the complete knowledge about a quantum system is 
described by the density matrix, so that microstates are arbitrary 
density matrces and a macrostate is simply a density matrix of a 
special form by which an arbitrary microstate (density matrix) can be 
well approximated when only macroscopic consequences are of interest. 
These special density matrices have the form $\rho=e^{-S/k_B}$ with a 
simple operator $S$ - in the equilibrium case a linear combination of 
1, $H$ (and various number operators $N_j$ if conserved), defining the 
canonical or grand canonical ensemble. This is consistent with all of 
statistical mechanics, and has the advantage of simplicity and 
completeness, compared to the ignorance interpretation, which needs 
the additional qualitative concept of ignorance and with it all sorts 
of questions that are too imprecise or too difficult to answer.


Thus I'd like to invite the defenders of orthodoxy to answer the 
following questions:

(i) Can the claim be checked experimentally that the density matrix 
(a canonical ensemble, say, which correctly describes a macroscopic 
system in equilibrium) describes ignorance? 
- If yes, how, and whose ignorance? 
- If not, why is this ignorance interpretation assumed though nothing 
at all depends on it?

(ii) In a though experiment, suppose Alice and Bob have different 
amounts of ignorance about a system. Thus Alice's knowledge amounts to 
a density matrix $\rho_A$, whereas Bob's knowledge amounts to 
a density matrix $\rho_B$. Given $\rho_A$ and $\rho_B$, how can one 
check in principle whether Bob's description is consistent 
with that of Alice? 

(iii) How does one decide whether a pure state $\psi$ is adequately 
represented by a statistical mechanics state $\rho_0$?
In terms of (ii), assume that Alice knows the true state of the 
system (according to the ignorance interpretation of statistical 
mechanics a pure state $\psi$, corresponding to $\rho_A=\psi\psi^*$), 
whereas Bob only knows the statistical mechanics description, 
$\rho_B=\rho_0$.

Presumably, there should be a kind of quantitative measure 
$M(\rho_A,\rho_B)\ge 0$ that vanishes when $\rho_A=\rho_B)$ and tells 
how compatible the two descriptions are. Otherwise, what can it mean 
that two descriptions are consistent? 

However, the mathematically natural candidate, the relative entropy 
(= Kullback-Leibler divergence) 
    $M(\rho_A,\rho_B)= Tr \rho_A\log\frac{\rho_A}{\rho_B}}$ 
apparently does not work. Indeed, in the situation (iii), 
$M(\rho_A,\rho_B)$ equals the expectation of $\beta H+\log Z$ in the 
pure state; this is minimal in the ground state of the Hamiltonian.
But this would say that the ground state would be most consistent with 
the density matrix of any temperature, an unacceptable condition.

In the terminology of p.5 of the paper
    http://bayes.wustl.edu/etj/articles/gibbs.paradox.pdf
by E.T. Jaynes, the density matrix $\rho_0$ represents a macrostate, 
while each wave function $\psi$ represents a microstate. The question 
is then: When may (or may not) a microstate $\psi$ be regarded as a 
macrostate $\rho_0$ without affecting the predictability of the 
macroscopic observations? In the above case, how do I compute the 
temperature of the macrostate corresponding to a particular microstate 
$\psi$ so that the macroscopic behavior is the same - if it is, and 
which criterion allows me to decide whether (given $\psi$) this 
approximation is reasonable? 

An example where it is not reasonable to regard $\psi$ as a microstate 
consistent with a canonical ensemble is if $\psi$ represents a 
composite system made of two pieces of the penny at different 
temperature. Clearly no canonical ensemble can describe this situation 
macroscopically correct. Thus the criterion sought must be able to 
decide between a state representing such a composite system and the 
state of a penny of uniform temperature, and in the latter case, must 
give a recipe how to assign a temperature to $\psi$, namely the 
temperature that nature allows me to measure.

The temperature of my penny is determined by Nature, hence must be 
determined by a microstate that claims to be a complete description of 
the penny.

I have never seen a discussion of such an identification criterion, 
although they are essential if one wants to maintain the idea - 
underlying the ignorance interpretation - that a completely specified 
quantum state must be a pure state.