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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.