Most of the literature says that for a quenched average over disorder, an average over the log of the partition function must be taken:
\begin{equation}
\langle \log Z \rangle,
\end{equation}
while for the annealed average, it's
\begin{equation}
\langle Z \rangle.
\end{equation}
But a while ago, I came across a book that said that the annealed average is not $\langle Z \rangle$, though I don't remember what it said should be calculated instead.
Does anyone know which book this is, or what they might want to calculate instead of $\langle Z \rangle$ for the annealed average?
Best Answer
Hopefully, I can clear this stuff up. The difference between annealed impurities and quenched impurities isn't that complicated, yet the literature out there sure makes it seem that way!
Annealed Impurity: When the system has come into thermal equilibrium, we are interested in determining the probability that the system is in a particular configuration. This probability is a JOINT probability that the impurity is located somewhere and the matrix is located somewhere. In the example of Ma, his system is a magnet with random annealed impurities. The joint probability in this case is the probability of observing a set of spins for the annealed impurities and a set of spins for the matrix. One does not multiply the two probabilities because the probability that the impurities have some set of spins could (and DOES) depend on the probability that the matrix adopts a certain set of spins. If x and y are the annealed and host variables respectively, the JOINT probability of observing a configuration is
$P[x,y]=(1/Z)exp(-H[x,y]/kT)$
$Z=\int dxdyexp(-H[x,y]/kT)$
Free Energy is $-kTlnZ$
Note we do not take an average of this free energy!
Quenched Impurity In this case we don't allow the system to TOTALLY relax in that suppose the impurities are frozen in. A charged polymer (such as a protein!) is a good example of this. The charged monomers remain charged and their positions on the chain are fixed, the molecule can't relax into a state where it rearranges its monomers...its stuck with the order because the location of the monomers are fixed or QUENCHED. Any probability, like the probability of finding the polymer in a particular configuration, is CONDITIONAL on the location of the charged monomers. If x and y are now the quenched and host variables (x could be the location of the charged monomer say), the CONDITIONAL probability is:
$P[y | x]=1/Zexp(-H[y|x]/kT)$
$Z[x]=\int dxexp(-H[y|x]/kT)$
The Free Energy for this particular configuration of quenched impurity is $F[x]=-kTln(Z[x])$. As you prepare different samples of the protein, the charged monomers may end up on different parts of the molecule. This probability distribution $P[x]$ of the charged monomers is related to the way that they were made...it does NOT encode anything about thermal equilibrium. Now let's say you measured the end-to-end distance of the polymer. Theoretically the answer is an average of the end-to-end distances of each configuration of the charged monomer. In an equation it is:
$R^{2}=\sum P[y]R[y]^2$
This last equation is a manifestation of doing an "average of the $ln(Z)$" I did not mention this because it is confusing.
Final Thoughts If we allowed the charged monomers to somehow redistribute themselves (now we are pretending the charged monomer is an annealed variable) so that the free energy of the polymer was a minimum we would find, in general, a DIFFERENT probability distribution of the charge monomer than the quenched case. Hope this helps and sorry for the bad notation!