some more writing

1 files changed, 55 insertions, 45 deletions

diff --git a/monte-carlo.tex b/monte-carlo.tex
index 50cd277..70d026b 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -417,57 +417,72 @@ is because in the low temperature phase the ergodic hypothesis---that the
 time-average value of observables is equal to their ensemble average---is
 violated. As the system size grows the likelihood of a fluctuation in any
 reasonable dynamics that flips the magnetization from one direction to the
-other becomes vanishingly small, and therefore it is inappropriate 
-
-For any finite system, the average magnetization at zero field is identically
-zero at all temperatures. However, people often want to use finite simulations
-to estimate the average magnetization of the Ising model in the thermodynamic
-limit, where due to a superselection principle there is one of two average
-magnetizations at zero field below the critical temperature. The ergodic
-principle that defends the esemble average replacing a time average is no
-longer valid, since once the system has taken a certain magnetization it has
-zero probability of ever changing direction. This is typically
-accomplished by using $\avg{|M|}$. But what is the best way to estimate the
-magnetization of a finite system in the presence of a field?
-
-$\tilde M$ is the true magnetization of the finite-size system. We can
-estimate the magnetization of system in the thermodynamic limit by enforcing
-the superselection principle:
-\[
-  S_\e=\{s\mid M(s)>0\}
-\]
-\[
-  \eavg{A}=\frac{\sum_{s\in S_\e}e^{-\beta\H(s)}A(s)}{\sum_{s\in
-  S_\e}e^{-\beta\H(s)}}=\frac1{Z_\e}\sum_{s\in S_\e}e^{-\beta\H(s)}A(s)
-\]
-\[
-  S_\m=\{s\mid M(s)<0\}
-\]
+other becomes vanishingly small, and therefore it is inappropriate to estimate
+expected values in the low temperature phase by averaging over the whole
+configuration space. Instead, values must be estimated by averaging over the
+portion of configuration space that is accessible to the dynamics.
+
+For finite size systems, like any we would simulate, dynamics at zero field or
+even small nonzero field do allow the whole configuration space to be
+explored. However, people usually want to use the results from finite size
+systems to estimate the expected values in the thermodynamic limit, where this
+is no longer true. At zero field, for instance, it is common practice to use
+$\avg{|M|}$ to estimate the expected value for the magnetization instead of
+$\avg M$. But what to do at finite field? Is this approach justified?
+
+Since, in the thermodynamic limit expected values are given by an average
+over a restricted configuration space, we can estimate those expected values
+at finite size by making the same restriction. Defining the reduced
+configuration spaces
+\begin{align}
+  S_\e^n&=\{s\in S^n\mid \sgn(H)M(s)>0\}
+  \\
+  S_\m^n&=\{s\in S^n\mid \sgn(H)M(s)<0\}
+  \\
+  S_0^n&=\{s\in S^n\mid \sgn(H)M(s)=0\}
+\end{align}
+where
 \[
-  \mavg{A}=\frac{\sum_{s\in S_\m}e^{-\beta\H(s)}A(s)}{\sum_{s\in
-  S_\m}e^{-\beta\H(s)}}=\frac1{Z_\m}\sum_{s\in S_\m}e^{-\beta\H(s)}A(s)
+  \sgn(H)=\begin{cases}1&H\geq0\\-1&H<0.\end{cases}
 \]
-For $H=0$, $\eavg M=\avg{|M|}$, defending the canonical measure of
-the ferromagnetic phase magnetization at zero field. This can be seen first by
-identifying the bijection $s\to-s$ that maps $S_\e$ to $S_\m$. Then, since
-$\H(s)=\H(-s)$, $M(s)=-M(-s)$ when $H=0$, $Z_\e=Z_\m$, $Z=2Z_\e+Z_0$, and
+Clearly, $S^n=S^n_\e\cup S^n_\m\cup S^n_0$, which implies that
+$Z=Z_\e+Z_\m+Z_0$. Since $S^n_\e=\{-s\mid s\in S^n_\m\}$, $|S^n_\e|=|S^n_\m|$.
+At zero field, $\H(s)=\H(-s)$, and therefore $Z_\e=Z_\m$.  This can be used to
+show the near-equivalence (at zero field) of taking expectation values
+$\avg{|M|}$ of the absolute value of the magnetization and taking expectation
+values $\avg M_\e$ of the magnetization on a reduced configuration space,
+since
 \begin{align}
   \avg{|M|}
-  &=\frac1{2Z_\e+Z_0}\sum_{s\in S}e^{-\beta\H(s)}M(s)\\
-  &=\frac1{Z_\e+\frac12Z_0}\sum_{s\in S_\e}e^{-\beta\H(s)}M(s)\\
+  &=\frac1Z\sum_{s\in S^n}e^{-\beta\H(s)}|M(s)|\\
+  &=\frac1{Z_\e+Z_\m+Z_0}\bigg(\sum_{s\in S^n_\e}e^{-\beta\H(s)}|M(s)|+
+  \sum_{s\in S^n_\m}e^{-\beta\H(s)}|M(s)|+\sum_{s\in
+  S^n_0}e^{-\beta\H(s)}|M(s)|\bigg)\\
+  &=\frac1{2Z_\e+Z_0}\bigg(\sum_{s\in S^n_\e}e^{-\beta\H(s)}M(s)+
+  \sum_{s\in S^n_\e}e^{-\beta\H(-s)}|M(-s)|\bigg)\\
+  &=\frac2{2Z_\e+Z_0}\sum_{s\in S^n_\e}e^{-\beta\H(s)}M(s)\\
   &=\frac1{1+\frac{Z_0}{2Z_\e}}\eavg M
 \end{align}
-At infinite temperature, $Z_0/Z_\e\simeq N^{-1/2}\sim L^{-1}$ for large $L$,
+At infinite temperature, $Z_0/Z_\e\simeq n^{-1/2}\sim L^{-1}$ for large $L$,
 $N$. At any finite temperature, especially in the ferromagnetic phase,
 $Z_0\ll Z_\e$ in a much more extreme way.
 
 If the ensemble average over only positive magnetizations can be said to
 converge to the equilibrium state of the Ising model in the thermodynamic
-limit, what of the average over only \emph{negative} magnetizations,
-ordinarily unphysical? 
-
-Collapse of the magnetizations $\avg M$ and $\eavg M$ at low and
-high temperature
+limit, what of the average over only \emph{negative} magnetizations, or the
+space $S^n_\m$,
+ordinarily unphysical?  This is, in one sense, precisely the definition of the
+metastable state of the Ising model. Expected values of observables in the
+metastable state can therefore be computed by considering this reduced
+ensemble.
+
+How, in practice, are these samplings over reduced ensembles preformed? There
+may be efficient algorithms for doing Monte Carlo sampling inside a particular
+reduced ensemble, but for finite-size systems with any of the algorithms
+described here the whole configuration space is available. Any of these can be
+used, however, to simultaneously sample all reduced configuration spaces by
+allowing them to sample the whole space and then only add a contribution to
+a given average if the associated state is in the reduced space of interest.
 
 \begin{acknowledgments}
   Thanks!
@@ -475,10 +490,5 @@ high temperature
 
 %\bibliography{monte-carlo}
 
-
-\begin{itemize}
-  \item metastable and stable state mixing at finite size?
-  \item tracking the wolff algorithm cluster size versus correlation length
-\end{itemize}
 \end{document}