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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} |