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@@ -4,7 +4,6 @@ *.blg *Notes.bib *.dvi -monte-carlo.pdf *.synctex.gz fig-*.tex diff --git a/monte-carlo.pdf b/monte-carlo.pdf Binary files differnew file mode 100644 index 0000000..78be27c --- /dev/null +++ b/monte-carlo.pdf diff --git a/monte-carlo.tex b/monte-carlo.tex index 283b609..7133461 100644 --- a/monte-carlo.tex +++ b/monte-carlo.tex @@ -222,7 +222,7 @@ field, $H=0$. This algorithm solves the problem of critical slowing-down by flipping carefully-constructed clusters of spins at once in a way that samples high-correlated states quickly while also always accepting prospective states. A random site $i$ is selected from the graph and its spin is flipped. Each of the -site's neighbors $j$ is also flipped with probability $1-e^{-2\beta|J_{ij}|}$ +site's neighbors $j$ is also flipped with probability $1-e^{-2\beta J_{ij}}$ if doing so would lower the energy of the bond $i$--$j$. The process is repeated with every neighbor that was flipped. While this algorithm successfully addresses the problems of critical slowing down at zero field, it @@ -233,7 +233,7 @@ then accepted or rejected Metropolis-style based on the change in energy same coupling to the external field, a strictly more efficient but exactly equivalent version of this hybrid is made by distributing the cluster acceptance or rejection across the spins in the cluster as they are -added. In this version, the cluster is abandoned with probability $1-e^{-\beta +added. In this version, the cluster is abandoned with probability $1-e^{-2\beta H}$ every time a spin is added to it. $z=0.29(1)$ \cite{wolff1989comparison,liu2014dynamic} $z=0.35(1)$ for @@ -443,6 +443,10 @@ bottom plot is for simulations at } \input{fig_correlation_collapse-hL} \end{figure} +Our algorithm for the Ising model in a field can be +generalized to run on the $q$-spin Potts or $O(n)$ models in exactly the same +way as the conventional Wolff algorithm. + \section{Magnetization Estimator} At any size, the ensemble average magnetization $\avg M$ is identically zero @@ -464,6 +468,46 @@ 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? +The provenance of the restricted configuration space in the thermodynamic +limit is the fact that, at criticality or for nonzero $H$ under the critical +temperature, an infinite cluster forms and can never be flipped by the +dynamics. In a finite system there is no infinite cluster, and even the +largest cluster has a nonzero probability of being flipped. However, we still +want to be able to use finite systems to estimate the quantities of infinite +ones. We approach this by thinking of a finite system as a piece of an +infinite one. Clearly, time-averages of quantities in the chunk are equal to +time-averages of the whole system, simply with much slower convergence. +However, simulation of finite systems present a problem: in each sample, we +are given a snapshot of a chunk of the system but do not know which direction +the infinite cluster in the surrounding infinite system is pointing. +Therefore, we must make a guess such that, in the limit of larger and larger +system size, the probability that our guess is wrong and that the infinite +cluster is pointed in a direction opposite the one we expected approaches +zero. To estimate the equilibrium values of quantities in the infinite system, +we only admit values resulting from states whose corresponding infinite +cluster is expected to point in the direction of our external field. If the +external field is zero, we choose one direction at random, say positive. + +Snapshots of the chunk what we suspect are likely to be a piece of an infinite +cluster facing opposite the external field are discarded for the purpose of +computing equilibrium values of the system, but still represent something +physical. When the model's state contains an infinite cluster oriented against +the external field, it is known as \emph{metastable}. Therefore in a finite +simulation one can make convergent estimates of quantities in both the +equilibrium and metastable states of the infinite system by dividing sampled +states into two sets: those that are likely snapshots of a finite block in an +infinite system whose infinite cluster is oriented with the field, and those +whose infinite cluster is oriented against the field. All we must do now is +specify how to distinguish between likely oriented with and likely oriented +against. Others have used the direction of the largest cluster in the finite +system to make this distinction. Far from the critical point in the +low-temperature phase this does a clear and consistent job. However, we would +like to study systems very near the critical point. We will take the likely direction of the +infinite cluster to be given by \emph{the direction in which the system is +magnetized}. Clearly, as our finite chunk grows larger and larger, the +probability that its magnetization is different from the direction of the +infinite cluster goes to zero. In the thermodynamic limite + 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 |