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diff --git a/referee_response_1.tex b/referee_response_1.tex new file mode 100644 index 0000000..511fc35 --- /dev/null +++ b/referee_response_1.tex @@ -0,0 +1,184 @@ +% +% test.tex - description +% +% Created by on Mon Oct 22 19:11:00 EDT 2018. +% Copyright (c) 2018 pants productions. All rights reserved. +% +\documentclass[fleqn]{article} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{amsmath,amssymb,latexsym,concmath,mathtools,fullpage} + +\mathtoolsset{showonlyrefs=true} + +\title{LS16406 Referee Response} +\author{Jaron Kent-Dobias \& James P Sethna} + +\begin{document} + +\def\[{\begin{equation}} +\def\]{\end{equation}} + +\maketitle + +We address the referee response to our first submission below. + +\begin{quote} +\begin{verbatim} +As mentioned before, other methods for carrying out cluster +simulations in a field have been proposed. See: + +V. Martin-Mayor, D. Yllanes Phys. Rev. E 80 (2009), 015701 + +In this paper it is shown how to construct a cluster algorithm with an +arbitrary conserved physical quantity, by working in a different +statistical ensemble. This is equivalent to fixing the conjugated +field: both are related by a Legendre transformation, dominated by a +saddle point, so there is a one-to-one relationship (fixing a +magnetization at a value m = c is equivalent to fixing the magnetic +field at a value h* such that <m>_h* = c). The method, Tethered Monte +Carlo, also works in situations where more than one order parameter is +fixed (or more than one field is applied): V. Martin-Mayor, B. Seoane, +D. Yllanes, J. Stat. Phys. 144 (2011) 554. + +Of course, the present approach is very different from that of the +above reference, but perhaps the authors could address the +differences. +\end{verbatim} +\end{quote} + +The indicated paper is indeed interesting, and indeed different in the ways +already outlined by the referee: the algorithm described operates in a +different statistical ensemble. Extracting values in the constant-field +ensemble is done by a numeric integral over results from simulations of many +constant-magnetization systems. Another difference is that the algorithm +described relies on a heat-bath method to update the clusters once formed, and +therefore spiritually belongs to a broad class of existing algorithms we +already cite that form clusters and decide to flip them using metropolis or +heat-bath methods directly in the constant field ensemble. Reference to this +work has now been made alongside these others. Notably, a constant-magnetization ensemble +cluster algorithm that uses clusters without the need for a separate auxiliary +update exists; see JR Heringa \& HWJ Blote, Phys Rev E 57 5 (1998), 4976. This +latter +work feels like a nearer analogue to our own. + +\begin{quote} +\begin{verbatim} +The part of the paper dealing with numerical tests of the method is +severely lacking in detail. First of all, the authors just say, after +eq. (12), that they measure tau "with standard methods", but cite a +paper from 1992 with a different approach to what is commonly done +nowadays. A useful reference could be + +G. Ossola, A.D. Sokal, "Dynamic critical behavior of the Swendsen–Wang +algorithm for the three-dimensional Ising model" Nucl. Phys. B 691 +(2004) 259, https://doi.org/10.1016/j.nuclphysb.2004.04.026 +\end{verbatim} +\end{quote} + +We found the suggested reference very helpful, and now use the methods +described therein for computation of correlation times and their uncertainties. + +\begin{quote} +\begin{verbatim} +In any case, more detailed is needed on the computation of tau, such +as showing some autocorrelation functions and explaining how the error +bars are estimated (this could be an appendix). +\end{verbatim} +\end{quote} + +Since the autocorrelation times and their uncertainties are now computed using +the method suggested above, explicit reference to that method seems sufficient +to explain how the data shown were processed. The autocorrelation functions +themselves appear unremarkable pure exponentials, as the energy +autocorrelation functions also were found to be in Ossola \& Sokal. Moreover, +we compute $\tau$ for six models at at least seven system sizes and at least +fifteen values of the field, meaning that there are hundreds of independent +autocorrelation functions, perhaps beyond the scope of even an appendix. + +\begin{quote} +\begin{verbatim} +A direct computation of z with their data would be much preferable +to the scaling collapses, which are semi-quantitative. Why has this +not been attempted? +\end{verbatim} +\end{quote} + +In the revised manuscript we provide rough estimates for $z$ in the models +studied, but reiterate that since the algorithm is identical to Wolff for +trivial fields, $z$ is simply that of the Wolff algorithm on each model. We +are principally interested in exploring the way the autocorrelation time +scales as one moves away from the zero field critical point---where the +dynamic behavior of the algorithm is already known---in the nonzero field +direction. Remeasuring $z$ for the Wolff algorithm does not accomplish this; +we believe that showing the scaling collapses, which in turn outline the form +of underlying universal scaling functions, does. + +\begin{quote} +\begin{verbatim} +As another general point, the authors should provide some technical +details of their simulations, such as the number of MC steps. For +systems other than the 2D Ising model not even the sizes simulated are +specified. +\end{verbatim} +\end{quote} + +Information about system sizes has been added. Since the work involves so +many separate data points, including such details for each would greatly +increase the size of the manuscript without adding much useful information. At +least $10^6$ runs were preformed for every data point involving +autocorrelation times. + +\begin{quote} +\begin{verbatim} +In Fig. 1, the authors show results for the 2D Ising model up to +sizes L = 256. This is a very small size for such a simple system, +especially considering that the point of these cluster algorithms is +that there is no critical slowing down. The figure should include a +legend saying which curve corresponds to which system size. +\end{verbatim} +\end{quote} + +A $512\times512$ curve has been added, along with system size labels. We +emphasize that unlike Ossola \& Sokal, this is not meant to be a precision study of any of these +models, for which extensive computer time might be dedicated to measuring +quantities for much larger systems, as we ourselves have done in another preprint +(arXiv:1707.03791 [cond-mat.stat-mech]). We believe the behavior we intend to +show---the way the algorithm scales as the critical point is departed in the +field direction---is demonstrated well by the system sizes used. + +\begin{quote} +\begin{verbatim} +Why is tau only computed for the Ising model? In Fig. 2 the +efficiency of the method is demonstrated via a more indirect method +for the other systems. In addition, this figure does not even say +which system sizes have been simulated. +\end{verbatim} +\end{quote} + +Autocorrelation times have been computed for the other models studied. System +size labels have been added to all figures. + +\begin{quote} +\begin{verbatim} +As the authors say, "the goal of statistical mechanics is to compute +expectation values of observables". In this sense, why don't the +authors compute some simple physical observable, such as the energy, +and show how much precision can be achieved for a given computational +effort? At the end of the day, this is the true measure of efficiency +for any Monte Carlo method. +\end{verbatim} +\end{quote} + +A great deal is known about the efficiency of the Wolff algorithm in this +regard. The algorithm described here is exactly the same as the Wolff +algorithm when there is no coupling to an external field. We hope that our +numeric experiments convincingly demonstrate that this algorithm's efficiency +scales from the already known zero field Wolff behavior into nonzero field as +an ordinary scaling analysis would predict. The supplied autocorrelation times +are already an indication of how much precision can be achieved for a given +computational effort. + +\end{document} + |