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diff --git a/monte-carlo.bib b/monte-carlo.bib index c2cf60d..6a0447e 100644 --- a/monte-carlo.bib +++ b/monte-carlo.bib @@ -564,6 +564,42 @@ random field Ising model and finally of quantum spin glasses.}, file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/MKA8WYZZ/Ossola and Sokal - 2004 - Dynamic critical behavior of the Swendsen–Wang alg.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/YHGX7CDT/S0550321304003098.html:text/html} } +@article{martin-mayor_cluster_2009, + title = {Cluster {Monte} {Carlo} algorithm with a conserved order parameter}, + volume = {80}, + url = {https://link.aps.org/doi/10.1103/PhysRevE.80.015701}, + doi = {10.1103/PhysRevE.80.015701}, + abstract = {We propose a cluster simulation algorithm for statistical ensembles with fixed order parameter. We use the tethered ensemble, which features Helmholtz’s effective potential rather than Gibbs’s free energy and in which canonical averages are recovered with arbitrary accuracy. For the D=2,3 Ising model our method’s critical slowing down is comparable to that of canonical cluster algorithms. Yet, we can do more than merely reproduce canonical values. As an example, we obtain a competitive value for the 3D Ising anomalous dimension from the maxima of the effective potential.}, + number = {1}, + urldate = {2018-09-19}, + journal = {Physical Review E}, + author = {Martin-Mayor, V. and Yllanes, D.}, + month = jul, + year = {2009}, + pages = {015701}, + file = {APS Snapshot:/home/pants/.zotero/data/storage/2USVICMH/PhysRevE.80.html:text/html;Martin-Mayor and Yllanes - 2009 - Cluster Monte Carlo algorithm with a conserved ord.pdf:/home/pants/.zotero/data/storage/7G4SJC85/Martin-Mayor and Yllanes - 2009 - Cluster Monte Carlo algorithm with a conserved ord.pdf:application/pdf} +} + +@article{martin-mayor_tethered_2011, + title = {Tethered {Monte} {Carlo}: {Managing} {Rugged} {Free}-{Energy} {Landscapes} with a {Helmholtz}-{Potential} {Formalism}}, + volume = {144}, + issn = {1572-9613}, + shorttitle = {Tethered {Monte} {Carlo}}, + url = {https://doi.org/10.1007/s10955-011-0261-4}, + doi = {10.1007/s10955-011-0261-4}, + abstract = {Tethering methods allow us to perform Monte Carlo simulations in ensembles with conserved quantities. Specifically, one couples a reservoir to the physical magnitude of interest, and studies the statistical ensemble where the total magnitude (system+reservoir) is conserved. The reservoir is actually integrated out, which leaves us with a fluctuation-dissipation formalism that allows us to recover the appropriate Helmholtz effective potential with great accuracy. These methods are demonstrating a remarkable flexibility. In fact, we illustrate two very different applications: hard spheres crystallization and the phase transition of the diluted antiferromagnet in a field (the physical realization of the random field Ising model). The tethered approach holds the promise to transform cartoon drawings of corrugated free-energy landscapes into real computations. Besides, it reduces the algorithmic dynamic slowing-down, probably because the conservation law holds non-locally.}, + language = {en}, + number = {3}, + urldate = {2018-09-19}, + journal = {Journal of Statistical Physics}, + author = {Martin-Mayor, V. and Seoane, B. and Yllanes, D.}, + month = aug, + year = {2011}, + keywords = {Barriers, Effective potential, Monte Carlo methods}, + pages = {554--596}, + file = {Martin-Mayor et al. - 2011 - Tethered Monte Carlo Managing Rugged Free-Energy .pdf:/home/pants/.zotero/data/storage/HEICZ4EE/Martin-Mayor et al. - 2011 - Tethered Monte Carlo Managing Rugged Free-Energy .pdf:application/pdf} +} + @article{ala-nissila_numerical_1994, title = {Numerical studies of the two-dimensional {XY} model with symmetry-breaking fields}, volume = {50}, diff --git a/monte-carlo.tex b/monte-carlo.tex index 4b9ba61..70cc952 100644 --- a/monte-carlo.tex +++ b/monte-carlo.tex @@ -89,17 +89,20 @@ extra degree of freedom, allowing the method to be used in a subcategory of interesting fields \cite{alexandrowicz_swendsen-wang_1989, wang_clusters_1989, ray_metastability_1990}. Static fields have also been applied by including a separate metropolis or heat bath update step after cluster formation -\cite{destri_swendsen-wang_1992, lauwers_critical_1989, ala-nissila_numerical_1994}, and other categories -of fields have been applied using replica methods -\cite{redner_graphical_1998,chayes_graphical_1998,machta_replica-exchange_2000}. -We show that the scaling of correlation time near the critical point of -several models suggests that the `ghost' approach is a natural one, e.g., that -it extends the celebrated scaling of dynamics in these algorithms at zero -field to various non-symmetric perturbations. We also show, by a redefinition -of the spin--spin coupling in a generic class of spin systems, -\emph{arbitrary} external fields can be treated using cluster methods. Rather -than the introduction of a `ghost spin,\!' our representation relies on -introducing a `ghost transformation.\!' +\cite{destri_swendsen-wang_1992, lauwers_critical_1989, +ala-nissila_numerical_1994}, and other categories of fields have been applied +using replica methods \cite{redner_graphical_1998, chayes_graphical_1998, +machta_replica-exchange_2000}. Monte Carlo techniques that involve cluster +updates at fixed magnetization have been used to examine quantities at fixed +field by integrating the associated thermodynamic functions +\cite{martin-mayor_cluster_2009, martin-mayor_tethered_2011}. We show that +the scaling of correlation time near the critical point of several models +suggests that the `ghost' approach is a natural one, e.g., that it extends the +celebrated scaling of dynamics in these algorithms at zero field to various +non-symmetric perturbations. We also show, by a redefinition of the spin--spin +coupling in a generic class of spin systems, \emph{arbitrary} external fields +can be treated using cluster methods. Rather than the introduction of a `ghost +spin,\!' our representation relies on introducing a `ghost transformation.\!' \section{Introduction} |