Monte Carlo Methods in Statistical Mechanics: Foundations and ...

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Note tothe Reader The following notes are based on my course \Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms" given at the Cours de Troisieme Cycle de laPhysique en Suisse Romande (Lausanne, Switzerland) in June 1989, and onmy course \Multi-Grid Monte Carlo for Lattice Field Theories" given at the Winter College on Multilevel Techniques in Computational Physics (Trieste, Italy) in January{February 1991. The reader is warned that some ofthis material is out-of-date (this is particularly true as regards reports ofnumerical work). For lack oftime, I have made noattempt to update the text, but Ihave added footnotes marked \Note Added 1996" that correct a few errors and give additional bibliography. My rst two lectures at Cargese 1996 were based on the material included here. My third lecture described the new nite-size-scaling extrapolation method of [97, 98, 99, 100, 101, 102, 103]. 1 Introduction The goal of these lectures is to giveanintroduction to current research on Monte Carlo methods in statistical mechanics and quantum eld theory, withanemphasis on: 1) the conceptual foundations of the method, including the possible dangers and misuses, and the correct use of statistical error analysis and 2) new Monte Carlo algorithms for problems in critical phenomena andquantum eld theory, aimed at reducing or eliminatingthe \critical slowing-down" found in conventional algorithms. These lectures are aimed at a mixed audience of theoretical, computational and mathematical physicists |someofwhom are currently doing orwant todoMonte Carlo studies themselves, others of whom wanttobeable toevaluatethereliabilityofpublished Monte Carlo work. Before embarking on9hours of lectures on Monte Carlo methods, let me o er a warning: Monte Carlo is an extremely bad method it should be used only when all alternative methods are worse. Why isthis so? Firstly, all numerical methods are potentially dangerous, compared to analytic methods there are more ways to make mistakes. Secondly, asnumerical methods go, Monte Carlo is one ofthe least e cient it should be used only on those intractable problems for which all other numerical methods are even less e cient. 1
Let me be more precise about this latter point. Virtually all Monte Carlo methods have the property that the statistical error behaves as error 1 p computational budget (or worse) this is an essentially universal consequence of the central limit theorem. It may be possible to improve the proportionality constant in this relation by afactor of 106 or more | this is one oftheprincipal subjects ofthese lectures | but the overall 1= p n behavior is inescapable. This should be contrasted with traditional deterministic numerical methods whose rate ofconvergence is typically something like1=n4 or e ;n or e ;2n.Therefore, Monte Carlo methods should be used only on those extremely di cult problems in which all alternative numerical methods behave even worse than 1= p n. Consider, for example, the problem of numerical integration in d dimensions, and let us compare Monte Carlo integration with a traditional deterministic method such as Simpson's rule. As is well known, the error in Simpson's rule with n nodal points behaves asymptotically as n ;4=d (for smooth integrands). In low dimension (d 8) it is much worse. So it is not surprising that MonteCarlo is the method of choice for performing high-dimensional integrals. It is still a bad method: with an error proportional to n ;1=2 , it is di cult toachieve more than 4 or 5 digits accuracy. Butnumerical integration in high dimension is very di cult though Monte Carlo is bad, all other known methods are worse. 1 In summary,MonteCarlo methods should be used only when neither analytic methods nor deterministic numerical methods are workable (or e cient). Onegeneral domain of application of Monte Carlo methods will be, therefore, to systems with many degrees of freedom, far from the perturbative regime. Butsuch systems are precisely the ones of greatest interest in statistical mechanics and quantumeld theory! It is appropriateto close this introduction with a general methodological observation, ably articulated by Wood and Erpenbeck [3]: :::these [Monte Carlo] investigations share some ofthe features of ordinary experimental work, in that they are susceptible to bothstatistical and systematic errors. With regardtothese matters, we believe that papers should meet muchthe samestandards as are normally required for experimental investigations. We have in mind the inclusion of estimates of statistical error, descriptions of experimental conditions (i.e. parameters of the calculation), 1 This discussion of numerical integration is grossly oversimpli ed. Firstly, there are deterministic methods better than Simpson's rule and there are also sophisticated Monte Carlo methods whose asymptotic behavior (on smooth integrands) behaves as n ;p with p strictly greater than 1=2 [1, 2]. Secondly, forallthese algorithms (except standard Monte Carlo), the asymptotic behavior as n !1 may be irrelevant in practice, because it is achieved only at ridiculously large values of n. For example, to carry out Simpson's rule with even 10 nodes per axis (a very coarse mesh) requires n =10 d ,which is unachievable for d > 10. 2
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[62] X.-J. Li and A.D. Sokal, Phys.
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[102] G. Mana, A. Pelissetto and A.
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