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Introduction to Monte Carlo Simulations and Their Statistical Analysis

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Abstract

This article is a tutorial on Markov chain Monte Carlo simulations and their statistical analysis. The theoretical concepts are illustrated through many numerical assignments from the author's book on the subject. Computer code (in Fortran) is available for all subjects covered and can be downloaded from the web.

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... For this, a multicanonical and PT algorithm were tested and the latter turned out to be more convenient. In the following, we briefly outline the PT scheme used within AMEA and refer to[106][107][108][109][110]for a thorough introduction to MCMC, simulated annealing, multicanonical sampling, and PT. As just stated, in a MCMC scheme, one typically samples from the Boltzmann distribution c b = ...
... This is done through an iteratively created chain of states { } x l , whereby one avoids the explicit calculation of the partition function Z. An effective and wellknown scheme for this is the Metropolis–Hastings algorithm[106,107]. One starts out with some state x l and proposes a new configuration x k , whereby it has to be ensured that every state of the system can be reached in order to achieve ergodicity. ...
... One starts out with some state x l and proposes a new configuration x k , whereby it has to be ensured that every state of the system can be reached in order to achieve ergodicity. The proposed state x k is accepted with probability[106,107] ...
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We present a general scheme to address correlated nonequilibrium quantum impurity problems based on a mapping onto an auxiliary open quantum system of small size. The infinite fermionic reservoirs of the original system are thereby replaced by a small number N B of noninteracting auxiliary bath sites whose dynamics are described by a Lindblad equation, which can then be exactly solved by numerical methods such as Lanczos or matrix-product states. The mapping becomes exponentially exact with increasing N B, and is already quite accurate for small N B. Due to the presence of the intermediate bath sites, the overall dynamics acting on the impurity site is non-Markovian. While in previous work we put the focus on the manybody solution of the associated Lindblad problem, here we discuss the mapping scheme itself, which is an essential part of the overall approach. On the one hand, we provide technical details together with an in-depth discussion of the employed algorithms, and on the other hand, we present a detailed convergence study. The latter clearly demonstrates the above-mentioned exponential convergence of the procedure with increasing N B. Furthermore, the influence of temperature and an external bias voltage on the reservoirs is investigated. The knowledge of the particular convergence behavior is of great value to assess the applicability of the scheme to certain physical situations. Moreover, we study different geometries for the auxiliary system. On the one hand, this is of importance for advanced manybody solution techniques such as matrix product states which work well for short-ranged couplings, and on the other hand, it allows us to gain more insights into the underlying mechanisms when mapping non-Markovian reservoirs onto Lindblad-type impurity problems. Finally, we present results for the spectral function of the Anderson impurity model in and out of equilibrium and discuss the accuracy obtained with the different geometries of the auxiliary system. In particular, we show that allowing for complex Lindblad couplings produces a drastic improvement in the description of the Kondo resonance.
... Grid-based methods [42] provide an alternative approach for approximating non-linear probability density functions, although they rapidly become computationally intractable in high dimensions. The Monte Carlo simulation-based techniques such as Sequential Monte Carlo (SMC) [43] and Markov Chain Monte Carlo (MCMC) [44] are among the most powerful and popular methods of approximating proabilities. They are also very flexible as they do not make any assumptions regarding the probability densities to be approximated. ...
... Here, a Markov chain is a sequence of random samples generated according to a transition probaility (kernel) function with Markovian property, i.e. the transition probabilities between different sample values in the state space depend only on the random samples' current state. It has been shown that one can always use a well-designed Markov chain that converges to a unique stationary density of interest (in terms of drawn samples) [44]. The convergence occurs after a sufficiently large number of iterations, called the burn-in period. ...
... The good look of the linear fits is deceptive as they have a rather large χ 2 and a small goodness of fit Q (see p. 111 of [53]) which can be explained by the small errors bars. Another potentially deceptive result is that the imaginary part of the lowest zero decreases like L −3.08 . ...
... It is questionable that two MUCA runs could lead to a reliable estimate of the errors. An error bar from just two independent measurements fluctuates strongly and reaches a 95% confidence range only at about 14 (instead of 2) error bars (see p.78 of [53]). We decided therefore to smoothen the error bars by assuming that the real relative error is the same for all four of our large lattices. ...
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We present high-accuracy calculations of the density of states using multicanonical methods for lattice gauge theory with a compact gauge group U(1) on 44, 64, and 84 lattices. We show that the results are consistent with weak and strong coupling expansions. We present methods based on Chebyshev interpolations and Cauchy theorem to find the (Fisher’s) zeros of the partition function in the complex β=1/g2 plane. The results are consistent with reweighting methods whenever the latter are accurate. We discuss the volume dependence of the imaginary part of the Fisher’s zeros, the width and depth of the plaquette distribution at the value of β where the two peaks have equal height. We discuss strategies to discriminate between first- and second-order transitions and explore them with data at larger volume but lower statistics. Higher statistics and even larger lattices are necessary to draw strong conclusions regarding the order of the transition.
... One of the most well-known generalized-ensemble algorithms is perhaps the multicanonical algorithm (MUCA) [42,43] (for reviews see, e.g., Refs. [44,45]). The method is also referred to as entropic sampling [46,47] and adaptive umbrella sampling [48] of the potential energy [49]. ...
... (The details of this process are described, for instance, in Refs. [44,45]). However, the iterative process can be non-trivial and very tedius for complex systems. ...
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... Grid-based methods [42] provide an alternative approach for approximating non-linear probability density functions, although they rapidly become computationally intractable in high dimensions. The Monte Carlo simulation-based techniques such as Sequential Monte Carlo (SMC) [43] and Markov Chain Monte Carlo (MCMC) [44] are among the most powerful and popular methods of approximating proabilities. They are also very flexible as they do not make any assumptions regarding the probability densities to be approximated. ...
... Here, a Markov chain is a sequence of random samples generated according to a transition probaility (kernel) function with Markovian property, i.e. the transition probabilities between different sample values in the state space depend only on the random samples' current state. It has been shown that one can always use a well-designed Markov chain that converges to a unique stationary density of interest (in terms of drawn samples) [44]. The convergence occurs after a sufficiently large number of iterations, called the burn-in period. ...
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... Kim, Shephard and Chib (1998), Meyer and Yu (2000) and Tse, Zhang and Yu (2004) noted that the mixing performance of the sample paths can be measured by simulation inefficiency factor (SIF), which is also known as the integrated autocorrelation time by Berg (2005). It is estimated as the sample mean from an sampler that draws iid observations from the posterior distribution, SIF is given by σ 2 /σ 2 . ...
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... Ref. [7]). In fact, the recently discovered [6] simple modification of FMC to effectively suppress critical slowing down [8] [9] [10] promises to promote FMC in the current top league of simulation algorithms from critical behavior. Yet, even though there is definitely interest in using the machinery of FMC, despite past efforts [11] [12] many potential users found the task of setting up the formulas and working out a concrete implementation to be too painful to seriously consider using it in practice. ...
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The Fourier Monte Carlo algorithm represents a powerful tool to study criticality in lattice spins systems. In par- ticular, the algorithm constitutes an interesting alternative to other simulation approaches for models with microscopic or effective long-ranged interactions. However, due to the somewhat involved implementation of the basic algorithmic machinery, many researchers still refrain from using Fourier Monte Carlo. It is the aim of the present article to lower this barrier. Thus, the basic Fourier Monte Carlo algorithm is presented in great detail with emphasis on providing ready-to-use formulas for the reader's own implementation.
... A bunch of other MCMC techniques are discussed later in this chapter but the discussion is by no means exhaustive. For a detailed discussion see again Gilks et al. (1996) or Berg (2004) and Hobson et al. (2010). ...
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... [26] in MCMC suggest that the number of iterations, NUM, for the MCMC sampler to settle in equilibrium should be, NUM ≫ τ. ...
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... ALGORITHM This section proposes a developed algorithm based on the path reputation based scheme being solved by the classical monte Carlo method (PRRMC) [15]. The misbehaving sensor nodes are both intentionally and unintentionally trouble making nodes for many reasons, e.g., unexpected node failures, overflow traffic injections in a vulnerable node, node malfunctions , etc. Theoretically, the monte Carlo uses the feedback experiences to evaluate a proper action to be taken at a particular state. ...
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... Various so-called approximate estimation methods have been developed, but unfortunately none has good properties for all possible models and data sets (e.g., for ungrouped binary data). For this reason, numerical methods involving the Markov Chain Monte Carlo method (Berg 4) have increasingly been used, as increasing computing power and advances in methods have made them more practical. However, drawbacks exist here too, as the underlying distribution needs to be specified in advance. ...
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... Otherwise, we stay in the present state, i.e. the next state is x. See [8] for a general introduction to MCMC methods. A simple choice for the proposal kernel is to sample from the prior distribution, i.e. k(x, y) = µ 0 (y). ...
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Several decision-analytic modeling techniques are in use for pharmacoeconomic analyses. Discretely integrated condition event (DICE) simulation is proposed as a unifying approach that has been deliberately designed to meet the modeling requirements in a straightforward transparent way, without forcing assumptions (e.g., only one transition per cycle) or unnecessary complexity. At the core of DICE are conditions that represent aspects that persist over time. They have levels that can change and many may coexist. Events reflect instantaneous occurrences that may modify some conditions or the timing of other events. The conditions are discretely integrated with events by updating their levels at those times. Profiles of determinant values allow for differences among patients in the predictors of the disease course. Any number of valuations (e.g., utility, cost, willingness-to-pay) of conditions and events can be applied concurrently in a single run. A DICE model is conveniently specified in a series of tables that follow a consistent format and the simulation can be implemented fully in MS Excel, facilitating review and validation. DICE incorporates both state-transition (Markov) models and non-resource-constrained discrete event simulation in a single formulation; it can be executed as a cohort or a microsimulation; and deterministically or stochastically.
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Critical engineering systems generally demand high reliability while considering uncertainties, which makes failure event identification and reliability analysis based on computer simulation codes computationally very expensive. Rare event can be defined as a failure event that has a small probability of failure value, normally less than 10−5. This paper presents a new approach, referred to as Accelerated Failure Identification Sampling (AFIS), for probability analysis of rare events, enabling savings of vast computational efforts. To efficiently identify rare failure sample points in probability analysis, the proposed AFIS technique will first construct a Gaussian process (GP) model for system performance of interest and then utilize the developed model to predict unknown responses of Monte Carlo sample points. Second, a new quantitative measure, namely “failure potential”, is developed to iteratively search sample points that have the best chance to be a failure sample point. Third, the identified sample points with highest failure potentials are evaluated for the true performance and then used to update the GP model. The failure identification process will be iteratively preceded and the Monte Carlo simulation will then be employed to estimate probabilities of rare events if the maximum failure potential of existing Monte Carlo samples falls below a given target value. Two case studies are used to demonstrate the effectiveness of the developed AFIS approach for rare events identification and probability analysis.
Chapter
In molecular simulations of complex systems with many degrees of freedom, conventional Monte Carlo and molecular dynamics simulations in canonical ensemble or isobaric-isothermal ensemble suffer from a great difficulty, in which simulations tend to get trapped in states of energy local minima. A simulation in generalized ensemble performs a random walk in specified variables and overcomes this difficulty. In this chapter, we review the generalized-ensemble algorithms. Replica-exchange method, multicanonical algorithm, and their extensions are described. Some simulation results based on these generalized-ensemble algorithms are also presented.
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The use of computer simulations as “virtual microscopes” is limited by sampling difficulties that arise fromthe large dimensionality and the complex energy landscapes of biological systems leading to poor convergences already in folding simulations of single proteins. In this chapter, we discuss a few strategies to enhance sampling in bimolecular simulations, and present some recent applications.
Chapter
Sensor-based localization has been found to be one of the most preliminary issues in the world of Mobile/Wireless Robotics. One can easily track a mobile robot using a Kalman Filter, which uses a Phase Locked Loop for tracing via averaging the values. Tracking has now become very easy, but one wants to proceed to navigation. The reason behind this is that tracking does not help one determine where one is going. One would like to use a more precise “Navigation” like Monte Carlo Localization. It is a more efficient and precise way than a feedback loop because the feedback loops are more sensitive to noise, making one modify the external loop filter according to the variation in the processing. In this case, the robot updates its belief in the form of a probability density function (pdf). The supposition is considered to be one meter square. This probability density function expands over the entire supposition. A door in a wall can be identified as peak/rise in the probability function or the belief of the robot. The mobile updates a window of 1 meter square (area depends on the sensors) as its belief. One starts with a uniform probability density function, and then the sensors update it. The authors use Monte Carlo Localization for updating the belief, which is an efficient method and requires less space. It is an efficient method because it can be applied to continuous data input, unlike the feedback loop. It requires less space. The robot does not need to store the map and, hence, can delete the previous belief without any hesitation.
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The term ‘time series’ refers to data that can be represented as a sequence. This includes for example financial data in which the sequence index indicates time, and genetic data (e.g. ACATGC …) in which the sequence index has no temporal meaning. In this tutorial we give an overview of discrete-time probabilistic models, which are the subject of most chapters in this book, with continuous-time models being discussed separately in Chapters 4, 6, 11 and 17. Throughout our focus is on the basic algorithmic issues underlying time series, rather than on surveying the wide field of applications.
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Network meta-analysis (NMA) is an extension of pairwise meta-analysis that facilitates comparisons of multiple interventions over a single analysis. It is the method in which multiple interventions (that is, three or more) are compared using both direct comparisons of interventions within randomized controlled trials and indirect comparisons across trials based on a common comparator. NMA is methodologically complex compared to simple pairwise meta-analysis as it accounts for a broader evidence base. Results from NMA are more useful to policy makers, service commissioners, and providers when making choices between multiple alternatives than those from multiple, separate pairwise meta-analyses. It can be an ideal choice to be extended to compare complex interventions that are multifaceted. Apart from the numer-ous benefits the NMA offers, it is prone to methodological complications that need to be understood, implemented, and finally reported correctly. This article is meant to provide a primer to the various methodological issues pertaining to NMA. The NMA can be as valid as a standard pairwise meta-analysis if these methodological issues are taken care of.
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Conference Paper
This paper presents a decision-centered lifetime and reliability prognostics using a generic Bayesian framework. This generic Bayesian framework models and updates sensory degradation data, remaining life, and reliability using non-conjugate Bayesian updating mechanism. Thus, it continuously updates lifetime distributions of degraded system in realtime. Furthermore, the generic Bayesian framework eliminates dependency of evolutionary updating process on a selection of distribution types for the parameters of a sensory degradation model. The Markov Chain Monte Carlo (MCMC) technique is employed as a numerical method of non-conjugate Bayesian updating framework. While accounting for variability in loading conditions, material properties, and manufacturing tolerances over the population of system samples, different reliabilities will be identified for different samples. So, reliability distribution for an engineering system can be obtained and updated in a Bayesian format. The proposed Bayesian methodology is generally applicable for different degradation models and prior distribution types. The proposed methodology is successfully demonstrated with 26 resistors for the lifetime and reliability prognostics. Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc.
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Multicanonical MCMC (Multicanonical Markov Chain Monte Carlo; Multicanonical Monte Carlo) is discussed as a method of rare event sampling. Starting from a review of the generic framework of importance sampling, multicanonical MCMC is introduced, followed by applications in random matrices, random graphs, and chaotic dynamical systems. Replica exchange MCMC (also known as parallel tempering or Metropolis-coupled MCMC) is also explained as an alternative to multicanonical MCMC. In the last section, multicanonical MCMC is applied to data surrogation; a successful implementation in surrogating time series is shown. In the appendix, calculation of averages and normalizing constant in an exponential family, phase coexistence, simulated tempering, parallelization, and multivariate extensions are discussed.
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We present and analyze in detail a test bench for random number sequences based on the use of physical models. The first two tests, namely the cluster test and the autocorrelation test, are based on exactly known properties of the two-dimensional Ising model. The other two, the random walk test and the n-block test, are based on random walks on lattices. We have applied these tests to a number of commonly used pseudorandom number generators. The cluster test is shown to be particularly efficient in detecting periodic correlations on bit level, while the autocorrelation, the random walk, and the n-block tests are very well suited for studies of weak correlations in random number sequences. Based on the test results, we demonstrate the reasons behind errors in recent high precision Monte Carlo simulations, and discuss how these could be avoided.
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A new approach to Monte Carlo simulations is presented, giving a highly efficient method of simulation for large systems near criticality. The algorithm violates dynamic universality at second-order phase transitions, producing unusually small values of the dynamical critical exponent.
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We present a new Monte Carlo algorithm that produces results of high accuracy with reduced simulational effort. Independent random walks are performed (concurrently or serially) in different, restricted ranges of energy, and the resultant density of states is modified continuously to produce locally flat histograms. This method permits us to directly access the free energy and entropy, is independent of temperature, and is efficient for the study of both 1st order and 2nd order phase transitions. It should also be useful for the study of complex systems with a rough energy landscape.
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Multicanonical ensemble simulations for the simulation of first-order phase transitions suffer from exponential slowing down. Monte Carlo autocorrelation times diverge exponentially with free energy barriers ΔF, which in L d boxes grow as L d−1. We exemplify the situation in a study of the 2D Ising-model at temperature T/T c =0.63 for two different lattice manifolds, toroidal lattices, and surfaces of cubes. For both geometries the effect is caused by discontinuous droplet shape transitions between various classical crystal shapes obeying geometrical constraints. We use classical droplet theory and numerical simulations to calculate transition points and barrier heights. On toroidal lattices we determine finite size corrections to the droplet free energy, which are given by a linear combination of Gibbs–Thomson corrections, capillary wave fluctuation corrections, constant terms, and logarithmic terms in the droplet volume. Tolman corrections are absent. In addition, we study the finite size effects on the condensation phase transition, which occurs in infinite systems at the Onsager value of the magnetization. We find that this transition is of discontinuous order also. A combination of classical droplet theory and Gibbs–Thomson corrections yields a fair description for the transition point and for the droplet size discontinuity for large droplets. We also estimate the nucleation barrier that has to be surmounted in the formation of the stable droplet at coexistence.
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Sampling, Statistics and Computer Code Error Analysis for Independent Random Variables Markov Chain Monte Carlo Error Analysis for Markov Chain Data Advanced Monte Carlo Parallel Computing Conclusions, History and Outlook.
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This is a tutorial review on the Potts model aimed at bringing out in an organized fashion the essential and important properties of the standard Potts model. Emphasis is placed on exact and rigorous results, but other aspects of the problem are also described to achieve a unified perspective. Topics reviewed include the mean-field theory, duality relations, series expansions, critical properties, experimental realizations, and the relationship of the Potts model with other lattice-statistical problems.
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Preface; 1. Overview; 2. Structure and scattering; 3. Thermodynamics and statistical mechanics; 4. Mean-field theory; 5. Field theories, critical phenomena, and the renormalization group; 6. Generalized elasticity; 7. Dynamics: correlation and response; 8. Hydrodynamics; 9. Topological defects; 10. Walls, kinks and solitons; Glossary; Index.
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The critical-point anomaly of a plane square m×n Ising lattice with periodic boundary conditions (a torus) is analyzed asymptotically in the limit n→∞ with ξ=m/n fixed. Among other results, it is shown that for fixed τ=n(T-Tc)/Tc, the specific heat per spin of a large lattice is given by Cmn(T)/kBmn=A0lnn+B(τ, ξ)+B1(τ)(lnn)/n+B2(τ, ξ)/n+O[(lnn)3/n2], where explicit expressions can be given for A0 and for the functions B, B1, and B2. It follows that the specific-heat peak of the finite lattice is rounded on a scale δ=ΔT/Tc∼1/n, while the maximum in Cmn(T) is displaced from Tc by ε=(Tc-Tmax)/Tc∼1/n. For ξ0>ξ>ξ0-1, where ξ0=3.13927⋯, the maximum lies above Tc; but for ξ>ξ0 or ξ<ξ0-1, the maximum is depressed below Tc; when ξ=∞, ξ0, or ξ0-1, the relative shift in the maximum from Tc is only of order (lnn)/n2. Detailed graphs and numerical data are presented, and the results are compared with some for lattices with free edges. Some heuristic arguments are developed which indicate the possible nature of finite-size critical-point effects in more general systems.
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This new and updated edition deals with all aspects of Monte Carlo simulation of complex physical systems encountered in condensed-matter physics, statistical mechanics, and related fields. After briefly recalling essential background in statistical mechanics and probability theory, it gives a succinct overview of simple sampling methods. The concepts behind the simulation algorithms are explained comprehensively, as are the techniques for efficient evaluation of system configurations generated by simulation. It contains many applications, examples, and exercises to help the reader and provides many new references to more specialized literature. This edition includes a brief overview of other methods of computer simulation and an outlook for the use of Monte Carlo simulations in disciplines beyond physics. This is an excellent guide for graduate students and researchers who use computer simulations in their research. It can be used as a textbook for graduate courses on computer simulations in physics and related disciplines. A broad and self-contained overview of Monte Carlo simulations Contains extensive cross-referencing between simulation and relevant theory and between applications of similar algorithms in different contexts Provides many applications, examples, `recipes', and specific case studies
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It is suggested that the interface free energy between bulk phases with a macroscopically flat interface can be estimated from the variation of certain probability distribution functions of finite blocks with block size. For a liquid-gas system the probability distribution of the density would have to be used. The method is particularly suitable for the critical region where other methods are hard to apply. As a test case, the two-dimensional lattice-gas model is treated and it is shown that already, from rather small blocks, one obtains results consistent with the exact solution of Onsager for the surface tension, by performing appropriate extrapolations. The surface tension of the three-dimensional lattice-gas model is also estimated and found to be reasonably consistent with the expected critical behavior. The universal amplitude of the surface tension of fluids near their critical point is estimated and shown to be in significantly better agreement with experimental data than the results of Fisk and Widom and the first-order 4-d renormalization-group expansion. Also the universal amplitude ratio used in nucleation theory near the critical point is estimated.
It is shown that the two-dimensional q-component Potts model is equivalent to a staggered ice-type model. It is deduced that the model has a first-order phase transition for q>4, and a higher-order transition for q<or=4. The free energy and latent heat at the transition are calculated.
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A general method, suitable for fast computing machines, for investigating such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two-dimensional rigid-sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four-term virial coefficient expansion.
Article
A general method, suitable for fast computing machines, for investigating such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two-dimensional rigid-sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four-term virial coefficient expansion. The Journal of Chemical Physics is copyrighted by The American Institute of Physics.
Book
Preface; 1. Introduction; 2. Some necessary background; 3. Simple sampling Monte Carlo methods; 4. Importance sampling Monte Carlo methods; 5. More on importance sampling Monte Carlo methods of lattice systems; 6. Off-lattice models; 7. Reweighting methods; 8. Quantum Monte Carlo methods; 9. Monte Carlo renormalization group methods; 10. Non-equilibrium and irreversible processes; 11. Lattice gauge models: a brief introduction; 12. A brief review of other methods of computer simulation; 13. Monte Carlo simulations at the periphery of physics and beyond; 14. Monte Carlo studies of biological molecules; 15. Outlook; Appendix; Index.
Article
We consider the exact correlation length calculations for the two-dimensional Potts model at the transition point $\beta_{\rm t}$ by Klümper, Schadschneider and Zittartz, and by Buffenoir and Wallon. We argue that the correlation length calculated by the latter authors is the correlation length in the disordered phase and then combine their result with duality and the assumption of complete wetting to give an explicit formula for the order-disorder interface tension $\sigma_{\rm od}$ of this model. The result is used to clarify a controversy stemming from different numerical simulations of $\sigma_{\rm od}$.
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This article describes an approach towards a random number generator that passes all of the stringent tests for randomness we have put to it, and that is able to produce exactly the same sequence of uniform random variables in a wide variety of computers, including TRS80, Apple, Macintosh, Commodore, Kaypro, IBM PC, AT, PC and AT clones, Sun, Vax, IBM , 3090, Amdahl, CDC Cyber and even 205 and ETA supercomputers.
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We present a new method for using the data from Monte Carlo simulations that can increase the efficiency by 2 or more orders of magnitude. A single Monte Carlo simulation is sufficient to obtain complete thermodynamic information over the entire scaling region near a phase transition. The accuracy of the method is demonstrated by comparison with exact results for the d=2 Ising model. New results for d=2 eight-state Potts model are also presented. The method is generally applicable to statistical models and lattice-gauge theories.
Article
A Monte Carlo algorithm is presented that updates large clusters of spins simultaneously in systems at and near criticality. We demonstrate its efficiency in the two-dimensional O(n) σ models for n=1 (Ising) and n=2 (x-y) at their critical temperatures, and for n=3 (Heisenberg) with correlation lengths around 10 and 20. On lattices up to 1282 no sign of critical slowing down is visible with autocorrelation times of 1-2 steps per spin for estimators of long-range quantities.
Article
Relying on the recently proposed multicanonical algorithm, we present a numerical simulation of the first order phase transition in the 2d 10-state Potts model on lattices up to sizes $100\times100$. It is demonstrated that the new algorithm $lacks$ an exponentially fast increase of the tunneling time between metastable states as a function of the linear size $L$ of the system. Instead, the tunneling time diverges approximately proportional to $L^{2.65}$. Thus the computational effort as counted per degree of freedom for generating an independent configuration in the unstable region of the model rises proportional to $V^{2.3}$, where $V$ is the volume of the system. On our largest lattice we gain more than two orders of magnitude as compared to a standard heat bath algorithm. As a first physical application we report a high precision computation of the interfacial tension.
Article
The low-temperature series expansion for the partition function of the two-dimensional Ising model on a square lattice can be determined exactly for finite lattices using Kaufman's generalization of Onsager's solution. The exact distribution function for the energy can then be determined from the coefficients of the partition function. This provides an exact solution with which one can compare energy histograms determined in Monte Carlo simulations. This solution should prove useful for detailed studies of statistical and systematic errors in histogram reweighting.
Article
The problem of calculating multicanonical parameters recursively is discussed. I describe in detail a computational implementation which has worked reasonably well in practice. Comment: 23 pages, latex, 4 postscript figures included (uuencoded Z-compressed .tar file created by uufiles), figure file corrected.
Article
For the Edwards-Anderson Ising spin-glass model in three and four dimensions (3d and 4d) we have performed high statistics Monte Carlo calculations of those free-energy barriers $F^q_B$ which are visible in the probability density $P_J(q)$ of the Parisi overlap parameter $q$. The calculations rely on the recently introduced multi-overlap algorithm. In both dimensions, within the limits of lattice sizes investigated, these barriers are found to be non-self-averaging and the same is true for the autocorrelation times of our algorithm. Further, we present evidence that barriers hidden in $q$ dominate the canonical autocorrelation times. Comment: 20 pages, Latex, 12 Postscript figures, revised version to appear in Phys. Rev. B
Article
The purpose of this article is to provide a starter kit for multicanonical simulations in statistical physics. Fortran code for the $q$-state Potts model in $d=2, 3,...$ dimensions can be downloaded from the Web and this paper describes simulation results, which are in all details reproducible by running prepared programs. To allow for comparison with exact results, the internal energy, the specific heat, the free energy and the entropy are calculated for the $d=2$ Ising ($q=2$) and the $q=10$ Potts model. % in a temperature range from $T=\infty$ down to sufficiently low % temperatures, such that the groundstates are included in the sampling. Analysis programs, relying on an all-log jackknife technique, which is suitable for handling sums of very large numbers, are introduced to calculate our final estimators.
Markov Chain Monte Carlo Simulations and Their Statistical Analysis Information on the web at http://www.hep.fsu.edu/~ berg
  • B A Berg
B.A. Berg, Markov Chain Monte Carlo Simulations and Their Statistical Analysis, World Scientific, Singapore, 2004. Information on the web at http://www.hep.fsu.edu/~ berg.
2d Crystal Shapes, Droplet Condensation and February 2, 2008 Master Review Vol. 9in x 6in – (for Lecture Note Series, IMS, NUS) article 54 B.A. Berg Supercritical Slowing Down in Simulations of First Order Phase Transitions
  • T Neuhaus
  • J S Hager
T. Neuhaus and J.S. Hager, 2d Crystal Shapes, Droplet Condensation and February 2, 2008 Master Review Vol. 9in x 6in – (for Lecture Note Series, IMS, NUS) article 54 B.A. Berg Supercritical Slowing Down in Simulations of First Order Phase Transitions, J. Stat. Phys. 113 (2003), 47–83.
  • B A Berg
  • Multicanonical Recursions
B.A. Berg, Multicanonical Recursions, J. Stat. Phys. 82 (1996), 323–342.
  • B A Berg
B.A. Berg, Multicanonical Simulations Step by Step, Comp. Phys. Commun. 153 (2003), 397–406.
Supercritical Slowing Down in Simulations of First Order Phase Transitions
Supercritical Slowing Down in Simulations of First Order Phase Transitions, J. Stat. Phys. 113 (2003), 47–83.