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We present a review of extended ensemble methods and ensemble optimization techniques. Extended ensemble methods, such as multicanonical sampling, broad histograms, or parallel tempering aim to accelerate the simulation of systems with large energy barriers, as they occur in the vicinity of first order phase transitions or in complex systems with rough energy landscapes, such as spin glasses or proteins. We present a recently developed feedback algorithm to iteratively achieve an optimal ensemble, with the fastest equilibration and shortest autocorrelation times. In the second part we review time-discretization free world line representations for quantum systems, and show how any algorithm developed for classical systems, such as local updates, cluster updates or the extended and optimized ensemble methods can also be applied to quantum systems. An overview over the methods is followed by a selection of typical applications. [ABSTRACT FROM AUTHOR]

We review a recent method we have developed for Fermion quantum Monte Carlo. By using combinatorial arguments to perform resummations over paths, we reformulate the stochastic problem of sampling paths in terms of sampling "graphs", which are much better behaved with regards sign-cancellation problems encountered in path-integral simulations of Fermions. Detailed mathematical derivations of the new results are presented. [ABSTRACT FROM AUTHOR]

Twenty years ago Car and Parrinello introduced an efficient method to perform Molecular Dynamics simulation for classical nuclei with forces computed on the "fly" by a Density Functional Theory (DFT) based electronic calculation [1]. Because the method allowed study of the statistical mechanics of classical nuclei with many-body electronic interactions, it opened the way for the use of simulation methods for realistic systems with an accuracy well beyond the limits of available effective force fields. In the last twenty years, the number of applications of the Car-Parrinello ab-initio molecular dynamics has ranged from simple covalent bonded solids, to high pressure physics, material science and biological systems. There have also been extensions of the original algorithm to simulate systems at constant temperature and constant pressure [2], finite temperature effects for the electrons [3], and quantum nuclei [4]. [ABSTRACT FROM AUTHOR]

We review recently developed approximate methods for computing quantum time correlation functions based on linearizing the phase of their path integral expressions in the difference between paths representing the forward and backward propagators. Our focus here will be on problems that can be partitioned into two subsystems: One that is best described by a few discrete quantum states such as the high frequency vibrations or electronic states of molecules, and the other subsystem, "the bath", composed of the remaining degrees of freedom that will be described by a continuous representation. The general theory will first be developed and applied to model condensed phase problems. Approximations to the theory will be then made enabling applications to large scale realistic systems. [ABSTRACT FROM AUTHOR]

Quantum mechanics provides us with the most fundamental description of natural phenomena. In many instances classical mechanics constitutes an adequate approximation and it is widely used in simulations of both static and dynamic properties of many-body systems. Often, however, quantum effects cannot be neglected and one is faced with the task of devising methods to simulate the behavior of the quantum system. [ABSTRACT FROM AUTHOR]

We report on a novel approach to the automatic identification of metastablestates from long term simulation of complex molecular systems. The new approachis based on a hierarchical concept of metastability: metastable statesare understood as subsets of state or configuration space from which the dynamicsexits only very rarely; subsets with the smallest exit probabilities areof most interest, their further decomposition then may reveal subsets fromwhich exiting is less but comparably difficult for the system under investigation.The article gives a survey of the theoretical foundation of the approachand its algorithmic realization that generalizes the well-known concept of HiddenMarkov Models. The performance of the resulting algorithm is illustratedby an application to a 100 ns simulation of penta-alanine with explicit water.We demonstrate that the resulting metastable states allow to reveal theconformation dynamics of the molecule. [ABSTRACT FROM AUTHOR]

The dynamical behavior of many systems arising in physics, chemistry, biology, etc. is dominated by rare but important transition events between long lived states. For over 70 years, transition state theory (TST) has provided the main theoretical framework for the description of these events [17,33,34]. Yet, while TST and evolutions thereof based on the reactive flux formalism [1, 5] (see also [30,31]) give an accurate estimate of the transition rate of a reaction, at least in principle, the theory tells very little in terms of the mechanism of this reaction. Recent advances, such as transition path sampling (TPS) of Bolhuis, Chandler, Dellago, and Geissler [3, 7] or the action method of Elber [15, 16], may seem to go beyond TST in that respect: these techniques allow indeed to sample the ensemble of reactive trajectories, i.e. the trajectories by which the reaction occurs. And yet, the reactive trajectories may again be rather uninformative about the mechanism of the reaction. This may sound paradoxical at first: what more than actual reactive trajectories could one need to understand a reaction? The problem, however, is that the reactive trajectories by themselves give only a very indirect information about the statistical properties of these trajectories. This is similar to why statistical mechanics is not simply a footnote in books about classical mechanics. What is the probability density that a trajectory be at a given location in state-space conditional on it being reactive? What is the probability current of these reactive trajectories? What is their rate of appearance? These are the questions of interest and they are not easy to answer directly from the ensemble of reactive trajectories. The right framework to tackle these questions also goes beyond standard equilibrium statistical mechanics because of the nontrivial bias that the very definition of the reactive trajectories imply - they must be involved in a reaction. The aim of this chapter is to introduce the reader to the probabilistic framework one can use to characterize the mechanism of a reaction and obtain the probability density, current, rate, etc. of the reactive trajectories. [ABSTRACT FROM AUTHOR]

An algorithm to compute classical trajectories using boundary value formulation is presented and discussed. It is based on an optimization of a functional of the complete trajectory. This functional can be the usual classical action, and is approximated by discrete and sequential sets of coordinates. In contrast to initial value formulation, the pre-specified end points of the trajectories are useful for computing rare trajectories. Each of the boundary-value trajectories ends at desired products. A difficulty in applying boundary value formulation is the high computational cost of optimizing the whole trajectory in contrast to the calculation of one temporal frame at a time in initial value formulation. [ABSTRACT FROM AUTHOR]

This chapter summarizes several computational strategies to study the kinetics of two-state protein folding using all atom models. After explaining the background of two state folding using energy landscapes I introduce common protein models and computational tools to study folding thermodynamics and kinetics. Free energy landscapes are able to capture the thermodynamics of two-state protein folding, and several methods for efficient sampling of these landscapes are presented. An accurate estimate of folding kinetics, the main topic of this chapter, is more difficult to achieve. I argue that path sampling methods are well suited to overcome the problems connected to the sampling of folding kinetics. Some of the major issues are illustrated in the case study on the folding of the GB1 hairpin. [ABSTRACT FROM AUTHOR]

Transition path sampling, based on a statistical mechanics in trajectory space, is a set of computational methods for the simulation of rare events in complex systems. In this chapter we give an overview of these techniques and describe their statistical mechanical basis as well as their application. [ABSTRACT FROM AUTHOR]

Computer simulations of complex polyatomic systems are nowadays routinely exploited in chemistry, physics and biophysics for extracting informations about equilibrium properties and predictions of long time behavior. As it is well known, a straightforward simulation by molecular dynamics or Monte Carlo can usually provide only limited information, due to the intrinsic complexity of most of the potential energy functions needed to describe real systems. For example, the folding time of most of the proteins is of the order of seconds, while direct simulation of these systems can access at most the microseconds time scale. For this reason the problem of sampling is of great importance in computational physics. One would like to exploit at best the available computer time in order to extract information about events that might happen on a long time scale ("rare events") and to predict the most probable state of a system. [ABSTRACT FROM AUTHOR]

Density functional theory DFT Kohn-Sham [16] is the method of choice for the calculation of electronic properties of large systems. This is due to the combination of accuracy and efficiency that has been achieved for the Kohn- Sham (KS) method in DFT [19]. DFT based electronic structure calculations are nowadays routinely used by chemists and physicists to support their research. Increasingly complex systems can be treated and the inclusion of environmental effects, through implicit or explicit solvent treatments, as well as the effects of different thermodynamic parameters (temperature, pressure) through first-principles molecular dynamics, opens the door for simulations close to experimental conditions. The accuracy of the method is such that many properties of systems of interest to chemistry, physics, material science, and biology can be predicted in a parameter free way. The success of the KS method makes it also the favorite framework for new developments to improve both, accuracy and efficiency. Better accuracy in this context can be achieved along two lines. On one hand the numerical limit of a given model should be reached and on the other hand more accurate models should be developed (i.e. exchange-correlation functional in DFT). The development of new functionals is an art on its own and will not concern us here. However, it is intimately related to the efficiency problem, as only numerically accurate tests on more and more complex systems can give unambiguous information on the performance of new functionals. The goal of improved algorithms is therefore, to provide methods to accurately and efficiently solve the KS equations. [ABSTRACT FROM AUTHOR]

Ab initio Molecular Dynamics (MD) on the contrary to empirical force field Molecular Dynamics simulations employs an electronic structure calculation at each time-step of the dynamics to determine the forces on the nuclei. This allows for the simulation of materials in a broad range of situations, including during chemical reactions, while chemical bonds are broken or formed. The last few years, use of ab initio MD has spread very rapidly to many fields and is now used by many groups. At the same time many new developments have been pursued including, e.g., the calculation of electronic properties. Ab initio MD is also an integrated part of a new range of techniques to bridge length and time scales: QM/MM, transition path sampling, metadynamics etc. many of whose are discussed in this book. [ABSTRACT FROM AUTHOR]

We give an introduction into the topic of how to compute efficiently long range interactions. We start with reviewing the traditional Ewald sum for 3D Coulomb systems, discuss then in some detail the P3M method of Hockney and Eastwood. We continue with explaining some strategies to perform the sum under partially periodic boundary conditions, where we present two recently developed methods, namely MMM2D and the ELC method for two dimensionally periodic boundary conditions, and the MMM1D method for systems with only one periodic coordinate. After this, we briefly mention alternative ways of dealing with the Coulomb sum, such as the MEMD method of Maggs. In the second part we present our recently developed MD simulation package, ESPResSo, that includes most of the introduced algorithms. We give a short introduction into the capabilities of ESPResSo and its usage. Finally we present some recent simulation results for polymer networks which were obtained by using ESPResSo. [ABSTRACT FROM AUTHOR]

Levinthal's paradox [1,2], first introduced in the 1960's (early in the childhood of simulations in Chemistry), serves as a good illustration of the limitations we still face in the application of molecular dynamics (MD). Levinthal reasoned that if we were to assume that every residue in a polypeptide has a least two stable conformations, then a small 100 residue polypeptide would have 2100 possible states. If we were to study such a protein using traditional, state of the art, MD techniques, the native state would only be deduced after a little more than a billion years. [ABSTRACT FROM AUTHOR]

The Metropolis (Markov Chain) Monte Carlo method is simple and powerful. Since 1953, many extensions of the original Markov Chain Monte Carlo method have been proposed, but they are all based on the original Metropolis prescription that only states belonging to the Markov Chain should be sampled. In particular, if trial moves to a potential target state are rejected, that state is not included in the sampling. I will argue that the efficiency of effectively all Markov Chain MC schemes can be improved by including the rejected states in the sampling procedure. Such an approach requires only a trivial (and cheap) extension of existing programs. I will demonstrate that the approach leads to improved estimates of the energy of a system and that it leads to better estimates of free-energy landscapes. [ABSTRACT FROM AUTHOR]

The study of phase transitions has played a central role in the study of condensed matter. Since the first applications of molecular simulations, which provided some of the first evidence in support of a freezing transition in hardsphere systems, to contemporary research on complex systems, including polymers, proteins, or liquid crystals, to name a few, molecular simulations are increasingly providing a standard against which to measure the validity of theoretical predictions or phenomenological explanations of experimentally observed phenomena. [ABSTRACT FROM AUTHOR]

The phenomenon of phase behavior is the organization of many-body systems into forms which reflect the interplay between constraints imposed macroscopically (through the prevailing external conditions) and microscopically (through the interactions between the elementary constituents). In this article we focus on generic computational strategies needed to address the problems of phase behavior, or more specifically the task of mapping equilibrium phase boundaries. [ABSTRACT FROM AUTHOR]

This chapter provides an introduction to cluster Monte Carlo algorithms for classical statistical-mechanical systems. A brief review of the conventional Metropolis algorithm is given, followed by a detailed discussion of the lattice cluster algorithm developed by Swendsen and Wang and the single-cluster variant introduced by Wolff. For continuum systems, the geometric cluster algorithm of Dress and Krauth is described. It is shown how their geometric approach can be generalized to incorporate particle interactions beyond hardcore repulsions, thus forging a connection between the lattice and continuum approaches. Several illustrative examples are discussed. [ABSTRACT FROM AUTHOR]

The basic laws of physics that govern the phenomena on the scales of length and energy relevant for condensed matter systems, ranging from simple fluids and solids to complex multicomponent materials and even problems of chemical biology, are well known and understood: one just deals with the Schrödinger equation for the quantum many-body problem of the nuclei and electrons interacting with Coulomb potentials (for simplicity, we disregard, here throughout, the need for relativistic corrections in electronic structure calculations of matter containing heavy atoms). Statistical mechanics then supplies the framework to extend this quantum many-body theory to provide a statistical description in terms of averages taken at nonzero temperature. [ABSTRACT FROM AUTHOR]