Enhanced Sampling Theory And Methods In Monte Carlo Simulation

Monte Carlo computing method is an important method of stochastic modeling, which iswidely used in many fields of Statistical Physics, Biophysics, and Materials Science and Engi-neering. The validity of the Monte Carlo method relies on the sampling process in simulation.The Monte Carlo method is able to construct kinds of target probability distributions, and theimportance sampling technique specifies a way to construct the probability distribution, so asto reduce the sample variance of statistic. The Markov Chain Monte Carlo (MCMC) methodis a common implementation of the importance sampling. It is an important mean of model-ing in the fields of Molecular Modeling and Statistical Physics. The equilibrium evolution ofMarkov chain in the MCMC modeling requires a variety of conditions, typically like the ergod-icity condition, however the local minima of the probability distribution in complex system orthe existence of meta-stable states in statistical system will results in such conditions within alimited time impossible or difficult to meet, leading to inefficient MCMC sampling or statis-tical errors, which becomes one of the important research topics of the Monte Carlo enhancedsampling study. Broadly, Monte Carlo enhanced sampling refers to the development of newmethods and means from the theories and applications of the Monte Carlo method, in order toenhance the validity of the simulation samples, to improve the statistical accuracy of samples,and to speed up the Monte Carlo modeling.On the other hand, with the availability of computer increasing, there has been great de-mand for large-scale Monte Carlo calculations, such as the modern molecular modeling of drugdesign and protein-polymer. Ones demand for Monte Carlo method in an easy computer imple-mentation and of effective in parallel computing over network, such that Monte Carlo methodhas potential significances in parallel sampling and the design of software system.This study mainly focuses on the theory and methods of Monte Carlo enhanced samplingand have completed the following key work and innovations:1. With the analysis of key issues in Monte Carlo calculation and simulation, the distribu-tion of generalized canonical ensemble is proposed to overcome the impact of local minima ormetastable states in order to improve the effectiveness of Monte Carlo sampling.TheunderstandingofkeyissuesinMonteCarlocalculationandsimulationlaysthefounda- tions of the study of Monte Carlo enhanced sampling. Monte Carlo calculation and simulation are closely related to the probability distribution of system. In term of the micro-state distri-bution in statistical physics, the distribution of generalized canonical ensemble is a generaliza-tion of the traditional canonical distribution. The Monte Carlo simulation using the generalized canonical ensemble, by adjusting the free parameters of the generalized canonical distribution, can regulate the sampling distribution in simulation, and helps to overcome the impact of the metastable states increasing the effectiveness of sampling, also helps to reduce the relaxation time and the tunneling time, to speed up the simulation progress.2. The Wolff algorithm of generalized canonical ensemble is presented, and is applied to the two-dimensional Potts model in large size. It makes further understanding of the conformation form of the phase transition.In the MCMC simulation of the lattice system, the random walks of the commonly used Metropolis-Hastings spin single-flipping lacks of the activity in dynamics, and while Wolff al-gorithms using cluster-flipping greatly enhances the activity of the Monte Carlo simulation. In combination of the advantages of generalized canonical ensemble, this work realizes the Wolff algorithm of generalized canonical ensemble, which has double advantages of distribution-adjustable and strong dynamics, and increases the detection ability of Monte Carlo simulation. The application in the large-size Potts model obtains a special phase transformation, which en-hances the understanding of the first-order phase transition of Potts model.3. By deducing the necessary and sufficient conditions of generating the flat histogram distribution, a flat-histogram (ensemble) method of linear prediction is proposed to estimate the density of states, which reduces the required random walks in the common flat ensemble methods to build a flat distribution.The class of flat-histogram ensemble methods are summarized, and the conditions of how to make a flat sampling distribution and how to stabilize a normal distribution are deduced in theoretically. Based on these investigations, a fast ensemble method is presented to estimate the density of states. This method adaptively predicts the derivative function of system entropy according to the energy distributions obtained, in which the order of random steps required to build a flat histogram is reduced from the common O (N2) to O (N3/2)～O (N1/2).4. The key means in Monte Carlo distributed parallel sampling, such as the sample fault- tolerance and iteratively-updating statistics, are investigated; It is on the theoretical and ex-perimental proof that the parallel tempering simulation of generalized canonical ensemble canstabilize the swap acceptance rate (SAR) between the replicas.At first, the parallel way of Monte Carlo computing is modeled, and then the key means ofthe sample fault-tolerance and iteratively-updating statistics in Monte Carlo distributed parallelsampling, based on the analysis of sample statistics, are investigated. The characteristics of theparallel tempering simulation of generalized canonical ensemble have been studied, and furtherthe function of stabilizing the SAR of replicas is conformed, which make the tempering processof Monte Carlo simulation more efficient.5. A Monte Carlo software system of classes in core is designed. It not only implementsthe algorithms in this work and related research applications, but also has a strong re-usabilityand scalability.ThisworkdesignsandimplementsasetofMonteCarlosoftware, whichfullyconsidersthecharacteristics of the Monte Carlo scientific computing, supports multi-language programming,and reaches the class-library level and even the component-level re-usability; meanwhile it isfacilitatory for the two-stage of sampling and statistical analysis. It provides a feasible andeffective solution of Monte Carlo computing software. The system uses the classes-core design,and also present a component design under the CCA framework. It enhances the ability ofdistributed computing of Monte Carlo applications. The system completes the independenceand scalability in overall of algorithms, physical models and programming languages, whichprovides a feasible software computing platform for further study and application of the MonteCarlo method.The above work covers the theory and methods of Monte Carlo enhanced sampling, andIt involves in not only the sampling theory but also the theory and practice in computer, whichdevelopsadepthandsystematicstudyinMonteCarloenhancedsampling. Theresearchachieve-ments in this work will help ones to solve the computing problem of complex multi-dimensionusing Monte Carlo method.