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Monte Carlo Method In Statistical Physics
Monte Carlo in statistical physics refers to the application of the Monte Carlo method to problems in statistical physics, or statistical mechanics. Overview The general motivation to use the Monte Carlo method in statistical physics is to evaluate a multivariable integral. The typical problem begins with a system for which the Hamiltonian is known, it is at a given temperature and it follows the Boltzmann statistics. To obtain the mean value of some macroscopic variable, say A, the general approach is to compute, over all the phase space, PS for simplicity, the mean value of A using the Boltzmann distribution: :\langle A\rangle=\int_ A_ \frac d\vec. where E(\vec)=E_ is the energy of the system for a given state defined by \vec - a vector with all the degrees of freedom (for instance, for a mechanical system, \vec = \left(\vec, \vec \right) ), \beta\equiv 1/k_bT and :Z= \int_ e^d\vec is the partition function. One possible approach to solve this multivariable integral is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monte Carlo Method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution. In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model, interacting particle systems, McKean–Vlasov processes, kinetic models of gases). Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of ris ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wang And Landau Algorithm
The Wang and Landau algorithm, proposed by Fugao Wang and David P. Landau, is a Monte Carlo method designed to estimate the density of states of a system. The method performs a non-Markovian random walk to build the density of states by quickly visiting all the available energy spectrum. The Wang and Landau algorithm is an important method to obtain the density of states required to perform a multicanonical simulation. The Wang–Landau algorithm can be applied to any system which is characterized by a cost (or energy) function. For instance, it has been applied to the solution of numerical integrals and the folding of proteins. The Wang–Landau sampling is related to the metadynamics algorithm.Christoph Junghans, Danny Perez, and Thomas Vogel. "Molecular Dynamics in the Multicanonical Ensemble: Equivalence of Wang–Landau Sampling, Statistical Temperature Molecular Dynamics, and Metadynamics." Journal of Chemical Theory and Computation 10.5 (2014): 1843-1847. doibr>10.1021/ct5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Chemistry
Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into computer programs, to calculate the structures and properties of molecules, groups of molecules, and solids. It is essential because, apart from relatively recent results concerning the hydrogen molecular ion (dihydrogen cation, see references therein for more details), the quantum many-body problem cannot be solved analytically, much less in closed form. While computational results normally complement the information obtained by chemical experiments, it can in some cases predict hitherto unobserved chemical phenomena. It is widely used in the design of new drugs and materials. Examples of such properties are structure (i.e., the expected positions of the constituent atoms), absolute and relative (interaction) energies, electronic charge density distributions, dipoles and higher multipole moments, vi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monte Carlo Molecular Modeling
Monte Carlo molecular modelling is the application of Monte Carlo methods to molecular problems. These problems can also be modelled by the molecular dynamics method. The difference is that this approach relies on equilibrium statistical mechanics rather than molecular dynamics. Instead of trying to reproduce the dynamics of a system, it generates states according to appropriate Boltzmann distribution. Thus, it is the application of the Metropolis Monte Carlo simulation to molecular systems. It is therefore also a particular subset of the more general Monte Carlo method in statistical physics. It employs a Markov chain procedure in order to determine a new state for a system from a previous one. According to its stochastic nature, this new state is accepted at random. Each trial usually counts as a move. The avoidance of dynamics restricts the method to studies of static quantities only, but the freedom to choose moves makes the method very flexible. These moves must only satisfy a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Monte Carlo
Quantum Monte Carlo encompasses a large family of computational methods whose common aim is the study of complex quantum systems. One of the major goals of these approaches is to provide a reliable solution (or an accurate approximation) of the quantum many-body problem. The diverse flavors of quantum Monte Carlo approaches all share the common use of the Monte Carlo method to handle the multi-dimensional integrals that arise in the different formulations of the many-body problem. Quantum Monte Carlo methods allow for a direct treatment and description of complex many-body effects encoded in the wave function, going beyond mean-field theory. In particular, there exist numerically exact and polynomially-scaling algorithms to exactly study static properties of boson systems without geometrical frustration. For fermions, there exist very good approximations to their static properties and numerically exact exponentially scaling quantum Monte Carlo algorithms, but none that are b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Importance Sampling
Importance sampling is a Monte Carlo method for evaluating properties of a particular distribution, while only having samples generated from a different distribution than the distribution of interest. Its introduction in statistics is generally attributed to a paper by Teun Kloek and Herman K. van Dijk in 1978, but its precursors can be found in statistical physics as early as 1949. Importance sampling is also related to umbrella sampling in computational physics. Depending on the application, the term may refer to the process of sampling from this alternative distribution, the process of inference, or both. Basic theory Let X\colon \Omega\to \mathbb be a random variable in some probability space (\Omega,\mathcal,P). We wish to estimate the expected value of ''X'' under ''P'', denoted E 'X;P'' If we have statistically independent random samples x_1, \ldots, x_n, generated according to ''P'', then an empirical estimate of E 'X;P''is : \widehat_ ;P= \frac \sum_^n x_i \quad \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simulated Annealing
Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. It is often used when the search space is discrete (for example the traveling salesman problem, the boolean satisfiability problem, protein structure prediction, and job-shop scheduling). For problems where finding an approximate global optimum is more important than finding a precise local optimum in a fixed amount of time, simulated annealing may be preferable to exact algorithms such as gradient descent or branch and bound. The name of the algorithm comes from annealing in metallurgy, a technique involving heating and controlled cooling of a material to alter its physical properties. Both are attributes of the material that depend on their thermodynamic free energy. Heating and cooling the material affects both the temperature and the the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnetic Susceptibility
In electromagnetism, the magnetic susceptibility (Latin: , "receptive"; denoted ) is a measure of how much a material will become magnetized in an applied magnetic field. It is the ratio of magnetization (magnetic moment per unit volume) to the applied magnetizing field intensity . This allows a simple classification, into two categories, of most materials' responses to an applied magnetic field: an alignment with the magnetic field, , called paramagnetism, or an alignment against the field, , called diamagnetism. Magnetic susceptibility indicates whether a material is attracted into or repelled out of a magnetic field. Paramagnetic materials align with the applied field and are attracted to regions of greater magnetic field. Diamagnetic materials are anti-aligned and are pushed away, toward regions of lower magnetic fields. On top of the applied field, the magnetization of the material adds its own magnetic field, causing the field lines to concentrate in paramagnetism, or be excl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ising Model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The model allows the identification of phase transitions as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition. The Ising model was invented by the physicist , who gave it as a prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Of States
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states in the system of volume V whose energies lie in the range from E to E+\delta E. It is mathematically represented as a distribution by a probability density function, and it is generally an average over the space and time domains of the various states occupied by the system. The density of states is directly related to the dispersion relations of the properties of the system. High DOS at a specific energy level means that many states are available for occupation. Generally, the density of states of matter is continuous. In isolated systems however, such as atoms or molecules in the gas phase, the density distribution is discrete, like a spectral density. Local variations, most often due to distortions of the original system, are often referr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Physics
Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the Mathematics, mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic nature. Its applications include many problems in the fields of physics, biology, chemistry, and neuroscience. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics develop the Phenomenology (particle physics), phenomenological results of thermodynamics from a probabilistic examination of the underlying microscopic systems. Historically, one of the first topics in physics where statistical methods were applied was the field of classical mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metropolis Algorithm
A metropolis () is a large city or conurbation which is a significant economic, political, and cultural center for a country or region, and an important hub for regional or international connections, commerce, and communications. A big city belonging to a larger urban agglomeration, but which is not the core of that agglomeration, is not generally considered a metropolis but a part of it. The plural of the word is ''metropolises'', although the Latin plural is ''metropoles'', from the Greek ''metropoleis'' (). For urban centers outside metropolitan areas that generate a similar attraction on a smaller scale for their region, the concept of the regiopolis ("regio" for short) was introduced by urban and regional planning researchers in Germany in 2006. Etymology Metropolis (μητρόπολις) is a Greek word, coming from μήτηρ, ''mḗtēr'' meaning "mother" and πόλις, ''pólis'' meaning "city" or "town", which is how the Greek colonies of antiquity referred to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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