|
Kinetic Monte Carlo
The kinetic Monte Carlo (KMC) method is a Monte Carlo method computer simulation intended to simulate the time evolution of some processes occurring in nature. Typically these are processes that occur with known transition rates among states. It is important to understand that these rates are inputs to the KMC algorithm, the method itself cannot predict them. The KMC method is essentially the same as the dynamic Monte Carlo method and the Gillespie algorithm. Algorithms One possible classification of KMC algorithms is as rejection-KMC (rKMC) and rejection-free-KMC (rfKMC). Rejection-free KMC A rfKMC algorithm, often only called KMC, for simulating the time evolution of a system, where some processes can occur with known rates r, can be written for instance as follows: # Set the time t = 0. # Choose an initial state ''k''. # Form the list of all N_k possible transition rates in the system r_, from state ''k'' into a generic state ''i''. States that do not communicate ... [...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] |
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] |
Monte Carlo Methods
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 risk in b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interstitial Defect
In materials science, an interstitial defect is a type of point crystallographic defect where an atom of the same or of a different type, occupies an interstitial site in the crystal structure. When the atom is of the same type as those already present they are known as a self-interstitial defect. Alternatively, small atoms in some crystals may occupy interstitial sites, such as hydrogen in palladium. Interstitials can be produced by bombarding a crystal with elementary particles having energy above the displacement threshold for that crystal, but they may also exist in small concentrations in thermodynamic equilibrium. The presence of interstitial defects can modify the physical and chemical properties of a material. History The idea of interstitial compounds was started in the late 1930s and they are often called Hagg phases after Hägg. Transition metals generally crystallise in either the hexagonal close packed or face centered cubic structures, both of which can be co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Impurity
In chemistry and materials science, impurities are chemical substances inside a confined amount of liquid, gas, or solid, which differ from the chemical composition of the material or compound. Firstly, a pure chemical should appear thermodynamically in at least one chemical phase and can also be characterized by its one-component- phase diagram. Secondly, practically speaking, a pure chemical should prove to be homogeneous (i.e., will show no change of properties after undergoing a wide variety of consecutive analytical chemical procedures). The perfect pure chemical will pass all attempts and tests of further separation and purification. Thirdly, and here we focus on the common chemical definition, it should not contain any trace of any other kind of chemical species. In reality, there are no absolutely 100% pure chemical compounds, as there is always some minute contamination. Indeed, as detection limits in analytical chemistry decrease, the number of impurities detected t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystallographic Defect
A crystallographic defect is an interruption of the regular patterns of arrangement of atoms or molecules in crystalline solids. The positions and orientations of particles, which are repeating at fixed distances determined by the unit cell parameters in crystals, exhibit a periodic crystal structure, but this is usually imperfect.Ehrhart, P. (1991Properties and interactions of atomic defects in metals and alloys, volume 25 of Landolt-Börnstein, New Series III, chapter 2, p. 88, Springer, Berlin Several types of defects are often characterized: point defects, line defects, planar defects, bulk defects. Topological homotopy establishes a mathematical method of characterization. Point defects Point defects are defects that occur only at or around a single lattice point. They are not extended in space in any dimension. Strict limits for how small a point defect is are generally not defined explicitly. However, these defects typically involve at most a few extra or missing atoms. La ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alloy
An alloy is a mixture of chemical elements of which at least one is a metal. Unlike chemical compounds with metallic bases, an alloy will retain all the properties of a metal in the resulting material, such as electrical conductivity, ductility, opacity, and luster, but may have properties that differ from those of the pure metals, such as increased strength or hardness. In some cases, an alloy may reduce the overall cost of the material while preserving important properties. In other cases, the mixture imparts synergistic properties to the constituent metal elements such as corrosion resistance or mechanical strength. Alloys are defined by a metallic bonding character. The alloy constituents are usually measured by mass percentage for practical applications, and in atomic fraction for basic science studies. Alloys are usually classified as substitutional or interstitial alloys, depending on the atomic arrangement that forms the alloy. They can be further classified as homo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacancy (chemistry)
In crystallography, a vacancy is a type of point defect in a crystal where an atom is missing from one of the lattice sites.Ehrhart, P. (1991) "Properties and interactions of atomic defects in metals and alloys", chapter 2, p. 88 in ''Landolt-Börnstein, New Series III'', Vol. 25, Springer, Berlin Crystals inherently possess imperfections, sometimes referred to as crystalline defects. Vacancies occur naturally in all crystalline materials. At any given temperature, up to the melting point of the material, there is an equilibrium concentration (ratio of vacant lattice sites to those containing atoms). At the melting point of some metals the ratio can be approximately 1:1000. This temperature dependence can be modelled by :N_ = N \exp(-Q_/k_ T) where is the vacancy concentration, is the energy required for vacancy formation, is the Boltzmann constant, is the absolute temperature, and is the concentration of atomic sites i.e. : N = m N_ / M where is mass, Avogadro con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystal Structure
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter. The smallest group of particles in the material that constitutes this repeating pattern is the unit cell of the structure. The unit cell completely reflects the symmetry and structure of the entire crystal, which is built up by repetitive translation of the unit cell along its principal axes. The translation vectors define the nodes of the Bravais lattice. The lengths of the principal axes, or edges, of the unit cell and the angles between them are the lattice constants, also called ''lattice parameters'' or ''cell parameters''. The symmetry properties of the crystal are described by the concept of space groups. All possible symmetric arrangements of par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Synopsys
Synopsys is an American electronic design automation (EDA) company that focuses on silicon design and verification, silicon intellectual property and software security and quality. Products include tools for logic synthesis and physical design of integrated circuits, simulators for development and debugging environments that assist in the design of the logic for chips and computer systems. In recent years, Synopsys has expanded its products and services to include application security testing. Synopsys has gained attention due to its relationship with various Chinese state entities. In 2018, Synopsys formed a partnership with the People's Liberation Army National Defence University and, in 2022, the company came under investigation by the United States Department of Justice for technology transfers to sanctioned entities in China. History Synopsys was founded by Aart J de Geus and David Gregory in 1986 in Research Triangle Park, North Carolina. The company was initi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-stationary Distribution
In probability a quasi-stationary distribution is a random process that admits one or several absorbing states that are reached almost surely, but is initially distributed such that it can evolve for a long time without reaching it. The most common example is the evolution of a population: the only equilibrium is when there is no one left, but if we model the number of people it is likely to remain stable for a long period of time before it eventually collapses. Formal definition We consider a Markov process (Y_t)_ taking values in \mathcal. There is a measurable set \mathcal^of absorbing states and \mathcal^a = \mathcal \setminus \mathcal^. We denote by T the hitting time of \mathcal^, also called killing time. We denote by \ the family of distributions where \operatorname_x has original condition Y_0 = x \in \mathcal. We assume that \mathcal^ is almost surely reached, i.e. \forall x \in \mathcal, \operatorname_x(T t) = \nu(B)where \operatorname_\nu = \int_ \operatorname_x \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Langevin Dynamics
In physics, Langevin dynamics is an approach to the mathematical modeling of the dynamics of molecular systems. It was originally developed by French physicist Paul Langevin. The approach is characterized by the use of simplified models while accounting for omitted degrees of freedom by the use of stochastic differential equations. Langevin dynamics simulations are a kind of Monte Carlo simulation. Overview A real world molecular system is unlikely to be present in vacuum. Jostling of solvent or air molecules causes friction, and the occasional high velocity collision will perturb the system. Langevin dynamics attempts to extend molecular dynamics to allow for these effects. Also, Langevin dynamics allows temperature to be controlled like with a thermostat, thus approximating the canonical ensemble. Langevin dynamics mimics the viscous aspect of a solvent. It does not fully model an implicit solvent; specifically, the model does not account for the electrostatic screening and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
