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Glauber Dynamics
In statistical physics, Glauber dynamics is a way to simulate the Ising model (a model of magnetism) on a computer. It is a type of Markov Chain Monte Carlo algorithm. The algorithm In the Ising model, we have say N particles that can spin up (+1) or down (-1). Say the particles are on a 2D grid. We label each with an x and y coordinate. Glauber's algorithm becomes: # Choose a particle \sigma_ at random. # Sum its four neighboring spins. S = \sigma_ + \sigma_ + \sigma_ + \sigma_. # Compute the change in energy if the spin x, y were to flip. This is \Delta E = 2\sigma_ S (see the Hamiltonian for the Ising model). # Flip the spin with probability e^/(1 + e^) where T is the temperature . # Display the new grid. Repeat the above N times. In Glauber algorithm, if the energy change in flipping a spin is zero, \Delta E = 0, then the spin would always gets flipped with probability p(0, T) = 0.5. Glauber V.S. Metropolis–Hastings algorithm Metropolis–Hastings algorithm gives ident ... [...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] |
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] |
Monte Carlo Algorithm
In computing, a Monte Carlo algorithm is a randomized algorithm whose output may be incorrect with a certain (typically small) probability. Two examples of such algorithms are Karger–Stein algorithm and Monte Carlo algorithm for minimum Feedback arc set. The name refers to the grand casino in the Principality of Monaco at Monte Carlo, which is well-known around the world as an icon of gambling. The term "Monte Carlo" was first introduced in 1947 by Nicholas Metropolis. Las Vegas algorithms are a dual of Monte Carlo algorithms that never return an incorrect answer. However, they may make random choices as part of their work. As a result, the time taken might vary between runs, even with the same input. If there is a procedure for verifying whether the answer given by a Monte Carlo algorithm is correct, and the probability of a correct answer is bounded above zero, then with probability, one running the algorithm repeatedly while testing the answers will eventually give a corr ... [...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] |
Roy J
Roy is a masculine given name and a family surname with varied origin. In Anglo-Norman England, the name derived from the Norman ''roy'', meaning "king", while its Old French cognate, ''rey'' or ''roy'' (modern ''roi''), likewise gave rise to Roy as a variant in the Francophone world. In India, Roy is a variant of the surname ''Rai'',. likewise meaning "king".. It also arose independently in Scotland, an anglicisation from the Scottish Gaelic nickname ''ruadh'', meaning "red". Given name * Roy Acuff (1903–1992), American country music singer and fiddler * Roy Andersen (born 1955), runner * Roy Andersen (South Africa) (born 1948), South African businessman and military officer * Roy Anderson (American football) (born 1980), American football coach * Sir Roy M. Anderson (born 1947), British scientific adviser * Roy Andersson (born 1943), Swedish film director * Roy Andersson (footballer) (born 1949), footballer from Sweden * Roy Chapman Andrews (1884–1960), American natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Detailed Balance
The principle of detailed balance can be used in kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions). It states that at equilibrium, each elementary process is in equilibrium with its reverse process. History The principle of detailed balance was explicitly introduced for collisions by Ludwig Boltzmann. In 1872, he proved his H-theorem using this principle.Boltzmann, L. (1964), Lectures on gas theory, Berkeley, CA, USA: U. of California Press. The arguments in favor of this property are founded upon microscopic reversibility. Tolman, R. C. (1938). ''The Principles of Statistical Mechanics''. Oxford University Press, London, UK. Five years before Boltzmann, James Clerk Maxwell used the principle of detailed balance for gas kinetics with the reference to the principle of sufficient reason. He compared the idea of detailed balance with other types of balancing (like cyclic balance) and found that "Now it is impossible to as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodicity
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored into smaller components. Ergodic theory is the study of systems possessing ergodicity. Ergodic systems occur in a broad range of systems in physics and in geometry. This can be roughly understood to be due to a common phenomenon: the motion of particles, that is, geodesics on a hyperbolic manifold are divergent; when that manifold is compact, that is, of finite size, those orbits return to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Markov Chain Monte Carlo
In statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain a sample of the desired distribution by recording states from the chain. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution. Various algorithms exist for constructing chains, including the Metropolis–Hastings algorithm. Application domains MCMC methods are primarily used for calculating numerical approximations of multi-dimensional integrals, for example in Bayesian statistics, computational physics, computational biology and computational linguistics. In Bayesian statistics, the recent development of MCMC methods has made it possible to compute large hierarchical models that require integrations over hundreds to thousands of unknown parameters. In rare even ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Lattice Ising Model
In statistical mechanics, the two-dimensional square lattice Ising model is a simple lattice model (physics), lattice model of interacting magnetic spins. The model is notable for having nontrivial interactions, yet having an analytical solution. The model was solved by Lars Onsager for the special case that the external magnetic field ''H'' = 0. An analytical solution for the general case for H \neq 0 has yet to be found. Defining the partition function Consider a 2D Ising model on a square lattice \Lambda with ''N'' sites and periodic boundary conditions in both the horizontal and vertical directions, which effectively reduces the topology of the model to a torus. Generally, the horizontal coupling J \neq the vertical one J^*. With \beta = \frac and absolute temperature T and Boltzmann's constant k, the partition function (statistical mechanics), partition function : Z_N(K \equiv \beta J, L \equiv \beta J^*) = \sum_ \exp \left( K \sum_ \sigma_i \sigma_j + L \sum_ \sigma_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boltzmann Distribution
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of that state's energy and the temperature of the system. The distribution is expressed in the form: :p_i \propto e^ where is the probability of the system being in state , is the energy of that state, and a constant of the distribution is the product of the Boltzmann constant and thermodynamic temperature . The symbol \propto denotes proportionality (see for the proportionality constant). The term ''system'' here has a very wide meaning; it can range from a collection of 'sufficient number' of atoms or a single atom to a macroscopic system such as a natural gas storage tank. Therefore the Boltzmann distribution can be used to solve a very wide variety of problems. The distribu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simulation
A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or process, whereas the simulation represents the evolution of the model over time. Often, computers are used to execute the computer simulation, simulation. Simulation is used in many contexts, such as simulation of technology for performance tuning or optimizing, safety engineering, testing, training, education, and video games. Simulation is also used with scientific modelling of natural systems or human systems to gain insight into their functioning, as in economics. Simulation can be used to show the eventual real effects of alternative conditions and courses of action. Simulation is also used when the real system cannot be engaged, because it may not be accessible, or it may be dangerous or unacceptable to engage, or it is being designed bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
