|
Mean-field Particle Methods
Mean-field particle methods are a broad class of ''interacting type'' Monte Carlo algorithms for simulating from a sequence of probability distributions satisfying a nonlinear evolution equation. These flows of probability measures can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depends on the distributions of the current random states. A natural way to simulate these sophisticated nonlinear Markov processes is to sample a large number of copies of the process, replacing in the evolution equation the unknown distributions of the random states by the sampled empirical measures. In contrast with traditional Monte Carlo and Markov chain Monte Carlo methods these mean-field particle techniques rely on sequential interacting samples. The terminology mean-field reflects the fact that each of the ''samples (a.k.a. particles, individuals, walkers, agents, creatures, or phenotypes)'' interacts with the empirical measures ... [...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. The name comes from the Monte Carlo Casino in Monaco, where the primary developer of the method, mathematician Stanisław Ulam, was inspired by his uncle's gambling habits. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from a probability distribution. They can also be used to model phenomena with significant uncertainty in inputs, such as calculating the risk of a nuclear power plant failure. Monte Carlo methods are often implemented using computer simulations, and they can provide approximate solutions to problems that are otherwise intractable or too complex to analyze mathematically. Monte Carlo methods are widely used in va ... [...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] |
Branching Process
In probability theory, a branching process is a type of mathematical object known as a stochastic process, which consists of collections of random variables indexed by some set, usually natural or non-negative real numbers. The original purpose of branching processes was to serve as a mathematical model of a population in which each individual in generation n produces some random number of individuals in generation n+1, according, in the simplest case, to a fixed probability distribution that does not vary from individual to individual. Branching processes are used to model reproduction; for example, the individuals might correspond to bacteria, each of which generates 0, 1, or 2 offspring with some probability in a single time unit. Branching processes can also be used to model other systems with similar dynamics, e.g., the spread of surnames in genealogy or the propagation of neutrons in a nuclear reactor. A central question in the theory of branchi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Population Genetics
Population genetics is a subfield of genetics that deals with genetic differences within and among populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as Adaptation (biology), adaptation, speciation, and population stratification, population structure. Population genetics was a vital ingredient in the emergence of the Modern synthesis (20th century), modern evolutionary synthesis. Its primary founders were Sewall Wright, J. B. S. Haldane and Ronald Fisher, who also laid the foundations for the related discipline of quantitative genetics. Traditionally a highly mathematical discipline, modern population genetics encompasses theoretical, laboratory, and field work. Population genetic models are used both for statistical inference from DNA sequence data and for proof/disproof of concept. What sets population genetics apart from newer, more phenotypic approaches to modelling evolution, such as evolutionary game theory and evolu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Biology
Computational biology refers to the use of techniques in computer science, data analysis, mathematical modeling and Computer simulation, computational simulations to understand biological systems and relationships. An intersection of computer science, biology, and data science, the field also has foundations in applied mathematics, molecular biology, cell biology, chemistry, and genetics. History Bioinformatics, the analysis of informatics processes in biological systems, began in the early 1970s. At this time, research in artificial intelligence was using network models of the human brain in order to generate new algorithms. This use of biological data pushed biological researchers to use computers to evaluate and compare large data sets in their own field. By 1982, researchers shared information via Punched card, punch cards. The amount of data grew exponentially by the end of the 1980s, requiring new computational methods for quickly interpreting relevant information. Per ... [...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 \exp\left(- \frac \right) where is the probability of the system being in state , is the exponential function, 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 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 sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schrödinger Equation
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, an Austrian physicist, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system. The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave. The equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and Microscopic scale, (optical) microscopic) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales. Quantum systems have Bound state, bound states that are Quantization (physics), quantized to Discrete mathematics, discrete values of energy, momentum, angular momentum, and ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Physics
Computational physics is the study and implementation of numerical analysis to solve problems in physics. Historically, computational physics was the first application of modern computers in science, and is now a subset of computational science. It is sometimes regarded as a subdiscipline (or offshoot) of theoretical physics, but others consider it an intermediate branch between theoretical and experimental physics — an area of study which supplements both theory and experiment. Overview In physics, different theories based on mathematical models provide very precise predictions on how systems behave. Unfortunately, it is often the case that solving the mathematical model for a particular system in order to produce a useful prediction is not feasible. This can occur, for instance, when the solution does not have a closed-form expression, or is too complicated. In such cases, numerical approximations are required. Computational physics is the subject that deals with these ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boltzmann Equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium; it was devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3. The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number. The equation arises not by analyzing the individual positions and momenta of each particle in the fluid but rather by considering a probability distribution for the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reaction–diffusion System
Reaction–diffusion systems are mathematical models that correspond to several physical phenomena. The most common is the change in space and time of the concentration of one or more chemical substances: local chemical reactions in which the substances are transformed into each other, and diffusion which causes the substances to spread out over a surface in space. Reaction–diffusion systems are naturally applied in chemistry. However, the system can also describe dynamical processes of non-chemical nature. Examples are found in biology, geology and physics (neutron diffusion theory) and ecology. Mathematically, reaction–diffusion systems take the form of semi-linear parabolic partial differential equations. They can be represented in the general form :\partial_t \boldsymbol = \underline \,\nabla^2 \boldsymbol + \boldsymbol(\boldsymbol), where represents the unknown vector function, is a diagonal matrix of diffusion coefficients, and accounts for all local reactions. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
McKean–Vlasov Process
In probability theory, a McKean–Vlasov process is a stochastic process described by a stochastic differential equation where the coefficients of the diffusion depend on the distribution of the solution itself. The equations are a model for Vlasov equation and were first studied by Henry McKean in 1966. It is an example of propagation of chaos, in that it can be obtained as a limit of a mean-field system of interacting particles: as the number of particles tends to infinity, the interactions between any single particle and the rest of the pool will only depend on the particle itself. Definition Consider a measurable function \sigma:\R^d \times \mathcal(\R^d)\to \mathcal_(\R) where \mathcal(\R^d) is the space of probability distributions on \R^d equipped with the Wasserstein metric W_2 and \mathcal_(\R) is the space of square matrices of dimension d. Consider a measurable function b:\R^d\times \mathcal(\R^d)\to \R^d. Define a(x,\mu) := \sigma(x,\mu)\sigma(x,\mu)^T. A stochastic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
