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Universality Classes
In statistical mechanics, a universality class is a collection of mathematical models which share a single scale invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite scales, their behavior will become increasingly similar as the limit scale is approached. In particular, asymptotic phenomena such as critical exponents will be the same for all models in the class. Some well-studied universality classes are the ones containing the Ising model or the percolation theory at their respective phase transition points; these are both families of classes, one for each lattice dimension. Typically, a family of universality classes will have a lower and upper critical dimension: below the lower critical dimension, the universality class becomes degenerate (this dimension is 2d for the Ising model, or for directed percolation, but 1d for undirected percolation), and above the upper critical dimension the critical expo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles. Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscopic parameters that fluctuate about average values and are characterized by probability distributions. This established the fields of statistical thermodynamics and statistical physics. The founding of the field of statistical mechanics is generally credited to three physicists: *Ludwig Boltzmann, who developed the fundamental interpretation of entropy in terms of a collection of microstates *James Clerk Maxwell, who developed models of probability distr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Molecular Beam Epitaxy
Molecular-beam epitaxy (MBE) is an epitaxy method for thin-film deposition of single crystals. MBE is widely used in the manufacture of semiconductor devices, including transistors, and it is considered one of the fundamental tools for the development of nanotechnologies. MBE is used to fabricate diodes and MOSFETs (MOS field-effect transistors) at microwave frequencies, and to manufacture the lasers used to read optical discs (such as CDs and DVDs). History Original ideas of MBE process were first established by Günther. Films he deposited were not epitaxial, but were deposited on glass substrates. With the development of vacuum technology, MBE process was demonstrated by Davey and Pankey who succeeded in growing GaAs epitaxial films on single crystal GaAs substrates using Günther's method. Major subsequent development of MBE films was enabled by J.R. Arthur's investigations of kinetic behavior of growth mechanisms and Alfred Y. Cho's in situ observation of MBE process usi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean-field Theory
In physics and probability theory, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of freedom (the number of values in the final calculation of a statistic that are free to vary). Such models consider many individual components that interact with each other. The main idea of MFT is to replace all interactions to any one body with an average or effective interaction, sometimes called a ''molecular field''. This reduces any many-body problem into an effective one-body problem. The ease of solving MFT problems means that some insight into the behavior of the system can be obtained at a lower computational cost. MFT has since been applied to a wide range of fields outside of physics, including statistical inference, graphical models, neuroscience, artificial intelligence, epidemic models, queueing theory, computer-network p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heisenberg Model (classical)
The Classical Heisenberg model, developed by Werner Heisenberg, is the n = 3 case of the n-vector model, one of the models used in statistical physics to model ferromagnetism, and other phenomena. Definition It can be formulated as follows: take a d-dimensional lattice, and a set of spins of the unit length :\vec_i \in \mathbb^3, , \vec_i, =1\quad (1), each one placed on a lattice node. The model is defined through the following Hamiltonian: : \mathcal = -\sum_ \mathcal_ \vec_i \cdot \vec_j\quad (2) with : \mathcal_ = \begin J & \mboxi, j\mbox \\ 0 & \mbox\end a coupling between spins. Properties * The general mathematical formalism used to describe and solve the Heisenberg model and certain generalizations is developed in the article on the Potts model. * In the continuum limit the Heisenberg model (2) gives the following equation of motion :: \vec_=\vec\wedge \vec_. :This equation is called the continuous classical Heisenberg ferromagnet equation or shortly Heisenberg model ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
XY Model
The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. In general, the XY model can be seen as a specialization of Stanley's ''n''-vector model for . Definition Given a -dimensional lattice , per each lattice site there is a two-dimensional, unit-length vector The ''spin configuration'', is an assignment of the angle for each . Given a ''translation-invariant'' interaction and a point dependent external field \mathbf_=(h_j,0), the ''configuration energy'' is : H(\mathbf) = - \sum_ J_\; \mathbf_i\cdot\mathbf_j -\sum_j \mathbf_j\cdot \mathbf_j =- \sum_ J_\; \cos(\theta_i-\theta_j) -\sum_j h_j\cos\theta_j The case in which except for nearest neighbor is called ''nearest neighbor'' case. The ''configuration probability'' is given by the Boltzmann distribution with inverse temperature : :P(\mathbf)=\frac \qquad Z=\int_ \prod_ d\theta_j\;e^. where is the normalization, or partition func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ising Critical Exponents
This article lists the critical exponents of the ferromagnetic transition in the Ising model. In statistical physics, the Ising model is the simplest system exhibiting a continuous phase transition with a scalar order parameter and \mathbb_2 symmetry. The critical exponents of the transition are universal values and characterize the singular properties of physical quantities. The ferromagnetic transition of the Ising model establishes an important universality class, which contains a variety of phase transitions as different as ferromagnetism close to the Curie point and critical opalescence of liquid near its critical point. From the quantum field theory point of view, the critical exponents can be expressed in terms of scaling dimensions of the local operators \sigma,\epsilon,\epsilon' of the conformal field theory describing the phase transition (In the Ginzburg–Landau description, these are the operators normally called \phi,\phi^2,\phi^4.) These expressions are given in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Protected Percolation
Protection is any measure taken to guard a thing against damage caused by outside forces. Protection can be provided to physical objects, including organisms, to systems, and to intangible things like civil and political rights. Although the mechanisms for providing protection vary widely, the basic meaning of the term remains the same. This is illustrated by an explanation found in a manual on electrical wiring: Some kind of protection is a characteristic of all life, as living things have evolved at least some protective mechanisms to counter damaging environmental phenomena, such as ultraviolet light. Biological membranes such as bark on trees and skin on animals offer protection from various threats, with skin playing a key role in protecting organisms against pathogens and excessive water loss. Additional structures like scales and hair offer further protection from the elements and from predators, with some animals having features such as spines or camouflage servin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conserved Directed Percolation
Conservation is the preservation or efficient use of resources, or the conservation of various quantities under physical laws. Conservation may also refer to: Environment and natural resources * Nature conservation, the protection and management of the environment and natural resources * Conservation biology, the science of protection and management of biodiversity * Conservation movement, political, environmental, or social movement that seeks to protect natural resources, including biodiversity and habitat * Conservation organization, an organization dedicated to protection and management of the environment or natural resources * Wildlife conservation, the practice of protecting wild species and their habitats in order to prevent species from going extinct * ''Conservation'' (magazine), published by the Society for Conservation Biology from 2000 to 2014 ** ''Conservation Biology'' (journal), scientific journal of the Society for Conservation Biology Physical laws * Conser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Percolation
In statistical physics, directed percolation (DP) refers to a class of models that mimic filtering of fluids through porous materials along a given direction, due to the effect of gravity. Varying the microscopic connectivity of the pores, these models display a phase transition from a macroscopically permeable (percolating) to an impermeable (non-percolating) state. Directed percolation is also used as a simple model for epidemic spreading with a transition between survival and extinction of the disease depending on the infection rate. More generally, the term directed percolation stands for a universality class of continuous phase transitions which are characterized by the same type of collective behavior on large scales. Directed percolation is probably the simplest universality class of transitions out of thermal equilibrium. Lattice models One of the simplest realizations of DP is bond directed percolation. This model is a directed variant of ordinary (isotropic) percol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Percolation Critical Exponents
In the context of the physical and mathematical theory of percolation, a percolation transition is characterized by a set of ''universal'' critical exponents, which describe the fractal properties of the percolating medium at large scales and sufficiently close to the transition. The exponents are universal in the sense that they only depend on the type of percolation model and on the space dimension. They are expected to not depend on microscopic details such as the lattice structure, or whether site or bond percolation is considered. This article deals with the critical exponents of random percolation. Percolating systems have a parameter p\,\! which controls the occupancy of sites or bonds in the system. At a critical value p_c\,\!, the mean cluster size goes to infinity and the percolation transition takes place. As one approaches p_c\,\!, various quantities either diverge or go to a constant value by a power law in , p - p_c, \,\!, and the exponent of that power law is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Potts Model
In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid-state physics. The strength of the Potts model is not so much that it models these physical systems well; it is rather that the one-dimensional case is exactly solvable, and that it has a rich mathematical formulation that has been studied extensively. The model is named after Renfrey Potts, who described the model near the end of his 1951 Ph.D. thesis. The model was related to the "planar Potts" or " clock model", which was suggested to him by his advisor, Cyril Domb. The four-state Potts model is sometimes known as the Ashkin–Teller model, after Julius Ashkin and Edward Teller, who considered an equivalent model in 1943. The Potts model is related to, and generalized by, several other models, including the XY model, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
