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Corner Transfer Matrix
In statistical mechanics, the corner transfer matrix describes the effect of adding a quadrant to a lattice. Introduced by Rodney Baxter in 1968 as an extension of the Kramers-Wannier row-to-row transfer matrix, it provides a powerful method of studying lattice models. Calculations with corner transfer matrices led Baxter to the exact solution of the hard hexagon model in 1980. Definition Consider an IRF (interaction-round-a-face) model, i.e. a square lattice model with a spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ... σ''i'' assigned to each site ''i'' and interactions limited to spins around a common face. Let the total energy be given by :E=\sum_\epsilon\left(\sigma_,\sigma_,\sigma_,\sigma_\right), where for each face the surrounding sites ''i'', ''j'', ''k'' and ''l ... [...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] |
Diagonal Form
In mathematics, a diagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving different indeterminates. That is, it is :\sum_^n a_i ^m\ for some given degree ''m''. Such forms ''F'', and the hypersurfaces ''F'' = 0 they define in projective space, are very special in geometric terms, with many symmetries. They also include famous cases like the Fermat curves, and other examples well known in the theory of Diophantine equations. A great deal has been worked out about their theory: algebraic geometry, local zeta-functions via Jacobi sums, Hardy-Littlewood circle method. Examples :X^2+Y^2-Z^2 = 0 is the unit circle in ''P''2 :X^2-Y^2-Z^2 = 0 is the unit hyperbola in ''P''2. :x_0^3+x_1^3+x_2^3+x_3^3=0 gives the Fermat cubic surface in ''P''3 with 27 lines. The 27 lines in this example are easy to describe explicitly: they are the 9 lines of the form (''x'' : ''ax'' : ''y'' : ''by'') where ''a'' and ''b'' are fixed numbers with cube −1, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Models
In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are quite popular in theoretical physics, for many reasons. Some models are exactly solvable, and thus offer insight into physics beyond what can be learned from perturbation theory. Lattice models are also ideal for study by the methods of computational physics, as the discretization of any continuum model automatically turns it into a lattice model. The exact solution to many of these models (when they are solvable) includes the presence of solitons. Techniques for solving these include the inverse scattering transform and the method of Lax pairs, the Yang–Baxter equation and quantum groups. The solution of these models has given i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exactly Solvable Models , a sniper rifl ...
Exact may refer to: * Exaction, a concept in real property law * ''Ex'Act'', 2016 studio album by Exo * Schooner Exact, the ship which carried the founders of Seattle Companies * Exact (company), a Dutch software company * Exact Change, an American independent book publishing company * Exact Editions, a content management platform Mathematics * Exact differentials, in multivariate calculus * Exact algorithms, in computer science and operations research * Exact colorings, in graph theory * Exact couples, a general source of spectral sequences * Exact sequences, in homological algebra * Exact functor, a function which preserves exact sequences See also * *Exactor (other) *XACT (other) *EXACTO EXACTO, an acronym of "Extreme Accuracy Tasked Ordnance", is a sniper rifle firing smart bullets being developed for DARPA (Defense Advanced Research Projects Agency) by Lockheed Martin and Teledyne Scientific & Imaging in November 2008. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transfer-matrix Method
In statistical mechanics, the transfer-matrix method is a Mathematical physics, mathematical technique which is used to write the Partition function (mathematics), partition function into a simpler form. It was introduced in 1941 by Hans Kramers and Gregory Wannier. In many one dimensional Lattice model (physics), lattice models, the partition function is first written as an ''n''-fold summation over each possible Microstate (statistical mechanics), microstate, and also contains an additional summation of each component's contribution to the energy of the system within each microstate. Overview Higher dimensional models contain even more summations. For systems with more than a few particles, such expressions can quickly become too complex to work out directly, even by computer. Instead, the partition function can be rewritten in an equivalent way. The basic idea is to write the partition function (mathematics), partition function in the form : \mathcal = \mathbf_0 \cdot \left ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thermodynamic Limit
In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blundell and K.M. Blundell, "Concepts in Thermal Physics", Oxford University Press (2009) The thermodynamic limit is defined as the limit of a system with a large volume, with the particle density held fixed. : N \to \infty,\, V \to \infty,\, \frac N V =\text In this limit, macroscopic thermodynamics is valid. There, thermal fluctuations in global quantities are negligible, and all thermodynamic quantities, such as pressure and energy, are simply functions of the thermodynamic variables, such as temperature and density. For example, for a large volume of gas, the fluctuations of the total internal energy are negligible and can be ignored, and the average internal energy can be predicted from knowledge of the pressure and temperature of the gas. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to end th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity matrix. This leads to the equivalent characterization: a matrix is orthogonal if its transpose is equal to its inverse: Q^\mathrm=Q^, where is the inverse of . An orthogonal matrix is necessarily invertible (with inverse ), unitary (), where is the Hermitian adjoint (conjugate transpose) of , and therefore normal () over the real numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the inner product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation, reflection or rotoreflection. In other words, it is a unitary transformation. The set of orthogonal matrices, under multiplication, forms the group , known as the orthogonal gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvector Matrix
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CTM - FTM Factorisation
CTM is an initialism that may stand for: Companies and organizations * Compagnie de Transports au Maroc, a Moroccan public bus transport company * Companhia de Telecomunicações de Macau, a Macau telecommunications company * Confederation of Mexican Workers, a confederation of labor unions Technology * Cell Transmission Model, a traffic prediction algorithm * Chemical transport model, a simulation of atmospheric chemistry and pollution * Close to Metal, a low-level programming interface * '' Concepts, Techniques, and Models of Computer Programming'', a 2004 textbook * Corner transfer matrix, a method in statistical mechanics * Critical thermal maximum, the temperature above which an organism cannot survive * Current Transformation Matrix, the transformation matrix currently applying in a graphics pipeline * Certified Technology Manager, an accreditation by ATMAE - Association of Technology, Management and Applied Engineering. Transport * Chatham railway station, Kent; Natio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rodney Baxter
Rodney James Baxter FRS FAA (born 8 February 1940 in London, United Kingdom) is an Australian physicist, specialising in statistical mechanics. He is well known for his work in exactly solved models, in particular vertex models such as the six-vertex model and eight-vertex model, and the chiral Potts model and hard hexagon model. A recurring theme in the solution of such models, the Yang–Baxter equation, also known as the "star–triangle relation", is named in his honour. Biography Baxter was educated at Bancroft's School and Trinity College, Cambridge (BA, MA), before relocating to the Australian National University in Canberra to complete his PhD. He was among the first doctoral graduates in theoretical physics from the ANU, graduating in 1964. Then, in 1964 and 1965, he worked for the Iraq Petroleum Company. He worked as an assistant professor at the Massachusetts Institute of Technology from 1968 until 1970, when he took up a position at the ANU, and served a term ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CTM - Full Lattice
CTM is an initialism that may stand for: Companies and organizations * Compagnie de Transports au Maroc, a Moroccan public bus transport company * Companhia de Telecomunicações de Macau, a Macau telecommunications company * Confederation of Mexican Workers, a confederation of labor unions Technology * Cell Transmission Model, a traffic prediction algorithm * Chemical transport model, a simulation of atmospheric chemistry and pollution * Close to Metal, a low-level programming interface * '' Concepts, Techniques, and Models of Computer Programming'', a 2004 textbook * Corner transfer matrix, a method in statistical mechanics * Critical thermal maximum, the temperature above which an organism cannot survive * Current Transformation Matrix, the transformation matrix currently applying in a graphics pipeline * Certified Technology Manager, an accreditation by ATMAE - Association of Technology, Management and Applied Engineering. Transport * Chatham railway station, Kent; Natio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
