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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 model (physics), 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 (physics), spin 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'' are arranged as follows: For a lattice with ''N'' sites, the Partition function (statistical mechanics), partition function is :Z_=\sum_\prod_w\left(\sigma_,\sigma_,\sigma_,\sigma_\right), where the sum is ... [...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. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in a wide variety of fields such as biology, neuroscience, computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ..., information theory and sociology. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. 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 microscop ... [...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 of the form :\sum_^n a_i ^m\ for some 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. Diagonalization Any degree-2 homogeneous polynomial can be transformed to a diagonal form by variable substitution. Higher-degree homogeneous polynomials can be diagonalized if and only if their catalecticant is non-zero. The process is particularly simple for degree-2 forms (quadratic forms), based on the eigenvalues of the symmetric matrix representing the quadratic f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Models
Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an ornamental pattern of crossing strips of pastry Companies * Lattice Engines, a technology company specializing in business applications for marketing and sales * Lattice Group, a former British gas transmission business * Lattice Semiconductor, a US-based integrated circuit manufacturer Science, technology, and mathematics Mathematics * Lattice (group), a repeating arrangement of points ** Lattice (discrete subgroup), a discrete subgroup of a topological group whose quotient carries an invariant finite Borel measure ** Lattice (module), a module over a ring that is embedded in a vector space over a field ** Lattice graph, a graph that can be drawn within a repeating arrangement of points ** Lattice-based cryptography, encryp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exactly Solvable Models
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, a sniper rifle {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transfer-matrix Method (statistical Mechanics)
In statistical mechanics, the transfer-matrix method is a mathematical technique which is used to write the partition function into a simpler form. It was introduced in 1941 by Hans Kramers and Gregory Wannier. In many one dimensional lattice models, the partition function is first written as an ''n''-fold summation over each possible 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 in the form : \mathcal = \mathbf_0 \cdot \left\ \cdot \mathbf_ where v0 and v''N''+1 are vectors of dimension ''p'' and the ''p'' × ''p'' matrices W''k'' are the so-called transfer ... [...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 (mathematics), 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 List of thermodynamic properties, 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 fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean, mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would expect to get in reality. 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 Integral, 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 a ... [...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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvector Matrix
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a constant factor \lambda when the linear transformation is applied to it: T\mathbf v=\lambda \mathbf v. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \lambda (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed. The e ... [...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), Kent; Nat ... [...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), Kent; Nat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
