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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] |
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] |
Ising Model
The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent Nuclear magnetic moment, magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a Graph (abstract data type), graph, usually a lattice (group), lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases.The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition. Though it is a highly simplified model of a magnetic material, the Ising model can sti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transfer Operator
In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. In all usual cases, the largest eigenvalue is 1, and the corresponding eigenvector is the invariant measure of the system. The transfer operator is sometimes called the Ruelle operator, after David Ruelle, or the Perron–Frobenius operator or Ruelle–Perron–Frobenius operator, in reference to the applicability of the Perron–Frobenius theorem to the determination of the eigenvalues of the operator. Definition The iterated function to be studied is a map f\colon X\rightarrow X for an arbitrary set X. The transfer operator is defined as an operator \mathcal acting on the space of functions \ as :(\mathcal\Phi)(x) = \sum_ g(y) \Phi(y) where g\colon X\rightarrow\mathbb is an auxiliary valuation function. When f has a Jacobian determinant , J, , then g is usually taken ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lars Onsager
Lars Onsager (November 27, 1903 – October 5, 1976) was a Norwegian American physical chemist and theoretical physicist. He held the Gibbs Professorship of Theoretical Chemistry at Yale University. He was awarded the Nobel Prize in Chemistry in 1968. Education and early life Lars Onsager was born in Kristiania (now Oslo), Norway. His father was a lawyer. After completing secondary school in Oslo, he attended the Norwegian Institute of Technology, Norwegian Institute of Technology (NTH) in Trondheim, graduating as a chemical engineering, chemical engineer in 1925. While there he worked through ''A Course of Modern Analysis'', which was instrumental in his later work. Career and research In 1925 he arrived at a correction to the Debye–Hückel equation, Debye-Hückel theory of electrolyte, electrolytic Solution (chemistry), solutions, to specify Brownian movement of ions in solution, and during 1926 published it. He traveled to Zürich, where Peter Debye was teaching, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lifson–Roig Model
In polymer science, the Lifson–Roig model is a helix-coil transition model applied to the alpha helix- random coil transition of polypeptides; it is a refinement of the Zimm–Bragg model that recognizes that a polypeptide alpha helix is only stabilized by a hydrogen bond only once three consecutive residues have adopted the helical conformation. To consider three consecutive residues each with two states (helix and coil), the Lifson–Roig model uses a 4x4 transfer matrix instead of the 2x2 transfer matrix of the Zimm–Bragg model, which considers only two consecutive residues. However, the simple nature of the coil state allows this to be reduced to a 3x3 matrix for most applications. The Zimm–Bragg and Lifson–Roig models are but the first two in a series of analogous transfer-matrix methods in polymer science that have also been applied to nucleic acids and branched polymers. The transfer-matrix approach is especially elegant for homopolymers, since the statistical mec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zimm–Bragg Model
In statistical mechanics, the Zimm–Bragg model is a helix-coil transition model that describes helix-coil transitions of macromolecules, usually polymer chains. Most models provide a reasonable approximation of the fractional helicity of a given polypeptide; the Zimm–Bragg model differs by incorporating the ease of propagation (self-replication) with respect to nucleation. It is named for co-discoverers Bruno H. Zimm and J. K. Bragg. Helix-coil transition models Helix-coil transition models assume that polypeptides are linear chains composed of interconnected segments. Further, models group these sections into two broad categories: ''coils'', random conglomerations of disparate unbound pieces, are represented by the letter 'C', and ''helices'', ordered states where the chain has assumed a structure stabilized by hydrogen bonding, are represented by the letter 'H'. Thus, it is possible to loosely represent a macromolecule as a string such as CCCCHCCHCHHHHHCHCCC and so forth. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Weight
In statistical mechanics, multiplicity (also called statistical weight) refers to the number of microstate (statistical mechanics), microstates corresponding to a particular macrostate of a thermodynamic system. Commonly denoted \Omega, it is related to the configuration entropy of an isolated system via Boltzmann's entropy formula S = k_\text \log \Omega, where S is the Boltzmann entropy, entropy and k_\text is the Boltzmann constant. Example: the two-state paramagnet A simplified model of the two-state paramagnetism, paramagnet provides an example of the process of calculating the multiplicity of particular macrostate. This model consists of a system of microscopic dipoles which may either be aligned or anti-aligned with an externally applied magnetic field . Let N_\uparrow represent the number of dipoles that are aligned with the external field and N_\downarrow represent the number of anti-aligned dipoles. The energy of a single aligned dipole is U_\uparrow = -\mu B, whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin (physics)
Spin is an Intrinsic and extrinsic properties, intrinsic form of angular momentum carried by elementary particles, and thus by List of particles#Composite particles, composite particles such as hadrons, atomic nucleus, atomic nuclei, and atoms. Spin is quantized, and accurate models for the interaction with spin require relativistic quantum mechanics or quantum field theory. The existence of electron spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum. The relativistic spin–statistics theorem connects electron spin quantization to the Pauli exclusion principle: observations of exclusion imply half-integer spin, and observations of half-integer spin imply exclusion. Spin is described mathematically as a vector for some particles such as photons, and as a spinor or bispinor for other particles such as electrons. Sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics, known as physical mathematics. Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical parts of our world. Classical mechanics Applying the techniques of mathematical physics to classical mechanics typically involves the rigorous, abstract, and advanced reformulation of Newtonian mechanics in terms of Lagrangian mechanics and Hamiltonian mechanics (including both approaches in the presence of constraints). Both formulations are embodied in analytical mechanics and lead ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenanalysis
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] |
