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Mori-Zwanzig Formalism
The Mori–Zwanzig formalism, named after the physicists Hajime Mori and Robert Zwanzig, is a method of statistical physics. It allows the splitting of the dynamics of a system into a relevant and an irrelevant part using projection operators, which helps to find closed equations of motion for the relevant part. It is used e.g. in fluid mechanics or condensed matter physics. Idea Macroscopic systems with a large number of microscopic degrees of freedom are often well described by a small number of relevant variables, for example the magnetization in a system of spins. The Mori–Zwanzig formalism allows the finding of macroscopic equations that only depend on the relevant variables based on microscopic equations of motion of a system, which are usually determined by the Hamiltonian. The irrelevant part appears in the equations as noise. The formalism does not determine what the relevant variables are, these can typically be obtained from the properties of the system. The observab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hajime Mori
is the Japanese word meaning . In the Japanese traditional martial arts such as karate, judo, aikido, Kūdō and kendo, it is a verbal command to "begin". Hajime is also a common Japanese given name for males. In the Amami Islands, Hajime (元) is a surname. Possible writings Hajime can be written using different kanji characters and can mean: *始め, "beginning" or "start" *初め, "beginning" or "first" ;as a given name *一, "first" *元, "beginning" or "origin" *始, "beginning" or "start" *肇, "beginning" *基, "fundamental" *創, "genesis" *孟, "beginning" or "chief" *朔, "first day of month" *甫, "beginning" or "great" The name can also be written in hiragana as はじめ and katakana as ハジメ People Given name *, Japanese politician *, Japanese musician, actor and comedian *, Japanese sumo wrestler *, Japanese politician *, Japanese football player *, first doctor to discover the Minamata disease *, Japanese voice actor *, Japanese manga artist; creator of m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Zwanzig
Robert Walter Zwanzig (born Brooklyn, New York, 9 April 1928 – died Bethesda, Maryland, May 15, 2014) was an American theoretical physicist and chemist who made important contributions to the statistical mechanics of irreversible processes, protein folding, and the theory of liquids and gases. Background Zwanzig received his bachelor's degree from Brooklyn Polytechnic Institute in 1948 and his master's degree from 1950 at the University of Southern California. In 1952 he completed a doctorate in physical chemistry at Caltech under the supervision of John G. Kirkwood. His thesis title was ''Quantum Hydrodynamics: a statistical mechanical theory of light scattering from simple non-polar fluids''. From 1951 to 1954 he worked as a post-doctoral researcher in theoretical chemistry at Yale University, and from 1954 to 1958 he was an assistant professor in chemistry at Johns Hopkins University. From 1958 to 1966 he was a physical chemist at the National Bureau of Standards and from 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Physics
Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the Mathematics, mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic nature. Its applications include many problems in the fields of physics, biology, chemistry, and neuroscience. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics develop the Phenomenology (particle physics), phenomenological results of thermodynamics from a probabilistic examination of the underlying microscopic systems. Historically, one of the first topics in physics where statistical methods were applied was the field of classical mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluid Mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and biomedical engineering, geophysics, oceanography, meteorology, astrophysics, and biology. It can be divided into fluid statics, the study of fluids at rest; and fluid dynamics, the study of the effect of forces on fluid motion. It is a branch of continuum mechanics, a subject which models matter without using the information that it is made out of atoms; that is, it models matter from a ''macroscopic'' viewpoint rather than from ''microscopic''. Fluid mechanics, especially fluid dynamics, is an active field of research, typically mathematically complex. Many problems are partly or wholly unsolved and are best addressed by numerical methods, typically using computers. A modern discipline, called computational fluid dynamics (CFD), is dev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Condensed Matter Physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the subject deals with "condensed" phases of matter: systems of many constituents with strong interactions between them. More exotic condensed phases include the superconducting phase exhibited by certain materials at low temperature, the ferromagnetic and antiferromagnetic phases of spins on crystal lattices of atoms, and the Bose–Einstein condensate found in ultracold atomic systems. Condensed matter physicists seek to understand the behavior of these phases by experiments to measure various material properties, and by applying the physical laws of quantum mechanics, electromagnetism, statistical mechanics, and other theories to develop mathematical models. The diversity of systems and phenomena available for study makes condensed matter phy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta''. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics. Overview Phase space coordinates (p,q) and Hamiltonian H Let (M, \mathcal L) be a mechanical system with the configuration space M and the smooth Lagrangian \mathcal L. Select a standard coordinate system (\boldsymbol,\boldsymbol) on M. The quantities \textstyle p_i(\boldsymbol,\boldsymbol,t) ~\stackrel~ / are called ''momenta''. (Also ''generalized momenta'', ''conjugate momenta'', and ''canonical momenta''). For a time instant t, the Legendre transformat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar Product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more). Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Operator
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent ''mixed states''. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations, and second when one wants to describe a physical system which is entangled with another, without describing their combined state. Density matrices are thus crucial tools in areas of quantum mechanics that deal with mixed states, such as quantum statistical mechanics, open quantum systems, quantum decoherence, and quantum information. Definition and m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nakajima–Zwanzig Equation
The Nakajima–Zwanzig equation (named after the physicists who developed it, Sadao Nakajima and Robert Zwanzig) is an integral equation describing the time evolution of the "relevant" part of a quantum-mechanical system. It is formulated in the density matrix formalism and can be regarded a generalization of the master equation. The equation belongs to the Mori-Zwanzig formalism within the statistical mechanics of irreversible processes (named after Hazime Mori). By means of a projection operator the dynamics is split into a slow, collective part (''relevant part'') and a rapidly fluctuating ''irrelevant'' part. The goal is to develop dynamical equations for the collective part. Derivation The starting pointA derivation analogous to that presented here is found, for instance, in Breuer, Petruccione ''The theory of open quantum systems'', Oxford University Press 2002, S.443ff is the quantum mechanical version of the von Neumann equation, also known as the Liouville equation: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zwanzig Projection Operator
The Zwanzig projection operator is a mathematical device used in statistical mechanics. It operates in the linear space of phase space functions and projects onto the linear subspace of "slow" phase space functions. It was introduced by Robert Zwanzig to derive a generic master equation. It is mostly used in this or similar context in a formal way to derive equations of motion for some "slow" collective variables. Slow variables and scalar product The Zwanzig projection operator operates on functions in the 6N-dimensional phase space \Gamma=\ of N point particles with coordinates \mathbf_i and momenta \mathbf_i. A special subset of these functions is an enumerable set of "slow variables" A(\Gamma)=\. Candidates for some of these variables might be the long-wavelength Fourier components \rho_k(\Gamma) of the mass density and the long-wavelength Fourier components \mathbf_\mathbf(\Gamma) of the momentum density with the wave vector \mathbf identified with n. The Zwanzig projection ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
