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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] |
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] |
Density Matrix
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Master Equation
In physics, chemistry and related fields, master equations are used to describe the time evolution of a system that can be modelled as being in a probabilistic combination of states at any given time and the switching between states is determined by a transition rate matrix. The equations are a set of differential equations – over time – of the probabilities that the system occupies each of the different states. Introduction A master equation is a phenomenological set of first-order differential equations describing the time evolution of (usually) the probability of a system to occupy each one of a discrete set of states with regard to a continuous time variable ''t''. The most familiar form of a master equation is a matrix form: : \frac=\mathbf\vec, where \vec is a column vector (where element ''i'' represents state ''i''), and \mathbf is the matrix of connections. The way connections among states are made determines the dimension of the problem; it is either *a d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Hazime 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] |
Von Neumann Equation
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] |
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] |
Reduced Density Matrix
Reduction, reduced, or reduce may refer to: Science and technology Chemistry * Reduction (chemistry), part of a reduction-oxidation (redox) reaction in which atoms have their oxidation state changed. ** Organic redox reaction, a redox reaction that takes place with organic compounds ** Ore reduction: see smelting Computing and algorithms * Reduction (complexity), a transformation of one problem into another problem * Reduction (recursion theory), given sets A and B of natural numbers, is it possible to effectively convert a method for deciding membership in B into a method for deciding membership in A? * Bit Rate Reduction, an audio compression method * Data reduction, simplifying data in order to facilitate analysis * Graph reduction, an efficient version of non-strict evaluation * L-reduction, a transformation of optimization problems which keeps the approximability features * Partial order reduction, a technique for reducing the size of the state-space to be searched ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Redfield Equation
In quantum mechanics, the Redfield equation is a Markovian master equation that describes the time evolution of the reduced density matrix of a strongly coupled quantum system that is weakly coupled to an environment. The equation is named in honor of Alfred G. Redfield, who first applied it, doing so for nuclear magnetic resonance spectroscopy. There is a close connection to the Lindblad master equation. If a so-called secular approximation is performed, where only certain resonant interactions with the environment are retained, every Redfield equation transforms into a master equation of Lindblad type. Redfield equations are trace-preserving and correctly produce a thermalized state for asymptotic propagation. However, in contrast to Lindblad equations, Redfield equations do not guarantee a positive time evolution of the density matrix. That is, it is possible to get negative populations during the time evolution. The Redfield equation approaches the correct dynamics for suff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves (wave–particle duality); and there are limits to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
