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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 as 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. The Nakajima-Zwanzig (NZ) generalized master equation is a formally exact approach for simulating quantum dynamics in condensed phases. This framework is particularly designed to address the dynamics of a reduced system interact with a larger environment, often represented as a system coupled to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Memory Kernel Methods
Memory is the faculty of the mind by which data or information is encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If past events could not be remembered, it would be impossible for language, relationships, or personal identity to develop. Memory loss is usually described as forgetfulness or amnesia. Memory is often understood as an informational processing system with explicit and implicit functioning that is made up of a sensory processor, short-term (or working) memory, and long-term memory. This can be related to the neuron. The sensory processor allows information from the outside world to be sensed in the form of chemical and physical stimuli and attended to various levels of focus and intent. Working memory serves as an encoding and retrieval processor. Information in the form of stimuli is encoded in accordance with explicit or implicit functions by the working memory processor. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Zwanzig
Robert Walter Zwanzig (9 April 1928 – 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 1966 to 1979 he was a research professor at the In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Matrix
In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed states. These arise in quantum mechanics in two different situations: # when the preparation of a system can randomly produce different pure states, and thus one must deal with the statistics of possible preparations, and # when one wants to describe a physical system that is entangled with another, without describing their combined state. This case is typical for a system interacting with some environment (e.g. decoherence). In this case, the density matrix of an entangled system differs from that of an ensemble of pure states that, combined, would give the same statistical results upon measurement. Density matrices are thus crucial tools in areas of quantum ... [...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 modeled 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. The name was proposed in 1940: 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, 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-dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mori-Zwanzig Formalism
The Mori–Zwanzig formalism, named after the physicists 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 observables describ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hazime Mori
is the Japanese word meaning . In 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. Written forms 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" *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 manga ''Attack on Titan'' *, Japa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Equation
In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed states. These arise in quantum mechanics in two different situations: # when the preparation of a system can randomly produce different pure states, and thus one must deal with the statistics of possible preparations, and # when one wants to describe a physical system that is entangled with another, without describing their combined state. This case is typical for a system interacting with some environment (e.g. decoherence). In this case, the density matrix of an entangled system differs from that of an ensemble of pure states that, combined, would give the same statistical results upon measurement. Density matrices are thus crucial tools in areas of quantum mecha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Dynamics
System dynamics (SD) is an approach to understanding the nonlinear behaviour of complex systems over time using stocks, flows, internal feedback loops, table functions and time delays. Overview System dynamics is a methodology and mathematical modeling technique to frame, understand, and discuss complex issues and problems. Originally developed in the 1950s to help corporate managers improve their understanding of industrial processes, SD is currently being used throughout the public and private sector for policy analysis and design. Convenient graphical user interface (GUI) system dynamics software developed into user friendly versions by the 1990s and have been applied to diverse systems. SD models solve the problem of simultaneity (mutual causation) by updating all variables in small time increments with positive and negative feedbacks and time delays structuring the interactions and control. The best known SD model is probably the 1972 ''The Limits to Growth''. This model fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Redfield Equation
Redfield may refer to: People * Redfield (surname) Places ;United Kingdom *Redfield, Bristol, an area within the City of Bristol ;United States * Mount Redfield, a mountain in Essex County, New York *Redfield, Arkansas, a city in northwestern Jefferson County *Redfield, Iowa, a city in Dallas County * Redfield, Kansas, a city in Bourbon County * Redfield, New York, a town in Oswego County *Redfield, South Dakota Redfield is a city in and the county seat of Spink County, South Dakota, United States. The population was 2,214 at the 2020 United States Census, 2020 census. The city was named for J. B. Redfield, a railroad official. Geography According to th ..., a city in and the county seat of Spink County * Redfield, Texas, a census-designated place in Nacogdoches County * Redfield School Historic District, a former school and historic district in Redfield, Arkansas * Redfield Township, Spink County, South Dakota, a township in Spink County, South Dakota * Redfield & West Str ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and Microscopic scale, (optical) microscopic) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales. Quantum systems have Bound state, bound states that are Quantization (physics), quantized to Discrete mathematics, discrete values of energy, momentum, angular momentum, and ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
