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Multiscale Green's Function
Multiscale Green's function (MSGF) is a generalized and extended version of the classical Green's function (GF) technique for solving mathematical equations. The main application of the MSGF technique is in modeling of nanomaterials. These materials are very small – of the size of few nanometers. Mathematical modeling of nanomaterials requires special techniques and is now recognized to be an independent branch of science. A mathematical model is needed to calculate the displacements of atoms in a crystal in response to an applied static or time dependent force in order to study the mechanical and physical properties of nanomaterials. One specific requirement of a model for nanomaterials is that the model needs to be multiscale and provide seamless linking of different length scales. Green's function (GF) was originally formulated by the British mathematical physicist George Green in the year 1828 as a general technique for solution of operator equations. It has been exte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green's Function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential operator, then * the Green's function G is the solution of the equation \operatorname G = \delta, where \delta is Dirac's delta function; * the solution of the initial-value problem \operatorname y = f is the convolution (G \ast f). Through the superposition principle, given a linear ordinary differential equation (ODE), \operatorname y = f, one can first solve \operatorname G = \delta_s, for each , and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of . Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Molecular Dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the system. In the most common version, the trajectories of atoms and molecules are determined by numerically solving Newton's equations of motion for a system of interacting particles, where forces between the particles and their potential energies are often calculated using interatomic potentials or molecular mechanical force fields. The method is applied mostly in chemical physics, materials science, and biophysics. Because molecular systems typically consist of a vast number of particles, it is impossible to determine the properties of such complex systems analytically; MD simulation circumvents this problem by using numerical methods. However, long MD simulations are mathematically ill-conditioned, generating cumulative errors in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propagator
In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the respective diagram. These may also be viewed as the inverse of the wave operator appropriate to the particle, and are, therefore, often called ''(causal) Green's functions'' (called "''causal''" to distinguish it from the elliptic Laplacian Green's function). Non-relativistic propagators In non-relativistic quantum mechanics, the propagator gives the probability amplitude for a particle to travel from one spatial point (x') at one time (t') to another spatial point (x) at a later time (t). Consider a system with Hamiltonian . The Green's function (fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Causality (physics)
Causality is the relationship between causes and effects. While causality is also a topic studied from the perspectives of philosophy and physics, it is operationalized so that causes of an event must be in the past light cone of the event and ultimately reducible to fundamental interactions. Similarly, a cause cannot have an effect outside its future light cone. As a physical concept In classical physics, an effect cannot occur ''before'' its cause which is why solutions such as the advanced time solutions of the Liénard–Wiechert potential are discarded as physically meaningless. In both Einstein's theory of special and general relativity, causality means that an effect cannot occur from a cause that is not in the back (past) light cone of that event. Similarly, a cause cannot have an effect outside its front (future) light cone. These restrictions are consistent with the constraint that mass and energy that act as causal influences cannot travel faster than the speed of li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microsecond
A microsecond is a unit of time in the International System of Units (SI) equal to one millionth (0.000001 or 10−6 or ) of a second. Its symbol is μs, sometimes simplified to us when Unicode is not available. A microsecond is equal to 1000 nanoseconds or of a millisecond. Because the next SI prefix is 1000 times larger, measurements of 10−5 and 10−4 seconds are typically expressed as tens or hundreds of microseconds. Examples * 1 microsecond (1 μs) – cycle time for frequency (1 MHz), the inverse unit. This corresponds to radio wavelength 300 m (AM medium wave band), as can be calculated by multiplying 1 μs by the speed of light (approximately ). * 1 microsecond – the length of time of a high-speed, commercial strobe light flash (see air-gap flash). * 1 microsecond – protein folding takes place on the order of microseconds. * 1.8 microseconds – the amount of time subtracted from the Earth's day as a result of the 2011 Japanese earthquake. * 2 m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nanosecond
A nanosecond (ns) is a unit of time in the International System of Units (SI) equal to one billionth of a second, that is, of a second, or 10 seconds. The term combines the SI prefix ''nano-'' indicating a 1 billionth submultiple of an SI unit (e.g. nanogram, nanometre, etc.) and ''second'', the primary unit of time in the SI. A nanosecond is equal to 1000 picoseconds or microsecond. Time units ranging between 10 and 10 seconds are typically expressed as tens or hundreds of nanoseconds. Time units of this granularity are commonly found in telecommunications, pulsed lasers, and related aspects of electronics. Common measurements * 0.001 nanoseconds – one picosecond * 0.5 nanoseconds – the half-life of beryllium-13. * 0.96 nanoseconds – 100 Gigabit Ethernet Interpacket gap * 1.0 nanosecond – cycle time of an electromagnetic wave with a frequency of 1 GHz (1 hertz). * 1.0 nanosecond – electromagnetic wavelength of 1 light-nanosecond. Equiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Femtosecond
A femtosecond is a unit of time in the International System of Units (SI) equal to 10 or of a second; that is, one quadrillionth, or one millionth of one billionth, of a second. For context, a femtosecond is to a second as a second is to about 31.71 million years; a ray of light travels approximately 0.3 μm (micrometers) in 1 femtosecond, a distance comparable to the diameter of a virus.Compared with overview in: Page 3 The word ''femtosecond'' is formed by the SI prefix ''femto'' and the SI unit ''second''. Its symbol is fs. A femtosecond is equal to 1000 attoseconds, or 1/1000 picosecond. Because the next higher SI unit is 1000 times larger, times of 10−14 and 10−13 seconds are typically expressed as tens or hundreds of femtoseconds. * Typical time steps for molecular dynamics simulations are on the order of 1 fs. * The periods of the waves of visible light have a duration of about 2 femtoseconds. = = 2.0 \times 10^~ The precise duration depends on the ener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scanning Probe Microscopy
Scan may refer to: Acronyms * Schedules for Clinical Assessment in Neuropsychiatry (SCAN), a psychiatric diagnostic tool developed by WHO * Shared Check Authorization Network (SCAN), a database of bad check writers and collection agency for bad checks * Space Communications and Navigation Program (SCaN) * Social Cognitive and Affective Neuroscience (journal) * Scientific content analysis (SCAN), also known as statement analysis Businesses * Scan Furniture, Washington, D.C., US chain * SCAN Health Plan, not-for-profit health care company based in Long Beach, California * Scan AB or Scan Foods UK Ltd, the Swedish and UK subsidiaries of the Finnish HKScan Oyj * Seattle Community Access Network, Seattle, Washington, US TV channel * Scan (company), a software company based in Provo, Utah, US Electronics or computer related * 3D scanning * Counter-scanning, in physical micro and nanotopography measuring instruments like scanning probe microscope * Elevator algorithm (also SC ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Delta Function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. The current understanding of the unit impulse is as a linear functional that maps every continuous function (e.g., f(x)) to its value at zero of its domain (f(0)), or as the weak limit of a sequence of bump functions (e.g., \delta(x) = \lim_ \frace^), which are zero over most of the real line, with a tall spike at the origin. Bump functions are thus sometimes called "approximate" or "nascent" delta distributions. The delta function was introduced by physicist Paul Dirac as a tool for the normalization of state vectors. It also has uses in probability theory and signal processing. Its validity was disputed until Laurent Schwartz developed the theory of distributions where it is defined as a linear form acting on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Introduction To Solid State Physics
''Introduction to Solid State Physics'', known colloquially as ''Kittel'', is a classic condensed matter physics textbook written by American physicist Charles Kittel in 1953. The book has been highly influential and has seen widespread adoption; Marvin L. Cohen remarked in 2019 that Kittel's content choices in the original edition played a large role in defining the field of solid-state physics. It was also the first proper textbook covering this new field of physics. The book is published by John Wiley and Sons and, as of 2018, it is in its ninth edition and has been reprinted many times as well as translated into over a dozen languages, including Chinese, French, German, Hungarian, Indonesian, Italian, Japanese, Korean, Malay, Romanian, Russian, Spanish, and Turkish. In some later editions, the eighteenth chapter, titled ''Nanostructures'', was written by Paul McEuen. Along with its rival ''Ashcroft and Mermin'', the book is considered a standard textbook in condensed matter p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
