 | This article needs additional citations for verification .(August 2016) |
The Wolf summation is a method for computing the electrostatic interactions of systems (e.g. crystals). This method is generally more computationally efficient than the Ewald summation. It was proposed by Dieter Wolf. [1]
In mathematics, convolution is a mathematical operation on two functions that produces a third function that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result. The integral is evaluated for all values of shift, producing the convolution function.
Statics is the branch of classical mechanics that is concerned with the analysis of force and torque acting on physical systems that do not experience an acceleration (a=0), but rather, are in static equilibrium with their environment. The application of Newton's second law to a system gives:
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Objects that tensors may map between include vectors and scalars, and even other tensors. There are many types of tensors, including scalars and vectors, dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system.
In numerical analysis, the Kahan summation algorithm, also known as compensated summation, significantly reduces the numerical error in the total obtained by adding a sequence of finite-precision floating-point numbers, compared to the obvious approach. This is done by keeping a separate running compensation, in effect extending the precision of the sum by the precision of the compensation variable.
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.
In mathematical analysis, Cesàro summation assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series.
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
In digital signal processing, downsampling, compression, and decimation are terms associated with the process of resampling in a multi-rate digital signal processing system. Both downsampling and decimation can be synonymous with compression, or they can describe an entire process of bandwidth reduction (filtering) and sample-rate reduction. When the process is performed on a sequence of samples of a signal or a continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a lower rate.
The Madelung constant is used in determining the electrostatic potential of a single ion in a crystal by approximating the ions by point charges. It is named after Erwin Madelung, a German physicist.
IC 1613 is an irregular dwarf galaxy, visible in the constellation Cetus near the star 26 Ceti. It was discovered in 1906 by Max Wolf, and is approaching Earth at 234 km/s.
In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). The general task of pattern analysis is to find and study general types of relations in datasets. For many algorithms that solve these tasks, the data in raw representation have to be explicitly transformed into feature vector representations via a user-specified feature map: in contrast, kernel methods require only a user-specified kernel, i.e., a similarity function over all pairs of data points computed using Inner products. The feature map in kernel machines is infinite dimensional but only requires a finite dimensional matrix from user-input according to the Representer theorem. Kernel machines are slow to compute for datasets larger than a couple of thousand examples without parallel processing.
Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions in periodic systems. It was first developed as the method for calculating electrostatic energies of ionic crystals, and is now commonly used for calculating long-range interactions in computational chemistry. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an equivalent summation in Fourier space. In this method, the long-range interaction is divided into two parts: a short-range contribution, and a long-range contribution which does not have a singularity. The short-range contribution is calculated in real space, whereas the long-range contribution is calculated using a Fourier transform. The advantage of this method is the rapid convergence of the energy compared with that of a direct summation. This means that the method has high accuracy and reasonable speed when computing long-range interactions, and it is thus the de facto standard method for calculating long-range interactions in periodic systems. The method requires charge neutrality of the molecular system in order to accurately calculate the total Coulombic interaction. A study of the truncation errors introduced in the energy and force calculations of disordered point-charge systems is provided by Kolafa and Perram.
In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval are discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points.
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.
Particle–Particle–Particle–Mesh (P3M) is a Fourier-based Ewald summation method to calculate potentials in N-body simulations.
The reaction field method is used in molecular simulations to simulate the effect of long range dipole-dipole interactions for simulations with periodic boundary conditions. Around each molecule there is a 'cavity' or sphere within which the Coulomb interactions are treated explicitly. Outside of this cavity the medium is assumed to have a uniform dielectric constant. The molecule induces polarization in this media which in turn creates a reaction field, sometimes called the Onsager reaction field. Although Onsager's name is often attached to the technique, because he considered such a geometry in his theory of the dielectric constant, the method was first introduced by Barker and Watts in 1973.
The net electrostatic force acting on a charged particle with index contained within a collection of particles is given as:
INTLAB is an interval arithmetic library using MATLAB and GNU Octave, available in Windows and Linux, macOS. It was developed by S.M. Rump from Hamburg University of Technology. INTLAB was used to develop other MATLAB-based libraries such as VERSOFT and INTSOLVER, and it was used to solve some problems in the Hundred-dollar, Hundred-digit Challenge problems.
 | This computational physics-related article is a stub. You can help Wikipedia by expanding it. |
 | This atomic, molecular, and optical physics–related article is a stub. You can help Wikipedia by expanding it. |