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Shifted Force Method
The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf(\mathbf) = \sum_F(r)\mathbf \, \, \,; \, \, F(r) = \frac where \mathbf is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_ and q_ are the charges of particles i and j, respectively. With the electrostatic force being proportional to r^, individual particle-particle interactions are long-range in nature, presenting a challenging computational problem in the simulation of particulate systems. To determine the net forces acting on particles, the Ewald or Lekner summation methods are generally employed. One alternative and usually computationally faster technique based on the notion that interactions over large distances (''e.g.'' > 1 nm) are insignificant to the ''net'' forces acting in certain systems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electrostatic
Electrostatics is a branch of physics that studies slow-moving or stationary electric charges. Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word (), meaning 'amber', was thus the root of the word ''electricity''. Electrostatic phenomena arise from the forces that electric charges exert on each other. Such forces are described by Coulomb's law. There are many examples of electrostatic phenomena, from those as simple as the attraction of plastic wrap to one's hand after it is removed from a package, to the apparently spontaneous explosion of grain silos, the damage of electronic components during manufacturing, and photocopier and laser printer operation. The electrostatic model accurately predicts electrical phenomena in "classical" cases where the velocities are low and the system is macroscopic so no quantum effects are involved. It also plays a role in quantum mechanics, where addition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Force
In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitude (mathematics), magnitude and Direction (geometry, geography), direction of a force are both important, force is a Euclidean vector, vector quantity. The SI unit of force is the newton (unit), newton (N), and force is often represented by the symbol . Force plays an important role in classical mechanics. The concept of force is central to all three of Newton's laws of motion. Types of forces often encountered in classical mechanics include Elasticity (physics), elastic, frictional, Normal force, contact or "normal" forces, and gravity, gravitational. The rotational version of force is torque, which produces angular acceleration, changes in the rotational speed of an object. In an extended body, each part applies forces on the adjacent pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle
In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from subatomic particles like the electron, to microscopic particles like atoms and molecules, to macroscopic particles like powders and other granular materials. Particles can also be used to create scientific models of even larger objects depending on their density, such as humans moving in a crowd or celestial bodies in motion. The term ''particle'' is rather general in meaning, and is refined as needed by various scientific fields. Anything that is composed of particles may be referred to as being particulate. However, the noun '' particulate'' is most frequently used to refer to pollutants in the Earth's atmosphere, which are a suspension of unconnected particles, rather than a connected particle aggregation. Conceptual properties ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Vector
In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''normalized vector'' is sometimes used as a synonym for ''unit vector''. The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e., :\mathbf = \frac=(\frac, \frac, ... , \frac) where ‖u‖ is the Norm (mathematics), norm (or length) of u and \, \mathbf\, = (u_1, u_2, ..., u_n). The proof is the following: \, \mathbf\, =\sqrt=\sqrt=\sqrt=1 A unit vector is often used to represent direction (geometry), directions, such as normal directions. Unit vectors are often chosen to form the basis (linear algebra), basis of a vector space, and every vector in the space may be written as a linear combination form of unit vectors. Orthogonal coordinates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ewald Summation
Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions (e.g. electrostatic interactions) in periodic systems. It was first developed as the method for calculating the 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 spee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discontinuity (mathematics)
Continuous functions are of utmost importance in mathematics, functions and applications. However, not all Function (mathematics), functions are continuous. If a function is not continuous at a limit point (also called "accumulation point" or "cluster point") of its Domain of a function, domain, one says that it has a discontinuity there. The Set theory, set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function. The Oscillation (mathematics), oscillation of a function at a point quantifies these discontinuities as follows: * in a removable discontinuity, the distance that the value of the function is off by is the oscillation; * in a jump discontinuity, the size of the jump is the oscillation (assuming that the value ''at'' the point lies between these limits of the two sides); * in an essential discontinuity (a.k.a. infinite discontinuity), oscillation measures the failure of a Limit of a function, limit to exist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electric-field Screening
In physics, screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying mediums, such as ionized gases (classical plasmas), electrolytes, and electronic conductors (semiconductors, metals). In a fluid, with a given permittivity , composed of electrically charged constituent particles, each pair of particles (with charges and ) interact through the Coulomb force as \mathbf = \frac\hat, where the vector is the relative position between the charges. This interaction complicates the theoretical treatment of the fluid. For example, a naive quantum mechanical calculation of the ground-state energy density yields infinity, which is unreasonable. The difficulty lies in the fact that even though the Coulomb force diminishes with distance as , the average number of particles at each distance is proportional to , assuming the fluid is fairly isotropic. As a result, a charge fluctuation at any on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anisotropic
Anisotropy () is the structural property of non-uniformity in different directions, as opposed to isotropy. An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit very different physical property, physical or list of materials properties#Mechanical properties, mechanical properties when measured along different axes, e.g. absorbance, refractive index, electrical resistivity and conductivity, conductivity, and tensile strength. An example of anisotropy is light coming through a polarizer. Another is wood, which is easier to split along its wood grain, grain than across it because of the directional non-uniformity of the grain (the grain is the same in one direction, not all directions). Fields of interest Computer graphics In the field of computer graphics, an anisotropic surface changes in appearance as it rotates about its normal (geometry), geometric normal, as is the case with velvet. Anisotropic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interface (chemistry)
In the physical sciences, an interface is the boundary between two spatial regions occupied by different matter, or by matter in different physical states. The interface between matter and air, or matter and vacuum, is called a surface, and studied in surface science. In thermal equilibrium, the regions in contact are called phases, and the interface is called a phase boundary. An example for an interface out of equilibrium is the grain boundary in polycrystalline matter. The importance of the interface depends on the type of system: the bigger the quotient area/volume, the greater the effect the interface will have. Consequently, interfaces are very important in systems with large interface area-to-volume ratios, such as colloids. Interfaces can be flat or curved. For example, oil droplets in a salad dressing are spherical but the interface between water and air in a glass of water is mostly flat. Surface tension is the physical property which rules interface process ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Energy
Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and light. Energy is a Conservation law, conserved quantity—the law of conservation of energy states that energy can be Energy transformation, converted in form, but not created or destroyed. The unit of measurement for energy in the International System of Units (SI) is the joule (J). Forms of energy include the kinetic energy of a moving object, the potential energy stored by an object (for instance due to its position in a Classical field theory, field), the elastic energy stored in a solid object, chemical energy associated with chemical reactions, the radiant energy carried by electromagnetic radiation, the internal energy contained within a thermodynamic system, and rest energy associated with an object's rest mass. These are not mutual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Observables
In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum mechanics, an observable is an operator, or gauge, where the property of the quantum state can be determined by some sequence of operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value. Physically meaningful observables must also satisfy transformation laws that relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations that preserve certain mathematical properties of the space in question. Quantum mechanics In quantum mechanics, observables manifest as self-adjoint operators on a separable complex Hilbert space representing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence (mathematics)
In mathematics, a series (mathematics), series is the summation, sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_1, a_2, a_3, \ldots) defines a series (mathematics), series that is denoted :S=a_1 + a_2 + a_3 + \cdots=\sum_^\infty a_k. The th partial sum is the sum of the first terms of the sequence; that is, :S_n = a_1 +a_2 + \cdots + a_n = \sum_^n a_k. A series is convergent (or converges) if and only if the sequence (S_1, S_2, S_3, \dots) of its partial sums tends to a limit of a sequence, limit; that means that, when adding one a_k after the other ''in the order given by the indices'', one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if and only if there exists a number \ell such that for every arbitrarily small positive number \varepsilon, there is a (sufficiently large) integer N such that for all n \ge N, :\left , S_n - \ell \right , 1 produce a convergent series: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
