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Smoluchowski Coagulation Equation
In statistical physics, the Smoluchowski coagulation equation is a population balance equation introduced by Marian Smoluchowski in a seminal 1916 publication, describing the time evolution of the number density of particles as they coagulate (in this context "clumping together") to size ''x'' at time ''t''. Simultaneous coagulation (or aggregation) is encountered in processes involving polymerization, coalescence of aerosols, emulsication, flocculation. Equation The distribution of particle size changes in time according to the interrelation of all particles of the system. Therefore, the Smoluchowski coagulation equation is an integrodifferential equation of the particle-size distribution. In the case when the sizes of the coagulated particles are continuous variables, the equation involves an integral: : \frac=\frac\int^x_0K(x-y,y)n(x-y,t)n(y,t)\,dy - \int^\infty_0K(x,y)n(x,t)n(y,t)\,dy. If ''dy'' is interpreted as a discrete measure, i.e. when particles join in discrete s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoluchowski Aggregation Kinetics
Marian Smoluchowski (; 28 May 1872 – 5 September 1917) was a Polish physicist who worked in the Polish territories of the Austro-Hungarian Empire. He was a pioneer of statistical physics, and an avid mountaineer. Life Born into an upper-class family in Vorder-Brühl, near Vienna, Smoluchowski studied physics at the University of Vienna. His teachers included Franz S. Exner and Joseph Stefan. Ludwig Boltzmann held a position at Munich University during Smoluchowski's studies in Vienna, and Boltzmann returned to Vienna in 1894 when Smoluchowski was serving in the Austrian army. They apparently had no direct contact, although Smoluchowski's work follows in the tradition of Boltzmann's ideas. After several years at other universities (Paris, Glasgow, Berlin), in 1899 Smoluchowski moved to Lwów (present-day Lviv), where he took a position at the University of Lwów. He was president of the Polish Copernicus Society of Naturalists, 1906–7. In 1913 Smoluchowski moved to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel Function
In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving integral operator equations. Since then, positive-definite functions and their various analogues and generalizations have arisen in diverse parts of mathematics. They occur naturally in Fourier analysis, probability theory, operator theory, complex function-theory, moment problems, integral equations, boundary-value problems for partial differential equations, machine learning, embedding problem, information theory, and other areas. This article will discuss some of the historical and current developments of the theory of positive-definite kernels, starting with the general idea and properties before considering practical applications. Definition Let \mathcal X be a nonempty set, sometimes referred to as the index set. A symmetri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffusion-limited Aggregation
Diffusion-limited aggregation (DLA) is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles. This theory, proposed by T.A. Witten Jr. and L.M. Sander in 1981, is applicable to aggregation in any system where diffusion is the primary means of transport in the system. DLA can be observed in many systems such as electrodeposition, Hele-Shaw flow, mineral deposits, and dielectric breakdown. The clusters formed in DLA processes are referred to as Brownian trees. These clusters are an example of a fractal. In 2D these fractals exhibit a dimension of approximately 1.71 for free particles that are unrestricted by a lattice, however computer simulation of DLA on a lattice will change the fractal dimension slightly for a DLA in the same embedding dimension. Some variations are also observed depending on the geometry of the growth, whether it be from a single point radially outward or from a plane or line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured. It has also been characterized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently from the space it is embedded in; a fractal dimension does not have to be an integer. The essential idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed ''fractional dimensions''. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used ( see Fig. 1). In terms of that notion, the fractal dimension of a coastline quantifies ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase (matter)
In the physical sciences, a phase is a region of space (a thermodynamic system), throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, magnetization and chemical composition. A simple description is that a phase is a region of material that is chemically uniform, physically distinct, and (often) mechanically separable. In a system consisting of ice and water in a glass jar, the ice cubes are one phase, the water is a second phase, and the humid air is a third phase over the ice and water. The glass of the jar is another separate phase. (See ) The term ''phase'' is sometimes used as a synonym for state of matter, but there can be several immiscible phases of the same state of matter. Also, the term ''phase'' is sometimes used to refer to a set of equilibrium states demarcated in terms of state variables such as pressure and temperature by a phase boundary on a phase diagram. Bec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Collision
In physics, a collision is any event in which two or more bodies exert forces on each other in a relatively short time. Although the most common use of the word ''collision'' refers to incidents in which two or more objects collide with great force, the scientific use of the term implies nothing about the magnitude of the force. Some examples of physical interactions that scientists would consider collisions are the following: * When an insect lands on a plant's leaf, its legs are said to collide with the leaf. * When a cat strides across a lawn, each contact that its paws make with the ground is considered a collision, as well as each brush of its fur against a blade of grass. * When a boxer throws a punch, their fist is said to collide with the opponents body. * When an astronomical object merges with a black hole, they are considered to collide. Some colloquial uses of the word collision are the following: * A traffic collision involves at least one automobile. * A mid-air ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Transition
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of matter: solid, liquid, and gas, and in rare cases, plasma. A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such as temperature or pressure. This can be a discontinuous change; for example, a liquid may become gas upon heating to its boiling point, resulting in an abrupt change in volume. The identification of the external conditions at which a transformation occurs defines the phase transition point. Types of phase transition At the phase transition point for a substance, for instance the boiling point, the two phases involved - liquid and vapor, have identic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scale Invariance
In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term for this transformation is a dilatation (also known as dilation), and the dilatations can also form part of a larger conformal symmetry. *In mathematics, scale invariance usually refers to an invariance of individual functions or curves. A closely related concept is self-similarity, where a function or curve is invariant under a discrete subset of the dilations. It is also possible for the probability distributions of random processes to display this kind of scale invariance or self-similarity. *In classical field theory, scale invariance most commonly applies to the invariance of a whole theory under dilatations. Such theories typically describe classical physical processes with no characteristic length scale. *In quantum field theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamic Scaling
Dynamic scaling (sometimes known as Family-Vicsek scaling) is a litmus test that shows whether an evolving system exhibits self-similarity. In general a function is said to exhibit dynamic scaling if it satisfies: :f(x,t)\sim t^\theta \varphi \left( \frac x \right). Here the exponent \theta is fixed by the dimensional requirement [f]=[t^\theta]. The numerical value of f/t^\theta should remain invariant despite the unit of measurement of t is changed by some factor since \varphi is a dimensionless quantity. Many of these systems evolve in a self-similar fashion in the sense that data obtained from the snapshot at any fixed time is similar to the respective data taken from the snapshot of any earlier or later time. That is, the system is similar to itself at different times. The litmus test of such self-similarity is provided by the dynamic scaling. History The term "dynamic scaling" as one of the essential concepts to describe the dynamics of critical phenomena seems to origina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Function
In number theory, a multiplicative function is an arithmetic function ''f''(''n'') of a positive integer ''n'' with the property that ''f''(1) = 1 and f(ab) = f(a)f(b) whenever ''a'' and ''b'' are coprime. An arithmetic function ''f''(''n'') is said to be completely multiplicative (or totally multiplicative) if ''f''(1) = 1 and ''f''(''ab'') = ''f''(''a'')''f''(''b'') holds ''for all'' positive integers ''a'' and ''b'', even when they are not coprime. Examples Some multiplicative functions are defined to make formulas easier to write: * 1(''n''): the constant function, defined by 1(''n'') = 1 (completely multiplicative) * Id(''n''): identity function, defined by Id(''n'') = ''n'' (completely multiplicative) * Id''k''(''n''): the power functions, defined by Id''k''(''n'') = ''n''''k'' for any complex number ''k'' (completely multiplicative). As special cases we have ** Id0(''n'') = 1(''n'') and ** Id1(''n'') = Id(''n''). * ''ε''(''n''): the function defined by ''ε''(''n'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Map
In algebra, an additive map, Z-linear map or additive function is a function f that preserves the addition operation: f(x + y) = f(x) + f(y) for every pair of elements x and y in the domain of f. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation. For a specific case of this definition, see additive polynomial. More formally, an additive map is a \Z-module homomorphism. Since an abelian group is a \Z-module, it may be defined as a group homomorphism between abelian groups. A map V \times W \to X that is additive in each of two arguments separately is called a bi-additive map or a \Z-bilinear map. Examples Typical examples include maps between rings, vector spaces, or modules that preserve the additive group. An additive map does not necessarily preserve any other structure of the object; for example, the product operation of a ring. If f and g are additive maps, then the map f + g (defined pointwise) is additiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant (mathematics)
In mathematics, the word constant conveys multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other value); as a noun, it has two different meanings: * A fixed and well-defined number or other non-changing mathematical object. The terms '' mathematical constant'' or '' physical constant'' are sometimes used to distinguish this meaning. * A function whose value remains unchanged (i.e., a constant function). Such a constant is commonly represented by a variable which does not depend on the main variable(s) in question. For example, a general quadratic function is commonly written as: :a x^2 + b x + c\, , where , and are constants (or parameters), and a variable—a placeholder for the argument of the function being studied. A more explicit way to denote this function is :x\mapsto a x^2 + b x + c \, , which makes the function-argument status of (and by extension the constancy of , and ) clear. In this example , and are co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
