Percolation thresholdThe percolation threshold is a mathematical concept in percolation theory that describes the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist; while above it, there exists a giant component of the order of system size. In engineering and coffee making, percolation represents the flow of fluids through porous media, but in the mathematics and physics worlds it generally refers to simplified lattice models of random systems or networks (graphs), and the nature of the connectivity in them.
Percolation critical exponentsIn the context of the physical and mathematical theory of percolation, a percolation transition is characterized by a set of universal critical exponents, which describe the fractal properties of the percolating medium at large scales and sufficiently close to the transition. The exponents are universal in the sense that they only depend on the type of percolation model and on the space dimension. They are expected to not depend on microscopic details such as the lattice structure, or whether site or bond percolation is considered.
Percolation theoryIn statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected clusters merge into significantly larger connected, so-called spanning clusters. The applications of percolation theory to materials science and in many other disciplines are discussed here and in the articles Network theory and Percolation (cognitive psychology).
PercolationIn physics, chemistry, and materials science, percolation () refers to the movement and filtering of fluids through porous materials. It is described by Darcy's law. Broader applications have since been developed that cover connectivity of many systems modeled as lattices or graphs, analogous to connectivity of lattice components in the filtration problem that modulates capacity for percolation.
Critical exponentCritical exponents describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on some of its general features. For instance, for ferromagnetic systems, the critical exponents depend only on: the dimension of the system the range of the interaction the spin dimension These properties of critical exponents are supported by experimental data.
RenormalizationRenormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian.
Directed percolationIn statistical physics, directed percolation (DP) refers to a class of models that mimic filtering of fluids through porous materials along a given direction, due to the effect of gravity. Varying the microscopic connectivity of the pores, these models display a phase transition from a macroscopically permeable (percolating) to an impermeable (non-percolating) state. Directed percolation is also used as a simple model for epidemic spreading with a transition between survival and extinction of the disease depending on the infection rate.
Stochastic processIn probability theory and related fields, a stochastic (stəˈkæstɪk) or random process is a mathematical object usually defined as a sequence of random variables, where the index of the sequence has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.
Fundamental interactionIn physics, the fundamental interactions or fundamental forces are the interactions that do not appear to be reducible to more basic interactions. There are four fundamental interactions known to exist: gravity electromagnetism weak interaction strong interaction The gravitational and electromagnetic interactions produce long-range forces whose effects can be seen directly in everyday life. The strong and weak interactions produce forces at minuscule, subatomic distances and govern nuclear interactions inside atoms.
Formula unitIn chemistry, a formula unit is the empirical formula of any ionic or covalent network solid compound used as an independent entity for stoichiometric calculations. It is the lowest whole number ratio of ions represented in an ionic compound. Examples include ionic and and covalent networks such as and C (as diamond or graphite). Ionic compounds do not exist as individual molecules; a formula unit thus indicates the lowest reduced ratio of ions in the compound.
Regularization (physics)In physics, especially quantum field theory, regularization is a method of modifying observables which have singularities in order to make them finite by the introduction of a suitable parameter called the regulator. The regulator, also known as a "cutoff", models our lack of knowledge about physics at unobserved scales (e.g. scales of small size or large energy levels). It compensates for (and requires) the possibility that "new physics" may be discovered at those scales which the present theory is unable to model, while enabling the current theory to give accurate predictions as an "effective theory" within its intended scale of use.
Weak interactionIn nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, and gravitation. It is the mechanism of interaction between subatomic particles that is responsible for the radioactive decay of atoms: The weak interaction participates in nuclear fission and nuclear fusion.
MeltingMelting, or fusion, is a physical process that results in the phase transition of a substance from a solid to a liquid. This occurs when the internal energy of the solid increases, typically by the application of heat or pressure, which increases the substance's temperature to the melting point. At the melting point, the ordering of ions or molecules in the solid breaks down to a less ordered state, and the solid melts to become a liquid. Substances in the molten state generally have reduced viscosity as the temperature increases.
Random walkIn mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line which starts at 0, and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, or the price of a fluctuating stock and the financial status of a gambler.
Chemical formulaIn chemistry, a chemical formula is a way of presenting information about the chemical proportions of atoms that constitute a particular chemical compound or molecule, using chemical element symbols, numbers, and sometimes also other symbols, such as parentheses, dashes, brackets, commas and plus (+) and minus (−) signs. These are limited to a single typographic line of symbols, which may include subscripts and superscripts. A chemical formula is not a chemical name since it does not contain any words.
Ising modelThe Ising model (ˈiːzɪŋ) (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors.
Data signaling rateIn telecommunication, data signaling rate (DSR), also known as gross bit rate, is the aggregate rate at which data passes a point in the transmission path of a data transmission system. The DSR is usually expressed in bits per second. The data signaling rate is given by where m is the number of parallel channels, ni is the number of significant conditions of the modulation in the i-th channel, and Ti is the unit interval, expressed in seconds, for the i-th channel.
Empirical formulaIn chemistry, the empirical formula of a chemical compound is the simplest whole number ratio of atoms present in a compound. A simple example of this concept is that the empirical formula of sulfur monoxide, or SO, would simply be SO, as is the empirical formula of disulfur dioxide, S2O2. Thus, sulfur monoxide and disulfur dioxide, both compounds of sulfur and oxygen, have the same empirical formula. However, their molecular formulas, which express the number of atoms in each molecule of a chemical compound, are not the same.
Ultraviolet divergenceIn physics, an ultraviolet divergence or UV divergence is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with unbounded energy, or, equivalently, because of physical phenomena at infinitesimal distances. Since an infinite result is unphysical, ultraviolet divergences often require special treatment to remove unphysical effects inherent in the perturbative formalisms. In particular, UV divergences can often be removed by regularization and renormalization.
Symbol rateIn a digitally modulated signal or a line code, symbol rate, modulation rate or baud rate is the number of symbol changes, waveform changes, or signaling events across the transmission medium per unit of time. The symbol rate is measured in baud (Bd) or symbols per second. In the case of a line code, the symbol rate is the pulse rate in pulses per second. Each symbol can represent or convey one or several bits of data. The symbol rate is related to the gross bit rate, expressed in bits per second.
