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Entropy (other)
Entropy, in thermodynamics, is a property originally introduced to explain the part of the internal energy of a thermodynamic system that is unavailable as a source for useful work. Entropy may also refer to: Thermodynamics and statistical mechanics *Entropy (classical thermodynamics), thermodynamic entropy in macroscopic terms, with less emphasis on the statistical explanation **Entropic force. The product of temperature and the gradient of the entropy density is viewed an effective force, yielding a gradient in the energy density of a system. **Entropy production. Development of entropy in a thermodynamic system. *Entropy (statistical thermodynamics), the statistical explanation of thermodynamic entropy based on probability theory **Configuration entropy, the entropy change due to a change in the knowledge of the position of particles, rather than their momentum **Conformational entropy, the entropy change due to a change in the "configuration" of a particle (e.g. a right-handed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the microscopic description of nature in statistical physics, and to the principles of information theory. It has found far-ranging applications in chemistry and physics, in biological systems and their relation to life, in cosmology, economics, sociology, weather science, climate change, and information systems including the transmission of information in telecommunication. The thermodynamic concept was referred to by Scottish scientist and engineer William Rankine in 1850 with the names ''thermodynamic function'' and ''heat-potential''. In 1865, German physicist Rudolf Clausius, one of the leading founders of the field of thermodynamics, defined it as the quotient of an infinitesimal amount of hea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entropy (information Theory)
In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable X, which takes values in the alphabet \mathcal and is distributed according to p: \mathcal\to , 1/math>: \Eta(X) := -\sum_ p(x) \log p(x) = \mathbb \log p(X), where \Sigma denotes the sum over the variable's possible values. The choice of base for \log, the logarithm, varies for different applications. Base 2 gives the unit of bits (or " shannons"), while base ''e'' gives "natural units" nat, and base 10 gives units of "dits", "bans", or " hartleys". An equivalent definition of entropy is the expected value of the self-information of a variable. The concept of information entropy was introduced by Claude Shannon in his 1948 paper "A Mathematical Theory of Communication",PDF archived froherePDF archived frohere and is also referred to as Shannon entropy. Shannon's theory defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entropy (anesthesiology)
Entropy monitoring is a method of assessing the effect of certain anaesthetic drugs on the brain's EEG. It was commercially developed by Datex-Ohmeda, which is now part of GE Healthcare. Entropy is a quantitative EEG device which captures a single-lead frontal EEG via a 3-electrode sensor applied to the patient's forehead. The system calculates the "spectral entropy" of the electroencephalogram (EEG) signals, which is a measure of the degree that the power spectrum is uniform. Increasing brain levels of anaesthetic drugs causes the predominant frequencies in the EEG to be lower than when awake, and this is reflected in a decrease in the spectral entropy. Entropy monitors generate two numbers that are derived from different frequency bands used. The State Entropy (SE) is calculated from the 0.8 Hz to 32 Hz range, whereas the Response Entropy (RE) uses frequencies up to 47 Hz. Electromyogram activity is more predominant in those higher frequencies, and so the Response ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entropy (computing)
In computing, entropy is the randomness collected by an operating system or application for use in cryptography or other uses that require random data. This randomness is often collected from hardware sources (variance in fan noise or HDD), either pre-existing ones such as mouse movements or specially provided randomness generators. A lack of entropy can have a negative impact on performance and security. Linux kernel The Linux kernel generates entropy from keyboard timings, mouse movements, and IDE timings and makes the random character data available to other operating system processes through the special files /dev/random and /dev/urandom. This capability was introduced in Linux version 1.3.30. There are some Linux kernel patches allowing one to use more entropy sources. Thaudio_entropydproject, which is included in some operating systems such as Fedora, allows audio data to be used as an entropy source. Also available arvideo_entropydwhich calculates random data from a v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entropy Encoding
In information theory, an entropy coding (or entropy encoding) is any lossless data compression method that attempts to approach the lower bound declared by Shannon's source coding theorem, which states that any lossless data compression method must have expected code length greater or equal to the entropy of the source. More precisely, the source coding theorem states that for any source distribution, the expected code length satisfies \mathbb E_ (d(x))\geq \mathbb E_ \log_b(P(x))/math>, where l is the number of symbols in a code word, d is the coding function, b is the number of symbols used to make output codes and P is the probability of the source symbol. An entropy coding attempts to approach this lower bound. Two of the most common entropy coding techniques are Huffman coding and arithmetic coding. If the approximate entropy characteristics of a data stream are known in advance (especially for signal compression), a simpler static code may be useful. These static codes in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Entropy
In information theory, the graph entropy is a measure of the information rate achievable by communicating symbols over a channel in which certain pairs of values may be confused. This measure, first introduced by Körner in the 1970s, has since also proven itself useful in other settings, including combinatorics. Definition Let G = (V, E) be an undirected graph. The graph entropy of G, denoted H(G) is defined as ::H(G) = \min_ I(X ; Y) where X is chosen uniformly from V, Y ranges over independent sets of G, the joint distribution of X and Y is such that X\in Y with probability one, and I(X ; Y) is the mutual information of X and Y.G. Simonyi, "Perfect graphs and graph entropy. An updated survey," Perfect Graphs, John Wiley and Sons (2001) pp. 293-328, Definition 2” That is, if we let \mathcal denote the independent vertex sets in G, we wish to find the joint distribution X,Y on V \times \mathcal with the lowest mutual information such that (i) the marginal distribution of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Entropy (other) , establishes the concept of entropy as a physical property of a thermodynamic system
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Maximum entropy may refer to: * Entropy, a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty. Physics * Maximum entropy thermodynamics * Maximum entropy spectral estimation Mathematics and statistics * Principle of maximum entropy * Maximum entropy probability distribution * Maximum entropy classifier, in regression analysis See also * Second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume Entropy
The volume entropy is an asymptotic invariant of a compact Riemannian manifold that measures the exponential growth rate of the volume of metric balls in its universal cover. This concept is closely related with other notions of entropy found in dynamical systems and plays an important role in differential geometry and geometric group theory. If the manifold is nonpositively curved then its volume entropy coincides with the topological entropy of the geodesic flow. It is of considerable interest in differential geometry to find the Riemannian metric on a given smooth manifold which minimizes the volume entropy, with locally symmetric spaces forming a basic class of examples. Definition Let (''M'', ''g'') be a compact Riemannian manifold, with universal cover \tilde. Choose a point \tilde_0\in \tilde. The volume entropy (or asymptotic volume growth) h = h(M, g) is defined as the limit : h(M,g) = \lim_ \frac, where ''B''(''R'') is the ball of radius ''R'' in \tilde centered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Entropy In Physics
The topological entanglement entropy or ''topological entropy'', usually denoted by \gamma, is a number characterizing many-body states that possess topological order. A non-zero topological entanglement entropy reflects the presence of long range quantum entanglements in a many-body quantum state. So the topological entanglement entropy links topological order with pattern of long range quantum entanglements. Given a topological order, topologically ordered state, the topological entropy can be extracted from the asymptotic behavior of the Von Neumann entropy measuring the quantum entanglement between a spatial block and the rest of the system. The entanglement entropy of a simply connected region of boundary length ''L'', within an infinite two-dimensional topologically ordered state, has the following form for large ''L'': : S_L \; \longrightarrow \; \alpha L -\gamma +\mathcal(L^) \; , \qquad \nu>0 \,\! where -\gamma is the topological entanglement entropy. The topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Entropy
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a ''topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure-preserving Dynamical System
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special case of conservative systems. They provide the formal, mathematical basis for a broad range of physical systems, and, in particular, many systems from classical mechanics (in particular, most non-dissipative systems) as well as systems in thermodynamic equilibrium. Definition A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system :(X, \mathcal, \mu, T) with the following structure: *X is a set, *\mathcal B is a σ-algebra over X, *\mu:\mathcal\rightarrow ,1/math> is a probability measure, so that \mu (X) = 1, and \mu(\varnothing) = 0, * T:X \rightarrow X is a measurable transformation which preserves the measure \mu, i.e., \forall A\in \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rényi Entropy
In information theory, the Rényi entropy is a quantity that generalizes various notions of entropy, including Hartley entropy, Shannon entropy, collision entropy, and min-entropy. The Rényi entropy is named after Alfréd Rényi, who looked for the most general way to quantify information while preserving additivity for independent events. In the context of fractal dimension estimation, the Rényi entropy forms the basis of the concept of generalized dimensions. The Rényi entropy is important in ecology and statistics as index of diversity. The Rényi entropy is also important in quantum information, where it can be used as a measure of entanglement. In the Heisenberg XY spin chain model, the Rényi entropy as a function of can be calculated explicitly because it is an automorphic function with respect to a particular subgroup of the modular group. In theoretical computer science, the min-entropy is used in the context of randomness extractors. Definition The Rényi entro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
