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Gibbs Algorithm
FILE:Josiah Willard Gibbs -from MMS-.jpg, 200px, Josiah Willard Gibbs In statistical mechanics, the Gibbs algorithm, introduced by J. Willard Gibbs in 1902, is a criterion for choosing a probability distribution for the statistical ensemble of microstate (statistical mechanics), microstates of a thermodynamic system by minimizing the average log probability : \langle\ln p_i\rangle = \sum_i p_i \ln p_i \, subject to the probability distribution satisfying a set of constraints (usually expectation values) corresponding to the known macroscopic quantities. in 1948, Claude E. Shannon, Claude Shannon interpreted the negative of this quantity, which he called entropy_(Information_theory), information entropy, as a measure of the uncertainty in a probability distribution. In 1957, E.T. Jaynes realized that this quantity could be interpreted as missing information about anything, and generalized the Gibbs algorithm to non-equilibrium systems with the principle of maximum entropy and m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Josiah Willard Gibbs -from MMS-
Josiah () or Yoshiyahu was the Kings of Judah, 16th king of Judah (–609 BCE). According to the Hebrew Bible, he instituted major religious reforms by removing official worship of gods other than Yahweh. Until the 1990s, the biblical description of Josiah’s reforms were usually considered to be more or less accurate, but that is now heavily debated. According to the Bible, Josiah became king of the Kingdom of Judah at the age of eight, after the assassination of his father, Amon of Judah, King Amon, and reigned for 31 years, from 641/640 to 610/609 BCE. Josiah is known only from biblical texts; no reference to him exists in other surviving texts of the period from ancient Egypt or Babylon, and no clear archaeological evidence, such as inscriptions bearing his name, has ever been found. However, a seal bearing the name "Nathan-melech," the name of an administrative official under King Josiah according to , dating to the 7th century BCE, was found in situ in an archeological sit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Entropy Thermodynamics
In physics, maximum entropy thermodynamics (colloquially, ''MaxEnt'' thermodynamics) views equilibrium thermodynamics and statistical mechanics as inference processes. More specifically, MaxEnt applies inference techniques rooted in Shannon information theory, Bayesian probability, and the principle of maximum entropy. These techniques are relevant to any situation requiring prediction from incomplete or insufficient data (e.g., image reconstruction, signal processing, spectral analysis, and inverse problems). MaxEnt thermodynamics began with two papers by Edwin T. Jaynes published in the 1957 ''Physical Review''. Maximum Shannon entropy Central to the MaxEnt thesis is the principle of maximum entropy. It demands as given some partly specified model and some specified data related to the model. It selects a preferred probability distribution to represent the model. The given data state "testable information" about the probability distribution, for example particular exp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in a wide variety of fields such as biology, neuroscience, computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ..., information theory and sociology. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Family
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a sense very natural sets of distributions to consider. The term exponential class is sometimes used in place of "exponential family", or the older term Koopman–Darmois family. Sometimes loosely referred to as ''the'' exponential family, this class of distributions is distinct because they all possess a variety of desirable properties, most importantly the existence of a sufficient statistic. The concept of exponential families is credited to E. J. G. Pitman, G. Darmois, and B. O. Koopman in 1935–1936. Exponential families of distributions provide a general framework for selecting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Entropy Probability Distribution
In statistics and information theory, a maximum entropy probability distribution has entropy that is at least as great as that of all other members of a specified class of probability distributions. According to the principle of maximum entropy, if nothing is known about a distribution except that it belongs to a certain class (usually defined in terms of specified properties or measures), then the distribution with the largest entropy should be chosen as the least-informative default. The motivation is twofold: first, maximizing entropy minimizes the amount of prior information built into the distribution; second, many physical systems tend to move towards maximal entropy configurations over time. Definition of entropy and differential entropy If X is a continuous random variable with probability density p(x), then the differential entropy of X is defined as H(X) = - \int_^\infty p(x) \log p(x) \, dx ~. If X is a discrete random variable with distribution given by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Function (mathematics)
The partition function or configuration integral, as used in probability theory, information theory and dynamical systems, is a generalization of the definition of a partition function in statistical mechanics. It is a special case of a normalizing constant in probability theory, for the Boltzmann distribution. The partition function occurs in many problems of probability theory because, in situations where there is a natural symmetry, its associated probability measure, the Gibbs measure, has the Markov property. This means that the partition function occurs not only in physical systems with translation symmetry, but also in such varied settings as neural networks (the Hopfield network), and applications such as genomics, corpus linguistics and artificial intelligence, which employ Markov networks, and Markov logic networks. The Gibbs measure is also the unique measure that has the property of maximizing the entropy for a fixed expectation value of the energy; this under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grand Canonical Ensemble
In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium (thermal and chemical) with a reservoir. The system is said to be open in the sense that the system can exchange energy and particles with a reservoir, so that various possible states of the system can differ in both their total energy and total number of particles. The system's volume, shape, and other external coordinates are kept the same in all possible states of the system. The thermodynamic variables of the grand canonical ensemble are chemical potential (symbol: ) and absolute temperature (symbol: . The ensemble is also dependent on mechanical variables such as volume (symbol: , which influence the nature of the system's internal states. This ensemble is therefore sometimes called the ensemble, as each of these three quantitie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gibbs Distribution
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of that state's energy and the temperature of the system. The distribution is expressed in the form: :p_i \propto \exp\left(- \frac \right) where is the probability of the system being in state , is the exponential function, is the energy of that state, and a constant of the distribution is the product of the Boltzmann constant and thermodynamic temperature . The symbol \propto denotes proportionality (see for the proportionality constant). The term ''system'' here has a wide meaning; it can range from a collection of 'sufficient number' of atoms or a single atom to a macroscopic system such as a natural gas storage tank. Therefore, the Boltzmann distribution can be used to sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principle Of Maximum Entropy
The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy, in the context of precisely stated prior data (such as a proposition that expresses testable information). Another way of stating this: Take precisely stated prior data or testable information about a probability distribution function. Consider the set of all trial probability distributions that would encode the prior data. According to this principle, the distribution with maximal information entropy is the best choice. History The principle was first expounded by E. T. Jaynes in two papers in 1957, where he emphasized a natural correspondence between statistical mechanics and information theory. In particular, Jaynes argued that the Gibbsian method of statistical mechanics is sound by also arguing that the entropy of statistical mechanics and the information entropy of information theory are the same ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in a wide variety of fields such as biology, neuroscience, computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ..., information theory and sociology. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entropy (Information Theory)
In information theory, the entropy of a random variable quantifies the average level of uncertainty or information associated with the variable's potential states or possible outcomes. This measures the expected amount of information needed to describe the state of the variable, considering the distribution of probabilities across all potential states. Given a discrete random variable X, which may be any member x within the set \mathcal and is distributed according to p\colon \mathcal\to[0, 1], the entropy is \Eta(X) := -\sum_ p(x) \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 "shannon (unit), shannons"), while base Euler's number, ''e'' gives "natural units" nat (unit), nat, and base 10 gives units of "dits", "bans", or "Hartley (unit), hartleys". An equivalent definition of entropy is the expected value of the self-information of a v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Claude E
Claude may refer to: People and fictional characters * Claude (given name), a list of people and fictional characters * Claude (surname), a list of people * Claude Callegari (1962–2021), English Arsenal supporter * Claude Debussy (1862–1918), French composer * Claude Kiambe (born 2003), Congolese-born Dutch singer * Claude Lévi-Strauss (1908–2009), French anthropologist and ethnologist * Claude Lorrain (c. 1600–1682), French landscape painter, draughtsman and etcher traditionally called just "Claude" in English * Claude Makélélé (born 1973), French football manager * Claude McKay (1890–1948), Jamaican-American writer and poet * Claude Monet (1840–1926), French painter * Claude Rains (1889–1967), British-American actor * Claude Shannon (1916–2001), American mathematician, electrical engineer and computer scientist * Madame Claude (1923–2015), French brothel keeper Fernande Grudet Places * Claude, Texas, a city * Claude, West Virginia, an unincorporated commu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
