an equilibrium ensemble. The defining expression for informationentropyentropy in the theory of informationtheoryinformation established by Claude E. Shannon in 1948 is of the form math H sum i ... entropy Thermodynamics Statistical mechanics Informationtheory Physical information Fluctuation theorem Black hole entropy Black hole information paradox EntropyinformationtheoryEntropy statistical ... And InformationTheory Category Thermodynamic entropy Category Entropy and information ... entropyinformation theoretic entropy , usually expressed as H , of Claude Elwood Shannon Claude Shannon ..., commented on it upon publicizing informationtheory in A Mathematical Theory of Communication . This article ... W The defining expression for entropy in the theory of statistical mechanics established by Ludwig ... where W is the number of microstates. If all the messages are equiprobable, the informationentropy ... can also be taken to the natural base in the case of informationentropy. This is equivalent ..., informationentropy is almost always calculated using base 2 logarithms, but this distinction ... density function, p. d. f., H represents the entropy average information, disorder , diversity ... a good measure of uncertainty or information. For example, the differential entropy .... ref Jaynes, E.T., http bayes.wustl.edu etj articles brandeis.pdf InformationTheory And Statistical ... statistical mechanics quantum informationtheory setting, see e.g. cite journal author1 Hong Qian ... the two quantities. The informationentropy H can be calculated for any probability distribution ..., informationentropy of any macroscopic event is so small as to be completely irrelevant. However ... system. The global entropy is not decreased, but information to energy conversion is possible ... entropy if they were then re set to a known state. This can only be achieved under information ... else i.e. if the entropy of the environment or the non information bearing degrees of freedom is increased ... more details
about the Shannon entropy in informationtheoryEntropy disambiguation In informationtheory , entropy ... and informationtheory The inspiration for adopting the word entropy in informationtheory came from ... should be seen as an application of Shannon s informationtheory the thermodynamic entropy ... also Redundancy informationtheory . Further properties The Shannon entropy satisfies the following ... or randomness of a system. Theil index History of informationtheory History of entropy Shannon ... How Could Information Equal Entropy? a similar discussion on the bionet.info theory FAQ. http www.rheingold.com ... date September 2010 DEFAULTSORT EntropyInformationTheory Category Entropy and information Category ... refers to the Shannon entropy , which quantifies the expected value of the self informationinformation ... of entropy is bit for b 2, Nat information nat for b e, and dit ... primer primer.pdf Informationtheory primer with an appendix on logarithms , National Cancer Institute ... informationentropy and information uncertainty can be used interchangeably. ref Cite journal last Jaynes first E.T. url http bayes.wustl.edu etj articles theory.1.pdf title InformationTheory and Statistical ... practical level the links between informationentropy and thermodynamic entropy are not close ... many years in the development of the theories of statistical mechanics and informationtheory to make ... the total entropy does not decrease which resolves the paradox . Entropy as information content ... of entropy, when applied to an information source, can determine the minimum channel capacity ... in a digit from the information source. See also Shannon Hartley theorem . Shannon s entropy ... L295 L301. ref See also Kolmogorov complexity . Limitations of entropy as information content There are a number of entropy related concepts that mathematically quantify information content in some way ... ... Although entropy is often used as a characterization of the information content of a data source ... more details
About entropy in thermodynamics entropy in informationtheoryEntropyinformationtheory other uses Entropy disambiguation Seeintro File Ice water.jpg thumb Ice melting in a warm room is a common Entropy Ice melting example example of increasing entropy, ref group note In complex systems of molecules ... quantum state quantum mechanical states , a model directly related to the informationentropy ..., which is based on a combination of thermodynamics and informationtheory arguments. Landsberg argues ... mechanics quantum theory , entropy changes have been described in terms of the mixing or spreading ... first Daniel, R. coauthors Wiley, E.O. title Evolution as Entropy Towards a Unified Theory of Biology ... first John title InformationTheory and Evolution publisher World Scientific year 2003 isbn 981 238 399 9 ref ref Cite book last Yockey first Hubert, P. title InformationTheory, Evolution, and the Origin ... water mixture, entropy can either increase or decrease depending on system parameters, such as temperature ... place under adiabatic conditions the entropy of the resulting crystal will be greater than that of the supercooled ..., however, when ice melts, the entropy of the two adjoined systems, the hot and cold bodies, increases. Here are some further tutorials http jchemed.chem.wisc.edu JCESoft CCA CCA3 MAIN ENTROPY PAGE1.HTM Ice melting JCE example http www.bartleby.com 64 C004 024.html Ice melting and Entropy Change example http www.ac.wwu.edu vawter PhysicsNet Topics ThermLaw2 Entropy InterptEntropy.html Ice melting and Entropy Change discussions ref described in 1862 by Rudolf Clausius as an increase in the disgregation ... Entropy is a thermodynamics thermodynamic property that is a measure of the energy not available ... maximum efficiency when converting energy to work. During this work entropy accumulates in the system, but has to be removed by dissipation in the form of waste heat. The concept of entropy is defined by the second law of thermodynamics , which states that the entropy of a closed system ... more details
ref A key measure of information is known as Entropyinformationtheoryentropy , which is usually .... With it came the ideas of the informationentropy and redundancy informationtheory redundancy of a source ... theory and statistics . The most important quantities of information are Entropyinformationtheoryentropy , the information in a random variable , and mutual information , the amount of information ... are equally probable, as in an unbiased coin toss. The Entropyinformationtheoryentropy , math ... entropy over math Y math ref name Ash cite book title InformationTheory author Robert B. Ash ... informationtheory Redundancy R nyi entropy Self information Unicity distance Variety cybernetics ... field informatics . Informationtheory is a branch of applied mathematics and electrical engineering involving the quantification of information . Informationtheory was developed by Claude E. Shannon ..., E. T. 1957 http bayes.wustl.edu InformationTheory and Statistical Mechanics , Phys. Rev. 106 620 ... of a fair coin flip two equally likely outcomes provides less information lower entropy than specifying ... topics of informationtheory include lossless data compression e.g. ZIP file format ZIP files , lossy ... theory are source coding , channel coding , algorithmic complexity theory , algorithmic informationtheory , information theoretic security , and measures of information. Overview The main concepts of informationtheory can be grasped by considering the most widespread means of human communication ... coding . Source coding and channel coding are the fundamental concerns of informationtheory ... while the latter may be more important and more meaningful in many contexts. Informationtheory .... Informationtheory is generally considered to have been founded in 1948 by Claude Elwood ... paradigm of classical informationtheory is the engineering problem of the transmission of information ... event is given by its informationentropyentropy and Shannon s noisy channel coding theorem ... more details
wiktionarypar entropy div class infobox sisterproject Additional relevant articles may be found in the following categories Category Thermodynamic entropy Thermodynamic entropy Category Entropy and informationEntropy and information Category Quantum mechanical entropy Quantum mechanical entropy div Entropy , in thermodynamics, is a measure of the energy in a thermodynamic system not available to do useful work. Entropy may also refer to Thermodynamics Entropy classical thermodynamics , the macroscopic approach to thermodynamic entropyEntropy energy dispersal , a variant perspective on thermodynamic entropyEntropy statistical thermodynamics , the microscopic approach to thermodynamic entropy Gibbs entropy , statistical entropy of a thermodynamic system Boltzmann entropy , an approximation to Gibbs entropy Tsallis entropy , a generalization of Boltzmann Gibbs entropy von Neumann entropy , entropy of a quantum mechanical system Entropy order and disorder , entropy s association with the amount of order, disorder, and or chaos in a thermodynamic system History of entropy , concept of entropy ... Introduction to entropyEntropy and life , research concerning the relationship between the thermodynamic quantity entropy and the evolution of life Entropy astrophysics , the adiabatic constant Informationtheory and mathematics Entropyinformationtheory , a measure of the amount of information contained in a message Entropy encoding , data compression strategies to produce a code length equal to the entropy of a message Kolmogorov Sinai entropy , the rate of information generation by a measure preserving dynamical system R nyi entropy , a family of diversity measures used to define fractal dimensions Topological entropy , a measure of exponential growth in the number of distinguishable orbits of a dynamical system Volume entropy , a Riemannian invariant measuring the exponential rate of volume growth of a Riemannian metric Computer science Entropy anonymous data store , a computer ... more details
Entropy is considered to be an extensive property, i.e., that its value depends on the amount of material present. Constantino Tsallis has proposed a nonextensive entropy , which is a generalization of the traditional Boltzmann Gibbs entropy. The rationale behind the theory is that Gibbs Boltzmann entropy leads to systems that have a strong dependence on initial conditions. In reality most materials behave quite independently of initial conditions. Nonextensive entropy leads to nonextensive statistical mechanics, whose typical functions are power laws, instead of the traditional exponentials. See also Tsallis entropy Statistical mechanics topics Category Probability theory Category Entropy and information Category Thermodynamic entropy Category Informationtheory condensedmatter stub ... more details
Citations missing date April 2007 Mergeto Entropy anesthesiology date March 2008 Entropy monitoring is a relatively new method of assessing anaesthetic depth. It was commercially developed by Datex Ohmeda, now part of GE Healthcare . It relies on a method of assessing the degree of irregularity in electroencephalogram EEG signals. The founding principle behind this theory is that the irregularity within an EEG signal decreases with increasing brain levels of anaesthetic drugs. If we relate the irregularity to the Informationentropyentropy within the signal, then an entropy scale can be assigned. The signal is captured via a forehead mounted sensor, in a similar way employed by bispectral index BIS . Entropy monitors produce two numbers RE Response Entropy, SE State Entropy that are related to frequency bandpass used. Response Entropy incorporates higher frequency components that include that of electromyogram activity. The reason for using higher frequency bandpass in response entropy is to allow faster response from the monitor in relation to clinical state. Published studies show that entropy scores do relate to clinical levels of anaesthetic depth. Most anaesthetic drugs are detectable by entropy monitoring, a notable exception being nitrous oxide , in common with BIS monitoring. Future studies may show reduced levels of intraoperative awareness when using this type of monitoring. Future studies may also look into possibilities of influence on outcome when using depth of anaesthesia monitoring. Other Vital signs such as pulse , heart rate , blood pressure , and movement are indirect indicators of consciousness, and when these are combined with expired gas analysis of inhalational anaesthetic agents, an experienced anaesthetist can be confident a patient is unconscious ..., synchronicity, and burst suppression , that, in the case of BIS Bispectral index or Entropy, are converted ... anaesthesia awareness References Bein B Entropy Best practice&research clinical anesthesiology.Vol ... more details
In physics, the Tsallis entropy is a generalization of the standard Entropy statistical thermodynamics Boltzmann Gibbs entropy . In the scientific literature is highly debated the physical relevance of the Tsallis entropy. It is in fact not clear if any system obeys, and in this case in which regime, to the statistical mechanics that can be derived from such approach. The Tsallis entropy is defined as math S q p 1 over q 1 left 1 int p x q , dx right , math or in the discrete case math S q p 1 over q 1 left 1 sum x p x q right , math where S denotes entropy, p the probability distribution of interest, and q is a real parameter. In the limit as q &rarr 1, the normal Boltzmann Gibbs entropy is recovered. The parameter q is a measure of the non extensive quantity extensitivity of the system of interest. There are continuous and discrete versions of this entropic measure. Various relationships The discrete Tsallis entropy satisfies math S q left D q sum i p i x right x 1 math where D sub q sub is the q derivative . Non additivity Given two independent systems A and B , for which the joint probability density function probability density satisfies math p A, B p A p B , , math the Tsallis entropy of this system satisfies math S q A,B S q A S q B 1 q S q A S q B . , math From this result, it is evident that the parameter math 1 q math is a measure of the departure from additivity. In the limit when q 1, math S A,B S A S B , , math which is what is expected for an additive system. See also R nyi entropy References Unreferenced date March 2011 External links http www.cscs.umich.edu crshalizi notabene tsallis.html Tsallis Statistics, Statistical Mechanics for Non extensive Systems and Long Range Interactions condensedmatter stub Category Probability theory Category Entropy and information Category Thermodynamic entropy Category Informationtheory Category Q analogs pt Entropia de Tsallis ... more details
In probability theory or informationtheory , the min entropy of a discrete random event x with possible states or outcomes 1... n and corresponding probabilities p sub 1 sub ... p sub n sub is math H infty X min i 1 n log p i max i log p i log max i p i math The base of the logarithm is just a scaling constant for a result in bit s, use a base 2 logarithm. Thus, a distribution has a min entropy of at least b bits if no possible state has a probability greater than 2 sup b sup . The min entropy is always less than or equal to the Shannon entropy it is equal when all the probabilities p sub i sub are equal. Min entropy is important in the theory of randomness extractor . The notation math H infty X math derives from a parameterized family of Shannon like entropy measures, R nyi entropy , math H k X log sqrt k 1 begin matrix sum i p i k end matrix math k 1 is the Shannon entropy. As k is increased, more weight is given to the larger probabilities, and in the limit as k , only the largest p i has any effect on the result. See also R nyi entropy Leftover hash lemma , Extractor References unreferenced date July 2008 Category Entropy and information Category Informationtheory probability stub eo Vikipedio Projekto matematiko Minimum entropio ... more details
In the mathematical theory of probability , the entropy rate or source information rate of a stochastic process is, informally, the time density of the average information in a stochastic process. For stochastic processes with a countable index, the entropy rate H X is the limit of the joint entropy of n members of the process X sub k sub divided by n , as n Limit mathematics tends to infinity math H X lim n to infty frac 1 n H X 1, X 2, dots X n math when the limit exists. An alternative, related quantity is math H X lim n to infty H X n X n 1 , X n 2 , dots X 1 math For strongly stationary stochastic processes, math H X H X math . Entropy rates for Markov chains Since a stochastic process defined by a Markov chain that is irreducible and aperiodic has a stationary distribution , the entropy rate is independent of the initial distribution. For example, for such a Markov chain Y sub k sub defined on a countable number of states, given the transition matrix P sub ij sub , H Y is given by math displaystyle H Y sum ij mu i P ij log P ij math where &mu sub i sub is the stationary distribution of the chain. A simple consequence of this definition is that the entropy rate of an independent and identically distributed i.i.d. stochastic process has an entropy rate that is the same as the entropy of any individual member of the process. See also Information source mathematics Markov information source References Cover, T. and Thomas, J. 1991 Elements of InformationTheory, John Wiley and Sons, Inc., ISBN 0471062596 http www3.interscience.wiley.com cgi bin bookhome 110438582?CRETRY 1&SRETRY 0 External links http www.eng.ox.ac.uk samp Systems Analysis, Modelling and Prediction SAMP , University of Oxford MATLAB code for estimating information theoretic quantities for stochastic processes. Category Informationtheory Category Entropy Category Markov models de Entropierate ... more details
In informationtheory an entropy encoding is a lossless data compression scheme that is independent of the specific characteristics of the medium. One of the main types of entropy coding creates and assigns a unique prefix free code to each unique symbol that occurs in the input. These entropyinformationtheoryentropy encoders then compress data by replacing each fixed length input symbol by the corresponding variable length prefix free output codeword. The length of each codeword is approximately Proportionality mathematics proportional to the negative logarithm of the probability . Therefore, the most common symbols use the shortest codes. According to Claude Shannon Shannon s source coding theorem , the optimal code length for a symbol is &minus log sub b sub P , where b is the number of symbols used to make output codes and P is the probability of the input symbol. Two of the most common entropy encoding 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 include Universal code data compression universal codes such as Elias gamma coding or Fibonacci coding and Golomb coding Golomb codes such as unary coding or Rice coding . Entropy as a measure of similarity Besides using entropy encoding as a way to compress digital data, an entropy encoder can also be used to measure the amount of similarity between streams of data. This is done by generating an entropy coder compressor for each class of data unknown data is then classified by feeding the uncompressed data to each compressor and seeing which compressor ... mackay itila book.html On line textbook InformationTheory, Inference, and Learning Algorithms , by David MacKay scientist David MacKay , gives an accessible introduction to Shannon theory ... . Compression Methods Category Lossless compression algorithms Category Entropy and information ... more details
Expert subject Physics date June 2009 Context date June 2009 In quantum mechanics , and especially quantum information quantum information theory , the linear entropy of a quantum state state is a scalar physics scalar defined as math S L , dot , 1 mbox Tr rho 2 , math where math rho , math is the density matrix of the state. The linear entropy can range between zero, corresponding to a completely pure state, and math 1 1 d , math , corresponding to a completely mixed state. Here, math d , math is the dimension of the density matrix. Linear entropy is trivially related to the purity quantum mechanics purity math gamma , math of a state by math S L , , 1 gamma , . math Motivation The linear entropy is an approximation to the Von Neumann entropy math S , math , which is defined as math S , dot , mbox Tr rho log 2 rho , . math The linear entropy is obtained by approximating math ln rho , math with the first order term math rho 1 , math in the Mercator series math mbox Tr rho log 2 rho , to mbox Tr rho rho 1 mbox Tr rho rho 2 1 mbox Tr rho 2 S L math where the unit trace property of the density matrix has been used to get the second to last equality. The linear entropy and Von Neumann entropy are similar measures of the mixedness of a state, although the linear entropy is easier to calculate because it does not require the Diagonalizable matrix diagonalization of the density matrix. Alternate definition Some authors ref cite journal author Nicholas A. Peters, Tzu Chieh Wei, Paul G. Kwiat title Mixed state sensitivity of several quantum information benchmarks year 2004 journal Physical Review A volume 70 pages 052309 doi 10.1103 PhysRevA.70.052309 arxiv archive quant ph id 0407172 ref define linear entropy with a different normalization math S L , dot , frac d d 1 1 mbox Tr rho 2 , . math This ensures that the quantity ranges from zero to unity. References references Category Quantum mechanics quantum stub ... more details
Orphan date January 2011 Infobox Journal title Entropy cover Image entropy journal logo.gif discipline Physics , Chemistry , Biology , Engineering , Computer sciences , Economics , Philosophy language English abbreviation Sensors publisher MDPI country Switzerland frequency Quarterly history 1999 openaccess Yes website http www.mdpi.com journal entropy ISSN 1099 4300 Entropy ISSN 1099 4300 , an International and Interdisciplinary Journal of Entropy and Information Studies, is published by MDPI in Basel , Switzerland , and is a peer reviewed Open access publishing Open Access journal. External links http www.mdpi.org Molecular Diversity Preservation Int. MDPI Category Engineering journals Category Physics journals Category Chemistry journals Category Open access journals Category Molecular Diversity Preservation International academic journals fr Entropy journal ... more details
Maximum entropy may refer to The principle of maximum entropy The maximum entropy probability distribution Maximum entropy spectral estimation Maximum entropy spectral analysis Maximum entropy thermodynamics The law of maximum entropy production Entropy maximization See also Second law of thermodynamics also known as the law of increasing entropy disambig ... more details
Unreferenced date December 2009 Mergeto Shannon index date March 2008 Ecological entropy is a measure of biodiversity in the study of biological ecology . Definition Assume that within a system of unlimited individuals, there exists math n math species of organism s math A 1, A 2, ..., A j, ..., A n math , and the probability that an individual belongs to species math A j math is math p j math , where math sum i 1 n p i 1 math If there are math q math individuals for each species, the population will be math q 1 q 2 ... q j ... q n math . The number of individuals is math q n 10 n log q math . Set math N math as constant math C math . Diversity within each species, such as male, female, large, small, etc. can be expressed as math C sum j p i log q j C sum j p i log p I C log q quad mbox since q j qp I math Hence the specific diversity can be derived by subtracting the diversities within the species from the diversity of the whole population math C log q C sum j p i log p i C log q C sum j p i log p i math The expression is the same as that of informationentropy, except that it is customary to use binary logarithm base 2 for informationentropy, the unit being bit , and to use common logarithm base 10 for ecological entropy, the unit being decibel bel . The thermodynamic entropy uses natural logarithm base math e math , so that the constant math C math becomes the Boltzmann constant , the ratio of universal gas constant and Avogadro s number . See also Entropy thermodynamics Entropyinformationtheory DEFAULTSORT Entropy Ecology Category Measurement of biodiversity Category Entropy and information zh ... more details
Unreferenced date December 2009 In informationtheory , the cross entropy between two probability distribution s measures the average number of bit s needed to identify an event from a set of possibilities, if a coding scheme is used based on a given probability distribution math q math , rather than the true distribution math p math . The cross entropy for two distributions math p math and math q ... mathrm H p D mathrm KL p q math , where math mathrm H p math is the informationentropyentropy of math ... p math also known as the relative entropy . For discrete random variable discrete math p math and math ... math mathrm H p,q math is sometimes used for both the cross entropy as well as the joint entropy of math p math and math q math . Estimation There are many situations where cross entropy needs to be measured ... a model is created based on a training set, and then its cross entropy is measured on a test set to assess .... Since the true distribution is unknown, cross entropy cannot be directly calculated. In these cases, an estimate of cross entropy is calculated using the following formula math H T,q sum ... is calculated over math N math . Cross entropy minimization Cross entropy minimization is frequently used in optimization and rare event probability estimation see the cross entropy method . When comparing a distribution math q math against a fixed reference distribution math p math , cross entropy ..., and math mathrm H p math for cross entropy. In the engineering literature, the principle ... information Principle of Minimum Discrimination Information is often called the Principle of Minimum Cross Entropy MCE , or Minxent . However, as discussed in the article Kullback Leibler divergence ... to resolve the inconsistency by redefining cross entropy to be D sub KL sub p q , rather than H p , q . See also Cross entropy method conditional entropy DEFAULTSORT Cross Entropy Category Entropy and information de Kreuzentropie fr Entropie crois e ja ru ... more details
on the field and the context in which it is being used. Informationentropy takes the mathematical ... views of entropy using probability theory to describe the increased molecular movement on the microscopic ... sciences developed the concept of informationentropy where a constant replaces the temperature ...Thermodynamics cTopic List of thermodynamic properties System properties Introductory article EntropyEntropy is a measure of how evenly energy is distributed in a system. In a physical system, entropy ... heat flows from a hot region to a cold region entropy increases, as heat is distributed throughout the system. The concept of entropy is central to the second law of thermodynamics . The second law ... as saying that the entropy of an isolated system always increases, and processes which increase entropy can occur spontaneously. Since entropy increases as uniformity increases, the second law says qualitatively that uniformity increases. The term entropy was coined in 1865 by the German physicist ... Cite web title etymonline.com entropy url http www.etymonline.com index.php?search entropy&searchmode none accessdate 2009 06 15 ref Explanation The concept of thermodynamic entropy arises from the second law of thermodynamics . It uses entropy to quantify the capacity of a system for change, namely ... whether a thermodynamic process may occur. Entropy is defined by two descriptions, first as macroscopic ... ed., Wiley 2008 , p. 125 ref math rm d S frac rm delta q T . math Where d S is the change in entropy ... must be integral integrated over the temperature path. This definition of entropy does not allow the determination of an absolute value, only of differences. The second definition of entropy comes from statistical mechanics . The entropy of a particular Microstate statistical mechanics macrostate ... S is the entropy, k sub B sub is the Boltzmann constant, and &Omega is the number of microstates ... in the internal energy of the thermodynamic system . Thermodynamic entropy provides a comparative ... more details
about which kind of molecule it is. Using Informationtheory Conditional entropy equivocation conditional ... Shannon entropy or Informationentropy Formal definitions compositional uncertainty Ref 2 math ... solution theory provides the entropy of mixing for polymer solutions, in which the macromolecule ... Shannon introduced Informationentropy Formal definitions this expression for use in information ...The entropy of mixing is the change in the configuration entropy , an extensive quantity extensive thermodynamics ... entropy of mixing is misleading, since it is not the intermingling of the particles that creates the entropy ... Naim, Arieh , On the So Called Gibbs Paradox, and on the Real Paradox, Entropy, 9, 132 136, 2007 http www.mdpi.com 1099 4300 9 3 132 Link ref This entropy change is positive when there is more Informationentropy Formal definitions uncertainty about the space spatial locations of the different kinds ... and pressures, there will, of course, be an additional entropy increase in the mixed substance ... temperature and pressure, the entropy increase will be entirely due to the entropy of mixing. The entropy of mixing may be calculated by Gibbs Theorem which states that when two different substances mix, the entropy increase upon mixing is equal to the entropy increase that would occur if the two substances were to expand alone into the mixing volume. In this sense, then the term entropy of mixing is a misnomer, since the entropy increase is not due to any mixing effect. Nevertheless, the two substances must be different for the entropy of mixing to exist. This is the Gibbs paradox which states that if the two substances are identical, there will be no entropy change, yet the slightest detectable difference between the two will yield a considerable entropy change, and this is just the entropy of mixing. In other words, the entropy of mixing is not a continuous function of the degree ... partition, the entropy of mixing math Delta S , math is given by math Delta S nR x 1 ln x 1 x ... more details
article Topological entropy Topological entropy at Scholarpedia planetmath id 6068 title Topological Entropy Category Entropy and information Category Ergodic theory Category Topological dynamics fr ...In mathematics , the topological entropy of a topological dynamical system is a nonnegative real number that measures the complexity of the system. Topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew. Their definition was modelled after the definition of the Kolmogorov Sinai entropy Kolmogorov Sinai , or metric, entropy. Later, Dinaburg and Rufus Bowen gave a different, equivalent definition reminiscent of the Hausdorff dimension . The second definition clarified the meaning of the topological entropy for a system given by an iterated function , the topological entropy represents the exponential growth rate of the number of distinguishable orbits of the iterates. An important variational principle relates the notions of topological and measure theoretic entropy. Definition ... entropy is a nonnegative real number that can be defined in various ways, which are known to be equivalent ... for an infinite cover C . The use of arbitrary covers yields the same value of entropy. ref For two ... frac 1 n H C vee f 1 C vee ldots vee f n 1 C . math Then the topological entropy of f , denoted h f ... of these iterates, as seen by the partition C . Thus the topological entropy is the average per iteration amount of information needed to describe long iterations of the map f . Definition of Bowen ... entropy of the map f is defined by math h f lim epsilon to 0 left limsup n to infty frac 1 n log N ... of topological entropy in a way which permits X to be noncompact. Notes references See also Milnor Thurston kneading theory For the measure of correlations in systems with topological order see Topological entanglement entropy References R. L. Adler, A. G. Konheim, M. H. McAndrew, 1965 , http links.jstor.org sici?sici 0002 9947 196502 114 3A2 3C309 3ATE 3E2.0.CO 3B2 N Topological Entropy , Transactions ... more details
variables . Generalization to quantum theory In quantum informationtheory , the conditional entropy is generalized to the conditional quantum entropy . Other properties For any math X math and math ... 3 540 41633 1 oclc doi See also Entropyinformationtheory Mutual information Conditional quantum entropy Variation of information Likelihood function Category Entropy and information Category Informationtheory bar Bedingte Entropie de Bedingte Entropie fr Entropie conditionnelle pl Entropia warunkowa ... mathcal X math and math Y math with support math mathcal Y math , the conditional entropy of math ... probability, the chain rule for conditional entropy is math H Y X , ,H Y,X H X , . math This is true ... system contains math H X,Y math bits of information we need math H X,Y math bits of information ... bits of information, and the system has math H Y X math bits of uncertainty remaining. math H Y X ... information between math X math and math Y math . math I X Y le H X math , where math I X Y math is the mutual information between math X math and math Y math . For independent math X math and math ... isbn 0 486 41147 8 oclc doi cite book author C. Arndt title Information Measures Information and its ... more details
leq H X H Y math math H X 1, ..., X n leq H X 1 ... H X n math Relations to Other Entropy Measures Joint entropy is used in the definition of conditional entropy math H X Y H X,Y H Y , math and mutual information math I X Y H X H Y H X,Y , math In quantum informationtheory , the joint entropy is generalized into the joint quantum entropy . Category Entropy and information de Blockentropie fr Entropie ... more details
Measures in informationtheory Many of the formulas in informationtheory have separate versions for continuous ... Quantities of information measures of the informationentropyinformation content of random variables and a measure theory measure over sets. Namely the joint entropy , conditional entropy , and mutual information can be considered as the measure of a union set theory set union , set difference , and intersection set theory set intersection , respectively Reza pp. 106 108 . If we associate the existence ... mathematics discrete random variable s X and Y , somehow representing the informationentropyinformation ... entropy, and bivariate mutual information of discrete random variables are all nonnegative, many basic inequalities in informationtheory among no more than two random variables can be derived ... way as the entropy of a joint distribution, and a multivariate mutual information math I X Y ... the mutual information of a single random variable is defined to be its entropy math I X H X math . Then for math ... W,Y . math References Fazlollah M. Reza. An Introduction to InformationTheory . New York McGraw Hill ... . Notre Dame EE 80653 InformationTheory Tutorials, Fall 2005. http www.nd.edu jnl ee80653 Fall2005 tutorials sunil.pdf PDF . contains errors in formulas 9 and 23 R. W. Yeung, On entropy, information ... Informationtheory Measure theory Set theory Category Informationtheory Category Measure theory ... using measure theory . For discrete random variables, probability mass functions can be considered ... mathematics for what can be considered, in a measure theory context, integration. Because the same ... entropy of a continuous random variable with probability density function math f x math ... Leibler divergence is always non negative due to Gibbs inequality . Entropy as a measure Image Conditional ... of information content satisfies all the postulates and basic properties of a formal measure over sets, as commonly illustrated in an information diagram . This can be a handy mnemonic device in some ... more details
of statistical mechanics . Informationtheory An analog to thermodynamic entropy is informationentropy ... to develop his ideas of lost energy, and coined the term entropy . Since the mid 20th century the concept of entropy has found application in the analogous field of data loss in information transmission ... line signals. To do this, Shannon developed the very general concept of informationentropy , a fundamental cornerstone of informationtheory . Although the story varies, initially it seems that Shannon ... to his new informationtheory, according to one source ref M. Tribus, E.C. McIrvine, Energy and information ... to another source, when von Neumann asked him how he was getting on with his informationtheory, Shannon replied ref Cite book author Avery, John title InformationTheory and Evolution publisher ... that he needed a good name for missing information . Why don t you call it entropy , von Neumann suggested ..., and Entropy. ref C.E. Shannon, A Mathematical Theory of Communication , Bell System Technical ... amounts to a choice of a unit of measurement, plays a central role in informationtheory as measures ... or even stating that they are exactly the same. Shannon s informationentropy is a much more general concept than statistical thermodynamic entropy. Informationentropy is present whenever there are unknown ... Informationtheory and statistical mechanics , Physical Review 106 620 ref ref E. T. Jaynes 1957 http bayes.wustl.edu etj articles theory.2.pdf Informationtheory and statistical mechanics II , Physical ... of Shannon s informationentropy to the probabilities of particular microstates of a system ...The concept of entropy developed in response to the observation that a certain amount of functional energy ... energy the result was the concept of entropy . In the early 1850s, Rudolf Clausius set forth the concept ... of transformation energy or entropy , i.e. energy lost to dissipation and friction. ref Cite ... to be less efficient. This lost caloric was a precursory form of entropy loss as we ... more details
In informationtheory , the R nyi entropy , a generalisation of Shannon entropy , is one of a family ... Diversity indices Tsallis entropy Generalized entropy index DEFAULTSORT Renyi entropy Category Informationtheory Category Entropy and information bar R nyi Entropie de R nyi Entropie fr Entropie de ... information from communication theory, may be very important in other applications, or entirely ... title On measures of information and entropy booktitle Proceedings of the 4th Berkeley Symposium ... based information tools. Journal of Neuroscience Methods 153 2006 163 182 R nyi entropy as a measure ... Alfr d R nyi . The R nyi entropy of order , where math geq math 0, math neq math 1 is defined ..., approaching infinity, give a R nyi entropy which is increasingly determined by consideration of only the highest probability events. Lower values of , approaching zero, give a R nyi entropy which increasingly ... case 1 gives the Shannon entropy , which has special properties. When 0, it is the maximum possible Shannon entropy, log n . The R nyi entropies are important in ecology and statistics as diversity indices indices of diversity . The R nyi entropy also important in quantum information , it can be used as a measure of entanglement . In math XY math Heisenberg spin chain the R nyi entropy was calculated ... entropy of X . In the limit that math alpha math approaches 1, it can be shown using L H pital ... entropy . Collision entropy , sometimes just called R nyi entropy, refers to the case math alpha ... H infty X log sup i 1..n p i math and this is called Min entropy , because it is the smallest value ... by math H infty H 2 2 H infty math . On the other hand the Shannon entropy math H 1 math can be arbitrarily high for a random variable X with fixed min entropy. math H 2 2 H infty math is because ... entropy and the Kullback Leibler divergence , is special because it is only when 1 that one can ... measure m x , a , and we acquire new information which only affects the distribution of a , then the distribution ... more details
The volume entropy is an asymptotic invariant mathematics invariant of a compact space compact Riemannian manifold that measures the exponential growth rate of the volume of metric ball s 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 space s forming a basic ... math tilde M . math Choose a point math tilde x 0 in tilde M math . The volume entropy or asymptotic ... of balls in the universal cover as a function of the radius. Properties Volume entropy h is always bounded above by the topological entropy h sub top sub of the geodesic flow on M . Moreover .... More generally, volume entropy equals topological entropy under a weaker assumption that M is a closed ... entropy when the volume is prescribed. This is a corollary of a very general result due to Besson ... in differential geometry of surfaces Katok s entropy inequality was recently exploited to obtain a tight ... localement sym triques de courbure strictement n gative. French Entropy and rigidity of locally symmetric spaces with strictly negative curvature Geom. Funct. Anal. 5 1995 , no. 5, 731 799 Katok, A. Entropy ... to the modern theory of dynamical systems. With a supplementary chapter by Katok and L. Mendoza ... Katz, M. Sabourau, S. Entropy of systolically extremal surfaces and asymptotic bounds. Erg. Th. Dyn. Sys. 25 2005 , 1209 1220 Manning, A. Topological entropy for geodesic flows. Ann. of Math. 2 110 1979 , no. 3, 567 573 Category Differential geometry Category Dynamical systems Category Entropy Category Ergodic theory Category Systolic geometry ... more details
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