for informationentropyentropy in the theory of informationtheoryinformation established by Claude ... name TOA http www.talkorigins.org faqs information shannon.html Entropy Classical InformationTheory ... of the concept of entropy. DEFAULTSORT Entropy In Thermodynamics And InformationTheory Category ... and J. Willard Gibbs in the 1870s, and the informationentropyinformation theoretic entropy usually ..., featuring his equation S k log W The defining expression for entropy in the theory ... the messages are equiprobable, the informationentropy reduces to the Hartley entropy math H log M ... also be taken to the natural base in the case of informationentropy. This is equivalent to choosing ... function, pdf, H represents the entropy average information, disorder , diversity, etc. of f x ... measure of uncertainty or information. For example, the differential entropy can be negative also ... journal author Jaynes, E.T. authorlink Edwin Thompson Jaynes title InformationTheory And Statistical ... mechanics quantum informationtheory setting, see e.g. cite journal author1 Hong Qian title Relative ... 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 ... chemical entropy in 1930, Gain in entropy always means loss of information, and nothing more . To be more ... in the article Maximum entropy thermodynamics . Information is physical Szilard s engine Image ... system. The global entropy is not decreased, but information to energy conversion is possible ... somewhere else i.e. if the entropy of the environment or the non information bearing degrees of freedom ... entropy increase. When information is physical, all processing of its representations ... Theory , Brillouin described the Negentropy Principle of Information or NPI, the gist of which is that acquiring information about a system s microstates is associated with a decrease in entropy ... more details
In informationtheory , entropy is a measure of the uncertainty associated with a random variable . ref cite book author Ihara, Shunsuke title Informationtheory for continuous systems publisher World ... to thermodynamic entropy Main Entropy in thermodynamics and informationtheory The inspiration for adopting the word entropy in informationtheory came from the close resemblance between Shannon ... symbols can be defined simply as being equal to its n ary entropy. See also Redundancy informationtheory ... of entropy History of informationtheory Joint entropy is the measure how much entropy is contained ... theory FAQ. http www.rheingold.com texts tft 6.html Description of informationentropy from Tools ... Methods Use dmy dates date September 2010 DEFAULTSORT EntropyInformationTheory Category Entropy ... author Brillouin, L on title Science & InformationTheory publisher Dover Publications year 2004 ..., the Shannon entropy is a measure of the average information content one is missing when one does ... bit of message. The entropy of a message is in a certain sense a measure of how much information it really ... content of X . ref cite book author Borda, Monica title Fundamentals in InformationTheory and Coding ... Informationtheory primer with an appendix on logarithms , National Cancer Institute, 14 April 2007 ... is quantified in a lower entropy on average each toss of the coin delivers less than a full 1 bit of information ... toss of the coin delivers no information. In this respect, entropy can be normalized by dividing it by information length. The measure is called metric entropy and allowed to measure the randomness of the information ... math H X , math in Eq. 1 . The above also explained why informationentropy and information ... etj articles theory.1.pdf title InformationTheory and Statistical Mechanics journal ... level the links between informationentropy and thermodynamic entropy are not evident. Physicists and chemists ..., which is central to the definition of informationentropy. But, at a multidisciplinary level, connections ... more details
About entropy in thermodynamics entropy in informationtheoryEntropyinformationtheory a comparison of entropy in informationtheory with entropy in thermodynamics Entropy in thermodynamics and informationtheory other uses Entropy disambiguation See introduction 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 ... about a system. The equilibrium state of a system maximizes the entropy because we have lost all information ... Peter Landsberg, based on a combination of thermodynamics and informationtheory arguments. He argues ... is the entropy of the parts contained in the permitted ensemble, C sub I sub is the information capacity ... theory , entropy changes have been described in terms of the mixing or spreading of the total energy ... in other fields of study, including informationtheory , psychodynamics , thermoeconomics , and evolution ... as Entropy Towards a Unified Theory of Biology publisher University of Chicago Press year 1988 isbn 0 226 07574 5 ref ref Cite book last Avery first John title InformationTheory and Evolution publisher .... title InformationTheory, Evolution, and the Origin of Life publisher Cambridge University Press year ... or when salt is added to an ice water mixture, entropy can either increase or decrease depending ... of a supercooled liquid takes place under adiabatic conditions the entropy of the resulting crystal ... Equilibrium , 4th Ed. . In general, however, when ice melts, the entropy of the two adjoined systems ... 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 ..., S. 2. vol. vii. p. 209. ref EntropySegments Entropy is a List of thermodynamic properties thermodynamic ... to work. During this work, entropy accumulates in the system, which then dissipation dissipates ... 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 Entropy arrow of time , thermodynamic arrow of time 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 ... science Entropy computing , a discussion of entropy generation and API s in computer systems Entropy ... more details
original research date March 2012 unreferenced date March 2012 Weighted entropy is a notion which extends the classical Shannon entropy. Category Informationtheory ... more details
format pdf accessdate 2010 06 23 ref A key measure of information is known as Entropyinformationtheory ... Landauer in the 1960s, are explored in Entropy in thermodynamics and informationtheory . In Shannon ... entropy and redundancy informationtheory redundancy of a source, and its relevance through the source ... important quantities of information are Entropyinformationtheoryentropy , the information in a random ... coin toss. The Entropyinformationtheoryentropy , math H math , of a discrete random variable ... entropy over math Y math ref name Ash cite book title InformationTheory author Robert B. Ash publisher ... theory. Rate This section is linked from Channel capacity InformationEntropy rate rate is the average ... isbn 1558603697 ref It is common in informationtheory to speak of the rate or entropy of a language ... Cryptography Cybernetics Entropy in thermodynamics and informationtheory Gambling Intelligence ... Information PMI Receiver informationtheory Redundancy informationtheory Redundancy R nyi entropy ...Dablink Not to be confused with Information science . Informationtheory is a branch of applied mathematics and electrical engineering involving the quantification of information . Informationtheory was developed ..., thermal physics, ref Jaynes, E. T. 1957 http bayes.wustl.edu InformationTheory and Statistical ... outcomes provides less information lower entropy than specifying the outcome from a roll of a dice six equally likely outcomes . Applications of fundamental topics of informationtheory include ... s, and numerous other fields. Important sub fields of informationtheory 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 ... coding are the fundamental concerns of informationtheory. Note that these concerns have nothing ... and more meaningful in many contexts. Informationtheory, however, does not consider message ... more details
unreferenced date September 2011 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 law s, instead of the traditional exponential s. See also Tsallis entropy Statistical mechanics topics Category Probability theory Category Entropy and information Category Thermodynamic entropy Category Informationtheory condensedmatter stub ... 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 ... more details
Orphan date January 2011 Infobox journal title Entropy cover Image Entropy logo.png discipline Physics , chemistry abbreviation Entropy publisher MDPI country frequency Monthly history 1999 present openaccess Yes website http www.mdpi.com journal entropy ISSN 1099 4300 Entropy is a Peer review peer reviewed open access scientific journal on entropy and information studies . It was established in 1999 and is published by MDPI . According to the Journal Citation Reports , the journal has a 2010 impact factor of 1.109. ref name WoS cite book year 2011 chapter Entropy title 2010 Journal Citation Reports publisher Thomson Reuters edition Science accessdate 2011 09 15 work Web of Science postscript . ref References Reflist External links Official http www.mdpi.com journal entropy Category Physics journals Category Chemistry journals Category Open access journals Category Multidisciplinary Digital Publishing Institute academic journals Category Monthly journals Category English language journals Category Publications established in 1999 fr Entropy journal ... more details
1 , the entropy vanishes. This formulation is analogous to that of Entropyinformationtheory Shannon s informationentropy . The mathematical field of combinatorics , and in particular the mathematics of combination s and permutation s is highly important in the calculation of configurational entropy ...In statistical mechanics , configuration entropy is the portion of a system s entropy that is related to the position of its constituent particles rather than to their velocity or momentum . It is physically related to the number of ways of arranging all the particle s of the system while maintaining some overall set of specified system properties, such as energy . The configurational entropy is also known as microscopic entropy or conformational entropy in the study of macromolecules . In general, configurational entropy is the foundation of statistical thermodynamics. ref http www.entropysite.com calpoly talk.html ref It can be shown ref name Young cite book last Young first Hugh coauthors ... of configuration entropy of thermodynamic systems e.g., ideal gas, and other systems ... of the macroscopic entropy defined as dS Q T , where Q is the heat exchanged between the system and the surrounding media, and T is temperature. Therefore configuration entropy is the same as macroscopic entropy. Calculation The configurational entropy is related to the number of possible configurations by Boltzmann s entropy formula math S k B , ln W, math where k sub B sub is the Boltzmann ... can be in states n with probabilities P sub n sub , the configurational entropy of the system ..., but also analytically to determine the configurational entropy is the Widom insertion method . See also Conformational entropy Combinatorics Entropic force Nanomechanics Entropy of mixing Notes references ... Category Thermodynamic entropy Category Philosophy of thermal and statistical physics Category Fundamental physics concepts Entropy it Entropia configurazionale ... more details
Citations missing date April 2007 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 and not aware of their surroundings. However ... of BIS Bispectral index or Entropy, are converted to a number expressing the likelihood of awareness. See also bispectral index evoked potentials anaesthesia awareness References Bein B Entropy ... more details
This is a list of informationtheory topics , by Wikipedia page. A Mathematical Theory of Communication algorithmic informationtheory arithmetic encoding channel capacity Communication Theory of Secrecy Systems conditional entropy conditional quantum entropy confusion and diffusion cross entropy data compression entropy encoding Fisher information Hick s law Hirchman uncertainty Huffman encoding information bottleneck method informationentropyinformation theoretic security informationtheory joint entropy Kullback Leibler divergence lossless data compression negentropy principle of maximum entropy quantum information science range encoding redundancy informationtheory redundancy R nyi entropy self information Shannon limit Shannon s law Shannon s theorem Category Mathematics related lists Informationtheory Category Informationtheory Category Indexes of mathematics topics Informationtheory topics ... more details
Unreferenced date December 2009 In informationtheory , the cross entropy between two probability distribution ... mathrm H p D mathrm KL p q math , where math mathrm H p math is the informationentropyentropy of math ..., with some authors attempting 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 es Entrop a cruzada fr Entropie ... the true distribution math p math . The cross entropy for two distributions math p math and math q ... 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 math T math , and then its cross entropy is measured on a test ... by the model. 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 ... that the sum is calculated over math N math . This is a Monte Carlo estimate of the true cross entropy, where the training set is treated as samples from math p x 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 Kullback Leibler divergence KL divergence are identical up ... math , which is math 0 math for KL divergence, and math mathrm H p math for cross entropy. In the engineering ... Principle of minimum discrimination information Principle of Minimum Discrimination Information is often called the Principle of Minimum Cross Entropy MCE , or Minxent . However, as discussed in the article ... 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 quant ph 0407172 bibcode 2004PhRvA..70e2309P issue 5 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
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 ... trained on the data that was most similar to the unknown data. See also Entropyinformationtheory External links http www.inference.phy.cam.ac.uk mackay itila book.html On line textbook InformationTheory ... Compression Methods Category Lossless compression algorithms Category Entropy and information ... introduction to Shannon theory and data compression, including the Huffman coding and arithmetic ... 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 Maximum entropy classifier See also Second law of thermodynamics also known as the law of increasing entropy disambig ... more details
Image Entropy mutual information relative entropy relation diagram.svg thumb 256px right Individual H X ,H Y , joint H X,Y , and conditional entropies for a pair of correlated subsystems X,Y with mutual information I X Y . Joint entropyinformationtheoryentropy is a measure of the uncertainty associated with a set of random variables variables . Definition The joint entropy of two variables math X math and math Y math is defined as math H X,Y sum x sum y P x,y log 2 P x,y math where math x math and math y math are particular values of math X math and math Y math , respectively, math P x,y math is the probability of these values occurring together, and math P x,y log 2 P x,y math is defined to be 0 if math P x,y 0 math . For more than two variables math X 1, ..., X n math this expands to math H X 1, ..., X n sum x 1 ... sum x n P x 1, ..., x n log 2 P x 1, ..., x n math where math x 1,...,x n math are particular values of math X 1,...,X n math , respectively, math P x 1, ..., x n math is the probability of these values occurring together, and math P x 1, ..., x n log 2 P x 1, ..., x n math is defined to be 0 if math P x 1, ..., x n 0 math . Properties Greater than individual entropies The joint entropy of a set of variables is greater than or equal to all of the individual entropies of the variables in the set. math H X,Y geq max H X ,H Y math math H X 1, ..., X n geq max H X 1 , ..., H X n math Less than sum of individual entropies The joint entropy of a set of variables is less than or equal to the sum of the individual entropies of the variables in the set. This is an example ... math Relations to Other Entropy Measures Joint entropy is used in the definition of conditional entropy math H X Y H Y,X 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 conjointe ja zh ... 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. Example In the Human genome , for chromosome 1, which consists of 247 Base pair Length measurements Mbp , there is an entropy rate of 1.82 bits per base pair. 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 ... information theoretic quantities for stochastic processes. Category Informationtheory Category Entropy Category Markov models de Entropierate es Ratio de entrop a ... more details
the very general concept of informationentropy , a fundamental cornerstone of informationtheory ... of the cornerstones of the theory of statistical mechanics . Informationtheory An analog to thermodynamic entropy is informationentropy . In 1948, while working at Bell Telephone Company Bell Telephone ... or attenuation in phone line signals with reference to his new informationtheory, according to one ... him how he was getting on with his informationtheory, Shannon replied ref Cite book author Avery, John title InformationTheory and Evolution publisher World Scientific year 2003 isbn 981 238 400 6 ref cquote The theory was in excellent shape, except that he needed a good name for missing information ..., plays a central role in informationtheory as measures of information, choice, and uncertainty .... Shannon s informationentropy is a much more general concept than statistical thermodynamic entropy. Informationentropy is present whenever there are unknown quantities that can be described only ... 1957 http bayes.wustl.edu etj articles theory.1.pdf Informationtheory and statistical mechanics ... Informationtheory and statistical mechanics II , Physical Review 108 171 ref the statistical thermodynamic entropy can be seen as just a particular application of Shannon s informationentropy to the probabilities ...The concept of entropy developed in response to the observation that a certain amount of functional energy ... was the concept of entropy . In the early 1850s, Rudolf Clausius set forth the concept of the thermodynamic ... his ideas of lost energy, and coined the term entropy . Since the mid 20th century the concept of entropy has found application in the field of informationtheory , describing an analogous loss of data in information transmission systems. Classical thermodynamic views Main classical thermodynamics In 1803 ... energy or entropy , i.e. energy lost to dissipation and friction. ref Cite book author ... engine ideal engine in which any heat i.e., Caloric theory caloric converted into work, could be reinstated ... more details
Adler, Tomasz Downarowicz, Micha Misiurewicz, http www.scholarpedia.org article Topological entropy Topological entropy at Scholarpedia PlanetMath attribution id 6068 title Topological Entropy Category Entropy and information Category Ergodic theory Category Topological dynamics fr Entropie topologique ... by the partition C . Thus the topological entropy is the average per iteration amount of information ...About entropy in geometry and topology other uses Entropy disambiguation 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 ... relates the notions of topological and measure theoretic entropy. Definition A topological dynamical ... and a continuous function topology continuous self map f . Its topological entropy is a nonnegative ... . The use of arbitrary covers yields the same value of entropy. ref For two covers C and D , let math ... ldots vee f n 1 C . math Then the topological entropy of f , denoted h f , is defined to be the supremum ... the maximum cardinality of an n , &epsilon separated set. The topological entropy of the map f is defined ... dynamical system X , f . Rufus Bowen extended this definition 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 ... 9947 196502 114 3A2 3C309 3ATE 3E2.0.CO 3B2 N Topological Entropy , Transactions of the American ... more details
phenomena, depending on the field and the context in which it is being used. Informationentropy ... views statistical views of entropy using probability theory to describe the increased molecular movement ..., information sciences developed the concept of informationentropy where a constant replaces ...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 a 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 ... 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 ... B ln Omega, math Where S is the entropy, k sub B sub is the Boltzmann constant, and &Omega is the number ... will result in a decrease in the internal energy of the thermodynamic system . Thermodynamic entropy ... to becoming dispersed or spread out, if it is not hindered from doing so. Entropy change is the quantitative ... 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
information I X Y . In informationtheory , the conditional entropy or equivocation quantifies the remaining informationentropyentropy i.e. uncertainty of a random variable math Y math given that the value of another random variable math X math is known. It is referred to as the entropy of math ... Y math and math X math are independent random variables . Generalization to quantum theory In quantum informationtheory , the conditional entropy is generalized to the conditional quantum entropy ... Berlin year 2001 pages 370 373 isbn 3 540 41633 1 oclc doi See also Entropyinformationtheory Mutual information Conditional quantum entropy Variation of informationEntropy power inequality Likelihood function Category Entropy and information Category Informationtheory bar Bedingte Entropie ...Image Entropy mutual information relative entropy relation diagram.svg thumb 256px right Individual H ... entropy is measured in bit s, nat information nat s, or ban information ban s. Definition More precisely, if math H Y X x math is the entropy of the variable math Y math conditional on the variable ... math mathcal Y math , the conditional entropy of math Y math given math X math is defined as math ... and Y. The last formula above is the Kullback Leibler divergence , also known as relative entropy. Relative entropy is always positive, and vanishes if and only if math p x,y p x math . This is when knowing ... entropy is math H Y X , ,H X,Y H X , . math This is true because math begin align H Y X & sum x in mathcal ...,Y H X . end align math Intuition Intuitively, the combined system contains math H X,Y math bits of information we need math H X,Y math bits of information to reconstruct its exact state. If we learn the value of math X math , we have gained math H X math bits of information, and the system has math ... I X Y math is the mutual 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 ... more details
of information estimates. In InformationTheory in Psychology Problems and Methods , pp. 95 100 ... Ruyter 2004 Entropy and information in neural spike trains Progress on the sampling problem. Physical ... distribution . References reflist DEFAULTSORT Entropy Estimation Category Entropy and information Category Informationtheory Category Statistical randomness Category Randomness de Entropiesch tzung ...Estimating the differential entropy of a system or process, given some observations, is useful in various ... Fast algorithms for mutual information based independent component analysis. In Signal Processing .... Du, Y. Wang, J. Guo, S. M. Thouin, P.D. 2006 Survey and comparative analysis of entropy and relative entropy thresholding techniques. In Vision, Image and Signal Processing , Volume 153, Issue 6, 837 ..., doi 10.1109 JSTSP.2008.923841 ref speech recognition , ref Gue Jun Jung Yung Hwan Oh 2008 Information ... 2004 , Geodesic entropic graphs for dimension and entropy estimation in manifold learning. In Signal .... ref Benesty, J. Yiteng Huang Jingdong Chen 2007 Time Delay Estimation via Minimum Entropy. In Signal ... references Entropy 20estimation.pdf Nonparametric entropy estimation An overview . In International ... T. Sch rmann, Bias analysis in entropy estimation. In J. Phys. A Math. Gen , 37 2004 , pp. L295 ... that the differential entropy, math H X int mathbb X f x log f x ,dx math can be approximated by producing a histogram of the observations, and then finding the discrete entropy math begin matrix H ... method, and then, from the pdf estimate, compute the entropy. A useful pdf estimate method is e.g. Gaussian ... can then be used to calculate the entropy estimate, in a similar way to that given above for the histogram .... However, using analogous methods, some multidimensional entropy estimators have been developed. ref name lm2003 E. G. Learned Miller 2003 A new class of entropy estimators for multi dimensional ... density and entropy estimators for spherically invariant multidimensional data, In Neural Computation ... more details
Tsallis Category Probability theory Category Entropy and information Category Thermodynamic entropy Category Informationtheory Category Q analogs pt Entropia de Tsallis ...In physics, the Tsallis entropy is a generalization of the standard Boltzmann Gibbs entropy. It was introduced ... relevance of the Tsallis entropy was occasionally debated. However, from the years 2000 on, an increasingly ... confirm the predictions and consequences that are derived from this nonadditive entropy, such as nonextensive ... Gibbs theory. Among the various experimental verifications and applications presently available ... entropy and associated statistics apply, the following ones can be selected Anomalous diffusion ref ... theorem ref Cite doi 10.1016 S0375 9601 00 00337 6 ref . Sensitivity to initial conditions and entropy ... PhysRevE.69.045202 ref . Probability sets which make the nonadditive Tsallis entropy to be extensive ... math sum i p i 1 math , the Tsallis entropy is defined as math S q p i 1 over q 1 ... . In the limit as math q to 1 math , the usual Boltzmann Gibbs entropy is recovered, namely math S BG S 1 p k sum i p i ln p i . math For continuous probability distributions, we define the entropy ... function . The Tsallis Entropy has been used along with the Principle of maximum entropy to derive the Tsallis distribution . Various relationships The discrete Tsallis entropy satisfies math S q ... be compared to the standard entropy formula math S lim x rightarrow 1 frac d dx sum i p i x ... function probability density satisfies math p A, B p A p B , , math the Tsallis entropy of this system ... distribution belongs to the statistical exponential families. Tsallis entropy for an exponential family .... For multivariate normal, term k is zero, and therefore the Tsallis entropy is in closed form. See also R nyi entropy Tsallis distribution References references Refbegin cite journal last1 Nielsen ... Nielsen and Richard Nock title A closed form expression for the Sharma Mittal entropy of exponential ... more details
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