Expert subject Statistics date November 2008 The issue of randomness is an important philosophical and theoretical question. Many random number generators in use today generate what are called random sequences but they are actually the result of prescribed algorithms and so they are called pseudo random number generators. These generators do not always generate sequences which are sufficiently random and generate very repetetive patterns such as the infamous RANDU which fails many randomness tests ... in invalid experiments, due to the lack of randomness. Tests for randomness are not restricted ... a set of data has a recognisable pattern to it. For example Wolfram used randomness tests on the output ... isbn 1579550088 ref Randomness tests or tests of randomness , in data evaluation, are used to analyze ... which does pass the tests for randomness. There are many practical measures of randomness for a binary ... , and complexity or a mixture of these. The use of Hadamard transform to measure randomness was proposed ... Marsaglia Marsaglia and Zaman. ref Terry Ritter, Randomness tests a literature survey. http ..., provide spectral measures of randomness. T. Beth and Z D. Dai ref Beth, T. and Z D. Dai. 1989. On the Complexity ... to compare and contrast the randomness of string computer science strings . On probabilistic grounds, all strings, say of length 64, have the same randomness. However, two strings such as those given ... Hadamard transform , the first of these sequences will be found to be of much less randomness than the second one, which agrees with intuition. See also Diehard tests TestU01 Randomness Statistical randomness Algorithmically random sequence Seven states of randomness External links George Marsaglia , Wai Wan Tsang, http www.jstatsoft.org v07 i03 paper Some Difficult to pass Tests of Randomness, Journal ... dieharder.php DieHarder A Random Number Test Suite http www.cacert.at random Online randomness test Notes div class references small references Category Algorithmic information theory Category Randomness ... more details
Merge from Taleb distribution date October 2010 Infobox Book name Fooled by Randomness The Hidden Role of Chance in Life and in the Markets author Nassim Nicholas Taleb image Image Fooled by Randomness Paperback.jpg border Fooled by Randomness genre Statistics , Philosophy , Finance language English publisher Random House release date 2001 pages 316 isbn 0 8129 7521 9 dewey 123 .3 22 congress HG4521 .T285 2005 oclc 60349198 preceded by followed by The Black Swan Taleb book The Black Swan Fooled by Randomness The Hidden Role of Chance in Life and in the Markets is a book written by Nassim Nicholas Taleb about the fallibility of human knowledge. Reaction The book was selected by Fortune magazine Fortune as one of the 75 Smartest Books of All Time. ref http money.cnn.com magazines fortune fortune archive 2005 03 21 8254826 index.htm ref The book s name, Fooled by Randomness , has also become an idiom in English used to describe when someone sees a pattern where there is just random noise. Citation needed date July 2010 Thesis Taleb sets forth the idea that modern humans are often unaware of the existence of randomness . They tend to explain random outcomes as non random. Human beings overestimate causality , e.g., they see elephants in the clouds instead of understanding that they are in fact randomly shaped clouds that appear to our eyes as elephants or something else tend to view the world as more explanation explainable than it really is. So they look for explanations even when there are none. Other misperceptions of randomness that are discussed include Survivorship bias . We see the winners and try to learn from them, while forgetting the huge number of losers. Skewed distributions ..., ref http www.fooledbyrandomness.com fooled by randomness ref and is reported to have sold over half ... phenomenon of perceiving pattern in randomness References references External links http www.fooledbyrandomness.com ... as many media reviews DEFAULTSORT Fooled By Randomness Category 2001 books Category Books by Nassim ... more details
See also algorithmic randomness A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities sequences such as the results of an ideal dice dice roll , or the digits of pi exhibit statistical randomness ref http news.uns.purdue.edu UNS html4ever 2005 050426.Fischbach.pi.html Pi seems a good random number generator but not always the best , Chad Boutin, Purdue University ref . Statistical randomness does not necessarily imply true randomness , i.e., objective unpredictability. Pseudorandomness is sufficient for many uses. Global randomness and local randomness are different. Most philosophical conceptions of randomness are global&mdash because they are based on the idea that in the long run a sequence looks truly random, even if certain sub sequences would not look random. In a truly random sequence of numbers of sufficient length, for example, it is probable there would be long sequences of nothing but zeros, though on the whole the sequence might be random. Local randomness refers to the idea that there can be minimum sequence lengths ... by truly random processes, would diminish the local randomness of a sample it might only be locally ... standards of statistical randomness to slot machine s. Tests The first tests for random numbers ... . Kendall and Smith differentiated local randomness from true randomness in that many sequences generated with truly random methods might not display local randomness to a given degree &mdash very ... tests as exclusive verifications for their randomness, as they are decidedly not produced by truly random ... is fair Normal number Randomness Random number Statistical hypothesis testing One time pad Randomness tests Seven states of randomness Complete spatial randomness References reflist M.G. Kendall and B. Babington Smith, Randomness and Random Sampling Numbers, Journal of the Royal Statistical Society ... Statistical randomness es Aleatoriedad estad stica ... more details
of chance and randomness were intertwined with that of fate. Many ancient peoples threw ... cultures used various methods of divination to attempt to circumvent randomness and fate. ref Handbook ... people to formalize odds and chance 3,000 years ago. The Greek philosophers discussed randomness ... calculus had a positive impact on the formal study of randomness. In the 19th century the concept of entropy was introduced in physics. and a proof of the randomness of the digits of the number ... of randomness, and mathematical foundations for probability were introduced, leading to its axiomatization ... new dimensions to the field via the concept of algorithmic randomness . Although randomness had ... scientists began to realize that the deliberate introduction of randomness into computations can ... , 1541 In ancient history, the concepts of chance and randomness were intertwined with that of fate ... and randomness until the 16th century. The development of the concept of chance throughout history has been very gradual. Historians have wondered why progress in the field of randomness was so slow ... people face an inherent difficulty in understanding randomness, although the concept is often taken ... people do not attend to the detail necessary to gain such knowledge. ref Randomness , Deborah J. Bennett ... the earliest Western thinkers to address chance and randomness. Around 400 BC, Democritus presented a view of the world as governed by the unambiguous laws of order and considered randomness as a subjective ... of necessary determinism. He viewed randomness as a genuine and widespread part of the world ... BC Epicurus proposed the concept that randomness exists by itself, independent of human knowledge ... randomness at higher levels. ref Epicurus an Introduction , John M. Rist, 1972 ISBN 0521084261 p. 52 ... between the belief in free will and its implied randomness, and the idea that God knows ... and free will, but Martin Luther argued against randomness and took the position that God ... more details
Cleanup date January 2010 A randomness extractor , often simply called an extractor, is a function which, when applied to a information entropy high entropy source such as radioactive decay , or thermal ... TRNG . A randomness extractor is an algorithm that converts a weakly random source and a truly random ... random seed s length is useful to minimize. No single randomness extractor currently exists that has ... to define the level of randomness it produces. For this is used min entropy , which is a measurement of the amount of randomness in the worst case. The definition uses the worst case randomness of min entropy and not the average case randomness described by Information entropy Shannon entropy . Definition .... 5. ref . Dispersers Another variant of the randomness extractor is the disperser . Strong extractors The two inputs taken by an extractor must be independent sources of randomness the actual random ... on math 0,1 m d math . Randomness extractors in cryptography This section is mainly based on ref Jesse ... for public key encryption. In other words, randomness extraction can be useful in the key derivation ... the pseudo random source or the seed, but not both . In other words, the randomness, and thus the secrecy , of the output is intact even if the input source is compromised. The high level of randomness ... properties of the hash function. Applications Randomness extractors are used widely in cryptographic ... result. Randomness extractors have played a part in recent developments in quantum cryptography , where photons are used by the randomness extractor to generate secure random bits. http newsroom.spie.org x4741.xml?highlight x535 Randomness extraction is also used in some branches of computational ... of extractors , Ronen Shaltiel http people.csail.mit.edu dodis ps hmac.ps Randomness Extraction and Key ... 2005 061.pdf Key Derivation and Randomness Extraction , Olivier Chevassut et al. http www.cs.utexas.edu ... Category Randomness Category Computational complexity theory Category Cryptographic algorithms ... more details
Cleanup date March 2010 Main RandomnessRandomness has many uses in gambling , statistics , cryptography , art , etc. These uses have different randomness requirements, which leads to the use of different randomization methods. For example, applications in cryptography have strict requirements, whereas other uses such as generating a quote of the day can use a looser standard of randomness. Early uses Games Unpredictable by the humans involved numbers usually taken to be randomness random numbers were first formally investigated in the context of gambling . Many randomizing devices such as dice , shuffling playing cards , and roulette wheels, seem to have been developed for use in game of chance games of chance . Electronic gambling equipment cannot use these and so theoretical problems are less ... been alleged that some gaming machines software is deliberately biased to prevent true randomness ..., founded on the concept of randomness . Many elements of statistical practice depend on randomness ... on randomness include choosing a representative statistical sample sample , disguising the Clinical ... Consciousness Project , monitor fluctuations in the randomness of numbers generated by many hardware ..., music and art Some aesthetic theories claim to be based on randomness in one way or another. Little testing is done in these sitautions, and so claims of reliance on and use of randomness are generally abstract. An example of a need for randomness sometimes occurs in arranging items in an art ... and letters, has attempted to accommodate and acknowledge randomness in various ways. Often people mistake order for randomness based on lack of information e.g., Jackson Pollock s drip paintings, Helen ... . Randomness in music is generally thought to be postmodern music postmodern , including John Cage ... which way a villain might move in a computer game . Weaker forms of randomness are also ... DEFAULTSORT Applications Of Randomness Category Randomness ... more details
Summary book cover fur Article Fooled by Randomness Use Infobox OPTIONAL FIELDS Title Fooled by Randomness The Hidden Role of Chance in Life and in the Markets Author Nassim Nicholas Taleb Publisher Random House Cover artist Website Owner Commentary Year 2001 OVERRIDE FIELDS Description Source Portion Low resolution Purpose Replaceability Other information Source http images.kalahari.net ann all lg 081 297 521 0812975219.jpg Book cover for Nassim Taleb s book Fooled by Randomness Licensing Non free book cover ... more details
File Tetraedre fractal spherique.jpg thumb A tetrahedral fractal . The seven states of randomness in probability theory , fractals and risk analysis are extensions of the concept of normal distribution . These seven states were first introduced in by Beno t Mandelbrot in his 1997 book Fractals and scaling in finance which applied fractal analysis to the study of risk and randomness. ref name Mandelbrot1977 Fractals and scaling in finance by Beno t Mandelbrot 1997 ISBN 0387983635 pages 136 142 ref These seven states build on earlier work of Mandelbrot in 1964, titled The two stages of indeterminism in which he argued that most statistical models approached only a first stage of dealing with indeterminism in science, and that they ignored many aspects of real world turbulance, in particular, most cases of financial modeling. ref Gaussian Self Affinity and Fractals by Benoit Mandelbrot, F.J. Damerau, M. Frame, and K. McCamy 2001 ISBN 0387989935 page 20 ref ref The effortless economy of science? by Philip Mirowski 2004 ISBN 0822333228 page 255 ref Intuitively speaking, Mandelbrot argued that the traditional ... forms of randomness that can be used to model extreme changes in risk and randomness. He observed that randomness can become quite wild if the requirements regarding finite mean and variance are abandoned. Wild randomness corresponds to situations in which a single observation, or a particular ... Mandelbrot1977 Proper mild randomness Borderline mild randomness Slow randomness with finite delocalized moments Slow randomness with finite and localized moments Pre wild randomness Wild randomness Extreme randomness Traditional normal distributions are at the mild end of the scale within this categorization ... just ten trading days represent 63 per cent of the returns of the past 50 years. Wild randomness has .... 422 p. ISBN 978 1 4020 4369 7 ref See also History of randomness Random sequence References Reflist Category Fractals Category Probability theory Category Randomness ... more details
In the field of information retrieval , divergence from randomness is one type of probabilistic model. External links http ir.dcs.gla.ac.uk terrier doc dfr description.html Terrier s DFR Web page http ir.dcs.gla.ac.uk wiki DivergenceFromRandomness Glasgow IR group Wiki DFR page Category Information retrieval Category Probabilistic models comp sci stub ... more details
Expert subject statistics date June 2009 Complete spatial randomness CSR describes a point process whereby point events occur within a given study area in a completely random fashion. Such a process is often modeled using only one parameter, i.e. the density of points, math rho math within the defined area. This is also called a spatial Poisson process . Data in the form of a set of points, irregularly distributed within a region of space, arise in many different contexts examples include locations of trees in a forest, of nests of birds, of nuclei in tissue, of ill people in a population at risk. We call any such data set a spatial point pattern and refer to the locations as events, to distinguish these from arbitrary points of the region in question. Model The hypothesis of complete spatial randomness for a spatial point pattern asserts that the number of events in any region follows a Poisson distribution with given mean count per uniform subdivision. The intensity of events does not vary over the plane. This implies that there are no interactions amongst the events. For example, the independence assumption would be violated if the existence of one event either encouraged or inhibited the occurrence of other events in the neighborhood. The study CSR is essential for the comparison of measured point data from experimental sources. As a statistical testing method, the test for CSR has many applications in the social sciences and in astronomical examinations. ref cite web url http www.google.com url?sa U&start 18&q http www.galaxy.gmu.edu interface I02 I2002Proceedings HauckSteven HauckSteven.presentation.ppt&ei j7WOScSuNpLSkAW6hbTGDA&usg AFQjCNEo99S WCn5xQ1Mc06UK1vH5Wd2Sw title Statistics on Venus Craters and Catastrophes ref Distribution The probability of finding exactly ... Distance Tests of Randomness in Spatial Point Processes Diggle, P. J. 2003 Statistical Analysis ... data analysis Category Point processes Category Statistical randomness Category Spatial processes ... more details
Lawlessness may be lack of law , in any of the various senses of that word see also law disambiguation chaos randomness antinomianism anomie anarchy anarchism . disambig ... more details
Extractor may refer to Extractor firearms Extractor mathematics Extractor screws , a tool used to removed broken screws Randomness extractor Soxhlet extractor disambig ... more details
Random number may refer to A number generated for or part of a set exhibiting statistical randomness . A random sequence obtained from a stochastic process. An algorithmically random sequence in algorithmic information theory. The output of a random number generation random number generator . The least random number 17 , according to the Hacker s Jargon File . See also Category Randomness mathdab id Angka acak ms Nombor rawak sr ... more details
avoids a definition of a random sequence. ref Inevitable Randomness in Discrete Mathematics by J zsef ... sequences assuming some definition of randomness. The Bourbaki Bourbaki school considered ... of the first mathematicians to formally address randomness in 1909. ref E. Borel, Les probabilites denombrables ... Richard von Mises gave the first definition of algorithmic randomness, which was inspired by the law ... is often called Mises Church randomness . Modern approaches In the mid 1960s, A. N. Kolmogorov ... Loveland randomness . But this method was considered too weak by Alexander Shen who showed that there is a Kolmogorov Loveland stochastic sequence which does not conform to the general notion of randomness ... notion of algorithmic randomness . His original definition involved measure theory, but it was later ... have now emerged ref R. Downey, Some Recent Progress in Algorithmic Randomness in Mathematical foundations ... Chaitin . For finite random sequences, Kolmogorov defined the randomness as the entropy, i.e. Kolmogorov ... page 176 ref See also Randomness History of randomness Random number generator Seven states of randomness Statistical randomness References Sergio B. Volchan What Is a Random Sequence? The American ... RES RANDTEST.HTM vonNeumann63 Randomness tests by Terry Ritter DEFAULTSORT Random Sequence Category Randomness Category Sequences and series Category Statistical randomness cs N hodn sekvence ... more details
An math N,M,D,K, epsilon math extractor is a bipartite graph with math N math nodes on the left and math M math nodes on the right such that each node on the left has math D math neighbors on the right , which has the added property that for any subset math A math of the left vertices of size at least math K math , the distribution on right vertices obtained by choosing a random node in math A math and then following a random graph theory edge to get a node x on the right side is math epsilon math close to the uniform distribution in terms of total variation distance . A disperser is a related graph. An equivalent way to view an extractor is as a bivariate function math E N times D rightarrow M math in the natural way. With this view it turns out that the extractor property is equivalent to for any source of randomness math X math that gives math n math bit s with min entropy math log K math , the distribution math E X,U D math is math epsilon math close to math U M math , where math U T math denotes the uniform distribution on math T math . Extractors are interesting when they can be constructed with small math K,D, epsilon math relative to math N math and math M math is as close to math KD math the total randomness in the input sources as possible. Extractor functions were originally researched as a way to extract randomness from weakly random sources. See randomness extractor . Using the probabilistic method it is easy to show that extractor graphs with really good parameters exist. The challenge is to find explicit or polynomial time computable examples of such graphs with good parameters. Algorithms that compute extractor and disperser graphs have found many applications in computer science . References Ronen Shaltiel, http www.cs.haifa.ac.il ronen online papers survey.ps Recent developments in extractors a survey Category Graph families Category Pseudorandomness Category Theoretical computer science ... more details
Drunkard s Walk may refer to Drunkard s Walk novel Drunkard s Walk novel , a 1960 science fiction novel by Frederik Pohl Drunkard s walk, a patchwork pattern made up of squares of fabric with a quarter circle of contrasting fabric in one corner Drunkard s walk, a type of a random walk in which termination conditions lead to a biased ending state The Drunkard s Walk How Randomness Rules Our Lives , a non fiction book by Leonard Mlodinow . disambig ... more details
time to algorithms which may ask questions of an oracle, there are different notions of randomness. The most common of these is known as Martin L f randomness or 1 randomness , but stronger and weaker forms of randomness also exist. The term random used to refer to a sequence without clarification ... von Mises had attempted to formalize the notion of a randomness test test for randomness in order to define a random sequence as one that passed all tests for randomness however, the precise notion of a randomness ... define the notion of a test for randomness. This contrasts with the idea of randomness in probability ... L f randomness has since been shown to admit many equivalent characterizations in terms of data compression compression , randomness tests, and gambling that bear little outward resemblance to the original ... ought to have random sequences should be incompressible, they should pass statistical tests for randomness ... definitions of Martin L f randomness, and the stability of these definitions under different models of computation, give evidence that Martin L f randomness is a fundamental property of mathematics and not an accident of Martin L f s particular model. The thesis that the definition of Martin L f randomness correctly captures the intuitive notion of randomness has been called the Martin L f Chaitin ... books?id EDoXdoz qYQC&pg PA145&source gbs toc r&cad 0 0 Randomness, Unpredictability and Absence ... . This means that all effective tests for randomness that is, constructive null covers are, in a sense, subsumed by this universal test for randomness, since any sequence that passes this single test for randomness will pass all tests for randomness. Martin L f 1966 There is a universal constructive ... degree . Relative randomness As each of the equivalent definitions of a Martin L f random sequence ... problem . An important result relating to relative randomness is Michiel van Lambalgen van Lambalgen ... relative to B if and only if B is random relative to A . Stronger than Martin L f randomness ... more details
wiktionary random Random can refer to Randomness , the property of lacking any sort of order Science and technology Random number Random variable dev random , a Unix device file See also Category Randomness Places Random Lake , Wisconsin Random Island , Canada The former name of Brighton, Vermont Music Random musician Random band Random Lady Sovereign song , a song by Lady Sovereign Random , a song by Gary Numan released as a bonus track on his album The Pleasure Principle Gary Numan album The Pleasure Principle Random , a song from the 311 album 311 album by the band 311 Other Robert Random , a Canadian actor Random comics , a fictional character in the Marvel Universe Random, Prince of The Chronicles of Amber Amber , in the novels of Roger Zelazny Random Dent Random Frequent Flyer Dent , the daughter of Arthur Dent in the Hitchhikers Guide book Mostly Harmless Random House , Book Publisher Random refers to things in no sense or order See also Non sequitur , for which random is a common ally with today s youth. disambig pt Random ... more details
Non free use rationale Article Accordion solitaire Description A game of Accordion screenshot from Solitaire Deluxe by Interplay . Source The game Solitaire Deluxe by Interplay . Portion This entire screenshot showing the initial layout of the game. Low resolution It is of low resolution. Purpose This screenshot is used only to illustrate the initial layout of the game in which, realistically, a few cards are layout as opposed to image Accordion v2.jpg , which has all cards are laid out . Replaceability This screenshot cannot be replicated under free conditions, especially because of the randomness of the card shuffle. other information Non free game screenshot ... more details
The Ljung Box test named for Greta M. Ljung and George E. P. Box is a type of statistical test of whether any of a group of autocorrelation s of a time series are different from zero. Instead of testing randomness at each distinct lag , it tests the overall randomness based on a number of lags, and is therefore a portmanteau test . Formal definition The Ljung Box test test can be defined as follows. H sub 0 sub The data is independently distributed i.e. the correlations in the population from which the sample is taken are 0, so that any observed correlations in the data result from randomness of the sampling process . H sub a sub The data is not independently distributed. The test statistic is math Q n left n 2 right sum k 1 h frac hat rho 2 k n k math where n is the sample size, math hat rho k math is the sample autocorrelation at lag k , and h is the number of lags being tested. For significance level , the critical region for rejection of the hypothesis of randomness is math Q chi 1 alpha,h 2 math where math chi 1 alpha,h 2 math is the quantile of the chi square distribution with h degrees of freedom. The Ljung Box test is commonly used in autoregressive integrated moving average ARIMA modeling. Note that it is applied to the errors and residuals in statistics residuals of a fitted ARIMA model, not the original series. See also the general term Portmanteau test the obsolete Box Pierce test the Wald Wolfowitz runs test References cite journal doi 10.1093 biomet 65.2.297 title On a Measure of a Lack of Fit in Time Series Models author G. M. Ljung coauthors G. E. P. Box journal Biometrika year 1978 volume 65 pages 297 303 cite book title Introduction to Time Series and Forecasting edition 2nd. author Peter Brockwell coauthors Richard Davis publisher Springer year 2002 page 36 isbn 0387947191 NIST PD Category Statistical tests Category Time series analysis ja Ljung Box ru Q uk ... more details
Unexplained means not explain ed and may refer to Unexplained EP , a 1992 rock music album by EMF Television The Unexplained , a 1990s documentary television series Unexplained Mysteries , a 2003 documentary television series Unexplained Canada , a 2006 documentary television series Unexplained , a 2008 television series examining paranormal events, with Tony Robinson Channel 4 See also Medically unexplained physical symptoms Sudden unexplained death syndrome Unexplained variation randomness variance, in statistics see Explained variation Fraction of variance unexplained Fort Prophet of the Unexplained , a 2002 comic book Unexplained disappearances The Unexplained magazine disamb Category Paranormal ... more details
refer Cosworth DFR , a 1987 race car engine Dean Forest Railway A heritage railway in Gloucestershire, England Dihydroflavonol 4 reductase , an enzyme in the lignin metabolic pathway Direction Forward Routing, an List of ad hoc routing protocols ad hoc routing protocol Divergence from randomness model , a probabilistic information retrieval model. Document Filing and Retrieval, an OSI protocols OSI protocol Dounreay Fast Reactor , the second reactor to achieve criticality in Dounreay, Scotland Digital Fault Recorder , measurement instrument recording power line Fault power engineering faults in electrical substation electrical substations disambiguation de DFR ... more details
math s bar Y s sqrt N math valid? Importance Randomness along with fixed model, fixed variation ... processes. The randomness assumption is critically important for the following three reasons br Most standard statistical test s depend on randomness. The validity of the test conclusions is directly linked to the validity of the randomness assumption. Many commonly used statistical formulae depend on the randomness assumption, the most common formula being the formula for determining the standard .... Although heavily used, the results from using this formula are of no value unless the randomness ... 1. If the correlogram is being used to test for randomness i.e., there is no time dependence in the data ... more details
In statistical mechanics , the metastate is a probability measure on the space of all thermodynamic states for a system with quenched randomness. The term metastate, in this context, was first used in ref name NS96 Newman, C. M. Stein, D. L. 1996 , Spatial inhomogeneity and thermodynamic chaos , Phys. Rev. Lett., Vol. 76 4821 4824. ref . Two different versions have been proposed 1 The Aizenman Wehr construction, a canonical ensemble approach, constructs the metastate through an ensemble of states obtained by varying the random parameters in the Molecular Hamiltonian Hamiltonian outside of the volume being considered ref name AW90 Aizenman, M. Wehr, J. 1990 , Rounding effects of quenched randomness on first order phase transitions , Commun. Math. Phys., Vol. 130 489 528. ref . 2 The Newman Stein metastate, a microcanonical ensemble approach, constructs an empirical average from a deterministic i.e., chosen independently of the randomness subsequence of finite volume Gibbs measure Gibbs distributions ref name NS96 ref name NS97 Newman, C. M. Stein, D. L. 1997 , Metastate approach to thermodynamic chaos , Phys. Rev. E, Vol. 55 5194 5211. ref ref name NSBerlin Newman, C. M. Stein, D. L. 1998 . Thermodynamic chaos and the structure of short range spin glasses. in A. Bovier and P. Picco. Mathematics of Spin Glasses and Neural Networks. Boston Birkhauser. pp. 243 247. ref . It was proved ref name NSBerlin for Euclidean lattices that there always exists a deterministic subsequence along which the Newman Stein and Aizenman Wehr constructions result in the same metastate. The metastate is especially useful in systems where deterministic sequences of volumes fail to converge to a thermodynamic state , and or there are many competing observable thermodynamic states. References do not reduce font sizes, the Wiki style sheets control presentation references Category Statistical mechanics Category Condensed matter physics ... more details
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