78-424: Mandelbrot may refer to: Benoit Mandelbrot (1924–2010), a mathematician associated with fractal geometry Mandelbrot set , a fractal popularized by Benoit Mandelbrot Mandelbrot Competition , a mathematics competition Mandelbrot (cookie) , dessert associated with Eastern European Jews Szolem Mandelbrojt , a Polish-French mathematician Topics referred to by
156-469: A "fractalist" and is recognized for his contribution to the field of fractal geometry , which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity " in nature. In 1936, at the age of 11, Mandelbrot and his family emigrated from Warsaw , Poland, to France. After World War II ended, Mandelbrot studied mathematics, graduating from universities in Paris and in
234-459: A "mess", or "chaotic", such as clouds or shorelines, actually had a "degree of order". His math- and geometry-centered research included contributions to such fields as statistical physics , meteorology , hydrology , geomorphology , anatomy , taxonomy , neurology , linguistics , information technology , computer graphics , economics , geology , medicine , physical cosmology , engineering , chaos theory , econophysics , metallurgy , and
312-909: A Big Bang, but would allow for a dark sky even if the Big Bang had not occurred. Mandelbrot's awards include the Wolf Prize for Physics in 1993, the Lewis Fry Richardson Prize of the European Geophysical Society in 2000, the Japan Prize in 2003, and the Einstein Lectureship of the American Mathematical Society in 2006. The small asteroid 27500 Mandelbrot was named in his honor. In November 1990, he
390-615: A Scientific Maverick , was published posthumously in 2012. Benedykt Mandelbrot was born in a Lithuanian Jewish family, in Warsaw during the Second Polish Republic . His father made his living trading clothing; his mother was a dental surgeon. During his first two school years, he was tutored privately by an uncle who despised rote learning : "Most of my time was spent playing chess, reading maps and learning how to open my eyes to everything around me." In 1936, when he
468-575: A close friend from Paris, Zina Morhange , a physician in a nearby county seat. Simply to eliminate the competition, another physician denounced her ... We escaped this fate. Who knows why? In 1944, Mandelbrot returned to Paris, studied at the Lycée du Parc in Lyon , and in 1945 to 1947 attended the École Polytechnique , where he studied under Gaston Julia and Paul Lévy . From 1947 to 1949 he studied at California Institute of Technology, where he earned
546-601: A few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level. Many physical phenomena involve quasi-thermodynamic processes out of equilibrium, for example: All of these processes occur over time with characteristic rates. These rates are important in engineering. The field of non-equilibrium statistical mechanics
624-455: A finite volume. These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics. For systems containing many particles (the thermodynamic limit ), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used. The Gibbs theorem about equivalence of ensembles
702-812: A master's degree in aeronautics. Returning to France, he obtained his PhD degree in Mathematical Sciences at the University of Paris in 1952. From 1949 to 1958, Mandelbrot was a staff member at the Centre National de la Recherche Scientifique . During this time he spent a year at the Institute for Advanced Study in Princeton, New Jersey , where he was sponsored by John von Neumann . In 1955 he married Aliette Kagan and moved to Geneva, Switzerland (to collaborate with Jean Piaget at
780-451: A much stronger and more fundamental idea—put simply, that there are some geometric shapes, which he called "fractals", that are equally "rough" at all scales. No matter how close you look, they never get simpler, much as the section of a rocky coastline you can see at your feet looks just as jagged as the stretch you can see from space. Wolfram briefly describes fractals as a form of geometric repetition, "in which smaller and smaller copies of
858-462: A new explanation of Olbers' paradox (the "dark night sky" riddle), demonstrating the consequences of fractal theory as a sufficient, but not necessary , resolution of the paradox. He postulated that if the stars in the universe were fractally distributed (for example, like Cantor dust ), it would not be necessary to rely on the Big Bang theory to explain the paradox. His model would not rule out
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#1732780038521936-402: A pattern are successively nested inside each other, so that the same intricate shapes appear no matter how much you zoom in to the whole. Fern leaves and Romanesque broccoli are two examples from nature." He points out an unexpected conclusion: One might have thought that such a simple and fundamental form of regularity would have been studied for hundreds, if not thousands, of years. But it
1014-424: A role in materials science, nuclear physics, astrophysics, chemistry, biology and medicine (e.g. study of the spread of infectious diseases). Analytical and computational techniques derived from statistical physics of disordered systems, can be extended to large-scale problems, including machine learning, e.g., to analyze the weight space of deep neural networks . Statistical physics is thus finding applications in
1092-407: A state with a balance of forces that has ceased to evolve.) The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems. The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics)
1170-482: A surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion. The founding of the field of statistical mechanics is generally credited to three physicists: In 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius , Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave
1248-499: Is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.) In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent,
1326-464: Is different from Wikidata All article disambiguation pages All disambiguation pages Benoit Mandelbrot Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness " of physical phenomena and "the uncontrolled element in life". He referred to himself as
1404-513: Is firmly entrenched. Shortly before his death, Gibbs published in 1902 Elementary Principles in Statistical Mechanics , a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous. Gibbs' methods were initially derived in the framework classical mechanics , however they were of such generality that they were found to adapt easily to
1482-417: Is however a disconnect between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and
1560-713: Is indeed one of the most astonishing discoveries in the entire history of mathematics. Who could have dreamed that such an incredibly simple equation could have generated images of literally infinite complexity?" Clarke also notes an "odd coincidence": the name Mandelbrot, and the word " mandala "—for a religious symbol—which I'm sure is a pure coincidence, but indeed the Mandelbrot set does seem to contain an enormous number of mandalas. In 1982, Mandelbrot expanded and updated his ideas in The Fractal Geometry of Nature . This influential work brought fractals into
1638-643: Is not smooth, nor does lightning travel in a straight line. —Mandelbrot, in his introduction to The Fractal Geometry of Nature Mandelbrot has been called an artist, and a visionary and a maverick. His informal and passionate style of writing and his emphasis on visual and geometric intuition (supported by the inclusion of numerous illustrations) made The Fractal Geometry of Nature accessible to non-specialists. The book sparked widespread popular interest in fractals and contributed to chaos theory and other fields of science and mathematics. Mandelbrot also put his ideas to work in cosmology. He offered in 1974
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#17327800385211716-449: Is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics. Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections. One approach to non-equilibrium statistical mechanics
1794-469: Is primarily concerned with thermodynamic equilibrium , statistical mechanics has been applied in non-equilibrium statistical mechanics to the issues of microscopically modeling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions and flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study
1872-400: Is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium . Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. (By contrast, mechanical equilibrium is
1950-431: Is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium , and the microscopic behaviours and motions occurring inside the material. Whereas statistical mechanics proper involves dynamics, here
2028-411: Is to incorporate stochastic (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes , a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within
2106-417: Is usual for probabilities, the ensemble can be interpreted in different ways: These two meanings are equivalent for many purposes, and will be used interchangeably in this article. However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in
2184-650: The Collège-lycée Jacques-Decour ) in Paris until the start of World War II , when his family moved to Tulle , France. He was helped by Rabbi David Feuerwerker , the Rabbi of Brive-la-Gaillarde , to continue his studies. Much of France was occupied by the Nazis at the time, and Mandelbrot recalls this period: Our constant fear was that a sufficiently determined foe might report us to an authority and we would be sent to our deaths. This happened to
2262-744: The complex plane . Building on previous work by Gaston Julia and Pierre Fatou , Mandelbrot used a computer to plot images of the Julia sets. While investigating the topology of these Julia sets, he studied the Mandelbrot set which was introduced by him in 1979. In 1975, Mandelbrot coined the term fractal to describe these structures and first published his ideas in the French book Les Objets Fractals: Forme, Hasard et Dimension , later translated in 1977 as Fractals: Form, Chance and Dimension . According to computer scientist and physicist Stephen Wolfram ,
2340-723: The social sciences . Toward the end of his career, he was Sterling Professor of Mathematical Sciences at Yale University , where he was the oldest professor in Yale's history to receive tenure. Mandelbrot also held positions at the Pacific Northwest National Laboratory , Université Lille Nord de France , Institute for Advanced Study and Centre National de la Recherche Scientifique . During his career, he received over 15 honorary doctorates and served on many science journals, along with winning numerous awards. His autobiography, The Fractalist: Memoir of
2418-432: The von Neumann equation . These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. These ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy
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2496-558: The "father of fractal geometry". Best-selling essayist-author Nassim Nicholas Taleb has remarked that Mandelbrot's book The (Mis)Behavior of Markets is in his opinion "The deepest and most realistic finance book ever published". Statistical physics In physics , statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics , its applications include many problems in
2574-560: The "interesting" information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped semiconductors (in transistors ), where the electrons are indeed analogous to a rarefied gas. Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations,
2652-432: The 1960s psychedelic art with forms hauntingly reminiscent of nature and the human body". He also saw himself as a "would-be Kepler", after the 17th-century scientist Johannes Kepler , who calculated and described the orbits of the planets. Mandelbrot, however, never felt he was inventing a new idea. He described his feelings in a documentary with science writer Arthur C. Clarke: Exploring this set I certainly never had
2730-663: The Greek root of its name [geo-] seemed to promise—truthful measurement, not only of cultivated fields along the Nile River but also of untamed Earth? In his paper " How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension ", published in Science in 1967, Mandelbrot discusses self-similar curves that have Hausdorff dimension that are examples of fractals , although Mandelbrot does not use this term in
2808-1116: The International Centre for Genetic Epistemology) and later to the Université Lille Nord de France . In 1958 the couple moved to the United States where Mandelbrot joined the research staff at the IBM Thomas J. Watson Research Center in Yorktown Heights, New York . He remained at IBM for 35 years, becoming an IBM Fellow, and later Fellow Emeritus . From 1951 onward, Mandelbrot worked on problems and published papers not only in mathematics but in applied fields such as information theory , economics, and fluid dynamics . Mandelbrot saw financial markets as an example of "wild randomness", characterized by concentration and long-range dependence. He developed several original approaches for modelling financial fluctuations. In his early work, he found that
2886-476: The United States and receiving a master's degree in aeronautics from the California Institute of Technology . He spent most of his career in both the United States and France, having dual French and American citizenship. In 1958, he began a 35-year career at IBM , where he became an IBM Fellow , and periodically took leaves of absence to teach at Harvard University . At Harvard, following
2964-459: The age of 85 in a hospice in Cambridge, Massachusetts , on 14 October 2010. Reacting to news of his death, mathematician Heinz-Otto Peitgen said: "[I]f we talk about impact inside mathematics, and applications in the sciences, he is one of the most important figures of the last fifty years." Chris Anderson , TED conference curator, described Mandelbrot as "an icon who changed how we see
3042-453: The area of medical diagnostics . Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems . In quantum mechanics, a statistical ensemble (probability distribution over possible quantum states ) is described by a density operator S , which is a non-negative, self-adjoint , trace-class operator of trace 1 on the Hilbert space H describing
3120-625: The attention is focussed on statistical equilibrium (steady state). Statistical equilibrium does not mean that the particles have stopped moving ( mechanical equilibrium ), rather, only that the ensemble is not evolving. A sufficient (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.). There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics. Additional postulates are necessary to motivate why
3198-405: The book was a "breakthrough" for Mandelbrot, who until then would typically "apply fairly straightforward mathematics ... to areas that had barely seen the light of serious mathematics before". Wolfram adds that as a result of this new research, he was no longer a "wandering scientist", and later called him "the father of fractals": Mandelbrot ended up doing a great piece of science and identifying
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3276-650: The characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities. There are some cases which allow exact solutions. Although some problems in statistical physics can be solved analytically using approximations and expansions, most current research utilizes
3354-631: The conductance of an electronic system is the use of the Green–Kubo relations, with the inclusion of stochastic dephasing by interactions between various electrons by use of the Keldysh method. The ensemble formalism can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in: Statistical physics explains and quantitatively describes superconductivity , superfluidity , turbulence , collective phenomena in solids and plasma , and
3432-536: The ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state. One special class of ensemble
3510-524: The ensemble for a given system should have one form or another. A common approach found in many textbooks is to take the equal a priori probability postulate . This postulate states that The equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate: Other fundamental postulates for statistical mechanics have also been proposed. For example, recent studies shows that
3588-448: The feeling of invention. I never had the feeling that my imagination was rich enough to invent all those extraordinary things on discovering them. They were there, even though nobody had seen them before. It's marvelous, a very simple formula explains all these very complicated things. So the goal of science is starting with a mess, and explaining it with a simple formula, a kind of dream of science. According to Clarke, "the Mandelbrot set
3666-629: The fields of physics, biology , chemistry , neuroscience , computer science , information theory and sociology . Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics arose out of the development of classical thermodynamics , a field for which it was successful in explaining macroscopic physical properties—such as temperature , pressure , and heat capacity —in terms of microscopic parameters that fluctuate about average values and are characterized by probability distributions . While classical thermodynamics
3744-440: The first time and turned them into essential tools for the long-stalled effort to extend the scope of science to explaining non-smooth, "rough" objects in the real world. His methods of research were both old and new: The form of geometry I increasingly favored is the oldest, most concrete, and most inclusive, specifically empowered by the eye and helped by the hand and, today, also by the computer ... bringing an element of unity to
3822-410: The fluctuation–dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics. A few of the theoretical tools used to make this connection include: An advanced approach uses a combination of stochastic methods and linear response theory . As an example, one approach to compute quantum coherence effects ( weak localization , conductance fluctuations ) in
3900-518: The large processing power of modern computers to simulate or approximate solutions. A common approach to statistical problems is to use a Monte Carlo simulation to yield insight into the properties of a complex system . Monte Carlo methods are important in computational physics , physical chemistry , and related fields, and have diverse applications including medical physics , where they are used to model radiation transport for radiation dosimetry calculations. The Monte Carlo method examines just
3978-409: The later quantum mechanics , and still form the foundation of statistical mechanics to this day. In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics . For both types of mechanics, the standard mathematical approach is to consider two concepts: Using these two concepts, the state at any other time, past or future, can in principle be calculated. There
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#17327800385214056-471: The mainstream of professional and popular mathematics, as well as silencing critics, who had dismissed fractals as " program artifacts ". Mandelbrot left IBM in 1987, after 35 years and 12 days, when IBM decided to end pure research in his division. He joined the Department of Mathematics at Yale , and obtained his first tenured post in 1999, at the age of 75. At the time of his retirement in 2005, he
4134-679: The paper, as he did not coin it until 1975. The paper is one of Mandelbrot's first publications on the topic of fractals. Mandelbrot emphasized the use of fractals as realistic and useful models for describing many "rough" phenomena in the real world. He concluded that "real roughness is often fractal and can be measured." Although Mandelbrot coined the term "fractal", some of the mathematical objects he presented in The Fractal Geometry of Nature had been previously described by other mathematicians. Before Mandelbrot, however, they were regarded as isolated curiosities with unnatural and non-intuitive properties. Mandelbrot brought these objects together for
4212-427: The practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in. Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble , which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of
4290-407: The price changes in financial markets did not follow a Gaussian distribution , but rather Lévy stable distributions having infinite variance . He found, for example, that cotton prices followed a Lévy stable distribution with parameter α equal to 1.7 rather than 2 as in a Gaussian distribution. "Stable" distributions have the property that the sum of many instances of a random variable follows
4368-570: The proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H -theorem . The term "statistical mechanics" was coined by the American mathematical physicist J. Willard Gibbs in 1884. According to Gibbs, the term "statistical", in the context of mechanics, i.e. statistical mechanics,
4446-477: The proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics. Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium. Five years later, in 1864, Ludwig Boltzmann , a young student in Vienna, came across Maxwell's paper and spent much of his life developing
4524-456: The publication of his study of U.S. commodity markets in relation to cotton futures, he taught economics and applied sciences. Because of his access to IBM's computers, Mandelbrot was one of the first to use computer graphics to create and display fractal geometric images, leading to his discovery of the Mandelbrot set in 1980. He showed how visual complexity can be created from simple rules. He said that things typically considered to be "rough",
4602-583: The publication of the first comprehensive papers on scaling law in finance. This law shows similar properties at different time scales, confirming Mandelbrot's insight of the fractal nature of market microstructure. Mandelbrot's own research in this area is presented in his books Fractals and Scaling in Finance and The (Mis)behavior of Markets . As a visiting professor at Harvard University , Mandelbrot began to study mathematical objects called Julia sets that were invariant under certain transformations of
4680-432: The response can be analysed in linear response theory . A remarkable result, as formalized by the fluctuation–dissipation theorem , is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in
4758-623: The same distribution but with a larger scale parameter . The latter work from the early 60s was done with daily data of cotton prices from 1900, long before he introduced the word 'fractal'. In later years, after the concept of fractals had matured, the study of financial markets in the context of fractals became possible only after the availability of high frequency data in finance. In the late 1980s, Mandelbrot used intra-daily tick data supplied by Olsen & Associates in Zurich to apply fractal theory to market microstructure. This cooperation led to
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#17327800385214836-595: The same term [REDACTED] This disambiguation page lists articles associated with the title Mandelbrot . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Mandelbrot&oldid=1176255712 " Categories : Disambiguation pages Disambiguation pages with surname-holder lists Surnames of Jewish origin Yiddish-language surnames Hidden categories: Short description
4914-408: The same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium. This provides an indirect avenue for obtaining numbers such as ohmic conductivity and thermal conductivity by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations,
4992-413: The simplest non-equilibrium situation of a steady state current flow in a system of many particles. In 1738, Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases . In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on
5070-445: The size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system. Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from
5148-419: The structural features of liquid . It underlies the modern astrophysics . In solid state physics, statistical physics aids the study of liquid crystals , phase transitions , and critical phenomena . Many experimental studies of matter are entirely based on the statistical description of a system. These include the scattering of cold neutrons , X-ray , visible light , and more. Statistical physics also plays
5226-460: The subject further. Statistical mechanics was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory . Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem , transport theory , thermal equilibrium , the equation of state of gases, and similar subjects, occupy about 2,000 pages in
5304-462: The system, or to correlations between the system and environment. These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier. The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where
5382-413: The system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinate axes. In quantum statistical mechanics, the ensemble is a probability distribution over pure states and can be compactly summarized as a density matrix . As
5460-403: The theory of statistical mechanics can be built without the equal a priori probability postulate. One such formalism is based on the fundamental thermodynamic relation together with the following set of postulates: where the third postulate can be replaced by the following: There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside
5538-488: The world". Nicolas Sarkozy , President of France at the time of Mandelbrot's death, said Mandelbrot had "a powerful, original mind that never shied away from innovating and shattering preconceived notions [... h]is work, developed entirely outside mainstream research, led to modern information theory." Mandelbrot's obituary in The Economist points out his fame as "celebrity beyond the academy" and lauds him as
5616-485: The worlds of knowing and feeling ... and, unwittingly, as a bonus, for the purpose of creating beauty. Fractals are also found in human pursuits, such as music, painting, architecture, and in the financial field. Mandelbrot believed that fractals, far from being unnatural, were in many ways more intuitive and natural than the artificially smooth objects of traditional Euclidean geometry : Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark
5694-514: Was 11, the family emigrated from Poland to France. The move, World War II , and the influence of his father's brother, the mathematician Szolem Mandelbrojt (who had moved to Paris around 1920), further prevented a standard education. "The fact that my parents, as economic and political refugees, joined Szolem in France saved our lives," he writes. Mandelbrot attended the Lycée Rollin (now
5772-502: Was Sterling Professor of Mathematical Sciences. Mandelbrot created the first-ever "theory of roughness", and he saw "roughness" in the shapes of mountains, coastlines and river basins ; the structures of plants, blood vessels and lungs ; the clustering of galaxies . His personal quest was to create some mathematical formula to measure the overall "roughness" of such objects in nature. He began by asking himself various kinds of questions related to nature: Can geometry deliver what
5850-446: Was developed into the theory of concentration of measure phenomenon, which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology. Important cases where the thermodynamic ensembles do not give identical results include: In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in
5928-496: Was first used by the Scottish physicist James Clerk Maxwell in 1871: "In dealing with masses of matter, while we do not perceive the individual molecules, we are compelled to adopt what I have described as the statistical method of calculation, and to abandon the strict dynamical method, in which we follow every motion by the calculus." "Probabilistic mechanics" might today seem a more appropriate term, but "statistical mechanics"
6006-633: Was made a Chevalier in France's Legion of Honour . In December 2005, Mandelbrot was appointed to the position of Battelle Fellow at the Pacific Northwest National Laboratory . Mandelbrot was promoted to an Officer of the Legion of Honour in January 2006. An honorary degree from Johns Hopkins University was bestowed on Mandelbrot in the May 2010 commencement exercises. A partial list of awards received by Mandelbrot: Mandelbrot died from pancreatic cancer at
6084-544: Was not. In fact, it rose to prominence only over the past 30 or so years—almost entirely through the efforts of one man, the mathematician Benoit Mandelbrot. Mandelbrot used the term "fractal" as it derived from the Latin word "fractus", defined as broken or shattered glass. Using the newly developed IBM computers at his disposal, Mandelbrot was able to create fractal images using graphics computer code, images that an interviewer described as looking like "the delirious exuberance of
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