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Darwin–Fowler Method
In statistical mechanics, the Darwin–Fowler method is used for deriving the distribution functions with mean probability. It was developed by Charles Galton Darwin and Ralph H. Fowler in 1922–1923. Distribution functions are used in statistical physics to estimate the mean number of particles occupying an energy level (hence also called occupation numbers). These distributions are mostly derived as those numbers for which the system under consideration is in its state of maximum probability. But one really requires average numbers. These average numbers can be obtained by the Darwin–Fowler method. Of course, for systems in the thermodynamic limit (large number of particles), as in statistical mechanics, the results are the same as with maximization. Darwin–Fowler method In most texts on statistical mechanics the statistical distribution functions f in Maxwell–Boltzmann statistics, Bose–Einstein statistics, Fermi–Dirac statistics) are derived by determining those ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles. 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. This established the fields of statistical thermodynamics and statistical physics. The founding of the field of statistical mechanics is generally credited to three physicists: *Ludwig Boltzmann, who developed the fundamental interpretation of entropy in terms of a collection of microstates *James Clerk Maxwell, who developed models of probability distr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theory: the Schrödinger equation provides a way to calculate the wave function of a system and how it changes dynamically in time. In addition, he wrote many works on various aspects of physics: statistical mechanics and thermodynamics, physics of dielectrics, colour theory, electrodynamics, general relativity, and cosmology, and he made several attempts to construct a unified field theory. In his book ''What Is Life?'' Schrödinger addressed the problems of genetics, looking at the phenomenon of life from the point of view of physics. He also paid great attention to the philosophical aspects of science, ancient, and oriental philosophical concepts, ethics, and religion. He also wrote on philosophy and theoretical biology. In popular culture, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Method Of Steepest Descent
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals. The integral to be estimated is often of the form :\int_Cf(z)e^\,dz, where ''C'' is a contour, and λ is large. One version of the method of steepest descent deforms the contour of integration ''C'' into a new path integration ''C′'' so that the following conditions hold: # ''C′'' passes through one or more zeros of the derivative ''g''′(''z''), # the imaginary part of ''g''(''z'') is constant on ''C′''. The method of steepest descent was first published by , who used it to estimate Bessel functions and pointed out that it occurred in the u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saddle Point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and at a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function f(x,y) = x^2 + y^3 has a critical point at (0, 0) that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the y-direction. The name derives from the fact that the prototypical example in two dimensions is a surface that ''curves up'' in one direction, and ''curves down'' in a different direction, resembling a riding saddle or a mountain pass between two peaks forming a landform saddle. In te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors. He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had a great influence over his contemporaries and successors; Hans Freudenthal stated: "More concepts and theorems have been named for Cauchy than for any other mathematician (in Elasticity (physics), elasticity alone there are sixteen concepts and theorems named for Cauchy)." Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residue Theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. From a geometrical perspective, it can be seen as a special case of the generalized Stokes' theorem. Statement The statement is as follows: Let be a simply connected open subset of the complex plane containing a finite list of points , , and a function defined and holomorphic on . Let be a closed rectifiable curve in , and denote the winding number of around by . The line integral of around is equal to times the sum of residues of at the points, each counted as many times as winds around the point: \oint_\gamma f(z)\, dz = 2\pi i \sum_^n \operatorname(\gamma, a_k) \operatorname( f, a_k ). If is a positively oriented simple closed curve, if i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bose–Einstein Condensation
{{disambiguation ...
Bose–Einstein may refer to: * Bose–Einstein condensate ** Bose–Einstein condensation (network theory) * Bose–Einstein correlations * Bose–Einstein statistics In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harald J
Harald or Haraldr is the Old Norse form of the given name Harold. It may refer to: Medieval Kings of Denmark * Harald Bluetooth (935–985/986) Kings of Norway * Harald Fairhair (c. 850–c. 933) * Harald Greycloak (died 970) * Harald Hardrada (1015–1066) * Harald Gille (reigned 1130–1136) Grand Dukes of Kiev * Mstislav the Great (1076–1132), known as Harald in Norse sagas King of Mann and the Isles * Haraldr Óláfsson (died 1248) Earls of Orkney * Harald Haakonsson (died 1131) * Harald Maddadsson (–1206) * Harald Eiriksson Others * Hagrold (fl. 944–954), also known as Harald, Scandinavian chieftain in Normandy * Harald Grenske (10th century), petty king in Vestfold in Norway * Harald Klak (–), king in Jutland * Harald Wartooth, legendary king of Sweden, Denmark and Norway * Harald the Younger, 9th-century Viking leader Modern name Royalty * Harald V of Norway (born 1937), present King of Norway * Prince Harald of Denmark (1876–1949) Arts and entertainm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kerson Huang
Kerson Huang (; 15 March 1928 – 1 September 2016) was a Chinese American theoretical physicist and translator. Huang was born in Nanning, China and grew up in Manila, Philippines. He earned a B.S. and a Ph.D. in physics from the Massachusetts Institute of Technology (MIT) in 1950 and 1953, respectively. He served as an instructor at MIT from 1953 to 1955, and subsequently spent two years as a fellow at the Institute for Advanced Study. After returning to the MIT faculty in 1957, Huang became an authority on statistical physics, and worked on Bose–Einstein condensation and quantum field theory. At MIT, he had many PhD students in theoretical physics including Raymond G. Vickson who became a professor in Operations Research at the University of Waterloo. After retiring in 1999, he wrote on biophysics and was also a visiting professor at Nanyang Technological University in Singapore. Huang was best known to Chinese readers as the translator of the ''Rubaiyat of Omar Khayyam''; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edward A
Edward is an English language, English given name. It is derived from the Old English, Anglo-Saxon name ''Ēadweard'', composed of the elements ''wikt:ead#Old English, ēad'' "wealth, fortune; prosperous" and ''wikt:weard#Old English, weard'' "guardian, protector”. History The name Edward was very popular in Anglo-Saxon England, but the rule of the House of Normandy, Norman and House of Plantagenet, Plantagenet dynasties had effectively ended its use amongst the upper classes. The popularity of the name was revived when Henry III of England, Henry III named his firstborn son, the future Edward I of England, Edward I, as part of his efforts to promote a cult around Edward the Confessor, for whom Henry had a deep admiration. Variant forms The name has been adopted in the Iberian Peninsula#Modern Iberia, Iberian peninsula since the 15th century, due to Edward, King of Portugal, whose mother was English. The Spanish/Portuguese forms of the name are Eduardo and Duarte (name), Duarte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermi–Dirac Distribution
Fermi–Dirac may refer to: * Fermi–Dirac statistics or Fermi–Dirac distribution * Fermi–Dirac integral (other) ** Complete Fermi–Dirac integral ** Incomplete Fermi–Dirac integral See also * Fermi (other) Enrico Fermi (1901–1954) was an Italian physicist who created the world's first nuclear reactor. Fermi or Enrico Fermi may also refer to: * Fermi (crater), a large lunar impact crater * Fermi (microarchitecture), a microarchitecture developed by ... * Dirac (other) {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distribution Function (physics)
:''This article describes the ''distribution function'' as used in physics. You may be looking for the related mathematical concepts of cumulative distribution function or probability density function.'' In molecular kinetic theory in physics, a system's distribution function is a function of seven variables, f(x,y,z,t;v_x,v_y,v_z), which gives the number of particles per unit volume in single-particle phase space. It is the number of particles per unit volume having approximately the velocity \mathbf=(v_x,v_y,v_z) near the position \mathbf=(x,y,z) and time t. The usual normalization of the distribution function is :n(x,y,z,t) = \int f \,dv_x \,dv_y \,dv_z, :N(t) = \int n \,dx \,dy \,dz, where, ''N'' is the total number of particles, and ''n'' is the number density of particles – the number of particles per unit volume, or the density divided by the mass of individual particles. A distribution function may be specialised with respect to a particular set of dimensions. E.g. t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
