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Paul Ehrenfest
Paul Ehrenfest (18 January 1880 – 25 September 1933) was an Austrian theoretical physicist, who made major contributions to the field of statistical mechanics and its relations with quantum mechanics, including the theory of phase transition and the Ehrenfest theorem. He bonded with Albert Einstein on a visit to Prague in 1912 and became a professor in Leiden, where he frequently hosted Einstein. Biography Paul Ehrenfest was born and grew up in Vienna to Jewish parents from Loštice in Moravia (now part of the Czech Republic). His parents, Sigmund Ehrenfest and Johanna Jellinek, ran a grocery store. Although the family was not overly religious, Paul studied Hebrew and the history of the Jewish people. Later, he always emphasized his Jewish roots. Ehrenfest excelled in grade school but did not do well at the Akademisches Gymnasium, his best subject being mathematics. After transferring to the Franz Josef Gymnasium, his marks improved. In 1899, he passed the final exams. He m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vienna
en, Viennese , iso_code = AT-9 , registration_plate = W , postal_code_type = Postal code , postal_code = , timezone = CET , utc_offset = +1 , timezone_DST = CEST , utc_offset_DST = +2 , blank_name = Vehicle registration , blank_info = W , blank1_name = GDP , blank1_info = € 96.5 billion (2020) , blank2_name = GDP per capita , blank2_info = € 50,400 (2020) , blank_name_sec1 = HDI (2019) , blank_info_sec1 = 0.947 · 1st of 9 , blank3_name = Seats in the Federal Council , blank3_info = , blank_name_sec2 = GeoTLD , blank_info_sec2 = .wien , website = , footnotes = , image_blank_emblem = Wien logo.svg , blank_emblem_size = Vienna ( ; german: Wien ; ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gregory Breit
Gregory Breit (russian: Григорий Альфредович Брейт-Шнайдер, ''Grigory Alfredovich Breit-Shneider''; July 14, 1899, Mykolaiv, Kherson Governorate – September 13, 1981, Salem, Oregon) was a Russian-born Jewish American physicist and professor at New York University (1929–1934), University of Wisconsin–Madison (1934–1947), Yale University (1947–1968), and University at Buffalo (1968–1973). In 1921, he was Paul Ehrenfest's assistant in Leiden University. Biography After completing his Ph.D. at age 22, he was from 1923 to 1924 an assistant professor at the University of Minnesota. In 1925, while at the Carnegie Institution of Washington, Breit joined with Merle Tuve in using a pulsed radio transmitter to determine the height of the ionosphere, a technique important later in radar development. Together with Eugene Wigner, Breit gave a description of particle resonant states with the relativistic Breit–Wigner distribution in 1929, and wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonradiation Condition
Classical nonradiation conditions define the conditions according to classical electromagnetism under which a distribution of accelerating charges will not emit electromagnetic radiation. According to the Larmor formula in classical electromagnetism, a single point charge under acceleration will emit electromagnetic radiation, i.e. light. In some classical electron models a distribution of charges can however be accelerated so that no radiation is emitted. The modern derivation of these nonradiation conditions by Hermann A. Haus is based on the Fourier components of the current produced by a moving point charge. It states that a distribution of accelerated charges will radiate if and only if it has Fourier components synchronous with waves traveling at the speed of light. History Finding a nonradiating model for the electron on an atom dominated the early work on atomic models. In a planetary model of the atom, the orbiting point electron would constantly accelerate towards the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ehrenfest–Tolman Effect
In general relativity, the Ehrenfest–Tolman effect (also known as the Tolman–Ehrenfest effect), created by Richard C. Tolman and Paul Ehrenfest, argues that temperature is not constant in space at thermal equilibrium, but varies with the spacetime curvature. Specifically, it depends on the spacetime metric. In a stationary spacetime with timelike Killing vector field \xi, the temperature T satisfies instead the Tolman-Ehrenfest relation: T\,, , \xi, , =\mathrm, where , , \xi, , =\sqrt is the norm of the timelike Killing vector field. This relationship leads to the concept of ''thermal time'' which has been considered as a possible basis for a fully general-relativistic thermodynamics. It has been shown that the Tolman–Ehrenfest effect can be derived by applying the equivalence principle In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Transition
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of matter: solid, liquid, and gas, and in rare cases, plasma. A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such as temperature or pressure. This can be a discontinuous change; for example, a liquid may become gas upon heating to its boiling point, resulting in an abrupt change in volume. The identification of the external conditions at which a transformation occurs defines the phase transition point. Types of phase transition At the phase transition point for a substance, for instance the boiling point, the two phases involved - liquid and vapor, have identic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultraviolet Catastrophe
The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of energy as wavelength decreased into the ultraviolet range. The term "ultraviolet catastrophe" was first used in 1911 by Paul Ehrenfest, but the concept originated with the 1900 statistical derivation of the Rayleigh–Jeans law. The phrase refers to the fact that the Rayleigh–Jeans law accurately predicts experimental results at radiative frequencies below 100 THz, but begins to diverge from empirical observations as these frequencies reach the ultraviolet region of the electromagnetic spectrum. Since the first use of this term, it has also been used for other predictions of a similar nature, as in quantum electrodynamics and such cases as ultraviolet divergence. Problem The Rayleigh-Jeans law is an approximation to the spectral radianc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation. Unlike vectors and tensors, a spinor transforms to its negative when the space is continuously rotated through a complete turn from 0° to 360° (see picture). This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms). It is also possible to associate a substantially similar notion of spinor to Minkowski space, in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ehrenfest Model
The Ehrenfest model (or dog–flea model) of diffusion was proposed by Tatiana and Paul Ehrenfest to explain the second law of thermodynamics. The model considers ''N'' particles in two containers. Particles independently change container at a rate ''λ''. If ''X''(''t'') = ''i'' is defined to be the number of particles in one container at time ''t'', then it is a birth–death process with transition rates * q_ = i\, \lambda for ''i'' = 1, 2, ..., ''N'' * q_ = (N-i\,) \lambda for ''i'' = 0, 1, ..., ''N'' – 1 and equilibrium distribution \pi_i = 2^ \tbinom Ni. Mark Kac proved in 1947 that if the initial system state is not equilibrium, then the entropy, given by :H(t) = -\sum_ P(X(t)=i) \log \left( \frac\right), is monotonically increasing (H-theorem). This is a consequence of the convergence to the equilibrium distribution. Interpretation of results Consider that at the beginning all the particles are in one of the containers. It is expected that over time the num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ehrenfest Equations
Ehrenfest equations (named after Paul Ehrenfest) are equations which describe changes in specific heat capacity and derivatives of specific volume in second-order phase transitions. The Clausius–Clapeyron relation does not make sense for second-order phase transitions,Sivuhin D.V. General physics course. V.2. ''Thermodynamics and molecular physics''. 2005 as both specific entropy and specific volume do not change in second-order phase transitions. Quantitative consideration Ehrenfest equations are the consequence of continuity of specific entropy s and specific volume v, which are first derivatives of specific Gibbs free energy – in second-order phase transitions. If one considers specific entropy s as a function of temperature and pressure, then its differential is: ds = \left( \right)_P dT + \left( \right)_T dP. As \left( \right)_P = , \left( \right)_T = - \left( \right)_P , then the differential of specific entropy also is: d = dT - \left( \right)_P dP, where i= ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ehrenfest Paradox
The Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity. In its original 1909 formulation as presented by Paul Ehrenfest in relation to the concept of Born rigidity within special relativity, it discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry. The radius ''R'' as seen in the laboratory frame is always perpendicular to its motion and should therefore be equal to its value R0 when stationary. However, the circumference (2''R'') should appear Lorentz-contracted to a smaller value than at rest, by the usual factor γ. This leads to the contradiction that ''R'' = ''R''0 ''and'' ''R'' < ''R''0. The has been deepened further by , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ehrenfest Theorem
The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of the force F=-V'(x) on a massive particle moving in a scalar potential V(x), The Ehrenfest theorem is a special case of a more general relation between the expectation of any quantum mechanical operator and the expectation of the commutator of that operator with the Hamiltonian of the system where is some quantum mechanical operator and is its expectation value. It is most apparent in the Heisenberg picture of quantum mechanics, where it amounts to just the expectation value of the Heisenberg equation of motion. It provides mathematical support to the correspondence principle. The reason is that Ehrenfest's theorem is closely related to Liouville's theorem of Hamiltonian mechanics, which involves the Poisson bracket instead of a com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gunnar Nordström
Gunnar Nordström (12 March 1881 – 24 December 1923) was a Finnish theoretical physicist best remembered for his theory of gravitation, which was an early competitor of general relativity. Nordström is often designated by modern writers as ''The Einstein of Finland'' due to his novel work in similar fields with similar methods to Einstein.Raimo Keskinen (1981): Gunnar Nordström 1881B1923. Arkhimedes 2/1981, s. 71B84. In Finnish, excerpt http://www.tieteessatapahtuu.fi/797/KESKINEN.pdf Education and career Nordström graduated high-school from '' Brobergska skolan'' in central Helsinki 1899. At first he went on to study mechanical engineering, graduating in 1903 from the Polytechnic institute in Helsinki, later renamed Helsinki University of Technology and today a part of the Aalto University. During his studies he developed an interest for more theoretical subjects, proceeding after graduation to further study for a master's degree in natural science, mathematics and econo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
