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Mayer F-function
The Mayer f-function is an auxiliary function that often appears in the series expansion of thermodynamic quantities related to classical many-particle systems.Donald Allan McQuarrie, ''Statistical Mechanics'' (HarperCollins, 1976), page 228 It is named after chemist and physicist Joseph Edward Mayer. Definition Consider a system of classical particles interacting through a pair-wise potential :V(\mathbf,\mathbf) where the bold labels \mathbf and \mathbf denote the continuous degrees of freedom associated with the particles, e.g., :\mathbf=\mathbf_i for spherically symmetric particles and :\mathbf=(\mathbf_i,\Omega_i) for rigid non-spherical particles where \mathbf denotes position and \Omega the orientation parametrized e.g. by Euler angles. The Mayer f-function is then defined as :f(\mathbf,\mathbf)=e^-1 where \beta=(k_T)^ the inverse absolute temperature in units of (Temperature times the Boltzmann constant k_)−1 . See also *Virial coefficient *Cluster expansion *Excluded vol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thermodynamic
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of thermodynamics which convey a quantitative description using measurable macroscopic physical quantities, but may be explained in terms of microscopic constituents by statistical mechanics. Thermodynamics applies to a wide variety of topics in science and engineering, especially physical chemistry, biochemistry, chemical engineering and mechanical engineering, but also in other complex fields such as meteorology. Historically, thermodynamics developed out of a desire to increase the efficiency of early steam engines, particularly through the work of French physicist Sadi Carnot (1824) who believed that engine efficiency was the key that could help France win the Napoleonic Wars. Scots-Irish physicist Lord Kelvin was the first to formulate a co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Many-particle System
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provide an accurate description of the system. ''Many'' can be anywhere from three to infinity (in the case of a practically infinite, homogeneous or periodic system, such as a crystal), although three- and four-body systems can be treated by specific means (respectively the Faddeev and Faddeev–Yakubovsky equations) and are thus sometimes separately classified as few-body systems. In general terms, while the underlying physical laws that govern the motion of each individual particle may (or may not) be simple, the study of the collection of particles can be extremely complex. In such a quantum system, the repeated interactions between particles create quantum correlations, or entanglement. As a consequence, the wave function of the system is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
HarperCollins
HarperCollins Publishers LLC is one of the Big Five English-language publishing companies, alongside Penguin Random House, Simon & Schuster, Hachette, and Macmillan. The company is headquartered in New York City and is a subsidiary of News Corp. The name is a combination of several publishing firm names: Harper & Row, an American publishing company acquired in 1987—whose own name was the result of an earlier merger of Harper & Brothers (founded in 1817) and Row, Peterson & Company—together with Scottish publishing company William Collins, Sons (founded in 1819), acquired in 1989. The worldwide CEO of HarperCollins is Brian Murray. HarperCollins has publishing groups in the United States, Canada, the United Kingdom, Australia, New Zealand, Brazil, India, and China. The company publishes many different imprints, both former independent publishing houses and new imprints. History Collins Harper Mergers and acquisitions Collins was bought by Rupert Murdoch's News Corpora ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Edward Mayer
Joseph Edward Mayer (February 5, 1904, New York City – October 15, 1983) was a chemist who formulated the Mayer expansion in statistical field theory. He was professor of chemistry at the University of California San Diego from 1960 to 1972, and previously at Johns Hopkins University, Columbia University and the University of Chicago. He was married to Nobel Prize-winning physicist Maria Goeppert Mayer from 1930 until her death in 1972. He went to work with James Franck in Göttingen, Germany in 1929, where he met Maria, a student of Max Born. He was a member of the United States National Academy of Sciences (1946), the American Academy of Arts and Sciences (1958), and the American Philosophical Society (1970). Joseph Mayer was president of the American Physical Society from 1973 to 1975. Scientific contributions He developed the cluster expansion method and Mayer-McMillan solution theory. See also *Mayer f-function The Mayer f-function is an auxiliary function that often appe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Angles
The Euler angles are three angles introduced by Leonhard Euler to describe the Orientation (geometry), orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref> They can also represent the orientation of a mobile frame of reference in physics or the orientation of a general Basis (linear algebra), basis in 3-dimensional linear algebra. Alternative forms were later introduced by Peter Guthrie Tait and George H. Bryan intended for use in aeronautics and engineering. Chained rotations equivalence Euler angles can be defined by elemental geometry or by composition of rotations. The geometrical definition demonstrates that three composed ''elemental rotations'' (rotations about the axes of a coordinate system) are always sufficient to reach any target frame. The three elemental rotations may be #Conventions by extrinsic rotations, extrinsic (rotations about the axes ''xyz'' of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied on various reference points and thermometric substances for definition. The most common scales are the Celsius scale with the unit symbol °C (formerly called ''centigrade''), the Fahrenheit scale (°F), and the Kelvin scale (K), the latter being used predominantly for scientific purposes. The kelvin is one of the seven base units in the International System of Units (SI). Absolute zero, i.e., zero kelvin or −273.15 °C, is the lowest point in the thermodynamic temperature scale. Experimentally, it can be approached very closely but not actually reached, as recognized in the third law of thermodynamics. It would be impossible to extract energy as heat from a body at that temperature. Temperature is important in all fields of natur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boltzmann Constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, and in Planck's law of black-body radiation and Boltzmann's entropy formula, and is used in calculating thermal noise in resistors. The Boltzmann constant has dimensions of energy divided by temperature, the same as entropy. It is named after the Austrian scientist Ludwig Boltzmann. As part of the 2019 redefinition of SI base units, the Boltzmann constant is one of the seven " defining constants" that have been given exact definitions. They are used in various combinations to define the seven SI base units. The Boltzmann constant is defined to be exactly . Roles of the Boltzmann constant Macroscopically, the ideal gas law states that, for an ideal gas, the product of pressure and volume is proportional to the product of amount of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virial Coefficient
Virial coefficients B_i appear as coefficients in the virial expansion of the pressure of a many-particle system in powers of the density, providing systematic corrections to the ideal gas law. They are characteristic of the interaction potential between the particles and in general depend on the temperature. The second virial coefficient B_2 depends only on the pair interaction between the particles, the third (B_3) depends on 2- and non-additive 3-body interactions, and so on. Derivation The first step in obtaining a closed expression for virial coefficients is a cluster expansion of the grand canonical partition function : \Xi = \sum_ = e^ Here p is the pressure, V is the volume of the vessel containing the particles, k_B is Boltzmann's constant, T is the absolute temperature, \lambda =\exp mu/(k_BT) is the fugacity, with \mu the chemical potential. The quantity Q_n is the canonical partition function of a subsystem of n particles: : Q_n = \operatorname e^ Here H(1,2,\ld ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cluster Expansion
In statistical mechanics, the cluster expansion (also called the high temperature expansion or hopping expansion) is a power series expansion of the partition function of a statistical field theory around a model that is a union of non-interacting 0-dimensional field theories. Cluster expansions originated in the work of . Unlike the usual perturbation expansion which usually leads to a divergent asymptotic series, the cluster expansion may converge within a non-trivial region, in particular when the interaction is small and short-ranged. Classical case General theory In statistical mechanics, the properties of a system of noninteracting particles are described using the partition function. For N noninteracting particles, the system is described by the Hamiltonian : \big. H_0=\sum_i^N \frac, and the partition function can be calculated (for the classical case) as :\big. Z_0 =\frac\int \prod_i d\vec_i\;d\vec_i \exp\left\ =\frac\left( \frac \right)^. From the partition functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excluded Volume
The concept of excluded volume was introduced by Werner Kuhn in 1934 and applied to polymer molecules shortly thereafter by Paul Flory. Excluded volume gives rise to depletion forces. In liquid state theory In liquid state theory, the 'excluded volume' of a molecule is the volume that is inaccessible to other molecules in the system as a result of the presence of the first molecule. The excluded volume of a hard sphere is eight times its volume—however, for a two-molecule system, this volume is distributed among the two particles, giving the conventional result of four times the volume; this is an important quantity in the Van der Waals equation of state. The calculation of the excluded volume for particles with non-spherical shapes is usually difficult, since it depends on the relative orientation of the particles. The distance of closest approach of hard ellipses and their excluded area has been recently considered. In polymer science In polymer science, excluded volume ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
