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Kac Ring
In statistical mechanics, the Kac ring is a toy model introduced by Mark Kac in 1956 to explain how the second law of thermodynamics emerges from T-symmetry, time-symmetric interactions between molecules (see reversibility paradox). Although artificial, the model is notable as a mathematically transparent example of coarse-grained modeling, coarse-graining and is used as a didactic tool in non-equilibrium thermodynamics. Formulation The Kac ring consists of equidistant points in a circle. Some of these points are ''marked''. The number of marked points is , where 0 < 2M < N. Each point represents a site occupied by a ball, which is ''black'' or ''white''. After a unit of time, each ball moves to a neighboring point counterclockwise. Whenever a ball leaves a marked site, it switches color from black to white and vice versa. (If, however, the starting point is not marked, the ball completes its move without changing color.) An imagined observer can only measure coarse-gr ... [...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. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in a wide variety of fields such as biology, neuroscience, computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied 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 microscop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Molecular Chaos
In the kinetic theory of gases in physics, the molecular chaos hypothesis (also called ''Stosszahlansatz'' in the writings of Paul and Tatiana Ehrenfest) is the assumption that the velocities of colliding particles are uncorrelated, and independent of position. This means the probability that a pair of particles with given velocities will collide can be calculated by considering each particle separately and ignoring any correlation between the probability for finding one particle with velocity and probability for finding another velocity in a small region . James Clerk Maxwell introduced this approximation in 1867 although its origins can be traced back to his first work on the kinetic theory in 1860. The assumption of molecular chaos is the key ingredient that allows proceeding from the BBGKY hierarchy to Boltzmann's equation, by reducing the 2-particle distribution function showing up in the collision term to a product of 1-particle distributions. This in turn leads to Bol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Avogadro Constant
The Avogadro constant, commonly denoted or , is an SI defining constant with an exact value of when expressed in reciprocal moles. It defines the ratio of the number of constituent particles to the amount of substance in a sample, where the particles in question are any designated elementary entity, such as molecules, atoms, ions, ion pairs. The numerical value of this constant is known as the Avogadro number, commonly denoted . The Avogadro ''number'' is an exact number equal to the number of constituent particles in one mole of any substance (by definition of the mole), historically derived from the experimental determination of the number of atoms in 12 grams of carbon-12 (12C) before the 2019 revision of the SI. Both the constant and the number are named after the Italian physicist and chemist Amedeo Avogadro. The Avogadro constant is used as a proportionality factor in relating the ''amount of substance'' , in a sample of a substance , to the corresponding n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thermodynamic Equilibrium
Thermodynamic equilibrium is a notion of thermodynamics with axiomatic status referring to an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermodynamic equilibrium, there are no net macroscopic flows of mass nor of energy within a system or between systems. In a system that is in its own state of internal thermodynamic equilibrium, not only is there an absence of macroscopic change, but there is an “absence of any ''tendency'' toward change on a macroscopic scale.” Systems in mutual thermodynamic equilibrium are simultaneously in mutual thermal, mechanical, chemical, and radiative equilibria. Systems can be in one kind of mutual equilibrium, while not in others. In thermodynamic equilibrium, all kinds of equilibrium hold at once and indefinitely, unless disturbed by a thermodynamic operation. In a macroscopic equilibrium, perfectly or almost perfectly ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Distribution
In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory), experiments, each asking a yes–no question, and each with its own Boolean-valued function, Boolean-valued outcome (probability), outcome: ''success'' (with probability ) or ''failure'' (with probability ). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., , the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the binomial test of statistical significance. The binomial distribution is frequently used to model the number of successes in a sample of size drawn with replacement from a population of size . If the sampling is carried out without replacement, the draws ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function (mathematics), function in which * the Domain of a function, domain is the set of possible Outcome (probability), outcomes in a sample space (e.g. the set \ which are the possible upper sides of a flipped coin heads H or tails T as the result from tossing a coin); and * the Range of a function, range is a measurable space (e.g. corresponding to the domain above, the range might be the set \ if say heads H mapped to -1 and T mapped to 1). Typically, the range of a random variable is a subset of the Real number, real numbers. Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kac Ring (Graph)
KAC or Kac may refer to: Organizations * Kenitra Athletic Club, a football club in Kenitra, Morocco * EC KAC or Klagenfurter Athletiksport Club, an ice hockey club in Klagenfurt, Austria * Knight's Armament Company, an American firearm manufacturer * Korea Airports Corporation * Kosciusko Alpine Club, an Australian ski club People * Eduardo Kac (born 1960), Brazilian-American artist * Mac Kac (1920–1987), French jazz drummer * Mark Kac (1914–1984), Polish-American mathematician * Victor Kac (born 1943), Russian-American mathematician Places * Kać, Novi Sad, South Bačka District, Serbia * Kenyon Athletic Center, Gambier, Knox County, Ohio, US * Kiaracondong railway station, Bandung, West Java, Indonesia, code Other * ICAO designator for Kuwait Airways * Jinghpaw language (ISO 639-3 code) See also * Kács Kács is a village in Borsod-Abaúj-Zemplén County in northeastern Hungary Hungary is a landlocked country in Central Europe. Spanning much of the Pannon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Recurrence Theorem
In mathematics and physics, the Poincaré recurrence theorem states that certain dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (for discrete state systems), their initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence. This time may vary greatly depending on the exact initial state and required degree of closeness. The result applies to isolated mechanical systems subject to some constraints, e.g., all particles must be bound to a finite volume. The theorem is commonly discussed in the context of ergodic theory, dynamical systems and statistical mechanics. Systems to which the Poincaré recurrence theorem applies are called conservative systems. The theorem is named after Henri Poincaré, who discussed it in 1890. A proof was presented by Constantin Carathéodory using measure theory in 1919. Precise formulation Any dynam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Function
A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called ''aperiodic''. Definition A function is said to be periodic if, for some nonzero constant , it is the case that :f(x+P) = f(x) for all values of in the domain. A nonzero constant for which this is the case is called a period of the function. If there exists a least positive constant with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) Often, "the" period of a function is used to mean its fundamental period. A funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boltzmann Equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium; it was devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3. The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number. The equation arises not by analyzing the individual positions and momenta of each particle in the fluid but rather by considering a probability distribution for the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microscopic Scale
The microscopic scale () is the scale of objects and events smaller than those that can easily be seen by the naked eye, requiring a lens (optics), lens or microscope to see them clearly. In physics, the microscopic scale is sometimes regarded as the scale between the macroscopic scale and the quantum scale. Microscopic units and measurements are used to classify and describe very small objects. One common microscopic length scale unit is the micrometre (also called a ''micron'') (symbol: μm), which is one millionth of a metre. History Whilst compound microscopes were first developed in the 1590s, the significance of the microscopic scale was only truly established in the 1600s when Marcello Malpighi, Marcello Malphigi and Antonie van Leeuwenhoek microscopically observed frog lungs and microorganisms. As microbiology was established, the significance of making scientific observations at a microscopic level increased. Published in 1665, Robert Hooke's book Micrographia details h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toy Model
A toy or plaything is an object that is used primarily to provide entertainment. Simple examples include toy blocks, board games, and dolls. Toys are often designed for use by children, although many are designed specifically for adults and pets. Toys can provide utilitarian benefits, including physical exercise, cultural awareness, or academic education. Additionally, utilitarian objects, especially those which are no longer needed for their original purpose, can be used as toys. Examples include children building a fort with empty cereal boxes and tissue paper spools, or a toddler playing with a broken TV remote. The term "toy" can also be used to refer to utilitarian objects purchased for enjoyment rather than need, or for expensive necessities for which a large fraction of the cost represents its ability to provide enjoyment to the owner, such as luxury cars, high-end motorcycles, gaming computers, and flagship smartphones. Playing with toys can be an enjoyable way of tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
