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Supersymmetric Theory Of Stochastic Dynamics
Supersymmetric theory of stochastic dynamics (STS) is a multidisciplinary approach to stochastic dynamics on the intersection of dynamical systems theory, topological field theories, stochastic differential equations (SDE), and the theory of pseudo-Hermitian operators. It can be seen as an algebraic dual to the traditional set-theoretic framework of the dynamical systems theory, with its added algebraic structure and an inherent topological supersymmetry (TS) enabling the generalization of certain concepts from deterministic to stochastic models. Using tools of topological field theory originally developed in high-energy physics, STS seeks to give a rigorous mathematical derivation to several universal phenomena of stochastic dynamical systems. Particularly, the theory identifies dynamical chaos as a spontaneous order originating from the TS hidden in all stochastic models. STS also provides the lowest level classification of stochastic chaos which has a potential to explai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: in simple terms, a function f of a variable y, where y itself is a function of another variable x, may be written as a function of x. This is the pullback of f by the function y. f(y(x)) \equiv g(x) It is such a fundamental process that it is often passed over without mention. However, it is not just functions that can be "pulled back" in this sense. Pullbacks can be applied to many other objects such as differential forms and their cohomology classes; see * Pullback (differential geometry) * Pullback (cohomology) Fiber-product The pullback bundle is an example that bridges the notion of a pullback as precomposition, and the notion of a pullback as a Cartesian square. In that example, the base space of a fiber bundle is pulled b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Dimensional Reduction
Dimensional reduction is the limit of a compactified theory where the size of the compact dimension goes to zero. In physics, a theory in ''D'' spacetime dimensions can be redefined in a lower number of dimensions ''d'', by taking all the fields to be independent of the location in the extra ''D'' − ''d'' dimensions. For example, consider a periodic compact dimension with period ''L''. Let ''x'' be the coordinate along this dimension. Any field \phi can be described as a sum of the following terms: : \phi_n(x) = A_n \cos \left( \frac\right) with ''A''''n'' a constant. According to quantum mechanics, such a term has momentum ''nh''/''L'' along ''x'', where ''h'' is the Planck constant. Therefore, as ''L'' goes to zero, the momentum goes to infinity, and so does the energy, unless ''n'' = 0. However ''n'' = 0 gives a field which is constant with respect to ''x''. So at this limit, and at finite energy, \phi will not depend on ''x' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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BRST Quantization
In theoretical physics, the BRST formalism, or BRST quantization (where the ''BRST'' refers to the last names of Carlo Becchi, Alain Rouet, Raymond Stora and Igor Tyutin) denotes a relatively rigorous mathematical approach to quantizing a field theory with a gauge symmetry. Quantization rules in earlier quantum field theory (QFT) frameworks resembled "prescriptions" or "heuristics" more than proofs, especially in non-abelian QFT, where the use of " ghost fields" with superficially bizarre properties is almost unavoidable for technical reasons related to renormalization and anomaly cancellation. The BRST global supersymmetry introduced in the mid-1970s was quickly understood to rationalize the introduction of these Faddeev–Popov ghosts and their exclusion from "physical" asymptotic states when performing QFT calculations. Crucially, this symmetry of the path integral is preserved in loop order, and thus prevents introduction of counterterms which might spoil renormaliz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Langevin Equation
In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Langevin equation typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation. One application is to Brownian motion, which models the fluctuating motion of a small particle in a fluid. Brownian motion as a prototype The original Langevin equation describes Brownian motion, the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid, m\frac=-\lambda \mathbf+\boldsymbol\left( t\right). Here, \mathbf is the velocity of the particle, \lambda is its damping coefficient, and m is its mass. The force acting on the particle is w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Giorgio Parisi
Giorgio Parisi (born 4 August 1948) is an Italian theoretical physicist, whose research has focused on quantum field theory, 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 applicati ... and complex systems. His best known contributions are the Quantum chromodynamics, QCD evolution equations for parton densities, obtained with Guido Altarelli, known as the Altarelli–Parisi or DGLAP equations, the exact solution of the Sherrington–Kirkpatrick model of spin glasses, the Kardar–Parisi–Zhang equation describing dynamic scaling of growing interfaces, and the study of whirling flocks of birds. He was awarded the 2021 Nobel Prize in Physics jointly with Klaus Hasselmann and Syukuro Manabe for groundbreaking contributions to theory of complex systems, in particular "for t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Edge Of Chaos
The edge of chaos is a transition space between order and disorder that is hypothesized to exist within a wide variety of systems. This transition zone is a region of bounded instability that engenders a constant dynamic interplay between order and disorder. Even though the idea of the edge of chaos is an abstract one, it has many applications in such fields as ecology, business management, psychology, political science, and other domains of the social sciences. Physicists have shown that adaptation to the edge of chaos occurs in almost all systems with feedback. History The phrase ''edge of chaos'' was coined in the late 1980s by chaos theory physicist Norman Packard. In the next decade, Packard and mathematician Doyne Farmer co-authored many papers on understanding how self-organization and order emerges at the edge of chaos. One of the original catalysts that led to the idea of the edge of chaos were the experiments with cellular automata done by computer scientist C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Pink Noise
Pink noise, noise, fractional noise or fractal noise is a signal (information theory), signal or process with a frequency spectrum such that the power spectral density (power per frequency interval) is inversely proportional to the frequency of the signal. In pink noise, each Octave (electronics), octave interval (halving or doubling in frequency) carries an equal amount of noise energy. Pink noise sounds like a waterfall. It is often used to tune loudspeaker systems in professional audio. Pink noise is one of the most commonly observed signals in biological systems. The name arises from the pink appearance of visible light with this power spectrum. This is in contrast with white noise which has equal intensity per frequency interval. Definition Within the scientific literature, the term "1/f noise" is sometimes used loosely to refer to any noise with a power spectral density of the form S(f) \propto \frac, where is frequency, and , with exponent usually close to 1. On ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Goldstone Theorem
In physics, Goldstone bosons or Nambu–Goldstone bosons (NGBs) are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries. They were discovered by Yoichiro Nambu within the context of the BCS superconductivity mechanism, and subsequently elucidated by Jeffrey Goldstone, and systematically generalized in the context of quantum field theory. In condensed matter physics such bosons are quasiparticles and are known as Goldstone modes or Anderson–Bogoliubov modes. These spinless bosons correspond to the spontaneously broken internal symmetry generators, and are characterized by the quantum numbers of these. They transform nonlinearly (shift) under the action of these generators, and can thus be excited out of the asymmetric vacuum by these generators. Thus, they can be thought of as the excitations of the field in the broken symmetry directions in group space—and are massless if the spontaneously broken symmetry is not also broken e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Complexity
Complexity characterizes the behavior of a system or model whose components interact in multiple ways and follow local rules, leading to non-linearity, randomness, collective dynamics, hierarchy, and emergence. The term is generally used to characterize something with many parts where those parts interact with each other in multiple ways, culminating in a higher order of emergence greater than the sum of its parts. The study of these complex linkages at various scales is the main goal of complex systems theory. The intuitive criterion of complexity can be formulated as follows: a system would be more complex if more parts could be distinguished, and if more connections between them existed. , a number of approaches to characterizing complexity have been used in science; Zayed ''et al.'' reflect many of these. Neil Johnson states that "even among scientists, there is no unique definition of complexity – and the scientific notion has traditionally been conveyed using partic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Spontaneous Symmetry Breaking
Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the Lagrangian obey symmetries, but the lowest-energy vacuum solutions do not exhibit that same symmetry. When the system goes to one of those vacuum solutions, the symmetry is broken for perturbations around that vacuum even though the entire Lagrangian retains that symmetry. Overview The spontaneous symmetry breaking cannot happen in quantum mechanics that describes finite dimensional systems, due to Stone-von Neumann theorem (that states the uniqueness of Heisenberg commutation relations in finite dimensions). So spontaneous symmetry breaking can be observed only in infinite dimensional theories, as quantum field theories. By definition, spontaneous symmetry breaking requires the existence of physical laws which are invariant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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State-space Representation
In control engineering and system identification, a state-space representation is a mathematical model of a physical system that uses state variables to track how inputs shape system behavior over time through first-order differential equations or difference equations. These state variables change based on their current values and inputs, while outputs depend on the states and sometimes the inputs too. The state space (also called time-domain approach and equivalent to phase space in certain dynamical systems) is a geometric space where the axes are these state variables, and the system’s state is represented by a state vector. For linear, time-invariant, and finite-dimensional systems, the equations can be written in matrix form, offering a compact alternative to the frequency domain’s Laplace transforms for multiple-input and multiple-output (MIMO) systems. Unlike the frequency domain approach, it works for systems beyond just linear ones with zero initial conditions. Thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |