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Replica Method
In the statistical physics of spin glasses and other systems with quenched disorder, the replica trick is a mathematical technique based on the application of the formula: \ln Z=\lim_ or: \ln Z = \lim_ \frac where Z is most commonly the partition function, or a similar thermodynamic function. It is typically used to simplify the calculation of \overline, the expected value of \ln Z, reducing the problem to calculating the disorder average \overline where n is assumed to be an integer. This is physically equivalent to averaging over n copies or ''replicas'' of the system, hence the name. The crux of the replica trick is that while the disorder averaging is done assuming n to be an integer, to recover the disorder-averaged logarithm one must send n continuously to zero. This apparent contradiction at the heart of the replica trick has never been formally resolved, however in all cases where the replica method can be compared with other exact solutions, the methods lead to the sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Physics
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, 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 microscopic parameters that fluctuate about average values and are characterized by probability distributions. While classical thermodynamics is primarily concerned with thermodynamic equilibrium, statistical mechanics has been applied in non-equilibrium stat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodicity Breaking
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored into smaller components. Ergodic theory is the study of systems possessing ergodicity. Ergodic systems occur in a broad range of systems in physics and in geometry. This can be roughly understood to be due to a common phenomenon: the motion of particles, that is, geodesics on a hyperbolic manifold are divergent; when that manifold is compact, that is, of finite size, those orbits return to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Keldysh Formalism
In non-equilibrium physics, the Keldysh formalism or Keldysh–Schwinger formalism is a general framework for describing the quantum mechanical evolution of a system in a non-equilibrium state or systems subject to time varying external fields (electrical field, magnetic field etc.). Historically, it was foreshadowed by the work of Julian Schwinger and proposed almost simultaneously by Leonid Keldysh and, separately, Leo Kadanoff and Gordon Baym. It was further developed by later contributors such as O. V. Konstantinov and V. I. Perel. Extensions to driven-dissipative open quantum systems coupled to Markovian quantum baths were developed for both bosonic and fermionic systems. The Keldysh formalism provides a systematic way to study non-equilibrium systems, usually based on the two-point functions corresponding to excitations in the system. The main mathematical object in the Keldysh formalism is the non-equilibrium Green's function (NEGF), which is a two-point function of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersymmetry
Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It proposes that for every known particle, there exists a partner particle with different spin properties. There have been multiple experiments on supersymmetry that have failed to provide evidence that it exists in nature. If evidence is found, supersymmetry could help explain certain phenomena, such as the nature of dark matter and the hierarchy problem in particle physics. A supersymmetric theory is a theory in which the equations for force and the equations for matter are identical. In theoretical physics, theoretical and mathematical physics, any theory with this property has the ''principle of supersymmetry'' (SUSY). Dozens of supersymmetric theories exist. In theory, supersymmetry is a type of Spacetime symmetries, spacetime symmetry betwe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A directed tree, oriented tree,See .See . polytree,See . or singly connected networkSee . is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an arborescence or out-tree� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cavity Method
The cavity method is a mathematical method presented by Marc Mézard, Giorgio Parisi and Miguel Angel Virasoro in 1987 to derive and solve some mean field-type models in statistical physics, specially adapted to disordered systems. The method has been used to compute properties of ground states in many condensed matter and optimization problems. Initially invented to deal with the Sherrington–Kirkpatrick model of spin glasses, the cavity method has shown wider applicability. It can be regarded as a generalization of the Bethe– Peierls iterative method in tree-like graphs, to the case of a graph with loops that are not too short. The cavity method can solve many problems also solvable using the replica trick but has the advantage of being more intuitive and less mathematically subtle than replica-based methods. The cavity method proceeds by perturbing a large system with the addition of a non-thermodynamic number of additional constituents and approximating the response of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Energy Model
In the statistical physics of disordered systems, the random energy model is a toy model of a system with quenched disorder, such as a spin glass, having a first-order phase transition. It concerns the statistics of a collection of N spins (''i.e.'' degrees of freedom \boldsymbol\sigma\equiv \_^N that can take one of two possible values \sigma_i=\pm 1) so that the number of possible states for the system is 2^N. The energies of such states are independent and identically distributed Gaussian random variables E_x \sim \mathcal(0,N/2) with zero mean and a variance of N/2. Many properties of this model can be computed exactly. Its simplicity makes this model suitable for pedagogical introduction of concepts like quenched disorder and replica symmetry. Thermodynamic quantities Critical energy per particle: h_c = \sqrt. Critical inverse temperature \beta_c = 2\sqrt. Partition function Z(\beta) = \sum_s e^, which at large N becomes 2^N \mathbb E_E ^/math> when \beta denotes the av ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antiferromagnetic
In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usually related to the spins of electrons, align in a regular pattern with neighboring Spin (physics), spins (on different sublattices) pointing in opposite directions. This is, like ferromagnetism and ferrimagnetism, a manifestation of ordered magnetism. The phenomenon of antiferromagnetism was first introduced by Lev Landau in 1933. Generally, antiferromagnetic order may exist at sufficiently low temperatures, but vanishes at and above the Néel temperature – named after Louis Néel, who had first identified this type of magnetic ordering. Above the Néel temperature, the material is typically paramagnetism, paramagnetic. Measurement When no external field is applied, the antiferromagnetic structure corresponds to a vanishing total magnetization. In an external magnetic field, a kind of ferrimagnetic behavior may be displayed in the antiferromagnetic phase, with the absolute value o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ferromagnetic
Ferromagnetism is a property of certain materials (such as iron) that results in a significant, observable magnetic permeability, and in many cases, a significant magnetic coercivity, allowing the material to form a permanent magnet. Ferromagnetic materials are noticeably attracted to a magnet, which is a consequence of their substantial magnetic permeability. Magnetic permeability describes the induced magnetization of a material due to the presence of an external magnetic field. For example, this temporary magnetization inside a steel plate accounts for the plate's attraction to a magnet. Whether or not that steel plate then acquires permanent magnetization depends on both the strength of the applied field and on the coercivity of that particular piece of steel (which varies with the steel's chemical composition and any heat treatment it may have undergone). In physics, multiple types of material magnetism have been distinguished. Ferromagnetism (along with the similar effec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-averaging
A self-averaging physical property of a disordered system is one that can be described by averaging over a sufficiently large sample. The concept was introduced by Ilya Mikhailovich Lifshitz. Definition Frequently in physics one comes across situations where quenched randomness plays an important role. Any physical property ''X'' of such a system, would require an averaging over all disorder realisations. The system can be completely described by the average 'X''where ..denotes averaging over realisations (“averaging over samples”) provided the relative variance ''R''''X'' = ''V''''X'' / 'X''sup>2 → 0 as ''N''→∞, where ''V''''X'' = 'X''2nbsp;− 'X''sup>2 and ''N'' denotes the size of the realisation. In such a scenario a single large system is sufficient to represent the whole ensemble. Such quantities are called self-averaging. Away from criticality, when the larger lattice is built from smaller blocks, then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order And Disorder (physics)
In physics, the terms order and disorder designate the presence or absence of some symmetry or correlation in a many-particle system. In condensed matter physics, systems typically are ordered at low temperatures; upon heating, they undergo one or several phase transitions into less ordered states. Examples for such an order-disorder transition are: * the melting of ice: solid–liquid transition, loss of crystalline order; * the demagnetization of iron by heating above the Curie temperature: ferromagnetic–paramagnetic transition, loss of magnetic order. The degree of freedom that is ordered or disordered can be translational (crystalline ordering), rotational (ferroelectric ordering), or a spin state (magnetic ordering). The order can consist either in a full crystalline space group symmetry, or in a correlation. Depending on how the correlations decay with distance, one speaks of long range order or short range order. If a disordered state is not in thermodynamic equili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
