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Replica Trick
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, 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 same results. (To prove that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Physics
Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the Mathematics, mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic nature. Its applications include many problems in the fields of physics, biology, chemistry, and neuroscience. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics develop the Phenomenology (particle physics), phenomenological results of thermodynamics from a probabilistic examination of the underlying microscopic systems. Historically, one of the first topics in physics where statistical methods were applied was the field of classical mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ground State
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. In quantum field theory, the ground state is usually called the vacuum state or the vacuum. If more than one ground state exists, they are said to be degenerate. Many systems have degenerate ground states. Degeneracy occurs whenever there exists a unitary operator that acts non-trivially on a ground state and commutes with the Hamiltonian of the system. According to the third law of thermodynamics, a system at absolute zero temperature exists in its ground state; thus, its entropy is determined by the degeneracy of the ground state. Many systems, such as a perfect crystal lattice, have a unique ground state and therefore have zero entropy at absolute zero. It is also possible for the highest excited state to have absolute zero temper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Keldysh Technique (1904–1976), mathematician, ...
Keldysh (Russian: ) may refer to: Science * Keldysh formalism, for studying non-equilibrium quantum systems * '' Akademik Mstislav Keldysh'', a 1980 Russian scientific research vessel renowned for its visits to the wreck of the RMS ''Titanic'' * Keldysh Institute of Applied Mathematics, a Russian research institute * Keldysh (crater), a crater on the Moon * 2186 Keldysh, an asteroid People * Leonid Keldysh (1931–2016), Russian physicist, former director of the Lebedev Physical Institute (1988–1994), later a member of the physics faculty at Texas A&M University * Mstislav Keldysh (1911–1978), Russian mathematician, president of the Soviet Academy of Sciences (1961–1978) * Lyudmila Keldysh Lyudmila Vsevolodovna Keldysh (russian: Людмила Всеволодовна Келдыш; 12 March 1904 – 16 February 1976) was a Soviet mathematician known for set theory and geometric topology. Biography Lyudmila Vsevolodovna Keldysh was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories exist. Supersymmetry is a spacetime symmetry between two basic classes of particles: bosons, which have an integer-valued spin and follow Bose–Einstein statistics, and fermions, which have a half-integer-valued spin and follow Fermi–Dirac statistics. In supersymmetry, each particle from one class would have an associated particle in the other, known as its superpartner, the spin of which differs by a half-integer. For example, if the electron exists in a supersymmetric theory, then there would be a particle called a ''"selectron"'' (superpartner electron), a bosonic partner of the electron. In the simplest supersymmetry theories, with perfectly " unbroken" supersymmetry, each pair of superpartners would share the same mass and intern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ..., a tree is an undirected graph in which any two Vertex (graph theory), vertices are connected by ''exactly one'' Path (graph theory), path, or equivalently a Connected graph, connected Cycle (graph theory), acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a Disjoint union of graphs, disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .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 undirecte ... [...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 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 different approximations that can be done with the cavity method are usually named after their equivalent with the different steps of the replica method which is mathematically more subtle and less intuitive than the cavity approach. The cavity method has proved useful in the solution of optimization problems such as k-satisfiability and graph coloring. It ... [...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. Comparison with other disordered systems The r-spin infinite-range model, in which all r-spin sets interact with a random, independent, identically distributed interaction constant, becomes the random energy model in a suitably defined ... [...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 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 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 of one of the sublattice magneti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ferromagnetic
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials are the familiar metals noticeably attracted to a magnet, a consequence of their large magnetic permeability. Magnetic permeability describes the induced magnetization of a material due to the presence of an ''external'' magnetic field, and it is this temporarily induced magnetization inside a steel plate, for instance, which accounts for its attraction to the permanent magnet. Whether or not that steel plate acquires a permanent magnetization itself, depends not only on the strength of the applied field, but on the so-called coercivity of that material, which varies greatly among ferromagnetic materials. In physics, several different types of material magnetism are distinguished. Ferromagnetism (along with the similar effect ferrimagneti ... [...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 d ... [...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 equilibrium, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
