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Mean-field theory, the Glossary

In physics and probability theory, Mean-field theory (MFT) or Self-consistent field theory studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of freedom (the number of values in the final calculation of a statistic that are free to vary).^[1]

60 relations: Amino acid, Artificial intelligence, Bethe lattice, Boltzmann distribution, Classical field theory, Combinatorics, Coordination number, Critical dimension, Critical exponent, Degrees of freedom (physics and chemistry), Degrees of freedom (statistics), Dynamical mean-field theory, Elasticity (physics), Ensemble (mathematical physics), Entropy, Feynman diagram, Flory–Huggins solution theory, Fundamental interaction, Ginzburg criterion, Graphical model, Hamiltonian mechanics, Helmholtz free energy, Hubbard model, Ising model, Lagrange multiplier, Landau theory, Laplace operator, Liquid crystal, Many-body problem, Mathematical modelling of infectious diseases, Mathematical optimization, Mean-field game theory, Network performance, Neuroscience, Partition function (mathematics), Partition function (statistical mechanics), Perturbation theory, Phase transition, Physics, Pierre Curie, Pierre Weiss, Polymer, Probability theory, Protein structure prediction, Protein tertiary structure, Quantal response equilibrium, Queueing theory, Random field, Scheutjens–Fleer theory, Self-consistent mean field (biology), ... Expand index (10 more) »
Electronic structure methods

Amino acid

Amino acids are organic compounds that contain both amino and carboxylic acid functional groups.

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Artificial intelligence

Artificial intelligence (AI), in its broadest sense, is intelligence exhibited by machines, particularly computer systems.

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Bethe lattice

In statistical mechanics and mathematics, the Bethe lattice (also called a regular tree) is an infinite connected cycle-free graph where all vertices have the same number of neighbors.

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Boltzmann distribution

In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of that state's energy and the temperature of the system. Mean-field theory and Boltzmann distribution are statistical mechanics.

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Classical field theory

A classical field theory is a physical theory that predicts how one or more fields in physics interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories.

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Combinatorics

Combinatorics is an area of mathematics primarily concerned with the counting, selecting and arranging of objects, both as a means and as an end in itself.

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Coordination number

In chemistry, crystallography, and materials science, the coordination number, also called ligancy, of a central atom in a molecule or crystal is the number of atoms, molecules or ions bonded to it.

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Critical dimension

In the renormalization group analysis of phase transitions in physics, a critical dimension is the dimensionality of space at which the character of the phase transition changes. Mean-field theory and critical dimension are statistical mechanics.

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Critical exponent

Critical exponents describe the behavior of physical quantities near continuous phase transitions.

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Degrees of freedom (physics and chemistry)

In physics and chemistry, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system.

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Degrees of freedom (statistics)

In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.

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Dynamical mean-field theory

Dynamical mean-field theory (DMFT) is a method to determine the electronic structure of strongly correlated materials. Mean-field theory and Dynamical mean-field theory are electronic structure methods.

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Elasticity (physics)

In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed.

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Ensemble (mathematical physics)

In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in.

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Entropy

Entropy is a scientific concept that is most commonly associated with a state of disorder, randomness, or uncertainty.

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Feynman diagram

In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. Mean-field theory and Feynman diagram are concepts in physics.

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Flory–Huggins solution theory

Flory–Huggins solution theory is a lattice model of the thermodynamics of polymer solutions which takes account of the great dissimilarity in molecular sizes in adapting the usual expression for the entropy of mixing. Mean-field theory and Flory–Huggins solution theory are statistical mechanics.

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Fundamental interaction

In physics, the fundamental interactions or fundamental forces are the interactions that do not appear to be reducible to more basic interactions.

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Ginzburg criterion

Mean field theory gives sensible results as long as one is able to neglect fluctuations in the system under consideration. Mean-field theory and Ginzburg criterion are statistical mechanics.

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Graphical model

A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables.

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Hamiltonian mechanics

In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833.

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Helmholtz free energy

In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal).

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Hubbard model

The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems.

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Ising model

The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. Mean-field theory and Ising model are statistical mechanics.

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Lagrange multiplier

In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables).

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Landau theory

Landau theory (also known as Ginzburg–Landau theory, despite the confusing name) in physics is a theory that Lev Landau introduced in an attempt to formulate a general theory of continuous (i.e., second-order) phase transitions. Mean-field theory and Landau theory are statistical mechanics.

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Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space.

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Liquid crystal

Liquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals.

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Many-body problem

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.

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Mathematical modelling of infectious diseases

Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic (including in plants) and help inform public health and plant health interventions.

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Mathematical optimization

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives.

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Mean-field game theory

Mean-field game theory is the study of strategic decision making by small interacting agents in very large populations.

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Network performance

Network performance refers to measures of service quality of a network as seen by the customer.

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Neuroscience

Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders.

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Partition function (mathematics)

The partition function or configuration integral, as used in probability theory, information theory and dynamical systems, is a generalization of the definition of a partition function in statistical mechanics.

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Partition function (statistical mechanics)

In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium.

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Perturbation theory

In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. Mean-field theory and perturbation theory are concepts in physics.

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Phase transition

In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another.

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Physics

Physics is the natural science of matter, involving the study of matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force.

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Pierre Curie

Pierre Curie (15 May 1859 – 19 April 1906) was a French physicist, a pioneer in crystallography, magnetism, piezoelectricity, and radioactivity.

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Pierre Weiss

Pierre-Ernest Weiss (25 March 1865, Mulhouse – 24 October 1940, Lyon) was a French physicist who specialized in magnetism.

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Polymer

A polymer is a substance or material consisting of very large molecules linked together into chains of repeating subunits.

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Probability theory

Probability theory or probability calculus is the branch of mathematics concerned with probability.

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Protein structure prediction

Protein structure prediction is the inference of the three-dimensional structure of a protein from its amino acid sequence—that is, the prediction of its secondary and tertiary structure from primary structure.

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Protein tertiary structure

Protein tertiary structure is the three-dimensional shape of a protein.

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Quantal response equilibrium

Quantal response equilibrium (QRE) is a solution concept in game theory.

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Queueing theory

Queueing theory is the mathematical study of waiting lines, or queues.

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Random field

In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb^n).

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Scheutjens–Fleer theory

Scheutjens–Fleer theory is a lattice-based self-consistent field theory that is the basis for many computational analyses of polymer adsorption. Mean-field theory and Scheutjens–Fleer theory are statistical mechanics.

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Self-consistent mean field (biology)

The self-consistent mean field (SCMF) method is an adaptation of mean field theory used in protein structure prediction to determine the optimal amino acid side chain packing given a fixed protein backbone.

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Side chain

In organic chemistry and biochemistry, a side chain is a chemical group that is attached to a core part of the molecule called the "main chain" or backbone.

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Statistic

A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose.

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Statistical inference

Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.

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Statistical mechanics

In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities.

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Stochastic

Stochastic refers to the property of being well-described by a random probability distribution.

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Superconductivity

Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic fields are expelled from the material.

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Thermal fluctuations

In statistical mechanics, thermal fluctuations are random deviations of an atomic system from its average state, that occur in a system at equilibrium. Mean-field theory and thermal fluctuations are statistical mechanics.

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Thermodynamic free energy

In thermodynamics, the thermodynamic free energy is one of the state functions of a thermodynamic system (the others being internal energy, enthalpy, entropy, etc.). The change in the free energy is the maximum amount of work that the system can perform in a process at constant temperature, and its sign indicates whether the process is thermodynamically favorable or forbidden.

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Two-body problem

In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles.

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Variational Bayesian methods

Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning.

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References

[1] https://en.wikipedia.org/wiki/Mean-field_theory

Also known as Mean Field Theory, Mean field, Mean field approximation, Mean field model, Mean-field approach, Mean-field approximation, Mean-field model, Molecular field, Self-consistent field theory.

, Side chain, Statistic, Statistical inference, Statistical mechanics, Stochastic, Superconductivity, Thermal fluctuations, Thermodynamic free energy, Two-body problem, Variational Bayesian methods.

Mean-field theory, the Glossary

Table of Contents

See also

Electronic structure methods

References