Universality (dynamical systems)
Encyclopedia
In statistical mechanics
, universality is the observation that there are properties for a large class of systems that are independent of the dynamical
details of the system. Systems display universality in a scaling limit, when a large number of interacting parts come together. The modern meaning of the term was introduced by Leo Kadanoff
in the 1960s, but a simpler version of the concept was already implicit in the van der Waals equation
and in the earlier Landau theory of phase transitions, which did not incorporate scaling correctly.
The term is slowly gaining a broader usage in several fields of mathematics, including combinatorics
and probability theory
, whenever the quantitative features of a structure (such as asymptotic behaviour) can be deduced from a few global parameters appearing in the definition, without requiring knowledge of the details of the system.
The renormalization group
explains universality. It classifies operators in a statistical field theory into relevant and irrelevant. Relevant operators are those perturbations to the free energy, the imaginary time Lagrangian, that will affect the continuum limit, and can be seen at long distances. Irrelevant operators are those that only change the short-distance details. The collection of scale-invariant statistical theories define the universality classes, and the finite dimensional list of coefficients of relevant operators parametrize the near critical behavior.
s in statistical mechanics. A phase transition occurs when a material changes its properties in a dramatic way: water, as it is heated boils and turns into vapor; or a magnet, when heated, loses its magnetism. Phase transitions are characterized by an order parameter, such as the density or the magnetization, that changes as a function of a parameter of the system, such as the temperature. The special value of the parameter at which the system changes its phase is the system's critical point
. For systems that exhibit universality, the closer the parameter is to its critical value
, the less sensitively the order parameter depends on the details of the system.
If the parameter β is critical at the value βc, then the order parameter a will be well approximated by
The exponent α is a critical exponent
of the system. The remarkable discovery made in the second half of the twentieth century was that very different systems had the same critical exponents.
In 1976, Mitchell Feigenbaum
discovered universality in iterated maps.
in the 1970s and the 1980s was the realization that statistical field theory, similar to quantum field theory, could be used to provide a microscopic theory of universality. The core observation was that, for all of the different systems, the behaviour at a phase transition
is described by a continuum field, and that the same statistical field theory will describe different systems. The scaling exponents in all of these systems can be derived from the field theory alone, and are known as critical exponents.
The key observation is that near a phase transition or critical point
, disturbances occur at all size scales, and thus one should look for an explicitly scale-invariant theory to describe the phenomena, as seems to have been put in a formal theoretical framework first by Pokrovsky
and Patashinsky in 1965. Universality is a by-product of the fact that there are relatively few scale-invariant theories. For any one specific physical system, the detailed description may have many scale-dependent parameters and aspects. However, as the phase transition is approached, the scale-dependent parameters play less and less of an important role, and the scale-invariant parts of the physical description dominate. Thus, a simplified, and often exactly solvable, model can be used to approximate the behaviour of these systems near the critical point.
Percolation may be modeled by a random electrical resistor network, with electricity flowing from one side of the network to the other. The overall resistance of the network is seen to be described by the average connectivity of the resistors in the network.
The formation of tears and cracks may be modeled by a random network of electrical fuses. As the electric current flow through the network is increased, some fuses may pop, but on the whole, the current is shunted around the problem areas, and uniformly distributed. However, at a certain point (at the phase transition) a cascade failure may occur, where the excess current from one popped fuse overloads the next fuse in turn, until the two sides of the net are completely disconnected and no more current flows.
To perform the analysis of such random-network systems, one considers the stochastic space of all possible networks (that is, the canonical ensemble
), and performs a summation (integration) over all possible network configurations. As in the previous discussion, each given random configuration is understood to be drawn from the pool of all configurations with some given probability distribution; the role of temperature in the distribution is typically replaced by the average connectivity of the network.
The expectation values of operators, such as the rate of flow, the heat capacity
, and so on, are obtained by integrating over all possible configurations. This act of integration over all possible configurations is the point of commonality between systems in statistical mechanics
and quantum field theory
. In particular, the language of the renormalization group
may be applied to the discussion of the random network models. In the 1990s and 2000s, stronger connections between the statistical models and conformal field theory
were uncovered. The study of universality remains a vital area of research.
(such as entropy
and master equation
s), universality has proven a useful construct for characterizing distributed systems at a higher level, such as multi-agent systems. The term has been applied to multi-agent simulations, where the system-level behavior exhibited by the system is independent of the degree of complexity of the individual agents, being driven almost entirely by the nature of the constraints governing their interactions.
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, universality is the observation that there are properties for a large class of systems that are independent of the dynamical
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
details of the system. Systems display universality in a scaling limit, when a large number of interacting parts come together. The modern meaning of the term was introduced by Leo Kadanoff
Leo Kadanoff
Leo Philip Kadanoff is an American physicist. He is a professor of physics at the University of Chicago and a former President of the American Physical Society . He has contributed to the fields of statistical physics, chaos theory, and theoretical condensed matter physics.-Biography:Kadanoff...
in the 1960s, but a simpler version of the concept was already implicit in the van der Waals equation
Van der Waals equation
The van der Waals equation is an equation of state for a fluid composed of particles that have a non-zero volume and a pairwise attractive inter-particle force It was derived by Johannes Diderik van der Waals in 1873, who received the Nobel prize in 1910 for "his work on the equation of state for...
and in the earlier Landau theory of phase transitions, which did not incorporate scaling correctly.
The term is slowly gaining a broader usage in several fields of mathematics, including combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
and probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, whenever the quantitative features of a structure (such as asymptotic behaviour) can be deduced from a few global parameters appearing in the definition, without requiring knowledge of the details of the system.
The renormalization group
Renormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
explains universality. It classifies operators in a statistical field theory into relevant and irrelevant. Relevant operators are those perturbations to the free energy, the imaginary time Lagrangian, that will affect the continuum limit, and can be seen at long distances. Irrelevant operators are those that only change the short-distance details. The collection of scale-invariant statistical theories define the universality classes, and the finite dimensional list of coefficients of relevant operators parametrize the near critical behavior.
Universality in statistical mechanics
The notion of universality originated in the study of phase transitionPhase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
s in statistical mechanics. A phase transition occurs when a material changes its properties in a dramatic way: water, as it is heated boils and turns into vapor; or a magnet, when heated, loses its magnetism. Phase transitions are characterized by an order parameter, such as the density or the magnetization, that changes as a function of a parameter of the system, such as the temperature. The special value of the parameter at which the system changes its phase is the system's critical point
Critical point (thermodynamics)
In physical chemistry, thermodynamics, chemistry and condensed matter physics, a critical point, also called a critical state, specifies the conditions at which a phase boundary ceases to exist...
. For systems that exhibit universality, the closer the parameter is to its critical value
Critical value
-Differential topology:In differential topology, a critical value of a differentiable function between differentiable manifolds is the image ƒ in N of a critical point x in M.The basic result on critical values is Sard's lemma...
, the less sensitively the order parameter depends on the details of the system.
If the parameter β is critical at the value βc, then the order parameter a will be well approximated by
The exponent α is a critical exponent
Critical exponent
Critical exponents describe the behaviour of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e...
of the system. The remarkable discovery made in the second half of the twentieth century was that very different systems had the same critical exponents.
In 1976, Mitchell Feigenbaum
Mitchell Feigenbaum
Mitchell Jay Feigenbaum is a mathematical physicist whose pioneering studies in chaos theory led to the discovery of the Feigenbaum constants.- Biography :...
discovered universality in iterated maps.
Examples
Universality gets its name because it is seen in a large variety of physical systems. Examples of universality include:- AvalancheAvalancheAn avalanche is a sudden rapid flow of snow down a slope, occurring when either natural triggers or human activity causes a critical escalating transition from the slow equilibrium evolution of the snow pack. Typically occurring in mountainous terrain, an avalanche can mix air and water with the...
s in piles of sand. The likelihood of an avalanche is in power-law proportion to the size of the avalanche, and avalanches are seen to occur at all size scales. This is termed "self-organized criticalitySelf-organized criticalityIn physics, self-organized criticality is a property of dynamical systems which have a critical point as an attractor. Their macroscopic behaviour thus displays the spatial and/or temporal scale-invariance characteristic of the critical point of a phase transition, but without the need to tune...
". - The formation and propagation of cracks and tears in materials ranging from steel (e.g., scissors) to rock to paper. The variations of the direction of the tear, or the roughness of a fractured surface, are in power-law proportion to the size scale.
- The electrical breakdownElectrical breakdownThe term electrical breakdown or electric breakdown has several similar but distinctly different meanings. For example, the term can apply to the failure of an electric circuit....
of dielectricDielectricA dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...
s, which resemble cracks and tears. - The percolationPercolationIn physics, chemistry and materials science, percolation concerns the movement and filtering of fluids through porous materials...
of fluids through disordered media, such as petroleumPetroleumPetroleum or crude oil is a naturally occurring, flammable liquid consisting of a complex mixture of hydrocarbons of various molecular weights and other liquid organic compounds, that are found in geologic formations beneath the Earth's surface. Petroleum is recovered mostly through oil drilling...
through fractured rock beds, or water through filter paper, such as in chromatographyChromatographyChromatography is the collective term for a set of laboratory techniques for the separation of mixtures....
. Power-law scaling connects the rate of flow to the distribution of fractures. - The diffusionDiffusionMolecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
of moleculeMoleculeA molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
s in solutionSolutionIn chemistry, a solution is a homogeneous mixture composed of only one phase. In such a mixture, a solute is dissolved in another substance, known as a solvent. The solvent does the dissolving.- Types of solutions :...
, and the phenomenon of diffusion-limited aggregationDiffusion-limited aggregationDiffusion-limited aggregation is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles. This theory, proposed by Witten and Sander in 1981, is applicable to aggregation in any system where diffusion is the primary means...
. - The distribution of rocks of different sizes in an aggregate mixture that is being shaken (with gravity acting on the rocks).
- The appearance of critical opalescenceCritical opalescenceCritical opalescence is a phenomenon which arises in the region of a continuous, or second-order, phase transition. Originally reported by Thomas Andrews in 1869 for the liquid-gas transition in carbon dioxide, many other examples have been discovered since. The phenomenon is most commonly...
in fluids near a phase transitionPhase transitionA phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
.
Theoretical overview
One of the important developments in materials scienceMaterials science
Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. This scientific field investigates the relationship between the structure of materials at atomic or molecular scales and their macroscopic properties. It incorporates...
in the 1970s and the 1980s was the realization that statistical field theory, similar to quantum field theory, could be used to provide a microscopic theory of universality. The core observation was that, for all of the different systems, the behaviour at a phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
is described by a continuum field, and that the same statistical field theory will describe different systems. The scaling exponents in all of these systems can be derived from the field theory alone, and are known as critical exponents.
The key observation is that near a phase transition or critical point
Critical point (thermodynamics)
In physical chemistry, thermodynamics, chemistry and condensed matter physics, a critical point, also called a critical state, specifies the conditions at which a phase boundary ceases to exist...
, disturbances occur at all size scales, and thus one should look for an explicitly scale-invariant theory to describe the phenomena, as seems to have been put in a formal theoretical framework first by Pokrovsky
Valery Pokrovsky
Valery Pokrovsky is a Soviet and Russian physicist. He is a member ofthe Landau Institute in Chernogolovka nearMoscow in Russia and a professor for Theoretical Physics at Texas A&M University....
and Patashinsky in 1965. Universality is a by-product of the fact that there are relatively few scale-invariant theories. For any one specific physical system, the detailed description may have many scale-dependent parameters and aspects. However, as the phase transition is approached, the scale-dependent parameters play less and less of an important role, and the scale-invariant parts of the physical description dominate. Thus, a simplified, and often exactly solvable, model can be used to approximate the behaviour of these systems near the critical point.
Percolation may be modeled by a random electrical resistor network, with electricity flowing from one side of the network to the other. The overall resistance of the network is seen to be described by the average connectivity of the resistors in the network.
The formation of tears and cracks may be modeled by a random network of electrical fuses. As the electric current flow through the network is increased, some fuses may pop, but on the whole, the current is shunted around the problem areas, and uniformly distributed. However, at a certain point (at the phase transition) a cascade failure may occur, where the excess current from one popped fuse overloads the next fuse in turn, until the two sides of the net are completely disconnected and no more current flows.
To perform the analysis of such random-network systems, one considers the stochastic space of all possible networks (that is, the canonical ensemble
Canonical ensemble
The canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system...
), and performs a summation (integration) over all possible network configurations. As in the previous discussion, each given random configuration is understood to be drawn from the pool of all configurations with some given probability distribution; the role of temperature in the distribution is typically replaced by the average connectivity of the network.
The expectation values of operators, such as the rate of flow, the heat capacity
Heat capacity
Heat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...
, and so on, are obtained by integrating over all possible configurations. This act of integration over all possible configurations is the point of commonality between systems in statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
and quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
. In particular, the language of the renormalization group
Renormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
may be applied to the discussion of the random network models. In the 1990s and 2000s, stronger connections between the statistical models and conformal field theory
Conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...
were uncovered. The study of universality remains a vital area of research.
Applications to other fields
Like other concepts from statistical mechanicsStatistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
(such as entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
and master equation
Master equation
In physics and chemistry and related fields, master equations are used to describe the time-evolution of a system that can be modelled as being in exactly one of countable number of states at any given time, and where switching between states is treated probabilistically...
s), universality has proven a useful construct for characterizing distributed systems at a higher level, such as multi-agent systems. The term has been applied to multi-agent simulations, where the system-level behavior exhibited by the system is independent of the degree of complexity of the individual agents, being driven almost entirely by the nature of the constraints governing their interactions.