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Instatistical mechanics,universality is the observation that there are properties for a large class of systems that are independent of thedynamical 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 byLeo Kadanoff in the 1960s,[citation needed] but a simpler version of the concept was already implicit in thevan der Waals equation and in the earlierLandau theory of phase transitions, which did not incorporate scaling correctly.[citation needed]
The term is slowly gaining a broader usage in several fields of mathematics, includingcombinatorics andprobability 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.
Therenormalization group provides an intuitively appealing, albeit mathematically non-rigorous, explanation of universality. It classifies operators in astatistical field theory into relevant and irrelevant. Relevant operators are those responsible for perturbations to the free energy, theimaginary time Lagrangian, that will affect thecontinuum 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 theuniversality classes, and the finite-dimensional list of coefficients of relevant operators parametrize the near-critical behavior.
The notion of universality originated in the study ofphase transitions in statistical mechanics.[citation needed] 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 anorder 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'scritical point. For systems that exhibit universality, the closer the parameter is to itscritical 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 parametera will be well approximated by[citation needed]
The exponent α is acritical 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 .[citation needed]
In 1975,Mitchell Feigenbaum discovered universality in iterated maps.[1][2][3]
Universality gets its name because it is seen in a large variety of physical systems. Examples of universality include:
One of the important developments inmaterials science 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 .[citation needed] The core observation was that, for all of the different systems, the behaviour at aphase 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 ascritical exponents.
The key observation is that near a phase transition orcritical point, disturbances occur at all size scales, and thus one should look for an explicitlyscale-invariant theory to describe the phenomena, as seems to have been put in a formal theoretical framework first byPokrovsky and Patashinsky in 1965[4].[citation needed] Universality is a by-product of the fact that there are relatively few scale-invariant theories. For any one specificphysical 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 oftenexactly solvable, model can be used to approximate the behaviour of these systems near the critical point.
Percolation may be modeled by a randomelectrical 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 .[citation needed]
The formation of tears and cracks may be modeled by a random network ofelectrical 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) acascade 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 .[citation needed]
To perform the analysis of such random-network systems, one considers the stochastic space of all possible networks (that is, thecanonical 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 .[citation needed]
The expectation values of operators, such as the rate of flow, theheat 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 instatistical mechanics andquantum field theory. In particular, the language of therenormalization group may be applied to the discussion of the random network models. In the 1990s and 2000s, stronger connections between the statistical models andconformal field theory were uncovered. The study of universality remains a vital area of research.
Like other concepts fromstatistical mechanics (such asentropy andmaster equations), universality has proven a useful construct for characterizing distributed systems at a higher level, such asmulti-agent systems. The term has been applied[5] 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. Innetwork dynamics, universality refers to the fact that despite the diversity of nonlinear dynamic models, which differ in many details, the observed behavior of many different systems adheres to a set of universal laws. These laws are independent of the specific details of each system.[6]