Theoretical Overview
One of the important developments in materials 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. 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.
Read more about this topic: Universality (dynamical Systems)
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