Random surface in the smooth phase
String theories are quantum field theories in which the fundamental particles are tiny one dimensional strings, rather than points with no dimension. There has been great interest in string theories, since they potentially provide a long-awaited quantum theory of gravity, as well as a framework that may incorporate the standard quantum model of the other fundamental forces of nature. They are thus a candidate Theory of Everything (TOE). However string calculations are usually analytically intractible, so methods are being developed to do calculations numerically using computer simulation.
String theory calculations involve integrating over all possible two dimensional surfaces swept out by the string in some higher dimensional space-time. In order to compute this integral numerically, the surfaces are discretized as a triangulated mesh. The integral is then approximated by a sum over a large number of different meshes, which are obtained by making random changes to the mesh throughout the calculation, using a Monte Carlo method. The mesh is thus referred to as a dynamically triangulated random surface.
A host of chemical and biological systems can also be modeled by theories of random (fluid) fluctuating surfaces. Red blood cells `flicker' when their protein sheaths (cytoskeletons) are removed, seemingly acting as random fluctuating surfaces. In nature (and industry) there exist many examples of micro-emulsions, which are formed when one mixes oil, water and a surfactant together. The surfactant typically consists of a molecule with a head that is attracted to water and a tail that is repelled by it. The surfactant molecules thus align themselves along the oil-water boundary and reduce the boundary surface tension so that it becomes essentially a fluid fluctuating membrane (this is what happens when one adds egg to a mixture of oil and water to make mayonaisse).
Random surface in the smooth phase
This is not Cubist art, but rather a two dimensional dynamically triangulated random surface with toroidal boundary conditions embedded in a three dimensional space-time, representing a discretization of a model of string theory and quantum gravity. This is an example of a surface in the smooth phase.
Random surface in the crumpled phase
A random surface in the crumpled phase, just below the crumpling transition.
Wireframe rendering of a random surface
A wireframe rendering showing the complexity of a large dynamically triangulated random surface.
SLIDESHOW of the crumpling transition
Random surface models of quantum gravity may undergo a phase transition (called a crumpling transition) from a phase where the surfaces are smooth to a phase where the surfaces are spiky and crumpled. Such a phase transition is required in order for discrete random surface models to give results relevant to continuum quantum gravity. This is currently the subject of much research effort, to try to determine whether the change from smooth to crumpled is gradual, or a true phase transition.
Miscellaneous monochrome images of random surfaces:
