# Homogenisation for 14-year-olds (as written on a recent occasion)

### WARNING: You must not read this if you are fifteen or older!

Many things around us, both nature and man made, look as "homogeneous" ("uniform"), yet they contain a "microstructure", often invisible by naked eye but determining the "visible" properties of the things. For example, metals appear to be "polycrystals" made of a large number of bonded differently oriented small crystals, with each crystal having a pretty regular pattern of atoms but also a large number of "defects" whose presence allows for the metals to "flow" (to deform "plastically") hence letting us making of the metals things of desirable shapes. Wood or human bone look in the microscope as porous materials with fancy patterns, "nature made" microstructures. An aircraft has to be made of materials, which are both strong and light, glass in the car's windscreen or just at somebody's nose has to be both transparent and impact proof. The best way for achieving these is to use "mixtures" of natural materials, the so-called "composites", by arranging the components in a special way for the desirable properties. The properties of the "mixtures" will depend not only on the properties of the components but also on the "microgeometry" of mixing: if we dissolve stiff round particles in soft medium and then other way round we will get two different composites with different properties, even if we took equal initial ingredients. The problem of predicting how the properties of mixtures will depend on the properties of the components and of the microgeometry is the topic of Homogenisation Theory which uses advanced mathematics. The properties of each component material are described by relevant "differential equations", but since those are mixed the "coefficients" of this equations vary rapidly, and so are their solutions. In practice we do not need knowing the "exact" solution but rather some "mean" about which it (rapidly) varies. The homogenisation theory concerns about making precise mathematical meaning of this mean, deriving equations for it (the "homogenised equations") and assessing the error (hopefully small!) resulting from the replacement of the "actual" solution by the mean. This project suggests the use of some advanced mathematics to construct "higher-order" homogenised equations and the "means" them solving, which can reduce the error quite dramatically. The homogenisation theory has many diverse applications. One more on which this project focusses is relevant to fibre optics and so called "photonic crystals" which allow to transmit the information long distances with little loss. The information is "coded" in electromagnetic waves, which go along the channel and are unable to escape since the surrounding medium is a composite (photonic crystal), which does not propagate ("localises") those particular waves. On the other hand, if the waves do propagate, their propagation in the composites may be very different from that in a "uniform" medium: for example, those can travel in small bunches (the "wave packages") with very different speeds, which effects the homogenisation theory can also explain and predict.