Critical states and minima for an energy with second–order gradients

Abstract

Coleman, Marcus & Mizel studied the thermodynamical equilibrium of second–order materials, i.e. materials for which the free–energy density depends not only on the concentration but also on its first and second gradients. They showed that in certain regimes such materials must exhibit equilibrium states that are non–uniform by proving that space–periodic solutions can have lower energy than space–uniform solutions. A dynamical systems approach is used here and it is shown that there is a critical value of the mean concentration. When the mean concentration crosses this critical value, the uniform state can lose its minimizing character and periodic or quasi–periodic states can have lower energy. Mathematically speaking, the presence of that critical value corresponds to a 1:1 resonance. The reversible 1:1 resonance has been studied extensively by Iooss & Peroueme, who showed that a reversible system having a 1:1 resonance singularity can be approximated as closely as desired by an integrable vector field. On this integrable field, all bounded solutions can be easily found and they are described by two integrals. However, in the present application, a zero eigenvalue is present in addition to the double imaginary eigenvalues. Bounded solutions are then described by three integrals. Moreover, since the average concentration must remain constant, a non–local constraint is added, which changes the results significantly. Finally, a selection principle for second–order materials stemming from energy considerations is introduced. The energy of all bounded solutions is compared with the energy of the uniform state. In some cases, non–uniform solutions have a lower energy and lead to non–uniform equilibrium states. More precisely, it is shown that space–periodic or space–quasi–periodic states are minima. Quite surprisingly, it may occur that non–uniform states appear by dilution, i.e. by decreasing the concentration.