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

Coleman, Marcus & Mizel studied the thermodynamical equilibrium of second-o rder materials, i.e. materials for which the free-energy density depends no t only on the concentration but also on its first and second gradients. The y showed that in certain regimes such materials must exhibit equilibrium st ates 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 conce ntration. When the mean concentration crosses this critical value, the unif orm 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 resonanc e has been studied extensively by Iooss & Peroueme, who showed that a rever sible system having a 1:1 resonance singularity can be approximated as clos ely as desired by an integrable vector field. On this integrable field, all bounded solutions can be easily found and they are described by two integr als. However, in the present application, a zero eigenvalue is present in a ddition to the double imaginary eigenvalues. Bounded solutions are then des cribed by three integrals. Moreover, since the average concentration must r emain constant, a non-local constraint is added, which changes the results significantly. Finally, a selection principle for second-order materials st emming from energy considerations is introduced. The energy of all bounded solutions is compared with the energy of the uniform state. In some cad;es, non-uniform solutions have a lower energy and lead to non-uniform equilibr ium states. More precisely, it is shown that space-periodic or space-quasi- periodic states are minima. Quite surprisingly, it may occur that Iron-unif orm states appear by dilution, i.e. by decreasing the concentration.