Energy and interaction forces in classical two-dimensional crystals with inhomogeneous goarse-grained density - PEREIRA, Paulo César do Nascimento (2016)
In this M.Sc. thesis, we concentrate on classical two-dimensional crystals with soft pairwise interactions at low temperatures. Typically, the triangular lattice is the configuration which minimizes the interaction potential energy. Such energy is calculated through a lattice sum and we show some analytical approximations to it. We will be interested in cases where the coarse-grained density slightly depends on position. This can be caused by an external force on particles. Then the softness of interactions will determine how the coarse-grained density must vary. At equilibrium, the density gradient generates an equal and opposite force, resultant from interactions. In the limit of small gradients, the system has few defects and locally conserves the triangular lattice symmetry. Although the system’s configuration has a huge number of freedom degrees, only the position dependence of coarse-grained density is our relevant information at scales much greater than the nearest-neighbors’ distance. We then investigate the calculation of the resultant interaction force due to such density variations with position. A simple and intuitive, but not rigorous, way to obtain the Dynamical Density Functional Theory (DDFT) force is showed. Also, a microscopic approach giving the same result is proposed. In equilibrium, this force gives a minimization of the total free energy and it has been successful in many nonequilibrium systems. We show that this force fails in the case of long wavelength longitudinal waves, giving a smaller result for the sound speed. Also, in recent computer simulations, we obtained equilibrium configurations where the same correction in the force is needed. We show that such correction can be obtained by adding a correction term in the free energy, calculated as a functional of coarse-grained density.