The existence of incommensurate structures proves that a crystalline ordering is not always the most stable one for non-quantum matter. Some properties of structures which are obtained by minimizing a free energy are investigated in the Frenkel-Kontorova and related models. It is shown that an incommensurate structure can be either quasi-sinusoidal with a phason mode or built out of a sequence of equidistant defects (discommensurations) which are locked to the lattice by the Peierls force. In that situation the variation of the commensurability ratio with physical parameters forms a « complete devil's staircase » with interesting physical consequences. Some general results for all structures which minimize a free energy are given. In addition to the known crystal and incommensurate structures, the existence of a new class of structures which have local order at all scale is predicted. Properties of the new class are described in physical terms and possible applications to certain amorphous or non-stoichiometric compounds are discussed.

Devil's staircase and order without periodicity in classical condensed matter

Frustration is introduced in randomly connected totalistic cellular automata via mixing rules leading to incompatible periods. As the respective concentration of rules is varied, these cellular automata go through eight phases, many of which with symmetries different from the two rules mixed in. The complex phase diagram so created is similar to those seen in frustrated systems in static equilibrium. It shows that a minimized free energy is not necessary for frustration to produce this rich behaviour.

Multiple phase changes induced by frustration in randomly connected cellular automata

It is well known that extrema of classical, one-dimensional systems can be viewed as trajectories of a nonlinear, volume-preserving map. However, in general, thermodynamically stable extrema (i.e., local minima) of the free energy are numerically unstable trajectories of the map and so difficult to study by this technique. Here we explore a recent idea involving the use of a dissipative map to study the same kind of problem. Such a map may be designed so that its attractors are equal to, or close to, local minima of the energy. We apply such a dissipative map to the mean-field ANNNI model. We find that the technique reliably locates many kinds of metastable state: ferromagnetic, paramagnetic, commensurate, and incommensurate. However, the map also has two notable failings: it (apparently) has no chaotic attractors; and it fails to find correctly metastable states in a region of the phase diagram which is dominated by incommensurate states.

A study of the ANNNI model using a dissipative map

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A study of the ANNNI model using a dissipative map