Crystal Symmetry and Time Scales

The relation between the notion of crystalline symmetry and characteristic time intervals when this symmetry could be observed is analyzed. Several time scales are shown to exist for a system of interacting particles. It is only when the observation time is much larger than the mesoscopic fluctuation time, the notion of crystalline symmetry becomes physically meaningful. The ideas are concretized by a two-phase lattice model. 1 Characteristic Time Scales Crystalline symmetry is a well-defined geometric notion. But a crystal as such is a physical object consisting of many moving particles. In what sense these particles form a crystalline lattice with a given symmetry? An intuitive answer to this question would be that it is the average positions of particles which form a crystalline lattice, with the averaging performed over sufficiently large time. But, to be mathematically correct, it is necessary to concretize what does mean ”sufficiently large” with respect to the observation time. The aim of this report is to demonstrate that the minimal time scale providing the lower threshold for the observation time is the mesoscopic fluctuation time. There are several characteristic time scales for a system of interacting particles [1]. The smallest typical time is the interaction time τint ∼ as v , (1) in which as is an effective interaction radius, or effective particle size, or scattering length, and v is a characteristic particle velocity, say, sound velocity. When observing