Abstract
We consider a system of anisotropic plates in the three-dimensional continuum, interacting via purely hard core interactions. We assume that the particles have a finite number of allowed orientations. In a suitable range of densities, we prove the existence of a uni-axial nematic phase, characterized by long range orientational order (the minor axes are aligned parallel to each other, while the major axes are not) and no translational order. The proof is based on a coarse graining procedure, which allows us to map the plate model into a contour model, and in a rigorous control of the resulting contour theory, via Pirogov-Sinai methods.
Highlights
The mathematical theory of liquid crystalline (LC) phases, even just of their equilibrium properties, is still in a primitive stage: most of the predictions on the phase diagram of systems of anisotropic molecules are based on density functional, or mean field theories
As in any equilibrium statistical mechanics problem, one would like to start from a microscopic model of interacting particles, described in terms of a grand-canonical partition function at inverse temperature β and activity z, and derive bounds on the large distance decay of correlations, both for the orientational and the translational degrees of freedom of the particles, for different choices of (β, z)
If the density is well within the range where uni-axial nematic ordering is expected, that is, more precisely, if k−3α log k ρ kα−3, the system is, in a uni-axial, plate-like, nematic (N−) phase: in particular, we prove the existence of long range orientational order for the minor axes of the plates, and the absence of translational order, namely, exponential decay of the truncated center-center correlations
Summary
The mathematical theory of liquid crystalline (LC) phases, even just of their equilibrium properties, is still in a primitive stage: most of the predictions on the phase diagram of systems of anisotropic molecules are based on density functional, or mean field theories. From the heuristic discussion above, it would be tempting to think of the ‘uniformly magnetized’ regions, where both the axes of the plates are mostly aligned in a common direction, as a union of elementary slabs, each of which is a translate and/or rotation of the region J in (1) Even if natural, this choice creates difficulties in the treatment of the ‘transition layers’ between different uniformly magnetized regions: these layers, which are the basic constituents of the ‘Peierls’ contours’ generically have a wild geometric shape, which does not allow us to derive simple bounds on their probability, depending only on their volume. The methods of this paper did not allow us to overcome these difficulties: we limited ourselves to a range of densities where paving the space in cubes allow us to derive effective bounds on the probabilities of the ‘transition layers’, that is, of the connected components of the union of bad cubes
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call