Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPlate-nematic phase in three dimensions
124
0
0.0
(
0
)
Added by
Alessandro Giuliani
Publication date
2018
fields
Physics
and research's language is
English
Ask ChatGPT about the research
No Arabic 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.
rate research
Read More
We consider a monomer-dimer system with a strong attractive dimer-dimer interaction that favors alignment. In 1979, Heilmann and Lieb conjectured that this model should exhibit a nematic liquid crystal phase, in which the dimers are mostly aligned, but do not manifest any translational order. We prove this conjecture for large dimer activity and strong interactions. The proof follows a Pirogov-Sinai scheme, in which we map the dimer model to a system of hard-core polymers whose partition function is computed using a convergent cluster expansion.
Using overdamped Brownian dynamics simulations we investigate the isotropic-nematic (IN) transition of self-propelled rods in three spatial dimensions. For two well-known model systems (Gay-Berne potential and hard spherocylinders) we find that turning on activity moves to higher densities the phase boundary separating an isotropic phase from a (nonpolar) nematic phase. This active IN phase boundary is distinct from the boundary between isotropic and polar-cluster states previously reported in two-dimensional simulation studies and, unlike the latter, is not sensitive to the system size. We thus identify a generic feature of anisotropic active particles in three dimensions.
In [BEI] we introduced a Levy process on a hierarchical lattice which is four dimensional, in the sense that the Greens function for the process equals 1/x^2. If the process is modified so as to be weakly self-repelling, it was shown that at the critical killing rate (mass-squared) beta^c, the Greens function behaves like the free one. - Now we analyze the end-to-end distance of the model and show that its expected value grows as a constant times sqrt{T} log^{1/8}T (1+O((log log T)/log T)), which is the same law as has been conjectured for self-avoiding walks on the simple cubic lattice Z^4. The proof uses inverse Laplace transforms to obtain the end-to-end distance from the Greens function, and requires detailed properties of the Greens function throughout a sector of the complex beta plane. These estimates are derived in a companion paper [math-ph/0205028].
In this paper, we consider nearest-neighbor oriented percolation with independent Bernoulli bond-occupation probability on the $d$-dimensional body-centered cubic (BCC) lattice $mathbb{L}^d$ and the set of non-negative integers $mathbb{Z}_+$. Thanks to the nice structure of the BCC lattice, we prove that the infrared bound holds on $mathbb{L}^dtimesmathbb{Z}_+$ in all dimensions $dgeq 9$. As opposed to ordinary percolation, we have to deal with the complex numbers due to asymmetry induced by time-orientation, which makes it hard to estimate the bootstrapping functions in the lace-expansion analysis from above. By investigating the Fourier-Laplace transform of the random-walk Green function and the two-point function, we drive the key properties to obtain the upper bounds and resolve a problematic issue in Nguyen and Yangs bound.
We consider the isotropic perimeter generating functions of three-choice, imperfect, and 1-punctured staircase polygons, whose 8th order linear Fuchsian ODEs are previously known. We derive simple relationships between the three generating functions, and show that all three generating functions are joint solutions of a common 12th order Fuchsian linear ODE. We find that the 8th order differential operators can each be rewritten as a direct sum of a direct product, with operators no larger than 3rd order. We give closed-form expressions for all the solutions of these operators in terms of $_2F_1$ hypergeometric functions with rational and algebraic arguments. The solutions of these linear differential operators can in fact be expressed in terms of two modular forms, since these $_2F_1$ hypergeometric functions can be expressed with two, rational or algebraic, pullbacks.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
