10,132 research outputs found
Nematic liquid crystals : from Maier-Saupe to a continuum theory
We define a continuum energy functional in terms of the mean-field Maier-Saupe free energy, that describes both spatially homogeneous and inhomogeneous systems. The Maier-Saupe theory defines the main macroscopic variable, the Q-tensor order parameter, in terms of the second moment of a probability distribution function. This definition requires the eigenvalues of Q to be bounded both from below and above. We define a thermotropic bulk potential which blows up whenever the eigenvalues tend to these lower and upper bounds. This is in contrast to the Landau-de Gennes theory which has no such penalization. We study the asymptotics of this bulk potential in different regimes and discuss phase transitions predicted by this model
The hybrid SZ power spectrum: Combining cluster counts and SZ fluctuations to probe gas physics
Sunyaev-Zeldovich (SZ) effect from a cosmological distribution of clusters
carry information on the underlying cosmology as well as the cluster gas
physics. In order to study either cosmology or clusters one needs to break the
degeneracies between the two. We present a toy model showing how complementary
informations from SZ power spectrum and the SZ flux counts, both obtained from
upcoming SZ cluster surveys, can be used to mitigate the strong cosmological
influence (especially that of sigma_8) on the SZ fluctuations. Once the strong
dependence of the cluster SZ power spectrum on sigma_8 is diluted, the cluster
power spectrum can be used as a tool in studying cluster gas structure and
evolution. The method relies on the ability to write the Poisson contribution
to the SZ power spectrum in terms the observed SZ flux counts. We test the toy
model by applying the idea to simulations of SZ surveys.Comment: 12 pages. 11 plots. MNRAS submitte
Tangent unit-vector fields: nonabelian homotopy invariants and the Dirichlet energy
Let O be a closed geodesic polygon in S 2 . Maps from O into S 2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S 2 , we compute the infimum Dirichlet energy, E(H), for continuous maps satisfying tangent boundary conditions of arbitrary homotopy type H. The expression for E(H) involves a topological invariant â the spelling length â associated with the (nonabelian) fundamental group of the n-times punctured two-sphere, Ï1 (S 2 â {s1 , . . . , sn }, â). The lower bound for E(H) is obtained from combinatorial group theory arguments, while the upper bound is obtained by constructing explicit representatives which, on all but an arbitrarily small subset of O, are alternatively locally conformal or anticonformal. For conformal and anticonformal classes (classes containing wholly conformal and anticonformal representatives respectively), the expression for E(H) reduces to a previous result involving the degrees of a set of regular values s1 , . . . , sn in the target S 2 space. These degrees may be viewed as invariants associated with the abelianization of Ï1 (S 2 â {s1 , . . . , sn }, â). For nonconformal classes, however, E(H) may be strictly greater than the abelian bound. This stems from the fact that, for nonconformal maps, the number of preimages of certain regular values may necessarily be strictly greater than the absolute value of their degrees.\ud
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This work is motivated by the theoretical modelling of nematic liquid crystals in confined polyhedral geometries. The results imply new lower and upper bounds for the Dirichlet energy (one-constant Oseen-Frank energy) of reflection-symmetric tangent unit-vector fields in a rectangular prism
Energies of S^2-valued harmonic maps on polyhedra with tangent boundary conditions
A unit-vector field n:P \to S^2 on a convex polyhedron P \subset R^3
satisfies tangent boundary conditions if, on each face of P, n takes values
tangent to that face. Tangent unit-vector fields are necessarily discontinuous
at the vertices of P. We consider fields which are continuous elsewhere. We
derive a lower bound E^-_P(h) for the infimum Dirichlet energy E^inf_P(h) for
such tangent unit-vector fields of arbitrary homotopy type h. E^-_P(h) is
expressed as a weighted sum of minimal connections, one for each sector of a
natural partition of S^2 induced by P. For P a rectangular prism, we derive an
upper bound for E^inf_P(h) whose ratio to the lower bound may be bounded
independently of h. The problem is motivated by models of nematic liquid
crystals in polyhedral geometries. Our results improve and extend several
previous results.Comment: 42 pages, 2 figure
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