2,266 research outputs found
Chebyshev, Legendre, Hermite and other orthonormal polynomials in D-dimensions
We propose a general method to construct symmetric tensor polynomials in the
D-dimensional Euclidean space which are orthonormal under a general weight. The
D-dimensional Hermite polynomials are a particular case of the present ones for
the case of a gaussian weight. Hence we obtain generalizations of the Legendre
and of the Chebyshev polynomials in D dimensions that reduce to the respective
well-known orthonormal polynomials in D=1 dimensions. We also obtain new
D-dimensional polynomials orthonormal under other weights, such as the
Fermi-Dirac, Bose-Einstein, Graphene equilibrium distribution functions and the
Yukawa potential. We calculate the series expansion of an arbitrary function in
terms of the new polynomials up to the fourth order and define orthonormal
multipoles. The explicit orthonormalization of the polynomials up to the fifth
order (N from 0 to 4) reveals an increasing number of orthonormalization
equations that matches exactly the number of polynomial coefficients indication
the correctness of the present procedure.Comment: 20 page
Fully dissipative relativistic lattice Boltzmann method in two dimensions
In this paper, we develop and characterize the fully dissipative Lattice
Boltzmann method for ultra-relativistic fluids in two dimensions using three
equilibrium distribution functions: Maxwell-J\"uttner, Fermi-Dirac and
Bose-Einstein. Our results stem from the expansion of these distribution
functions up to fifth order in relativistic polynomials. We also obtain new
Gaussian quadratures for square lattices that preserve the spatial resolution.
Our models are validated with the Riemann problem and the limitations of lower
order expansions to calculate higher order moments are shown. The kinematic
viscosity and the thermal conductivity are numerically obtained using the
Taylor-Green vortex and the Fourier flow respectively and these transport
coefficients are compared with the theoretical prediction from Grad's theory.
In order to compare different expansion orders, we analyze the temperature and
heat flux fields on the time evolution of a hot spot
Active nematics on flat surfaces: from droplet motility and scission to active wetting
We consider the dynamics of active nematics droplets on flat surfaces, based
on the continuum hydrodynamic theory. We investigate a wide range of dynamical
regimes as a function of the activity and droplet size on surfaces
characterized by strong anchoring and a range of equilibrium contact angles.
The activity was found to control a variety of dynamical regimes, including the
self-propulsion of droplets on surfaces, scission, active wetting and droplet
evaporation. Furthermore, we found that on a given surface (characterized by
the anchoring and the equilibrium contact angle) the dynamical regimes may be
controlled by the active capillary number of suspended droplets. We also found
that the active nematics concentration of the droplets varies with the
activity, affecting the wetting behaviour weakly but ultimately driving droplet
evaporation. Our analysis provides a global description of a wide range of
dynamical regimes reported for active nematics droplets and suggests a unified
description of droplets on surfaces. We discuss the key role of the finite size
of the droplet and comment on the suppression of these regimes in the infinite
size limit, where the active nematics is turbulent at any degree of activity
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