Extended Czjzek model applied to NMR parameter distributions in sodium metaphosphate glass

The Extended Czjzek Model (ECM) is applied to the distribution of NMR parameters of a simple glass model (sodium metaphosphate, $\mathrm{NaPO_3}$) obtained by Molecular Dynamics (MD) simulations. Accurate NMR tensors, Electric Field Gradient (EFG) and Chemical Shift Anisotropy (CSA), are calculated from Density Functional Theory (DFT) within the well-established PAW/GIPAW framework. Theoretical results are compared to experimental high-resolution solid-state NMR data and are used to validate the considered structural model. The distributions of the calculated coupling constant $C_Q\propto |V_{zz}|$ and of the asymmetry parameter $\eta_Q$ that characterize the quadrupolar interaction are discussed in terms of structural considerations with the help of a simple point charge model. Finally, the ECM analysis is shown to be relevant for studying the distribution of CSA tensor parameters and gives new insight into the structural characterization of disordered systems by solid-state NMR.Comment: 17 pages, 12 figures to be published in J. Phys.: Condens. Matte

Neutron powder diffraction and Mössbauer spectroscopy (119Sn and 155Gd) studies of the CeScSi-type GdMgSn and GdMgPb compounds

International audienceThe magnetic structures of the CeScSi-type GdMgSn and GdMgPb compounds have been studied by both neutron powder diffraction and Mössbauer spectroscopy (119Sn and 155Gd). The neutron diffraction results show that the two compounds adopt incommensurate antiferromagnetic structures at 5.4 K with propagation vectors k⃗ =[0.910,0.077,0] for GdMgSn and k⃗ =[0.892,0,0] for GdMgPb. The magnetic moments lie in the basal plane, which is confirmed by both 119Sn and 155Gd Mössbauer spectroscopy. Mössbauer spectroscopy refinements and simulations reveal that the magnetic structure of GdMgSn is cycloidal at low temperature and undergoes a transition to a modulated magnetic structure above T∼40 K. A similar magnetic transition is inferred for GdMgPb. The magnetic structures of GdMgSn and GdMgPb are compared with those of other CeScSi-type compounds

A Pearson-Dirichlet random walk

A constrained diffusive random walk of n steps and a random flight in Rd, which can be expressed in the same terms, were investigated independently in recent papers. The n steps of the walk are identically and independently distributed random vectors of exponential length and uniform orientation. Conditioned on the sum of their lengths being equal to a given value l, closed-form expressions for the distribution of the endpoint of the walk were obtained altogether for any n for d=1, 2, 4 . Uniform distributions of the endpoint inside a ball of radius l were evidenced for a walk of three steps in 2D and of two steps in 4D. The previous walk is generalized by considering step lengths which are distributed over the unit (n-1) simplex according to a Dirichlet distribution whose parameters are all equal to q, a given positive value. The walk and the flight above correspond to q=1. For any d >= 3, there exist, for integer and half-integer values of q, two families of Pearson-Dirichlet walks which share a common property. For any n, the d components of the endpoint are jointly distributed as are the d components of a vector uniformly distributed over the surface of a hypersphere of radius l in a space Rk whose dimension k is an affine function of n for a given d. Five additional walks, with a uniform distribution of the endpoint in the inside of a ball, are found from known finite integrals of products of powers and Bessel functions of the first kind. They include four different walks in R3 and two walks in R4. Pearson-Liouville random walks, obtained by distributing the total lengths of the previous Pearson-Dirichlet walks, are finally discussed.Comment: 33 pages 1 figure, the paper includes the content of a recently submitted work together with additional results and an extended section on Pearson-Liouville random walk

A New Family of Solvable Pearson-Dirichlet Random Walks

International audienceAn n-step Pearson-Gamma random walk in ℝ d starts at the origin and consists of n independent steps with gamma distributed lengths and uniform orientations. The gamma distribution of each step length has a shape parameter q>0. Constrained random walks of n steps in ℝ d are obtained from the latter walks by imposing that the sum of the step lengths is equal to a fixed value. Simple closed-form expressions were obtained in particular for the distribution of the endpoint of such constrained walks for any d≥ d 0 and any n≥2 when q is either q = d/2 - 1 ( d 0=3) or q= d-1 ( d 0=2) (Le Caër in J. Stat. Phys. 140:728-751, 2010). When the total walk length is chosen, without loss of generality, to be equal to 1, then the constrained step lengths have a Dirichlet distribution whose parameters are all equal to q and the associated walk is thus named a Pearson-Dirichlet random walk. The density of the endpoint position of a n-step planar walk of this type ( n≥2), with q= d=2, was shown recently to be a weighted mixture of 1+ floor( n/2) endpoint densities of planar Pearson-Dirichlet walks with q=1 (Beghin and Orsingher in Stochastics 82:201-229, 2010). The previous result is generalized to any walk space dimension and any number of steps n≥2 when the parameter of the Pearson-Dirichlet random walk is q= d>1. We rely on the connection between an unconstrained random walk and a constrained one, which have both the same n and the same q= d, to obtain a closed-form expression of the endpoint density. The latter is a weighted mixture of 1+ floor( n/2) densities with simple forms, equivalently expressed as a product of a power and a Gauss hypergeometric function. The weights are products of factors which depends both on d and n and Bessel numbers independent of d

Local fluctuations of eigenvalues of random matrices from the beta-Hermite ensemble

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Two-step Dirichlet random walks

International audienceRandom walks of n steps taken into independent uniformly random directions in a d-dimensional Euclidean space (d>1) , which are characterized by a sum of step lengths which is fixed and taken to be 1 without loss of generality, are named “Dirichlet” when this constraint is realized via a Dirichlet law of step lengths. The latter continuous multivariate distribution, which depends on n positive parameters, generalizes the beta distribution (n=2) . It is simply obtained from n independent gamma random variables with identical scale factors. Previous literature studies of these random walks dealt with symmetric Dirichlet distributions whose parameters are all equal to a value q which takes half-integer or integer values. In the present work, the probability density function of the distance from the endpoint to the origin is first made explicit for a symmetric Dirichlet random walk of two steps. It is valid for any positive value of q and for all d>1 . The latter pdf is used in turn to express the related density of a random walk of two steps whose length is distributed according to an asymmetric beta distribution which depends on two parameters, namely q and q+s where s is a positive integer

Distribution of the determinant of a random real-symmetric matrix from the Gaussian orthogonal ensemble

International audienceThe Mellin transform of the probability density of the determinant of N X N random real-symmetric matrices from the Gaussian orthogonal ensemble is calculated. The determinant probability density is given by a single Meijer G function for odd N. The distribution of the potential at the origin, within the Coulomb gas interpretation, is investigated from the Mellin transform of the determinant distribution and is shown to be asymptotically Gaussian

The distributions of the determinant of fixed-trace ensembles of real-symmetric and of Hermitian random matrices

International audienceProbability densities of the determinant are obtained for fixed-trace ensembles of N x N Hermitian and real-symmetric matrices from the exact determinant densities of the Gaussian orthogonal ensemble (GOE) and of the Gaussian unitary ensemble (GUE), respectively

The fixed-trace β-Hermite ensemble of random matrices and the low temperature distribution of the determinant of an N × N β-Hermite matrix

International audienceThe β-Hermite ensemble (β-HE) of tridiagonal N × N random matrices of Dumitriu and Edelman (2002 J. Math. Phys. 43 5830) is a continuum of ensembles in which β, the reciprocal of the temperature in the 2D electrostatic interpretation of the eigenvalue characteristics, can take any value. The eigenvalue distributions coincide with those of the classical Gaussian ensembles (GOE, GUE, GSE) for β = 1, 2, 4. A fixed-trace β-Hermite ensemble (β-FTHE) is defined from the β-HE and is used to extend the spherical ensembles of classical symmetries to β-spherical ensembles. At low temperature, when β → ∞, for a fixed value of N, the asymptotic distributions of reduced determinants DN,β of random N × N β-H and β-FTH matrices are shown to be standard Gaussians. Accordingly, the fluctuations of the potential at the origin, -ln |DN,β|, have a generalized Gumbel distribution at low temperature. For large N and large β, a ln(N) variance results from the strongly correlated fluctuations of eigenvalues around their equilibrium positions