Uniform convergence to equilibrium for granular media

We study the long time asymptotics of a nonlinear, nonlocal equation used in the modelling of granular media. We prove a uniform exponential convergence to equilibrium for degenerately convex and non convex interaction or confinement potentials, improving in particular results by J. A. Carrillo, R. J. McCann and C. Villani. The method is based on studying the dissipation of the Wasserstein distance between a solution and the steady state

Invariant densities for dynamical systems with random switching

We consider a non-autonomous ordinary differential equation on a smooth manifold, with right-hand side that randomly switches between the elements of a finite family of smooth vector fields. For the resulting random dynamical system, we show that H\"ormander type hypoellipticity conditions are sufficient for uniqueness and absolute continuity of an invariant measure.Comment: 16 pages; we replaced our original article to point out and close a gap in the discussion of the Lorenz system in Section 7 (see Remark 2); this gap is only present in the journal version of this article --- it wasn't present in the previous arxiv versio

Self‐consistent intermediate Hamiltonians : A coupled cluster type formulation of the singles and doubles configuration interaction matrix dressing

This paper presents a new self‐consistent dressing of a singles and doubles configuration interaction matrix which insures size‐consistency, separability into closed‐shell subsystems if localized molecular orbitals (MOs) are used, and which includes all fourth order corrections. This method yields, among several schemes, a reformulation of the coupled cluster method, including fully the cluster operators of single and double excitations, and partially those of the triples (Bartlett’s algorithm named CCSDT‐1a). Further improvement can be easily included by adding exclusion principle violating corrections. Since it leads to a matrix diagonalization, the method behaves correctly in case of near degeneracies between the reference determinant and some doubles. Due to its flexibility this formulation offers the possibility of consistent combination with less expensive treatments for the study of very large [email protected] ; [email protected]

Direct generation of local orbitals for multireference treatment and subsequent uses for the calculation of the correlation energy

We present a method that uses the one-particle density matrix to generate directly localized orbitals dedicated to multireference wave functions. On one hand, it is shown that the definition of local orbitals making possible physically justified truncations of the CAS ~complete active space! is particularly adequate for the treatment of multireference problems. On the other hand, as it will be shown in the case of bond breaking, the control of the spatial location of the active orbitals may permit description of the desired physics with a smaller number of active orbitals than when starting from canonical molecular orbitals. The subsequent calculation of the dynamical correlation energy can be achieved with a lower computational effort either due to this reduction of the active space, or by truncation of the CAS to a shorter set of references. The ground- and excited-state energies are very close to the current complete active space self-consistent field ones and several examples of multireference singles and doubles calculations illustrate the interest of the procedur

Local character of magnetic coupling in ionic solids

Magnetic interactions in ionic solids are studied using parameter-free methods designed to provide accurate energy differences associated with quantum states defining the Heisenberg constant J. For a series of ionic solids including KNiF3, K2NiF4, KCuF3, K2CuF4, and high- Tc parent compound La2CuO4, the J experimental value is quantitatively reproduced. This result has fundamental implications because J values have been calculated from a finite cluster model whereas experiments refer to infinite solids. The present study permits us to firmly establish that in these wide-gap insulators, J is determined from strongly local electronic interactions involving two magnetic centers only thus providing an ab initio support to commonly used model Hamiltonians

On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures

This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the consequence of the log-Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics 225

The McKean-Vlasov Equation in Finite Volume

We study the McKean--Vlasov equation on the finite tori of length scale $L$ in $d$--dimensions. We derive the necessary and sufficient conditions for the existence of a phase transition, which are based on the criteria first uncovered in \cite{GP} and \cite{KM}. Therein and in subsequent works, one finds indications pointing to critical transitions at a particular model dependent value, $\theta^{\sharp}$ of the interaction parameter. We show that the uniform density (which may be interpreted as the liquid phase) is dynamically stable for $\theta < \theta^{\sharp}$ and prove, abstractly, that a {\it critical} transition must occur at $\theta = \theta^{\sharp}$. However for this system we show that under generic conditions -- $L$ large, $d \geq 2$ and isotropic interactions -- the phase transition is in fact discontinuous and occurs at some \theta\t < \theta^{\sharp}. Finally, for H--stable, bounded interactions with discontinuous transitions we show that, with suitable scaling, the \theta\t(L) tend to a definitive non--trivial limit as $L\to\infty$

Many-body-QED perturbation theory: Connection to the Bethe-Salpeter equation

The connection between many-body theory (MBPT)--in perturbative and non-perturbative form--and quantum-electrodynamics (QED) is reviewed for systems of two fermions in an external field. The treatment is mainly based upon the recently developed covariant-evolution-operator method for QED calculations [Lindgren et al. Phys. Rep. 389, 161 (2004)], which has a structure quite akin to that of many-body perturbation theory. At the same time this procedure is closely connected to the S-matrix and the Green's-function formalisms and can therefore serve as a bridge between various approaches. It is demonstrated that the MBPT-QED scheme, when carried to all orders, leads to a Schroedinger-like equation, equivalent to the Bethe-Salpeter (BS) equation. A Bloch equation in commutator form that can be used for an "extended" or quasi-degenerate model space is derived. It has the same relation to the BS equation as has the standard Bloch equation to the ordinary Schroedinger equation and can be used to generate a perturbation expansion compatible with the BS equation also for a quasi-degenerate model space.Comment: Submitted to Canadian J of Physic

Proposal of an extended t-J Hamiltonian for high-Tc cuprates from ab initio calculations on embedded clusters

A series of accurate ab initio calculations on Cu_pO-q finite clusters, properly embedded on the Madelung potential of the infinite lattice, have been performed in order to determine the local effective interactions in the CuO_2 planes of La_{2-x}Sr_xCuO_4 compounds. The values of the first-neighbor interactions, magnetic coupling (J_{NN}=125 meV) and hopping integral (t_{NN}=-555 meV), have been confirmed. Important additional effects are evidenced, concerning essentially the second-neighbor hopping integral t_{NNN}=+110meV, the displacement of a singlet toward an adjacent colinear hole, h_{SD}^{abc}=-80 meV, a non-negligible hole-hole repulsion V_{NN}-V_{NNN}=0.8 eV and a strong anisotropic effect of the presence of an adjacent hole on the values of the first-neighbor interactions. The dependence of J_{NN} and t_{NN} on the position of neighbor hole(s) has been rationalized from the two-band model and checked from a series of additional ab initio calculations. An extended t-J model Hamiltonian has been proposed on the basis of these results. It is argued that the here-proposed three-body effects may play a role in the charge/spin separation observed in these compounds, that is, in the formation and dynamic of stripes.Comment: 24 pages, 4 figures, submitted to Phys. Rev.

Ab initio evaluation of local effective interactions in α′NaV2O5\alpha^\prime NaV_2O_5

We will present the numerical evaluation of the hopping and magnetic exchange integrals for a nearest-neighbor $t-J$ model of the quarter-filled $\alpha^\prime NaV_2O_5$ compound. The effective integrals are obtained from valence-spectroscopy {\em ab initio} calculations of embedded crystal fragments (two $VO_5$ pyramids in the different geometries corresponding to the desired parameters). We are using a large configurations interaction (CI) method, where the CI space is specifically optimized to obtain accurate energy differences. We show that the $\alpha^\prime NaV_2O_5$ system can be seen as a two-dimensional asymmetric triangular Heisenberg lattice where the effective sites represent delocalized $V-O-V$ rung entities supporting the magnetic electrons.Comment: 24 pages, 5 figure