159 research outputs found
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
A self-consistent perturbative evaluation of ground state energies: application to cohesive energies of spin lattices
The work presents a simple formalism which proposes an estimate of the ground
state energy from a single reference function. It is based on a perturbative
expansion but leads to non linear coupled equations. It can be viewed as well
as a modified coupled cluster formulation. Applied to a series of spin lattices
governed by model Hamiltonians the method leads to simple analytic solutions.
The so-calculated cohesive energies are surprisingly accurate. Two examples
illustrate its applicability to locate phase transition.Comment: Accepted by Phys. Rev.
A new approach to quantitative propagation of chaos for drift, diffusion and jump processes
This paper is devoted the the study of the mean field limit for many-particle
systems undergoing jump, drift or diffusion processes, as well as combinations
of them. The main results are quantitative estimates on the decay of
fluctuations around the deterministic limit and of correlations between
particles, as the number of particles goes to infinity. To this end we
introduce a general functional framework which reduces this question to the one
of proving a purely functional estimate on some abstract generator operators
(consistency estimate) together with fine stability estimates on the flow of
the limiting nonlinear equation (stability estimates). Then we apply this
method to a Boltzmann collision jump process (for Maxwell molecules), to a
McKean-Vlasov drift-diffusion process and to an inelastic Boltzmann collision
jump process with (stochastic) thermal bath. To our knowledge, our approach
yields the first such quantitative results for a combination of jump and
diffusion processes.Comment: v2 (55 pages): many improvements on the presentation, v3: correction
of a few typos, to appear In Probability Theory and Related Field
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
in --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, of the interaction parameter. We show that
the uniform density (which may be interpreted as the liquid phase) is
dynamically stable for and prove, abstractly, that a
{\it critical} transition must occur at . However for
this system we show that under generic conditions -- large, 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
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.
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
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