Single-site Potential Research Articles

Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential

We consider a noninteracting unbounded spin system with conservation of the mean spin. We derive a uniform logarithmic Sobolev inequality (LSI) provided the single-site potential is a bounded perturbation of a strictly convex function. The scaling of the LSI constant is optimal in the system size. The argument adapts the two-scale approach of Grunewald, Villani, Westdickenberg and the second author from the quadratic to the general case. Using an asymmetric Brascamp-Lieb-type inequality for covariances, we reduce the task of deriving a uniform LSI to the convexification of the coarse-grained Hamiltonian, which follows from a general local Cram\'{e}r theorem.

Asymmetric covariance estimates of Brascamp–Lieb type and related inequalities for log-concave measures

Une inegalite de Brascamp et Lieb donne une estimation sur la covariance entre deux fonctions par rapport a une mesure log-concave, qui est bornee par le produit des normes $L^{2}$ des gradients des fonctions, ou l’amplitude du gradient est calculee en utilisant un produit scalaire egal a l’inverse de la matrice Hessienne du potentiel de la mesure log-concave. Menz et Otto [Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. (2011) Preprint] ont prouve une variante de ce resultat ou les normes $L^{2}$ sont remplacees par des normes $L^{1}$ et $L^{\infty}$, mais seulement dans $\mathbb{R}^{1}$. Nous prouvons une generalisation de ces deux resultats, avec une extension de ces inegalites a des normes $L^{p}$ et $L^{q}$ dans $\mathbb{R}^{n}$, pour tout $n\geq1$. Nous prouvons aussi une inegalite pour des integrales de differences divisees de fonctions a l’aide des integrales de leurs gradients.

Thermodynamic state-dependent structure-based coarse-graining of confined water

We develop thermodynamic state-dependent single-site isotropic coarse-grained potentials to predict the structure of water confined inside graphene slit-like channels by two multiscale simulation approaches: the coarse-grained molecular dynamics (CG-MD) and the empirical potential-based quasi-continuum theory (EQT). The structurally-consistent coarse-grained potentials for the CG-MD and EQT are systematically determined from the reference all-atom SPC/E water MD (AA-MD) results. For optimization of the CG-MD potentials, the relative entropy based coarse-graining method is used, and for determination of the EQT potentials, we develop a potential of mean force matching scheme. The optimized coarse-grained potentials are found to be dependent on the thermodynamic state. They are evaluated for their ability to predict the density profile of confined water, and it is found that the results obtained by the CG-MD and EQT simulations are in good agreement with the reference AA-MD results.

The Weak Localization for the Alloy-Type Anderson Model on a Cubic Lattice

We consider alloy type random Schr\"odinger operators on a cubic lattice whose randomness is generated by the sign-indefinite single-site potential. We derive Anderson localization for this class of models in the Lifshitz tails regime, i.e. when the coupling parameter $\lambda$ is small, for the energies $E \le -C \lambda^2$.

Generalized Pauli principle for particles with distinguishable traits

The s=3/2 Ising spin chain with uniform nearest-neighbor coupling, quadratic single-site potential, and magnetic field is shown to be equivalent to a system of 17 species of particles with internal structure. The same set of particles (with different energies) is shown to generate the spectrum of the s=1/2 Ising chain with dimerized nearest-neighbor coupling. The particles are free of interaction energies even at high densities. The mutual exclusion statistics of particles from all species is determined by their internal structure and encoded in a generalized Pauli principle. The exact statistical mechanical analysis can be performed for thermodynamically open or closed systems and with arbitrary energies assigned to all particle species. Special circumstances make it possible to merge two or more species into a single species. All traits that distinguish the original species become ignorable. The particles from the merged species are effectively indistinguishable and obey modified exclusion statistics. Different mergers may yield the same end product, implying that the inverse process (splitting any species into subspecies) is not unique. In a macroscopic system of two merged species at thermal equilibrium, the concentrations of the original species satisfy a functional relation governed by their mutual statistical interaction. That relation is derivable from an extremum principle. In the Ising context the system is open and the particle energies depend on the Hamiltonian parameters. Simple models of polymerization and solitonic paramagnetism each represent a closed system of two species that can transform into each other. Here they represent distinguishable traits with different energies of the same physical particle.

Simplicity of eigenvalues in Anderson-type models

We show almost sure simplicity of eigenvalues for several models of Anderson-type random Schrödinger operators, extending methods introduced by Simon for the discrete Anderson model. These methods work throughout the spectrum and are not restricted to the localization regime. We establish general criteria for the simplicity of eigenvalues which can be interpreted as separately excluding the absence of local and global symmetries, respectively. The criteria are applied to Anderson models with matrix-valued potential as well as with single-site potentials supported on a finite box.

Localization for Random Operators with Non-monotone Potentials with Exponentially Decaying Correlations

I consider random Schrodinger operators with exponentially decaying single site potential, which is allowed to change sign. For this model, I prove Anderson localization both in the sense of exponentially decaying eigenfunctions and dynamical localization. Furthermore, the results imply a Wegner-type estimate strong enough to use in classical forms of multi-scale analysis.

Anderson Localization for a Class of Models with a Sign-Indefinite Single-Site Potential via Fractional Moment Method

A technically convenient signature of Anderson localization is exponential decay of the fractional moments of the Green function within appropriate energy ranges. We consider a random Hamiltonian on a lattice whose randomness is generated by the sign-indefinite single-site potential, which is however sign-definite at the boundary of its support. For this class of Anderson operators we establish a finite-volume criterion which implies that above mentioned the fractional moment decay property holds. This constructive criterion is satisfied at typical perturbative regimes, e. g. at spectral boundaries which satisfy 'Lifshitz tail estimates' on the density of states and for sufficiently strong disorder. We also show how the fractional moment method facilitates the proof of exponential (spectral) localization for such random potentials.

Spectral properties of attractive bosons in a ring lattice including a single-site potential

The ground-state properties of attractive bosons trapped in a ring lattice including a single attractive potential well with an adjustable depth are investigated. The energy spectrum is reconstructed both in the strong-interaction limit and in the superfluid regime within the Bogoliubov picture. The analytical results thus obtained are compared with those found numerically from the exact Hamiltonian, in order to identify the regions in the parameter space where this picture is effective. The single potential introduced is the simplest way to break the translational symmetry and to observe, through a completely analytical approach, how the absence of symmetry affects the properties of the low-excited eigenstates of the system. This model gives a first insight into the properties of systems including more complex potentials.

Localization via fractional moments for models on with single-site potentials of finite support**Dedicated to the memory of Pierre Duclos.

One of the fundamental results in the theory of localization for discrete Schrödinger operators with random potentials is the exponential decay of Green's function and the absence of a continuous spectrum. In this paper, we provide a new variant of these results for one-dimensional alloy-type potentials with finitely supported sign-changing single-site potentials using the fractional moment method.

Lifshitz tails for alloy-type models in a constant magnetic field**Dedicated to the memory of Pierre Duclos.

In this paper, we study Lifshitz tails for a 2D Landau Hamiltonian perturbed by a random alloy-type potential constructed with single-site potentials decaying at least at a Gaussian speed. We prove that, if the Landau level stays preserved as a band edge for the perturbed Hamiltonian, at the Landau levels, the integrated density of states has a Lifshitz behavior of the type .RésuméDans cette note, nous démontrons qu'en dimension 2, la densité d'états intégrée d'un opérateur de Landau avec un potentiel aléatoire non négatif de type Anderson dont le potentiel de simple site décroît au moins aussi vite qu'une fonction gaussienne admet en chaque niveau de Landau, disons, 2bq, si celui-ci est un bord du spectre, une asymptotique de Lifshitz du type .

Wegner Estimate for Discrete Alloy-type Models

We study discrete alloy-type random Schr\"odinger operators on $\ell^2(\mathbb{Z}^d)$. Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. If the single site potential is compactly supported and the distribution of the coupling constant is of bounded variation a Wegner estimate holds. The bound is polynomial in the volume of the box and thus applicable as an ingredient for a localisation proof via multiscale analysis.

Lifshitz tails for generalized alloy-type random Schrödinger operators

We study Lifshitz tails for random Schrodinger operators where the random potential is alloy-type in the sense that the single site potentials are independent, identically distributed, but they may have various function forms. We suppose the single site potentials are distributed in a finite set of functions, and we show that under suitable symmetry conditions, they have a Lifshitz tail at the bottom of the spectrum except for special cases. When the single site potential is symmetric with respect to all the axes, we give a necessary and sufficient condition for the existence of Lifshitz tails. As an application, we show that certain random displacement models have a Lifshitz singularity at the bottom of the spectrum, and also complete our previous study (2009) of continuous Anderson type models.

Classical and Quantum Behavior of the Integrated Density of States for a Randomly Perturbed Lattice

The asymptotic behavior of the integrated density of states for a randomly perturbed lattice at the infimum of the spectrum is investigated. The leading term is determined when the decay of the single site potential is slow. The leading term depends only on the classical effect from the scalar potential. To the contrary, the quantum effect appears when the decay of the single site potential is fast. The corresponding leading term is estimated and the leading order is determined. In the multidimensional cases, the leading order varies in different ways from the known results in the Poisson case. The same problem is considered for the negative potential. These estimates are applied to investigate the long time asymptotics of Wiener integrals associated with the random potentials.

Wegner Estimates for Sign-Changing Single Site Potentials

We study Anderson and alloy type random Schr\"odinger operators on $\ell^2(\ZZ^d)$ and $L^2(\RR^d)$. Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. For a certain class of models we prove a Wegner estimate which is linear in the volume of the box and the length of the considered energy interval. The single site potential of the Anderson/alloy type model does not need to have fixed sign, but it needs be of a generalised step function form. The result implies the Lipschitz continuity of the integrated density of states.

Localization for the Random Displacement Model at Weak Disorder

This paper is devoted to the study of the random displacement model on \({\mathbb{R}^d}\) . We prove that, in the weak displacement regime, Anderson and dynamical localization hold near the bottom of the spectrum under a generic assumption on the single-site potential and a fairly general assumption on the support of the possible displacements. This result follows from the proof of the existence of Lifshitz tails and of a Wegner estimate for the model under scrutiny.

A new generic model potential for mesogenic systems: Square well line potential of variable range

A single-site pair potential is derived to approximate the linear n-site square well interaction. The resulting square well line (SWL) potential is analytical, fairly smooth, and reproduces the distance and orientation dependence of the multisite pair energy. It contains only three control parameters n, L, and s(2), in addition to the units of length s(1) and energy epsilon. The advantages of the new model over the traditional potentials such as Gay-Berne and Kihara are that n, L, and s(2) are physically meaningful quantities and that no additional adjustable parameters are introduced. With the SWL potential even very long square well chain molecules may be treated in Monte Carlo (MC) simulations; moreover the model is well suited for perturbation theory. Using Onsager-like theories we test the effect of molecular elongation, temperature, and the range of the square well potential on the vapor-liquid and nematic-smectic A (NS) phase transitions. We find that the vapor-liquid binodal of the SWL fluid is in good agreement with MC results for square well dumbbells. For repulsive SWL particles, varying the interaction range s(2) results in a similar effect on the NS transition as the change in the ionic strength in a real suspension of fd viruses.

Minimizing the Ground State Energy of an Electron in a Randomly Deformed Lattice

We provide a characterization of the spectral minimum for a random Schrödinger operator of the form \({H = -\Delta + \sum_{i \in \mathbb{Z}^d}q(x - i - \omega_i)}\) in \({L^2(\mathbb{R}^d)}\) , where the single site potential q is reflection symmetric, compactly supported in the unit cube centered at 0, and the displacement parameters ω i are restricted so that adjacent single site potentials do not overlap. In particular, we show that a minimizing configuration of the displacements is given by a periodic pattern of densest possible 2d-clusters of single site potentials.The main tool to prove this is a quite general phenomenon in the spectral theory of Neumann problems, which we dub “bubbles tend to the boundary.” How should a given compactly supported potential be placed into a bounded domain so as to minimize or maximize the first Neumann eigenvalue of the Schrödinger operator on this domain? For square or rectangular domains and reflection symmetric potentials, we show that the first Neumann eigenvalue is minimized when the potential sits in one of the corners of the domain and is maximized when it sits in the center of the domain. With different methods we also show a corresponding result for smooth strictly convex domains.

Markov Random Field-Based Statistical Character Structure Modeling for Handwritten Chinese Character Recognition

This paper proposes a statistical-structural character modeling method based on Markov random fields (MRFs) for handwritten Chinese character recognition (HCCR). The stroke relationships of a Chinese character reflect its structure, which can be statistically represented by the neighborhood system and clique potentials within the MRF framework. Based on the prior knowledge of character structures, we design the neighborhood system that accounts for the most important stroke relationships. We penalize the structurally mismatched stroke relationships with MRFs using the prior clique potentials, and derive the likelihood clique potentials from Gaussian mixture models, which encode the large variations of stroke relationships statistically. In the proposed HCCR system, we use the single-site likelihood clique potentials to extract many candidate strokes from character images, and use the pairsite clique potentials to determine the best structural match between the input candidate strokes and the MRF-based character models by relaxation labeling. The experiments on the KAIST character database demonstrate that MRFs can statistically model character structures, and work well in the HCCR system.

The modulus of continuity of Wegner estimates for random Schrödinger operators on metric graphs

We consider an alloy type potential on an infinite metric graph. We assume a covering condition on the single site potentials. For random Schr\"odingers operator associated with the alloy type potential restricted to finite volume subgraphs we prove a Wegner estimate which reproduces the modulus of continuity of the single site distribution measure. The Wegner constant is independent of the energy.