On the lattice oscillator-type Kirkwood–Salsburg equation with attractive many-body potentials

We consider a lattice oscillator-type Kirkwood–Salsburg (KS) equation with general one-body phase measurable space and many-body interaction potentials. For special choices of the measurable space, its solutions describe grand-canonical equilibrium states of lattice equilibrium classical and quantum linear oscillator systems. We prove the existence of the solution of the symmetrized KS equation for manybody interaction potentials which are either attractive (nonpositive) and finite-range or infinite-range and repulsive (positive). The proposed procedure of symmetrization of the KS equation is new and based on the superstability of many-body potentials.     

Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: A semigroup approach in L1‐spaces

We investigate the large time behavior of solutions to the spatially homogeneous linear Boltzmann equation from a semigroup viewpoint. Our analysis is performed in some (weighted) L¹‐spaces. We deal with both the cases of hard and soft potentials (with angular cut‐off). For hard potentials, we provide a new proof of the fact that, in weighted L¹‐spaces with exponential or algebraic weights, the solutions converge exponentially fast towards equilibrium. Our approach uses weak‐compactness arguments combined with recent results of the second author on positive semigroups in L¹‐spaces. For soft potentials, in L¹‐spaces, we exploit the convergence to ergodic projection for perturbed substochastic semigroup to show that, for very general initial datum, solutions to the linear Boltzmann equation converges to equilibrium in large time. Moreover, for a large class of initial data, we also prove that the convergence rate is at least algebraic. Notice that, for soft potentials, no exponential rate of convergence is expected because of the absence of spectral gap.     

Existence and concentration of ground states of coupled nonlinear Schrödinger equations

In this paper we consider the existence and concentration of ground states of coupled nonlinear Schrödinger equations with trap potentials. When the interaction between two states is repulsive, we prove the existence of ground states. Then concentration phenomenon of these ground states is studied as the small perturbed parameter (Planck constant) approaches zero. Roughly speaking, we prove that components of the ground states concentrate at the unique global minimum points of their potentials. Moreover, we prove the existence of ground states when the interaction is attractive.     

Construction, ergodicity and rate of convergence of N-particle Langevin dynamics with singular potentials

We construct N-particle Langevin dynamics in ${\mathbb{R}^d}$ or in a cuboid region with periodic boundary for a wide class of N-particle potentials Φ and initial distributions which are absolutely continuous w.r.t. Lebesgue measure. The potentials are in particular allowed to have singularities and discontinuous gradients (forces). An important point is to prove an L ^p -uniqueness of the associated non-symmetric, non-sectorial degenerate elliptic generator. Analyzing the associated functional analytic objects, we also give results on the long-time behaviour of the dynamics, when the invariant measure is finite: Firstly, we prove the weak mixing property whenever it makes sense (i.e. whenever {Φ < ∞} is connected). Secondly, for a still quite large class of potentials we also give a rate of convergence of time averages to equilibrium when starting in the equilibrium distribution. In particular, all results apply to N-particle systems with pair interactions of Lennard–Jones type.     

Infinite-dimensional stochastic differential equations related to random matrices

We solve infinite-dimensional stochastic differential equations (ISDEs) describing an infinite number of Brownian particles interacting via two-dimensional Coulomb potentials. The equilibrium states of the associated unlabeled stochastic dynamics are the Ginibre random point field and Dyson’s measures, which appear in random matrix theory. To solve the ISDEs we establish an integration by parts formula for these measures. Because the long-range effect of two-dimensional Coulomb potentials is quite strong, the properties of Brownian particles interacting with two-dimensional Coulomb potentials are remarkably different from those of Brownian particles interacting with Ruelle’s class interaction potentials. As an example, we prove that the interacting Brownian particles associated with the Ginibre random point field satisfy plural ISDEs.     

On Exact Controllability of Networks of Nonlinear Elastic Strings in 3-Dimensional Space

This paper concerns a system of nonlinear wave equations describing the vibrations of a 3-dimensional network of elastic strings.The authors derive the equations and appropriate nodal conditions,determine equilibrium solutions,and,by using the methods of quasilinear hyperbolic systems,prove that for tree networks the natural initial,boundary value problem has classical solutions existing in neighborhoods of the "stretched" equilibrium solutions.Then the local controllability of such networks near such equilibrium configurations in a certain specified time interval is proved.Finally,it is proved that,given two different equilibrium states satisfying certain conditions,it is possible to control the network from states in a small enough neighborhood of one equilibrium to any state in a suitable neighborhood of the second equilibrium over a suffciently large time interval.     

Non-perturbative positive Lyapunov exponent of Schrödinger equations and its applications to skew-shift mapping

We first study the discrete Schrödinger equations with analytic potentials given by a class of transformations. It is shown that if the coupling number is large, then the Lyapunov exponent equals approximately to the logarithm of this coupling number. When the transformation becomes the skew-shift mapping, we prove that the Lyapunov exponent is weak Hölder continuous, and the spectrum satisfies Anderson Localization and contains large intervals. Moreover, all of these conclusions are non-perturbative.     

Equilibrium states,pressure and escape for multimodal maps with holes

For a class of non-uniformly hyperbolic interval maps, we study rates of escape with respect to conformal measures associated with a family of geometric potentials. We establish the existence of physically relevant conditionally invariant measures and equilibrium states and prove a relation between the rate of escape and pressure with respect to these potentials. As a consequence, we obtain a Bowen formula: we express the Hausdorff dimension of the set of points which never exit through the hole in terms of the relevant pressure function. Finally, we obtain an expression for the derivative of the escape rate in the zero-hole limit.     

Punktprozesse mit Wechselwirkung

We consider classical, continuous systems of particles in r dimensions described by infinite system equilibrium states which have been defined by Dobrushin [5] and Lanford/Ruelle [24]. For a large class of potentials we prove the theorem of Lee/Yang [43] together with a variational characterizafor these equilibrium states. The main idea stems from Föllmer [9] who showed that in the case of lattice systems, the theorem of Lee/Yang is intimately related to Birkhoff's ergodic theorem and McMillan's theorem (ergodic theorem of information theory). Following this idea we obtain as main results an r-dimensional ergodic theorem for random measures in ^r , limit theorems concerning energy and entropy and an r-dimensional version of Breiman's theorem showing that there is almost sure convergence behind McMillan's theorem.

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Danken möchten wir Klaus Krickeberg, der diese Arbeit durch eine Fülle wertvoller Hinweise und Anregungen gefördert hat.     

Hydrodynamic limit for the Ginzburg–Landau ∇ϕ interface model with non-convex potential

The hydrodynamic limit of the Ginzburg–Landau $??$ interface model was derived in Funaki and Spohn (1997) and Nishikawa (2003) for strictly convex potentials. This paper deals with non-convex potentials under suitable assumptions on the free energy and identification of the extremal Gibbs measures which have been recently established at sufficiently high temperature in Cotar and Deuschel (2012). Because of the non-convexity, many difficulties arise, especially, on the identification of equilibrium states. We show the equivalence between the stationarity and the Gibbs property under quite general settings, and we complete the identification of equilibrium states. We also establish some uniform estimates for variances of extremal Gibbs measures.     

Geometric Gibbs theory

We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials. We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials. We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures. We prove that with this complex Banach manifold structure, the space is complete and, moreover, is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures. There is a maximum metric on the space, which is incomplete. We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same. We prove that a geometric Gibbs measure is an equilibrium state, and the infimum of the metric entropy function on the space is zero.

     

Rates of mixing for potentials of summable variation

It is well known that for subshifts of finite type and equilibrium measures associated to Hölder potentials we have exponential decay of correlations. In this article we derive explicit rates of mixing for equilibrium states associated to more general potentials.

     

Long-time stabilization of solutions to a phase-field model with memory

We prove that any global bounded solution of a phase field model with memory terms tends to a single equilibrium state for large times. Because of the memory effects, the energy is not a Lyapunov function for the problem and the set of equilibria may contain a nontrivial continuum of stationary states. The method we develop is applicable to a more general class of equations containing memory terms. Received August 11, 2000; accepted September 25, 2000.     

Geometric Gibbs theory In Memory of Professor Shantao Liao

We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials. We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials. We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures. We prove that with this complex Banach manifold structure, the space is complete and, moreover, is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures. There is a maximum metric on the space,which is incomplete. We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same. We prove that a geometric Gibbs measure is an equilibrium state, and the infimum of the metric entropy function on the space is zero.     

Inverse Scattering Problem for the Schrödinger Operator with External Yang–Mills Potentials in Two Dimensions at Fixed Energy

ABSTRACT

We consider the Schrödinger equation in ?², with external Yang–Mills potentials that decay exponentially as |x| → ∞. We prove that the scattering amplitude at fixed positive energy determines the potentials uniquely modulo a gauge transformation, assuming that potentials are small.     

Long‐time convergence of solutions to a phase‐field system

We prove that any global bounded solution of a phase field model tends to a single equilibrium state for large times though the set of equilibria may contain a nontrivial continuum of stationary states. The problem has a partial variational structure, specifically, only the elliptic part of the first equation represents an Euler–Lagrange equation while the second does not. This requires some modifications in comparison with standard methods used to attack this kind of problems. Copyright © 2001 John Wiley & Sons, Ltd.     

Finiteness of the discrete spectrum of the Schrödinger operator of three particles on a lattice

We consider a system of three quantum particles interacting by pairwise short-range attraction potentials on a three-dimensional lattice (one of the particles has an infinite mass). We prove that the number of bound states of the corresponding Schrödinger operator is finite in the case where the potentials satisfy certain conditions, the two two-particle sub-Hamiltonians with infinite mass have a resonance at zero, and zero is a regular point for the two-particle sub-Hamiltonian with finite mass.     

A transformational property of the Husimi function and its relation to the wigner function and symplectic tomograms

We consider the Husimi Q-functions, which are quantum quasiprobability distributions in the phase space, and investigate their transformation properties under a scale transformation (q, p) → (λq, λp). We prove a theorem that under this transformation, the Husimi function of a physical state is transformed into a function that is also a Husimi function of some physical state. Therefore, the scale transformation defines a positive map of density operators. We investigate the relation of Husimi functions to Wigner functions and symplectic tomograms and establish how they transform under the scale transformation. As an example, we consider the harmonic oscillator and show how its states transform under the scale transformation.     

一类微生物连续培养竞争系统的定性分析

研究一类微生物连续培养竞争系统的解的结构,分析了平衡点的稳定性及平衡点附近极限环存在唯一性,证明了该系统存在正向不变集.     

On pseudocyclic table algebras and applications to pseudocyclic association schemes

We effect a complete study of the thermodynamic formalism, the entropy spectrum of Birkhoff averages, and the ergodic optimization problem for a family of parabolic horseshoes. We consider a large class of potentials that are not necessarily regular, and we describe both the uniqueness of equilibrium measures and the occurrence of phase transitions for nonregular potentials in this class. Our approach consists in reducing the problems to the study of renewal shifts. We also describe applications of this approach to hyperbolic horseshoes as well as to noninvertible maps, both parabolic (with the Manneville-Pomeau map) and uniformly expanding. This allows us to recover in a unified manner several results scattered in the literature. For the family of hyperbolic horseshoes, we also describe the dimension spectrum of equilibrium measures of a class of potentials that are not necessarily regular. In particular, the dimension spectra need not be strictly convex.