Possible Loss and Recovery of Gibbsianness¶During the Stochastic Evolution of Gibbs Measures

We consider Ising-spin systems starting from an initial Gibbs measure ν and evolving under a spin-flip dynamics towards a reversible Gibbs measure μ≠ν. Both ν and μ are assumed to have a translation-invariant finite-range interaction. We study the Gibbsian character of the measure νS(t) at time t and show the following: (1) For all ν and μ, νS(t) is Gibbs for small t. (2) If both ν and μ have a high or infinite temperature, then νS(t) is Gibbs for all t > 0. (3) If ν has a low non-zero temperature and a zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t and non-Gibbs for large t. (4) If ν has a low non-zero temperature and a non-zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t, non-Gibbs for intermediate t, and Gibbs for large t. The regime where μ has a low or zero temperature and t is not small remains open. This regime presumably allows for many different scenarios. Received: 26 April 2001 / Accepted: 10 October 2001     

Stochastic Dissipative PDE's and Gibbs Measures

We study a class of dissipative nonlinear PDE's forced by a random force η^{omega( t} , ^x ), with the space variable x varying in a bounded domain. The class contains the 2D Navier–Stokes equations (under periodic or Dirichlet boundary conditions), and the forces we consider are those common in statistical hydrodynamics: they are random fields smooth in t and stationary, short-correlated in time t. In this paper, we confine ourselves to “kick forces” of the form

On the envelope soliton near critical density in a plasma with cold ions and two-temperature electrons

Summary We have analysed the formation of envelope soliton near critical density in a plasma consisting of two-temperature electrons and cold ions. The non-linear Schr?dinger-like equation obtained isiφ _t +pφ _xx +q|φ|⁴φ=0 which we call the modified non-linear Schr?dinger equation. It is also observed that this approach leads to a physical situation where a linear combination of both the modified and usual NLS equations holds, in the formiφ _t +pφ _xx +q ₁|φ|²φ +q ₂|φ|⁴φ=0. It is demonstrated through graphical analysis thatq ₁,q ₂, thought of as a function of β(=T _el/T _eh), behave in opposite way. That is, whenq ₁ grows,q ₂ decays, or vice versa. Lastly we demonstrate that this equation can sustain a type of solution other than the usual solitary profile. The form of such a wave is also depicted graphically.     

Short-Time Gibbsianness for Infinite-Dimensional Diffusions with Space-Time Interaction

We consider a class of infinite-dimensional diffusions where the interaction between the components has a finite extent both in space and time. We start the system from a Gibbs measure with a finite-range uniformly bounded interaction. Under suitable conditions on the drift, we prove that there exists t ₀>0 such that the distribution at time t≤t ₀ is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion of both the initial interaction and certain time-reversed Girsanov factors coming from the dynamics.     

Improvement of the pinning properties of (Bi/Pb)-2223 textured ceramics by columnar defects introduced by 6 GeV Pb ions

Pinning properties of (Bi/Pb)-2223 textured ceramics have been improved by 6 GeV Pb ions irradiation. Two samples were irradiated at fluencesφ _t=5 × 10¹⁰ ions/cm² and 2 × 10¹¹ ions/cm², respectively. The magnetic irreversibility is clearly improved in the intermediate range of temperature: (30–35) K<T<60 K for magnetic fields lower than the “fluence equivalent field”B _φ =φ _t.φ ₀. Moreover the irreversibility line is shifted towards higher fields. The results reported here show that the introduction of columnar defects or other extended defects is a route to improve the screening properties of tubes or cans based on (Bi/Pb)-2223 material and operating in liquid nitrogen.     

On the Spectrum of the Generator of an Infinite System of Interacting Diffusions

We study the spectrum of the operator

Theoretical study of incoherent φ photoproduction on a deuteron target

We study the photoproduction of φ mesons in deuteron, paying attention to the modification of the cross-section from bound protons to the free ones. For this purpose we take into account Fermi motion in single scattering and rescattering of φ to account for φ absorption on a second nucleon as well as the rescattering of the proton on the neutron. We find that the contribution of the double scattering for φ is much smaller than the typical cross-section of γp → φp in free space, which implies a very small screening of the φ production in deuteron. The contribution from the proton rescattering, on the other hand, is found to be not negligible compared to the cross-section of γp → φp in free space, and leads to a moderate reduction of the φ photoproduction cross-section on a deuteron at forward angles if the LEPS set-up is taken into account. The Fermi motion allows contribution of the single scattering in regions forbidden by phase-space in the free case. In particular, we find that for momentum transfer squared close to the maximum value, the Fermi motion changes drastically the shape of dσ/dt, to the point that the ratio of this cross-section to the free one becomes very sensitive to the precise value of t chosen, or the size of the bin used in an experimental analysis. Hence, this particular region of t does not seem to be the most indicated to find effects of a possible φ absorption in the deuteron. This reaction is studied theoretically as a function of t and the results are contrasted with recent experiments at LEPS and Jefferson Lab. The effect of the experimental angular cuts at LEPS is also discussed, providing guidelines for future experimental analyses of the reaction.     

The Variational Principle for a Class of Asymptotically Abelian C*-Algebras

Let (A,α) be a C^*-dynamical system. We introduce the notion of pressure P _α(H) of the automorphism α at a self-adjoint operator H∈A. Then we consider the class of AF-systems satisfying the following condition: there exists a dense α-invariant *-subalgebra ? of A such that for all pairs a,b∈? the C^*-algebra they generate is finite dimensional, and there is p=p(a,b)∈ℕ such that [α ^j (a),b]= 0 for |j|≥p. For systems in this class we prove the variational principle, i.e. show that P _α(H) is the supremum of the quantities h _φ(α) −φ(H), where h _φ(α) is the Connes–Narnhofer–Thirring dynamical entropy of α with respect to the α-invariant state φ. If H∈A, and P _α(H) is finite, we show that any state on which the supremum is attained is a KMS-state with respect to a one-parameter automorphism group naturally associated with H. In particular, Voiculescu's topological entropy is equal to the supremum of h _φ(α), and any state of finite maximal entropy is a trace. Received: 19 April 2000 / Accepted: 14 June 2000     

Analysis on configuration spaces and Gibbs cluster ensembles

The distribution μ of a Gibbs cluster point process in χ = ℝ^d (with n-point clusters) is studied via the projection of an auxiliary Gibbs measure defined on the space of configurations in χ × χ ⁿ. We show that μ is quasi-invariant with respect to the group Diff₀(χ) of compactly supported diffeomorphisms of χ and prove an integration-by-parts formula for μ. The corresponding equilibrium stochastic dynamics is then constructed by using the method of Dirichlet forms. Dedicated to the memory of Vladimir Geyler Research supported in part by DFG Grant 436 RUS 113/722.     

Stable Ergodic Properties of Cocycles¶over Hyperbolic Attractors

We consider a hyperbolic flow φ ^t defined on an attracting basic set Λ. A map from the first (Čech) cohomology group of Λ into the dynamic cohomology group is constructed. This map is used to discuss the stable ergodicity and mixing of compact Lie group extensions and velocity changes of φ ^t . Received: 17 June 1998 / Accepted: 24 February 1999     

3d Monte Carlo simulation of site-bond continuum percolation of spheres

We present off-lattice Monte Carlo simulations of site-bond percolation of semi-penetrable spheres or, equivalently, of hard spheres with a finite bond range. We will show that the crucial parameter is the effective volume fraction ( φ_e), i.e. the volume that is occupied or within the bond range of at least one particle. For the equivalent system of semi-penetrable spheres 1 - φ_e is the porosity. The bond percolation threshold (p _b) can be described in terms of φ_e by a simple analytical expression: log(φ_e)/log(φ_ec) + log(p _b)/log(p _bc) = 1, with p _bc = 0.12 independent of the bond range and φ_ec a constant that decreases with increasing bond range. Received: 10 March 2003 / Accepted: 23 April 2003 / Published online: 21 May 2003 RID="a" ID="a"e-mail: jean-christophe.gimel@univ-lemans.fr     

Phase Diagram of Ising Systems with Additional Long Range Forces

We consider ferromagnetic Ising systems where the interaction is given by the sum of a fixed reference potential and a Kac potential of intensity λ≥0 and scaling parameter γ>0$. In the Lebowitz Penrose limit γ→0⁺$ the phase diagram in the (T,λ) positive quadrant is described by a critical curve λ^mf(T), which separates the regions with one and two phases, respectively below and above the curve. We prove that if $λ>^mf(T), i.e. above the curve, there are at least two Gibbs states for small values of γ. If instead λ<λ^mf(T) and if the reference Gibbs state (i.e. without the Kac potential) satisfies a mixing condition at the temperature T, then, at the same temperature the full interaction (i.e. with also the Kac potential) satisfies the Dobrushin Shlosman uniqueness condition for small values of γ so that there is a unique Gibbs state. Received: 9 April 1996 / Accepted: 26 November 1996     

Cosmological Spacetimes Balanced by a Weyl Geometric Scale Covariant Scalar Field

A Weyl geometric approach to cosmology is explored, with a scalar field φ of (scale) weight −1 as crucial ingredient besides classical matter. Its relation to Jordan-Brans-Dicke theory is analyzed; overlap and differences are discussed. The energy-stress tensor of the basic state of the scalar field consists of a vacuum-like term Λg _{μ ν} with Λ depending on the Weylian scale connection and, indirectly, on matter density. For a particularly simple class of Weyl geometric models (called Einstein-Weyl universes) the energy-stress tensor of the φ-field can keep space-time geometries in equilibrium. A short glance at observational data, in particular supernovae Ia (Riess et al. in Astrophys. J. 659:98ff, 2007), shows encouraging empirical properties of these models.     

On the Motion of a Charged Particle Interacting with an Infinitely Extended System

 We study the time evolution of a charged particle moving in a medium under the action of a constant electric field E. In the framework of fully Hamiltonian models, we discuss conditions on the particle/medium interaction which are necessary for the particle to reach a finite limit velocity. We first consider the case when the charged particle is confined in an unbounded tube of ℝ³. The electric field E is directed along the symmetry axis of the tube and the particle also interacts with an infinitely many particle system. The background system initial conditions are chosen in a set which is typical for any reasonable thermodynamic (equilibrium or non-equilibrium) state. We prove that, for large E and bounded interactions between the charged particle and the background, the velocity v(t) of the charged particle does not reach a finite limit velocity, but it increases to infinite as: |v(t)−Et|≤C ₀ (1+t), where C ₀ is a constant independent of E. As a corollary we obtain that, if the initial conditions of the background system are distributed according to any Gibbs state, then the average velocity of the charged particle diverges as time goes to infinite. This result is obtained for E large enough in comparison with the mean energy of the Gibbs state. We next study the one-dimensional case, in which the estimates can be improved. We finally discuss, at an heuristic level, the existence of a finite limit velocity for unbounded interactions, and give some suggestions about the case of small electric fields. Received: 7 March 2002 / Accepted: 23 September 2002 Published online: 8 January 2003 RID="*" ID="*" Work partially supported by the GNFM-INDAM and the Italian Ministry of the University. Communicated by J.L. Lebowitz     

On Convergence to Equilibrium Distribution,I.¶The Klein–Gordon Equation with Mixing

Consider the Klein–Gordon equation (KGE) in ℝ ⁿ , n≥ 2, with constant or variable coefficients. We study the distribution μ _t of the random solution at time t∈ℝ. We assume that the initial probability measure μ₀ has zero mean, a translation-invariant covariance, and a finite mean energy density. We also assume that μ₀ satisfies a Rosenblatt- or Ibragimov–Linnik-type mixing condition. The main result is the convergence of μ _t to a Gaussian probability measure as t→∞ which gives a Central Limit Theorem for the KGE. The proof for the case of constant coefficients is based on an analysis of long time asymptotics of the solution in the Fourier representation and Bernstein's “room-corridor” argument. The case of variable coefficients is treated by using an “averaged” version ofthe scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay. Received: 4 January 2001 / Accepted: 2 July 2001     

Relaxation of Disordered Magnets in the Griffiths' Regime

We study the relaxation to equilibrium of discrete spin systems with random finite range (not necessarily ferromagnetic) interactions in the Griffiths' regime. We prove that the speed of convergence to the unique reversible Gibbs measure is almost surely faster than any stretched exponential, at least if the probability distribution of the interaction decays faster than exponential (e.g. Gaussian). Furthermore, if the interaction is uniformly bounded, the average over the disorder of the time–autocorrelation function, goes to equilibrium as (in d > 1), in agreement with previous results obtained for the dilute Ising model. Received: 12 June 1996 / Accepted: 23 January 1997     

Fluctuations of Matrix Elements of Regular Functions of Gaussian Random Matrices

We find the limit of the variance and prove the Central Limit Theorem (CLT) for the matrix elements φ _jk (M), j,k=1,…,n of a regular function φ of the Gaussian matrix M (GOE and GUE) as its size n tends to infinity. We show that unlike the linear eigenvalue statistics Tr φ(M), a traditional object of random matrix theory, whose variance is bounded as n→∞ and the CLT is valid for Tr φ(M)−E{Tr φ(M)}, the variance of φ _jk (M) is O(1/n), and the CLT is valid for . This shows the role of eigenvectors in the forming of the asymptotic regime of various functions (statistics) of random matrices. Our proof is based on the use of the Fourier transform as a basic characteristic function, unlike the Stieltjes transform and moments, used in majority of works of the field. We also comment on the validity of analogous results for other random matrices.     

On the asymptotical normality of statistical solutions for harmonic crystals in half-space

We consider the dynamics of a harmonic crystal in the half-space with zero boundary condition. It is assumed that the initial date is a random function with zero-mean, finite-mean energy density which also satisfies a mixing condition of Rosenblatt or Ibragimov type. We study the distribution μ _t of the solution at time t ∈ ℝ. The main result is the convergence of μ _t as t → ∞ to a Gaussian measure which is time stationary with a covariance inherited from the initial measure (non-Gaussian in general). Supported partly by research grant of RFBR (06-01-00096).     

Driven Tracer Particle in One Dimensional Symmetric Simple Exclusion

Consider an infinite system of particles evolving in a one dimensional lattice according to symmetric random walks with hard core interaction. We investigate the behavior of a tagged particle under the action of an external constant driving force. We prove that the diffusively rescaled position of the test particle εX(ε^-2 t), t > 0, converges in probability, as ε→ 0, to a deterministic function v(t). The function v(⋅) depends on the initial distribution of the random environment through a non-linear parabolic equation. This law of large numbers for the position of the tracer particle is deduced from the hydrodynamical limit of an inhomogeneous one dimensional symmetric zero range process with an asymmetry at the origin. An Einstein relation is satisfied asymptotically when the external force is small. Received: 5 December 1996 / Accepted: 30 June 1997