Composition dependence of the energy gap and thermal diffusivity in bulk As-Se glasses

We have measured the composition dependence of the optical energy gap and thermal diffusivity in bulk As _x Se_1−x (0.10⩽x⩽0.50) glasses using photoacoustic technique. The energy gap shows a threshold minimum value and thermal diffusivity has a threshold maximum value at the stoichiometric composition As₂Se₃ corresponding tox=0.40. The decrease in energy gap is explained on the basis of chemical bonding. It is argued that the threshold percolation of rigidity in the random network is responsible for the peaking of the thermal diffusivity at the stoichiometric composition.     

Originalarbeiten / Travaux originaux / Original Papers

An account is given of a variational calculation to estimate the amplitudes of resonance factorsɛ in a recent theory to describe the enhancement of crystalline field potentials by conduction electrons in heavy rare earth metals. It is demonstrated that the values ofɛ obtained by minimising the energy of interaction between conduction electrons and rare earth ions are consistent with those previously used to form a comparison with experiment. These latter values were obtained by maximising theA ₂ ⁰ crystal field coefficient with respect toɛ. Consistency is exhibited in both the sign and order of magnitude of the resonance amplitudes and renders the theory parameterless. The values ofɛ show an approximate linear dependence with the number of electrons in the incompletef shell of the rare earth ions.     

Energy and critical ionic-bond parameter of a 3D large-radius bipolaron

A theory of a strong-coupling large-radius bipolaron has been developed. The possibility of the formation of 3D bipolarons in high-temperature superconductors is discussed. For the bipolaron energy, the lowest variational estimate has been obtained at α > 8, where α is the electron-phonon coupling constant. The critical ionic-bond parameter η _c = ɛ_∞/ɛ₀, where ɛ_∞ and ɛ₀ are the high-frequency and static dielectric constants, has been found to be η _c = 0.2496.     

Why one needs a functional renormalization group to survive in a disordered world

In this paper, we discuss why functional renormalization is an essential tool to treat strongly disordered systems. More specifically, we treat elastic manifolds in a disordered environment. These are governed by a disorder distribution, which after a finite renormalization becomes non-analytic, thus overcoming the predictions of the seemingly exact dimensional reduction. We discuss how a renormalizable field theory can be constructed even beyond 2-loop order. We then consider an elastic manifold embedded inN dimensions, and give the exact solution forN →ɛ This is compared to predictions of the Gaussian replica variational ansatz, using replica symmetry breaking. Finally, the effective action at order 1/N is reported.     

Influence of local spin polarization to the Kondo effect

We use the spin non-degenerate single impurity Anderson model to investigate the influence of the local spin polarization to the Kondo effect. By using the Schrieffer-Wolff transformation, we obtain a generalized s-d exchange Hamiltonian, which describes the interaction between a polarized local spin and conduction electrons. In this case, the singlet is no longer an eigenstate as shown by variational calculations where the splitting of the local energy Δ = ɛ _d↑ − ɛ _d↓ can be arbitrarily small. The local spin polarization generates the instability of the singlet ground state of the S = 1/2 s-d exchange model.      

Critical Behavior of Massless Free Field at Depinning Transition

We consider the d-dimensional massless free field localized by a δ-pinning of strength ɛ. We study the asymptotics of the variance of the field (when d= 2), and of the decay-rate of its 2-point function (when d≥ 2), as ɛ goes to zero, for general Gaussian interactions. Physically speaking, we thus rigorously obtain the critical behavior of the transverse and longitudinal correlation lengths of the corresponding d+ 1-dimensional effective interface model in a non-mean-field regime. We also describe the set of pinned sites at small ɛ, for a broad class of d-dimensional massless models. Received: 1 November 2000 / Accepted: 15 June 2001     

Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group

We showed in Part I that the Hopf algebra ℋ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group G and that the renormalized theory is obtained from the unrenormalized one by evaluating at ɛ= 0 the holomorphic part γ₊(ɛ) of the Riemann–Hilbert decomposition γ₋(ɛ)^{− 1}γ₊(ɛ) of the loop γ(ɛ)∈G provided by dimensional regularization. We show in this paper that the group G acts naturally on the complex space X of dimensionless coupling constants of the theory. More precisely, the formula g ₀=gZ ₁ Z ₃ ^−3/2 for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra ℋ. This allows first of all to read off directly, without using the group G, the bare coupling constant and the renormalized one from the Riemann–Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter ɛ. It also allows to lift both the renormalization group and the β-function as the asymptotic scaling in the group G. This exploits the full power of the Riemann–Hilbert decomposition together with the invariance of γ₋(ɛ) under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group G for the full higher pole structure of minimal subtracted counterterms in terms of the residue. Received: 21 March 2000 / Accepted: 3 October 2000     

The Low-Temperature Expansion of the Wulff Crystal in the 3D Ising Model

We compute the expansion of the surface tension of the 3D random cluster model for q≥ 1 in the limit where p goes to 1. We also compute the asymptotic shape of a plane partition of n as n goes to ∞. This same shape determines the Wulff crystal to order o(ɛ) in the 3D Ising model (and more generally in the 3D random cluster model for q≥ 1) at temperature ɛ. Received: 15 February 2001/ Accepted: 11 May 2001     

An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows

Precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection-diffusion by a steady incompressible velocity field are developed here. Under these conditions, a rigorous Stieltjes integral representation for effective diffusivity in turbulent transport is derived. This representation is valid for all Péclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the Péclet number. One consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all Peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series. Explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all Péclet numbers. A nonlocal variational principle for effective diffusivity is developed along with applications to advection-diffusion by random arrays of vortices. A new class of rigorous examples is introduced. These examples have an explicit Stieltjes measure for the effective diffusivity; furthermore, the effective diffusivity behaves likek ₀(Pe)^1/2 in the limit of large Péclet numbers wherek ₀ is the molecular diffusivity. Formal analogies with the theory of composite materials are exploited systematically.Research partially supported by NSF DMS 90-05799 and ARO DAAL 03-89-K-0039 and AFOSR-90-0090Research partially supported by NSF DMS 87-02864, ARO DAAL 03-89-K-0013 and ONR N 00014-89-J-1044     

Random Cluster Models on the Triangular Lattice

We study percolation and the random cluster model on the triangular lattice with 3-body interactions. Starting with percolation, we generalize the star–triangle transformation: We introduce a new parameter (the 3-body term) and identify configurations on the triangles solely by their connectivity. In this new setup, necessary and sufficient conditions are found for positive correlations and this is used to establish regions of percolation and non-percolation. Next we apply this set of ideas to the q > 1 random cluster model: We derive duality relations for the suitable random cluster measures, prove necessary and sufficient conditions for them to have positive correlations, and finally prove some rigorous theorems concerning phase transitions.     

Couette Flows over a Rough Boundary and Drag Reduction

 We consider the Couette flow between two plates. The lower plate is fixed and has periodically placed riblets of the characteristic size ɛ on it. In the limit ɛ → 0 we find the effective Couette-Navier flow as an O(ɛ ² ) approximation for the effective mass flow and an O(ɛ ² )L ¹ -approximation for the velocity. In the effective solution the effect of roughness enters through the Navier slip condition with the matrix coefficient in front of the effective shear stress, calculated using a boundary layer problem. Furthermore, an O(ɛ ² ) approximation for the tangential drag force is found. In all estimates explicit dependence on the kinematic viscosity ν, the velocity of the upper plate and the distance between the plates L ₃ is kept. Also the uniqueness of the solution is expressed through a non-linear algebraic condition linking and L ₃ . Then the result is applied to the viscous sub-layers around immersed bodies, strictly containing the surface riblets. It is found that for the riblets of the characteristic size ɛ, being of the order smaller or equal to , the approximation obtained for the tangential drag could be applied. We compare ɛ and for realistic data and our results lead to the conclusion that the riblets reduce significantly tangential drag, which may explain their presence on the skin of Nektons. Received: 14 December 2001 / Accepted: 1 August 2002 Published online: 7 November 2002     

Slow Motion of Charges¶Interacting Through the Maxwell Field

We study the Abraham model for N charges interacting with the Maxwell field. On the scale of the charge diameter, R _ϕ, the charges are a distance ɛ^-1 R _ϕ apart and have a velocity with ɛ a small dimensionless parameter. We follow the motion of the charges over times of the order ɛ^-3/2 R _ϕ/c and prove that on this time scale their motion is well approximated by the Darwin Lagrangian. The mass is renormalized. The interaction is dominated by the instantaneous Coulomb forces, which are of the order ɛ². The magnetic fields and first order retardation generate the Darwin correction of the order ɛ³. Radiation damping would be of the order ɛ^7/2. Received: 13 January 2000 / Accepted: 4 February 2000     

Dynamical and thermodynamical stability of two-dimensional flows: variational principles and relaxation equations

We review and connect different variational principles that have been proposed to settle the dynamical and thermodynamical stability of two-dimensional incompressible and inviscid flows governed by the 2D Euler equation. These variational principles involve functionals of a very wide class that go beyond the usual Boltzmann functional. We provide relaxation equations that can be used as numerical algorithms to solve these optimization problems. These relaxation equations have the form of nonlinear mean field Fokker-Planck equations associated with generalized “entropic” functionals [P.H. Chavanis, Eur. Phys. J. B 62, 179 (2008)].     

Renormalization Group Approach to Interacting Polymerised Manifolds

We propose to study the infrared behaviour of polymerised (or tethered) random manifolds of dimension D interacting via an exclusion condition with a fixed impurity in d-dimensional Euclidean space in which the manifold is embedded. In this paper we take D=1, but modify the underlying free Gaussian covariance (thereby changing the canonical scaling dimension of the Gaussian random field) so as to simulate a polymerised manifold with fractional dimension . The canonical dimension of the coupling constant is , where −β/2 is the canonical scaling dimension of the Gaussian embedding field. β is held strictly positive and sufficiently small. For ɛ>0, sufficiently small, we prove for this model that the iterations of Wilson's renormalisation group transformations converge to a non-Gaussian fixed point. Although ɛ is small, our analysis is non-perturbative in ɛ. A similar model was studied earlier [CM] in the hierarchical approximation. Received: 7 January 1999 / Accepted: 20 August 1999     

Mott Law as Lower Bound for a Random Walk in a Random Environment

We consider a random walk on the support of an ergodic stationary simple point process on ℝ^d, d≥2, which satisfies a mixing condition w.r.t. the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott's law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem.     

Central Limit Theorem¶for Stochastic Hamilton–Jacobi Equations

We study the asymptotic behavior of , where u solves the Hamilton–Jacobi equation u _t +H(x,u _x ) ≡ 0 with H a stationary ergodic process in the x-variable. It was shown in Rezakhanlou–Tarver [RT] that u ^ɛ converges to a deterministic function provided H(x,p) is convex in p and the convex conjugate of H in the p-variable satisfies certain growth conditions. In this article we establish a central limit theorem for the convergence by showing that for a class of examples, u ^ɛ(x,t) can be (stochastically) represented as , where Z(x,t) is a suitable random field. In particular we establish a central limit theorem when the dimension is one and , where ω is a random function that enjoys some mild regularity. Received: 15 February 1999 / Accepted: 14 December 1999     

Dielectric properties of single-crystal deuterated triglycine sulfate (DTGS) in ultra-weak low-and infra-low-frequency electric fields

The complex permittivity ɛ* is studied with separate readings for ɛ′ and ɛ″ at low and infralow frequencies and ultraweak fields. The effective conductivity λ is determined. An Arrhenius dependence is observed for ln ɛ′(1/T), ln ɛ″(1/T), and ln λ(1/T), both in the paraphase and in the polar phase. It is proposed that the conductivity of crystalline DTGS in the paraphase is an ion jump conductivity. Fiz. Tverd. Tela (St. Petersburg) 41, 1073–1075 (June 1999)     

Surface Tension in the Dilute Ising Model. The Wulff Construction

We study the surface tension and the phenomenon of phase coexistence for the Ising model on with ferromagnetic but random couplings. We prove the convergence in probability (with respect to random couplings) of surface tension and analyze its large deviations: upper deviations occur at volume order while lower deviations occur at surface order. We study the asymptotics of surface tension at low temperatures and relate the quenched value τ ^q of surface tension to maximal flows (first passage times if d =  2). For a broad class of distributions of the couplings we show that the inequality –where τ ^a is the surface tension under the averaged Gibbs measure – is strict at low temperatures. We also describe the phenomenon of phase coexistence in the dilute Ising model and discuss some of the consequences of the media randomness. All of our results hold as well for the dilute Potts and random cluster models.     

Random Homogenization and Singular Perturbations¶in Perforated Domains

The paper considers the singularly perturbed Dirichlet problem −ɛΔu ^ɛ+u ^ɛ=f in a randomly perforated domain Ω^ɛ, which is obtained from a bounded open set Ω in R ^N after removing many holes of size ɛ ^q . The perforated domain is described in terms of an ergodic dynamical system acting on a probability space. Imposing certain conditions on the domain, the behaviour of u ^ɛ when ɛ→ 0 in Lebesgue spaces L ⁿ (Ω) is studied. Test functions together with the Birkhoff ergodic theorem are the main tools of analysis. The Poisson distribution of holes of size ɛ ^p with the intensity λɛ^{− r} is then considered. The above results apply in some cases; other cases are treated by the Wiener sausage approach. Received: 15 December 1999 / Accepted: 14 April 2000     

Percolation with excluded small clusters and Coulomb blockade in a granular system

We consider dc-conductivity σ of a mixture of small conducting and insulating grains slightly below the percolation threshold, where finite clusters of conducting grains are characterized by a wide spectrum of sizes. The charge transport is controlled by tunneling of carriers between neighboring conducting clusters via short “links“ consisting of one insulating grain. Upon lowering temperature small clusters (up to some T-dependent size) become Coulomb blockaded, and are avoided, if possible, by relevant hopping paths. We introduce a relevant percolational problem of next-nearest-neighbors (NNN) conductivity with excluded small clusters and demonstrate (both numerically and analytically) that σ decreases as power law of the size of excluded clusters. As a physical consequence, the conductivity is a power-law function of temperature in a wide intermediate temperature range. We express the corresponding index through known critical indices of the perco lation theory and confirm this relation numerically.