Scaling laws in the vortex lattice model of turbulence

The lattice vortex model of the intertial range in turbulence theory is reviewed; the model consists of an array of vortex tubes whose axes coincide with the bonds on a regular lattice, subjected to random stretching and successive scaling, and constrained by conservation laws for energy, specific volume, circulation, helicity, and an energy/vorticity relation. The scaling laws for vorticity are examined in detail, a Hausdorff dimension for the active portion of the vortex array is calculated, the origin of intermittency is exhibited, and it is pointed out that the Kolmogorov — 5/3 power law already accounts for intermittency effects.Partially supported by the Office of Energy Research, U.S. Department of Energy, under contract DE-AC03-76SF0098, and in part by the Office of Naval Research under contract N00014-76-C-0316     

Bicovariant quantum algebras and quantum Lie algebras

A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun toU _q g, given by elements of the pure braid group. These operators—the reflection matrixYL ⁺ SL ^– being a special case—generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation forY inSO _q (N).This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY90-21139     

A numerical method to compute exactly the partition function with application toZ(n) theories in two dimensions

I present a new method to exactly compute the partition function of a class of discrete models in arbitrary dimensions. The time for the computation for ann-state model on anL ^d lattice scales like . I show examples of the use of this method by computing the partition function of the 2D Ising and 3-state Potts models for maximum lattice sizes 10×10 and 8×8, respectively. The critical exponentsv and and the critical temperature one obtains from these are very near the exactly known values. The distribution of zeros of the partition function of the Potts model leads to the conjecture that the ratio of the amplitudes of the specific heat below and above the critical temperature is unity.     

Nonabelian orbifolds and the boson-fermion correspondence

For a positive integerl divisible by 8 there is a (bosonic) holomorphic vertex operator algebra (VOA) associated to the spin lattice _l . For a broad class of finite groupsG of automorphisms of we prove the existence and uniqueness of irreducibleg-twisted -modules and establish the modular-invariance of the partition functionsZ(g, h, ) for commuting elements inG. In particular, for any finite group there are infinitely many holomorphic VOAs admittingG for which these properties hold. The proof is facilitated by a boson-fermion correspondence which gives a VOA isomorphism between and a certain fermionic construction, and which extends work of Frenkel and others.Supported by NSA grant MDA904-92-H-3099.Supported by NSF grant DMS-9122030.     

The evolution of a turbulent vortex

We examine numerically the evolution of a perturbed vortex in a periodic box. The fluid is inviscid. We find that the vorticity blows up. The support of theL ₂ norm of the vorticity converges to a set of Hausdorff dimension 2.5. The distribution of the vorticity seems to converge to a lognormal distribution. We do not observe a convergence of the higher statistics towards universal statistics, but do observe a strong temporal intermittency.This work was supported in part by the Director, Office of Energy Research, Office of Basic Energy Sciences, Engineering, Mathematical and Geosciences Division of the U.S. Department of Energy, under contract W-7404-ENG-48, and in part by the Office of Naval Research, under contract N00014-76-C-0316     

Diagonalization of theXXZ Hamiltonian by vertex operators

We diagonalize the anti-ferroelectricXXZ-Hamiltonian directly in the thermodynamic limit, where the model becomes invariant under the action of . Our method is based on the representation theory of quantum affine algebras, the related vertex operators and KZ equation, and thereby bypasses the usual process of starting from a finite lattice, taking the thermodynamic limit and filling the Dirac sea. From recent results on the algebraic structure of the corner transfer matrix of the model, we obtain the vacuum vector of the Hamiltonian. The rest of the eigenvectors are obtained by applying the vertex operators, which act as particle creation operators in the space of eigenvectors. We check the agreement of our results with those obtained using the Bethe Ansatz in a number of cases, and with others obtained in the scaling limit—thesu(2)-invariant Thirring model.Dedicated to Professors Huzihiro Araki and Noboru Nakanishi on the occasion of their sixtieth birthdays     

Finite-size effects and scaling for the thermal QCD deconfinementphase transition within the exact color-singlet partition function

We study the finite-size effects for the thermal quantum chromodynamics (QCD) deconfinement phase transition, and use a numerical finite-size scaling analysis to extract the scaling exponents characterizing its scaling behavior when approaching the thermodynamic limit . For this, we use a simple model of coexistence of hadronic gas and color-singlet quark gluon plasma (QGP) phases in a finite volume. The color-singlet partition function of the QGP cannot be exactly calculated and is usually derived within the saddle-point approximation. When we try to do calculations with such an approximate color-singlet partition function, a problem arises in the limit of small temperatures and/or volumes , requiring additional approximations if we want to carry out calculations. We propose in this work a method for an accurate calculation of any quantity of the finite system, without any approximation. By probing the behavior of some useful thermodynamic response functions on the whole range of temperature, it turns out that, in a finite-size system, all singularities in the thermodynamic limit are smeared out and the transition point is shifted away. A numerical finite-size scaling (FSS) analysis of the obtained data allows us to determine the scaling exponents of the QCD deconfinement phase transition. Our results expressing the equality between their values and the space dimensionality is a consequence of the singularity characterizing a first-order phase transition and agree very well with the predictions of other FSS theoretical approaches to a first-order phase transition and with the results of calculations using Monte Carlo methods in both lattice QCD and statistical physics models. Received: 11 January 2005, Revised: 7 July 2005, Published online: 30 August 2005     

Logarithmic Sobolev inequality for lattice gases with mixing conditions

Let denote the grand canonical Gibbs measure of a lattice gas in a cube of sizeL with the chemical potential and a fixed boundary condition. Let be the corresponding canonical measure defined by conditioning on . Consider the lattice gas dynamics for which each particle performs random walk with rates depending on near-by particles. The rates are chosen such that, for everyn andL fixed, is a reversible measure. Suppose that the Dobrushin-Shlosman mixing conditions holds for forall chemical potentials . We prove that for any probability densityf with respect to ; here the constant is independent ofn orL andD denotes the Dirichlet form of the dynamics. The dependence onL is optimal.Research partially supported by U.S. National Science Foundations grant 9403462, Sloan Foundation Fellowship and David and Lucile Packard Foundation Fellowship.     

A note on the covariant anomaly as an equivariant momentum mapping

We show that there is a natural gauge invariant presymplectic structure on the spaceA of all vector potentials. The covariant axial anomaly is found to be the essentially unique infinitestimally equivariant momentum mapping for the action of the group of gauge transformations on (A,). The infinitesimal equivariance of is shown to be equivalent to the Wess-Zumino consistency condition for the consistent axial anomalyG. We also show that theX operator of Bardeen and Zumino, which relatesG and , corresponds to the one-form (onA) of the presymplectic structure.Research supported in part by NSF grant MCS 81-08814(A03)Research supported by the U.S. Department of Energy under contract number DE AC02 76 ER 02220     

Monte Carlo simulations concerning zero field host NMR in aCuMn spin glass

Own recent Monte Carlo calculations have been improved and extended. A three dimensionalCuMn spin glass with 3 at % Mn has been simulated. 6.912 lattice points and 207 classical Heisenberg spins with have been employed. Only RKKY and dipole interaction are introduced. For both interactions experimentally measured values of coupling strengths at 18 discrete near neighbour sites have been used. The distribution of hyperfine fieldsH _L at host nuclei sites and the distribution of exchange fieldsH _j for Mn spins have been calculated. Anisotropy fields of a field cooled sample have been evaluated beingH _a=1.0 kOe (H _c=2.5 kOe);H _a=0.7 kOe (H _c=5.5 kOe) andH _a=0.5 kOe (H _c=7.5 kOe). Two complete hysteresis cycles of two different statistical alloys were calculated showing both a sharp magnetization step one atH _d=–0.36 kOe and the other atH _d=–0.19 kOe. The details of spin orientation during a hysteresis cycle have been viewed. For small external fields the calculations are in favour of Allouls one domain model. The magnetization step occurs within the system of single spins. The magnetization of clusters remains constant during the whole hysteresis cycle. The calculations of the NMR enhancement factor are in general agreement with experiment.Tempelmann, C., Brömer, H.: Europhysics Conference Abstracts 9A, PWc-6-117 (1985)     

A note on γ traces for the Wilson action in the continuum limit

In d=4 and d=2 dimensions we calculate averages of certain products of matrices with respect to closed lattice paths of length L. The approach to the asymptotic behaviour for large L is considered and found to be quite different in d=4 and d=2 dimensions.Institute für Theoretische Physik der Universität Hamburg, F.R.G.     

A Note on the Time Development of Quantum Lattice Systems

The time development of quantum lattice systems is studied without any restrictions on the growth condition of the potential . A thermodynamic limit of quantum Gibbs state, a *-algebra and an automorphism group _t for which is a KMS state are constructed.     

Tunneling resistivity of a one-dimensional random lattice and the petersburg problem

The resistivity of a one-dimensional lattice consisting of randomly distributed conducting and insulating sites is considered. Tunneling resistance of the form ₀ ne ^bn is assumed for a cluster ofn adjacent insulating sites. In the thermodynamic limit, the mean resistance per site diverges at the critical filling fractionp _c =e^–b, while the mean square resistivity fluctuations diverge at the lower filling fraction =p _c ² . Computer simulations of large but finite systems, however, show only a very weak divergence of resistivity atp _c and no divergence of the fluctuations at . For finite lattices, calculation of the resistivity at the critical filling is shown to be simply related to the Petersburg problem. Analytic expressions for the resistivity and resistivity fluctuations are obtained in agreement with the results of computer simulations.     

Contribution of Sea Quarks and Gluons to the Proton Spin

Contributions of sea quark and gluon spins to the proton spin in Drell-Jahn processes and direct production of photons in proton-proton and proton-antiproton collisions are studied in the present work. Analytical expressions for two-spin asymmetries and are derived. In both processes, these asymmetries are studied and analyzed as functions of the kinematic variables , x _T, and x _F. Measurements of two-spin asymmetries and make it possible to determine the individual contributions of sea quark and gluon spins to the proton spin.     

Exact Potts Model Partition Functions for Strips of the Square Lattice

We present exact calculations of the Potts model partition function Z(G, q, v) for arbitrary q and temperature-like variable v on n-vertex square-lattice strip graphs G for a variety of transverse widths L _t and for arbitrarily great length L , with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These have the form Z(G, q, v)= . We give general formulas for N _{Z, G, j} and its specialization to v=–1 for arbitrary L _t for both types of boundary conditions, as well as other general structural results on Z. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. It is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the Potts model on the infinite square lattice. Considering the full generalization to arbitrary complex q and v, we determine the singular locus , arising as the accumulation set of partition function zeros as L , in the q plane for fixed v and in the v plane for fixed q.     

Self-avoiding walks on percolation clusters at criticality and lattice animals

We use the real-space renormalization group method to study the critical behavior of self-avoiding walks (SAWs) on both site percolation clusters at percolation threshold and site lattice animals in a square lattice. The correlation length exponents of SAWs are found to be on the percolation clusters atp _c and _SAW ^LA =0.804 on lattice animals. These results are compared with Kremer's suggestion of modified Flory formula where is the fractal dimension of the fractal object.     

The construction of a representation of loop algebras

By considering the cohomology of the loop algebraL , a representation ofL is constructed. the construction is based on a derivation ofL and a two-dimensional closed cochain ofl with coefficients in real numbersR ¹. In the case of =0, the differential of the energy representation of the corresponding loop groupLG is derived.This work was supported in part by the National Natural Science Foundation of China.     

Critical behaviour of a one-dimensional crystal at the displacive limit

The partition function of a one-dimensional, one-component model is calculated exactly by means of a transfer-operator method, and the critical behaviour at the displacive limit is evaluated analytically. We prove a generalized scaling hypothesis and discuss the scaling behaviour in the critical region. We find critical exponents for the specific heat,=2/3 for the susceptibility, and a crossover exponent = 2/3. The results satisfy the scaling relations which do not involve the dimensionality but violate those which contain the dimensionality. Scaling functions for the susceptibility and the order parameter are calculated.Supported by Schweizerischer Nationalfonds.     

Complex fragment emission from low energy compound nucleus decay to multifragmentation

In the first of these lectures, the experimental emission probabilities of complex fragments by low energy compound nuclei and their dependence upon energy and Z value are compared to the transtion state rates. In the second part, the high energy multi-fragment emission probabilities are shown to be reducible to the single fragment emission probability through the binomial distribution. The extracted one-fragment emission probabilities have a thermal dependence of the formp=e ^–B/T . This suggests that multifragmentation is a sequence of thermal binary decays.Invited lecture given at the International School-Workshop Relativistic Heavy-Ion Physics, Prague (Czech Republic), 19–23 September 1994.This work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Nuclear Physics Division of the US Department of Energy, under contract DE-AC03-76SF00098.     

Density functional approach to quantum lattice systems

For quantum lattice systems, we consider the problem of characterizing the set of single-particle densities,, which come from the ground-state eigenspace of someN-particle Hamiltonian of the form whereH ₀ is a fixed, bounded operator representing the kinetic and interaction energies. We show that the conditions on are that it be strictly positive, properly normalized, and consistent with the Pauli principle. Our results are valid for both finite and infinite lattices and for either bosons or fermions. The Coulomb interaction may be included inH ₀ if the lattice dimension is 2. We also characterize those single-particle densities which come from the Gibbs states of such Hamiltonians at finite temperature. In addition to the conditions stated above, must satisfy a finite entropy condition.Research supported by the National Science Foundation under grant No. PHY-82-03669.Research supported by Office of Naval Research under grant No. 0014-80-G-0084.On leave from Department of Mathematics, University of Lowell, Massachusetts 01854.