Spherical harmonics and cartesian tensor scalar product expansions of the distribution function

On the basis of the expansion of the distribution function in a sum of the spherical harmonics, the distribution functionf(v, r, t) is expanded in a series of scalar products of two Cartesian tensors term by term, i.e. The tensors and ^(l) (l=2, 3) are constructed in dependence on the spherical harmonic expansion coefficients (the tensors and ^(l) (l=0, 1) have been constructed by Jancel and Kahan [3]). On the basis of the knowledge of the analytic form off ₂ andf ₃ the equations forf ₁ f ₂ andf ₃ for the case of the Boltzmann's equation are determined.Technická 2, Praha 6, Czechoslovakia.     

Metrics on test function spaces for canonical field operators

In a canonical field theory, the field (f) and momentum (g) are assumed defined for test functionsf andg which are elements of linear vector spaces and, respectively. Generally, the continuity of the map onto the unitary Weyl operatorsU(f),V(g) is taken as ray continuity, the barest minimum to recover the field operators as their generators, i.e.,U(f)=e ^i(f) ,V(g)=e ^i(g) . This leaves open the question of whether any wider continuity properties follow and what form they would take. We show that much richer continuity properties do follow in a natural fashion for every cyclic representation of the canonical commutation relations. In particular, we show that the test function space may be taken as a metric space, that the space may be uniquely completed in this topology, and that the map into the unitary Weyl operators is strongly continuous in this topology. The topology induced by this metric is minimal in the sense that it is the weakest vector topology for which the mapsfU(f),gV(g) are strongly continuous. An expression for a suitable metric can easily be given in terms of a simple integral over a state on the Weyl operators.     

Vanishing of the Kontsevich Integrals of the Wheels

We prove that the Kontsevich integrals (in the sense of the formality theorem) of all even wheels are equal to zero. These integrals appear in the approach to the Duflo formula via the formality theorem. The result means that for any finite-dimensional Lie algebra g, and for invariant polynomials f, g [S ^·(g)]^g one has f · g = f * g, where * is the Kontsevich star product, corresponding to the Kirillov–Poisson structure on g^*. We deduce this theorem form the result contained in math.QA/0010321 on the deformation quantization with traces.     

Commuting Operators and Separation of Variables for Laplacians of Projectively Equivalent Metrics

Let two Riemannian metrics g and g on one manifold M ⁿ have the same geodesics (considered as unparameterized curves). Then we can construct invariantly n commuting differential operators of second order. The Laplacian _g of the metric g is one of these operators. For any x M ⁿ , consider the linear transformation G of T _x M ⁿ given by the tensor g ^Ig_j . If all eigenvalues of G are different at one point of the manifold then they are different at almost every point; the operators are linearly independent and their symbols are functionally independent. If all eigenvalues of G are different at each point of a closed manifold then it can be covered by the n-torus and we can globally separate the variables in the equation _g f = f on this torus.     

On nonlinear stationary half-space problems in discrete kinetic theory

We show that every steady discrete velocity model of the Boltzmann equation on the real line, _i·(d/dx)f ⁱ=C ⁱ(f), which satisfies anH-theorem and for which all _i0, has solutions on the half-line (0, ) which take prescribed non-negativef ⁱ(O) if _i>0 and approach a certain manifold of Maxwellians asx. Such solutions give the density distribution in a Knudsen boundary layer in the discrete velocity case.     

Entropy andP-particle observables. II. The two-particle entropyS 2

This paper defines, and then evaluates perturbatively, an information-theoretic notion of entropyS ₂ for a system ofN interacting particles which assesses an observer's limited knowledge of the state of the system, assuming that he or she can measure with arbitrary precision all one-particle observables and correlations involving pairs of particles, but is completely ignorant of the form of any higher-order correlations involving three or more particles. By construction, thisS ₂(t) involves only the reduced two-particle distribution functions, or density matrices,f ₂(i,j) at timet, and, though the implementation of a subdynamics,dS ₂ (t)/dt can be realized in terms of thef ₂(i, j)'s at retarded timest–. A similar line of reasoning demonstrates that the most probable three-particlef ₃(i,j, k) consistent with a knowledge of thef ₂'s is precisely thatf ₃ suggested by the Kirkwood, or cluster, decomposition.     

On the semiboundedness of the (ø4)2 Hamiltonian

An elementary alternate proof of the semiboundedness of the locally correct HamiltonianH ₀+:ø⁴(x):g(x)dx of the (ø⁴)₂ quantum field theory model. The interaction operator is expressed as the sum of a positive operator and operators which are tiny relative to LN for any >0, whereN is the number operator.Supported by the National Research Council of Canada.     

New proofs of the existence of the Feigenbaum functions

A new proof of the existence of analytic, unimodal solutions of the Cvitanovi-Feigenbaum functional equation g(x) = –g(g(–x)),g(x) 1 - const|x|^r at 0, valid for all in (0, 1), is given, and the existence of the Eckmann-Wittwer functions [8] is recovered. The method also provides the existence of solutions for certain given values ofr, and in particular, forr=2, a proof requiring no computer.     

Some properties of an “aesthetic” field theory

We continue our study of the Lorentz-invariant field theory based on the equations _jk;l ⁱ =0 and g_ij;k=0. To first order in a perturbation expansion, we find _jk;l ⁱ =0 reduces to the wave equation. In orders higher than the first, we find that _jk;l ⁱ =0 cannot be linearized. We also find that the simple wave-type equation g^ij2g/xⁱx^j=0 is contained in the theory when an appropriate choice is made for the parameters at the origin point.     

The effect of grain size on the deformation resistance of nickel

The effect of the grain sizel of commerical nickel on the lower yield point, _y, and flow stress, _f , has been investigated. From the relationship between _y andl ^–1/2 and between _f andl ^–1/2, and also by extrapolation, the parameters ₁( _i ^f ) and k_y(kf), which occur in Petch's well-known expression, were determined. It was found that the values of these parameters depend on the previous history of the samples. It is suggested that the more marked dependence of the deformation resistance of nickel on grain size arising from certain thermal treatments is due to the segregation of impurities to the grain boundaries. It is shown that this is in accord with the presence of grain-boundary hardening and with its dependence on quenching temperature.     

The number of bound states of singular oscillating potentials

For a spherically symmetric potential such that rVL ¹(a, ), a>0, and is such that, if we define W=– _r V(t) d(t), W belongs to L ¹ (0, ) and rW0 as r0, we show that the number of bound states in any partial-wave satisfies the bound n2 ₀ r W ² dr. It was shown in a previous paper [1] that this class of potentials is regular from the point of view of abstract scattering theory as well as from the time-independent theory and the Jost function approach. We show also that, for large values of the coupling constant, n(gV) has the asymptotic behaviour C _±g ₀ W(r) dr as g±.     

Growth of clusters in a first-order phase transition

The results of computer simulations of phase separation kinetics in a binary alloy quenched from a high temperature are analyzed in detail, using the ideas of Lifshitz and Slyozov. The alloy was modeled by a three-dimensional Ising model with Kawasaki dynamics. The temperature after quenching was 0.59T _c, whereT _c is the critical temperature, and the concentration of minority atoms was=0.075, which is about five times their largest possible single-phase equilibrium concentration at that temperature. The time interval covered by our analysis goes from about 1000 to 6000 attempted interchanges per site. The size distribution of small clusters of minority atoms is fitted approximately byc ₁(1-)³ w(t),c ₁ (1–)⁴ Q _l w(t)^l(2l10); wherec _l is the concentration of clusters of sizel;Q ₂,...,Q ₁₀ are known constants, the cluster partition functions;t is the time; andw(t)=0.015(1+7.17t ^–1/3). The distribution of large clusters (l20) is fitted approximately by the type of distribution proposed by Lifshitz and Slyozov,c _l ,(t)=–(d/dl) [lnt+p (l/t)], where is a function given by those authors and is defined by(x)=C _o e–x-C ₁ e ^–4x/3-C ₂ e ^–5x/3;C ₀,C ₁,C ₂ are constants determined by considering how the total number of particles in large clusters changes with time.Supported by the U.S. Air Force Office of Scientific Research under Grant No. 78-3522 and by the U.S. Department of Energy under Contract No. EY-76-C-02-3077*000.     

Letter: On the Newtonian Limit in Gravity Models with Inverse Powers of R

I reconsider the problem of the Newtonian limit in nonlinear gravity models in the light of recently proposed models with inverse powers of R. Expansion around a maximally symmetric local background with curvature scalar R ₀ > 0 gives the correct Newtonian limit on length scales R ₀ ^–1/2 if the gravitational Lagrangian satisfies f(R ₀)f(R₀) 1, and I propose two models with f(R ₀) = 0.     

On the high spin expansion in the SYM theory

We study the form of the high spin expansion of the minimal anomalous dimension for long operators belonging to the sl(2) sector of SYM. Keeping fixed the ratio j between the twist and the logarithm of the spin, the minimal anomalous dimension expands as γ(g,j,s)=f(g,j)lns+f⁽⁰⁾(g,j)+O(1/lns). This particular double scaling limit is efficiently described, including the desired accuracy O((lns)⁰), in terms of a linear integral equation. By its use, we are able to evaluate both at weak and strong coupling the subleading scaling function f⁽⁰⁾(g,j) as a series in j, up to the order j⁵. Thanks to these results, the possible extension of the liaison with the O(6) non-linear sigma model may be tackled on a solid ground.     

Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groupsU _q (g). They have the same FRT generatorsl ^± but a matrix braided-coproductL=LL, whereL=l ⁺ Sl ^–, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matricesBM _q(2); it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum doubleD(U _q (sl ₂)) (also known as the quantum Lorentz group) is the semidirect product as an algebra of two copies ofU _q (sl ₂), and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-classical limits as doubles of the Lie algebras of Poisson Lie groups.     

Dynamics of the generalized Glauber-Ising chain in a magnetic field

A one-dimensional kinetic Ising model with nearest neighbor interactionJ and magnetic fieldH 0 is treated in both linear and nonlinear response, using the most general single spin-flip transition probabilities that depend on nearest neighbor states only. The dynamics is reformulated in terms of kinetic equations for the concentration n_l ⁺(t) [@#@ n_l(t) of clusters containingl up- [or down-] spins, which is exact in the homogeneous case. The initial relaxation time ^* of the magnetization is obtained rigorously for arbitraryJ, H, and temperatureT. The relaxation function is found by numerical integration forJ/T < 2. It is shown that coagulation of minus-clusters becomes negligible for bothJ/T andH/T large, and the resulting set of equations is solved exactly in terms of an eigenvalue problem. A perturbation theory is developed to take into account the neglected coagulation terms. The relaxation function is found to be non-Lorentzian in general, in contrast to the Glauber results atH = 0, which are recovered as a special case. In addition, nonlinear and linear relaxation functions differ forH 0. Consequences for the application to biopolymers are briefly mentioned.Supported in part by the Deutsche Forschungsgemeinschaft (SFB 130).     

A diffusion model of metal surface modification during plasma nitriding

A diffusion model of metal surface modification by plasma nitriding has been developed. This model takes into account the erosion effects at the plasma/solid interface occurring due to the ion bombardment of the surface. For constant sputtering rate, which is the usual situation during plasma nitriding, the growth of the sub-layers is well described by the analytical expressiong(t) =g ₀,f ^–1 (t/t ₀), whereg(t) is the sub-layer thickness at timet,g ₀ andt ₀ are parameters which depend on the treated material and plasma characteristics,f ^–1 is the inverse of the function — In(1 - x) + x), 0 x 1. Under negligible erosion effects, the expression forg(t) reduces to the parabolic law. The diffusion zone (substratum) growth does not follow the parabolic law as well. However, the deviation occurs after long plasma nitriding time. The model can be used for experimentally determining the effective diffusion coefficients and the erosion rate during plasma nitriding of metal surfaces.     

Fundamental representations of quantum groups at roots of 1

To every finite-dimensional irreducible representation V of the quantum group U(g) where is a primitive lth root of unity (l odd) and g is a finite-dimensional complex simple Lie algebra, de Concini, Kac and Procesi have associated a conjugacy class C _V in the adjoint group G of g. We describe explicitly, when g is of type A _n , B _n , C _n , or D _n , the representations associated to the conjugacy classes of minimal positive dimension. We call such representations fundamental and prove that, for any conjugacy class, there is an associated representation which is contained in a tensor product of fundamental representations.     

Modified dispersion relations and sum rules

A systematic treatment of modifications to the usual form of dispersion relations and sum rules is given. The treatment is based on the fact that if a function g(z) has analyticity properties analogous to those of a scattering amplitudef (z), the usual dispersion relation for the productf (z)g(z) holds. After having described the general form of the weight function g(z) we concentrate on cases wheng(z) is large in a pre-chosen region and small elsewhere. Weight functions of this type are useful in practical applications where a weight functiong(z) may enhance a contribution of regions where accurate data are available and suppress those of regions with inaccurate or lacking data.Dedicated to Professor V. Votruba on his sixtieth birthday.     

An algebraic approach to quantum field theory and commutation relations for energy momentum in the framework of general relativity

We present a consistent set of commutation relations (C.R.) for a quantum system immersed in a classical gravitational field. The gravity field is described by metric tensorg _ik (x) andg ₀₀(x) with coordinate gaugeg _i0=0. The Hamiltonian of the system is found to be a linear function of [–g ₀₀(x)]^1/2. Its properties we define by C.R. avoiding explicit expression in terms of fields, as well as its splitting into free and interaction parts. In this way a consistent set of C.R., which are equally simple for a flat and curvilinear space, can be established. To stress the main idea of our approach, we consider the simple but still nontrivial example of a scalar electrodynamics immersed in a gravity field. The electromagnetic current operator we define by its C.R. and not explicitly. An interesting feature of this approach is that the Poisson equation follows from the consistency of the C.R. The C.R. for the energy and momentum operators of the system in a gravity field are established which generalize the usual Poincare group generators C.R. For example, we find (i/hc ²)[H _(x) ,H _(x) ]=P _– , whereH _(x) is the Hamiltonian of the system, which is a linear functional of (x)[–g ₀₀(x)]^1/2 andP _s(x) represents the momentum-density operator [averaged with the classical functions(x)].