Multiparticle fractal aggregation

Kinetic fractal aggregation in a particle bath where a fractionf of the sites are initially occupied is studied withd=2 computer simulations. Independent particles diffusing to a fixed cluster produce an aggregate with fractal dimensionD 1.7 up to a correlation length(f). At larger lengthsD2.(f) asf 0. When the particles remain fixed but the cluster undergoes a rigid random walkD appears constant at larger scales but varies withf. D 1.95 at largef andD 1.7 asf 0. In both cases, the aggregate sizeN(t) grows with timet ^(f) . Aggregation on a surface by independently diffusing particles produces shapes reminiscent of electrochemical dendritic growth. The dependence of growth rate and geometry is studied as a function of particle concentration and sticking probability.     

Non-commutative spheres

LetA be the irrational rotation algebra, i.e. theC ^*-algebra generated by two unitariesU, V satisfyingVU=e ²ⁱ UV, with irrational, and consider the fixed point subalgebraB under the flip automorphismUU ^–1,VV ^–1. We prove thatB is an AF-algebra.Dedicated to Professor Huzihiro Araki on the occasion of his 60'th birthday     

π-Meson absorption by three-particle nuclei

The reactions ⁺ +³He 3p and ^– +³H 3n are considered with both two- and three-nucleon pion-absorption taken into account. The pion-absorption with all nucleons of the nucleus involved and the interference of the two absorption mechanisms are shown to be of the same importance as a pure two-particle absorption in nuclei. The latter cannot be considered dominating as it used to be.     

Rate equations in the hopping transport

The usual kinetic equations for the site occupation probabilities in an external field are solved exactly in a simple one-dimensional periodic model with two kinds of atoms using a) free boundary conditions and order of limitsN, 0 needed for a proper treatment of the dc conductivity here b) boundary conditions with metallic contacts and order of limitsN, 0 and c) the same boundary conditions but reversed order of limiting processes 0,N typical of e.g. numerical and percolation treatments. (N and are the number of sites and frequency.) It is demonstrated that though the bulk dc conductivity is the same in all three cases, local bulk properties of the material are strongly dependent on the régime used. The role of the order of all three limiting processes 0,N+ andn+ (N–n+) for local shifts of the chemical potential _n in the dc limit is examined (n is the number of the relevant site calculated from a boundary of the chain). It is shown especially that the rate equation treatment (régime a) on the one hand and numerical or percolation treatments (régime c) on the other hand never yield the same bulk values of _r.     

Scalar particle with polarizability in a uniform external magnetic field

The model of a scalar structured particle is considered, which possesses polarizability in an external electromagnetic field. The expression for the 4-dimensional current density is found. The exact solution of the equations describing a scalar particle with polarizability in a uniform external magnetic field is obtained. Up to the terms of order O(H²), the energy spectrum can be formally obtained by the substitution of the particle mass in the expression for a pointlike scalar particle: mm–H²/2, where is the magnetic polarizability of the particle. It is shown that the rms radius of a trajectory can be obtained by the substitution of the charge in the well-known formula for a structureless scalar particle: ee(1{-H²/m)^1/2 (where is the electric polarizability).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 91–94, January, 1991.I thank A. I. L'vov for discussions.     

Phase diagram and scattering intensity of binary amphiphilic systems

A Ginzburg-Landau model with a scalar and a vector order parameter, which describe the concentration and orientation of the amphiphile, respectively, is used to study the phase diagram and the scattering intensity of binary amphiphilic systems. With increasing amphiphile concentration, the calculated phase diagram shows the typical sequence of ordered phases observed experimentally, that is micellar liquid cubic micellar hexagonal lamellar cubic bicontinuous invers hexagonal. The scattering intensity in the homogeneous phase is calculated in the oneloop approximation. In the vicinity of a phase transition to an ordered phase, the intensity is found to show a 1/q behavior for not too small wave vectorsq, followed by a small peak, and a 1/q ² decay for large wave vectors, in agreement with experimental observations in theL ₃-(or sponge-)phase.Dedicated to Prof. H. Wagner on the occasion of his 60th birthday     

Semiclassical quantum mechanics

We consider the 0 limit of the quantum dynamics generated by the HamiltonianH()=–(²/2m)+V. We prove that the evolution of certain Gaussian states is determined asymptotically as 0 by classical mechanics. For suitable potentialsV inn3 dimensions, our estimates are uniform in time and our results hold for scattering theory.Supported in part by the National Science Foundation under Grant PHY 78-08066     

The free energy of a class of Hopfield models

We show that in the limitp ,N 0,=p/N 0 the limit free energy of the Hopfield model equals in probability the Curie-Weiss free energy. We prove also that the free energy of the Hopfield model is self-averaging for any finite .     

Conductance distributions in random resistor networks. Self-averaging and disorder lengths

The self-averaging properties of the conductanceg are explored in random resistor networks (RRN) with a broad distribution of bond strengthsP(g)g ^–1. The RRN problem is cast in terms of simple combinations of random variables on hierarchical lattices. Distributions of equivalent conductances are estimated numerically on hierarchical lattices as a function of sizeL and the distribution tail strength parameter . For networks above the percolation threshold, convergence to a Gaussian basin is always the case, except in the limit 0. Adisorder length _D is identified, beyond which the system is effectively homogeneous. This length scale diverges as _Dµ^–v ( is the regular percolation correlation length exponent) when the microscopic distribution of conductors is exponentially wide (0). This implies that exactly the same critical behavior can be induced by geometrical disorder and by strong bond disorder with the bond occupation probabilityp. We find that only lattices at the percolation threshold have renormalized probability distributions in aLevy-like basin. At the percolation threshold the disorder length diverges at a critical tail strength µ_c as µ–^–z withz3.2±0.1, a new exponent.Critical path analysis is used in a generalized form to give the macroscopic conductance in the case of lattices abovep _c.     

Long-range correlations at depinning transitions

Semi-infinite systems are considered with long-range surface fields B _z ^–(1+r) for large distancesz from the surface. The influence of such fields on the global phase diagram and on the critical singularities of depinning transitions is studied within Landau theory. For |B|0, the correlation length diverges as b ^–1/2 withb=|Bln|B^–(1+r). For finiteB, t ^v withv =(2+r)/(2+2r) wheret measures the distance from bulk coexistence. In the latter case, a Ginzburg criterion leads to the upper critical dimensiond ^*=(2+3r)/(2+r).     

μSR studies in the heavy fermion systems UNi2Al3, UPd2Al3 and U2PtC2

We have performed Muon Spin Relaxation (SR) measurements of several heavy fermion systems: UNi₂Al₃ (single crystal), UPd₂Al₃ (single crystal) and U₂PtC₂ (polycrystal). ZF -SR measurements of UNi₂Al₃ show magnetic order below the Néel temperatureT _N=4.6K, with the muon precession frequency(T 0)=2.5MHz at low temperatures. The local field distribution obtained at low temperatures in UNi₂Al₃ resembles that of the incommensurate spin density wave (SDW) system (TMTSF)₂PF₆, suggesting an incommensurate spin arrangement in UNi₂Al₃. TF -SR measurements give the penetration depth(T 0)=6100Å for UPd₂Al₃ and(T 0)=7000Å for U₂PtC₂ respectively. Since UPd₂Al₃ and U₂PtC₂ are clean superconductors (/l-0), we can combine n _s/m^* with the Sommerfeld constant n _s ^1/3 m^* to calculate the Fermi temperatureT _F ^3/4–1/4, which represents the characteristic energy scale of the superconducting carriers. We obtainT _C/T_F 0.01 for both UPd₂Al₃ and U₂PtC₂, comparable to other exotic superconductors, but quite different from simple metal BCS superconductors, whereT _C/T_F 0.01.     

Spherically Symmetric Space-Time with Two Cosmological Constants

We present the analytic spherically symmetric solution of the Einstein equations, which has de Sitter asymptotics for both r and r 0. This two-lambda spherically symmetric solution is globally regular. At the range of mass parameter M_cr1 < M < M_cr2 it has three horizons and describes a neutral black hole whose singularity is replaced by a cosmological constant of Planck or GUT scale, at the background of small . Global structure of space-time contains an infinite sequence of black and white holes, de Sitter-like past and future regular cores (with + at r 0) replacing singularities, asymptotically de Sitter external universes (with for r ), and spacelike infinities. In the range of mass parameter M < M_cr1 we have a one-horizon solution describing recovered selfgravitating particle-like structure at the background of small , and for M > M_cr2 another one-horizon configuration which can be called de Sitter bag. The solutions with M = M_cr1 and M = M_cr2 represent two extreme states of a neutral nonsingular cosmological black hole.     

R andRτ ratios at the five-loop level of perturbative QCD

We study perturbative QCD at the five-loop level. In particular we considerR = _tot(e ⁺ e ^– hadrons)/(e ⁺ e ^{– + –}) andR = ( v+hadrons)/( ev). We use our method to estimate the five-loop coefficients. As a result, we obtain _s (M _z ) = 0.1186(11) and _s (34 GeV) = 0.1396(16), which are accurate at the 1% level. We also findR = 3.8350(18), which is consistent withR and is accurate to 0.05%.     

The infinite coupling limit and the asymptotics of coefficients of the perturbation expansion

For some functions in quantum field theory and quantum mechanics, presented by the asymptotic expansion in coupling constant g, the method for finding the approximate asymptotics when g is proposed.Such an estimation requires the first exact terms of the perturbation theory, their high-order asymptotics and supposition on the Borel summability. This method for the Gell-Mann-Low function (g) in the (4²/3)g ₍₄₎ ⁴ model gives approximately (g) 1.8 g ².     

The van der Waals limit for classical systems

For a -dimensional system of particles with the two-body potentialq(r)+ ^v K(r) and density , it is proved under fairly weak conditions onq andK that the canonical pressure (, ) and chemical potential (, ) tend to definite limits when 0. The limiting functions are absolutely continuous and are given in terms of the derivative of the limiting free energy density which was found in Part I.     

The Micro-Canonical Point Vortex Ensemble: Beyond Equivalence

The fluid limit N is constructed for a sequence of ensembles of N classical point vortices in a finite domain ² whose ensemble densities (w.r.t. Liouville measure) are Gaussian approximations to (E-H). Letting the variance 0 after N has been taken, one recovers the special class of nonlinear stationary Euler flows that is expected from the micro-canonical ensemble. The construction improves over previous ones which either had to regularize the logarithmic singularities of the point vortex Hamiltonian or had to assume equivalence of ensembles. In particular, nonequivalence between micro-canonical and canonical ensemble prevails for certain geometries where conditionally stable configurations with negative 'global vortex pair-specific heat' can and do exist in the micro-canonical but not in the canonical ensemble.     

Construction and Borel summability of planar 4-dimensional euclidean field theory

We use the methods of [1] to show that the planar part of the renormalized perturbation theory for ₄ ⁴ -euclidean field theory is Borel-summable on the asymptotically free side of the theory. The Borel sum can therefore be taken as a rigorous definition of theN limit of a massiveN×N matrix model with a +trg ⁴ interaction, hence with wrong sign ofg. Our construction is relevant for a solution of the ultra-violet problem for planar QCD. We also propose a program for studying the structure of the renormalons singularities within the planar world.     

Four-baryon weak interactions with ΔS = 1 and SU (6) symmetry

Relations are found among the amplitudes for four-baryon processes with S = 1 under the assumption that the Lagrangian for s-wave weak interactions has the transformation properties of a component of the SU(6) 35-plet. In particular, it is found that the amplitudes for the¹S₀ ¹S₀ and³S₁ ³S₁ transitions for the p pn reaction are the same, within a sign.Translated from Izvestiya VUZ. Fizika, No. 10, pp. 70–77, October, 1969.     

Leptonic decays of vector mesons and a possible μ+μ− resonance

The possible existence of a vector leptonic resonance (⁰) in the ^+– system is considered. We discuss the effect such a resonance would have on the g factor of the muon and also on the ratio of the partial widths of the muonic and electronic decay modes (R_V = (V ^+–)/G (V e⁺ e^–)) of the neutral vector mesons ⁰,,,, and. From the experimental values of R and R, the following values are obtained for the mass, coupling constant, and partial decay widths of the resonance: M = 872 ± 60 MeV, f²/4 = 4 ± 2) ·10^–4, ( ^{0 + –}) = 0.12±0.06 MeV, and (^{0 0 +–}) = 0.13±0.06 MeV.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 39–43, May, 1977.     

Temperature Dependence of the Gibbs State in the Random Energy Model

We consider the problem of temperature dependence of the Gibbs states in two spin-glass models: Derrida's Random Energy Model and its analogue, where the random variables in the Hamiltonian are replaced by independent standard Brownian motions. For both of them we compute in the thermodynamic limit the overlap distribution ^N _{i=1 i i} /N[–1,1] of two spin configurations , under the product of two Gibbs measures, which are taken at temperatures T,T respectively. If TT are fixed, then at low temperature phase the results are different for these models: for the first one this distribution is D _{0 0}+D _{1 1}, with random weights D ₀, D ₁, while for the second one it is ₀. We compute consequently the overlap distribution for the second model whenever T–T0 at different speeds as N.