Particle Weights and Their Disintegration II

The first article in this series presented a thorough discussion of particle weights and their characteristic properties. In this part a disintegration theory for particle weights is developed which yields pure components linked to irreducible representations and exhibiting features of improper energy-momentum eigenstates. This spatial disintegration relies on the separability of the Hilbert space as well as of the C*-algebra. Neither is present in the GNS-representation of a generic particle weight so that we use a restricted version of this concept on the basis of separable constructs. This procedure does not entail any loss of essential information insofar as under physically reasonable assumptions on the structure of phase space the resulting representations of the separable algebra are locally normal and can thus be continuously extended to the original quasi-local C*-algebra.     

The Regular Truncation Scheme of the BBGKY Hierarchy Equations for the Multi-Component Weakly Non-Ideal Plasma

A regular truncation scheme in which the divergencies characteristic of the Debye truncation are removed by taking into account the shall-range correlations. As a result, the two radii of screening the Debye radius D and √2 D appear in plasma which is not symmetrical as to its particle charges. The criterion of the applicability of this approach relates the plasma parameter to the potential depth of unlike-charged particles interaction.     

Probability Collectives for Unstable Particles

Unstable particles, together with their stable decay products, constitute probability collectives that are defined as Hilbert spaces with dimension higher than one, nondecomposable in a particle basis. Their structure is considered in the framework of Birkhoff-von Neumann's Hilbert subspace lattices. Bases with particle states are related to bases with a diagonal scalar product by a Hilbert-bein involving the characteristic decay parameters (in some analogy to the n-bein structures of metrical manifolds). Probability predictions as expectation values, involving unstable particles, have to take into account all members of the higher dimensional collective. For example, the unitarity structure of the S-matrix for an unstable particle collective can be established by a transformation with its Hilbert-bein.     

Quantum Equilibrium Distributions of Displacements and Momenta of Particles in Molecules and Solids

The quantum equilibrium distribution, ?Qm, of an arbitrary number, m, of momentum or displacement components is determined for atoms that are part of a polyatomic molecule or a solid. This is shown to be a multidimensional Gaussian distribution. Two cases are considered: (1) the motion of the system as a whole is given, (2) it is in itself determined by the statistical equilibrium conditions. In the first case we obtain distributions for the vibrational momenta and displacements and in the second for the total momenta, including the momenta of vibrational, translational and rotational motions. The distributions for momenta and displacements of one particle and for the maximum number of linearly independent components of momenta and displacements of all particles of the system are considered as particular cases. It is shown that the averaging of any function, F_m, depending on an arbitrary number, m, of components of displacements or momenta of particles, over the canonical ensemble is reduced to the integration of this function weighted by ?Qm over all its arguments between infinite limits.     

Stationarity of Lagrangian Velocity¶in Compressible Environments

We study the transport of a passive tracer particle by a random d-dimensional, Gaussian, compressible velocity field. It is well known, since the work of Lumley, see [13], and Port and Stone, see [20], that the observations of the velocity field from the moving particle, the so-called Lagrangian velocity process, are statistically stationary when the field itself is incompressible. In this paper we study the question of stationarity of Lagrangian observations in compressible environments. We show that, given sufficient temporal decorrelation of the velocity statistics, there exists a transformation of the original probability measure, under which the Lagrangian velocity process is time stationary. The transformed probability is equivalent to the original measure. As an application of this result we prove the law of large numbers for the particle trajectory. Received: 1 May 2001 / Accepted: 4 December 2001     

Coulomb repulsion versus Hubbard repulsion in a disordered chain

We study the difference between on site Hubbard and long range Coulomb repulsions for two interacting particles in a disordered chain. The system size L (in units of the lattice spacing) is of the order of the one particle localization length and the energies are taken near the band center. In the two cases, the limits of weak and strong interactions are characterized by uncorrelated energy levels and are separated by a crossover regime where the states are more extended and the spectra more rigid. U denoting the interaction strength and t the kinetic energy scale, the crossovers take place for interaction energy to kinetic energy ratios U/t and U/(2tL) of order one, for Hubbard and Coulomb repulsions respectively. While Hubbard repulsion can only yield weak critical chaos with intermediate spectral statistics, Coulomb repulsion can drive the two particle system to quantum chaos with Wigner-Dyson spectral statistics. The interaction matrix elements are studied to explain this difference. Received 21 March 2000 and Received in final form 5 February 2001     

Assembly of Nano-Biocatalyst for the Tandem Hydrolysis and Reduction of p-Nitrophenol Esters

Hybrid nano-biomaterials are exploited in the design and performance of chemo-enzymatic cascades. In this study, lipase is immobilized from Candida antarctica fraction B (CALB) and gold nanoparticles (Au NPs) on magnetic particles coated with silica (MNP@SiO₂) to stepwise hydrolyze and reduce p-nitrophenyl esters in tandem reaction. The assembly of the two catalysts at the interface of the MNP@SiO₂ particles and the temporal control of the reaction turns out to be the most determinant parameters for the cascade kinetics. When both CALB and Au NPs are co-immobilized at the MNP@SiO₂ particle, the tandem reactions take place significantly faster than when both catalysts are physically segregated by their immobilization on different MNP@SiO₂ particles. Herein, it is demonstrated that the co-immobilization of biocatalysts and nanocatalysts in solid materials creates hybrid interfaces that accelerated chemo-enzymatic tandem reactions.     

Review of Particle Physics

This biennial Review summarizes much of Particle Physics. Using data from previous editions, plus 1600 new measurements from 550 papers, we list, evaluate, and average measured properties of gauge bosons, leptons, quarks, mesons, and baryons. We also summarize searches for hypothetical particles such as Higgs bosons, heavy neutrinos, and supersymmetric particles. All the particle properties and search limits are listed in Summary Tables. We also give numerous tables, figures, formulae, and reviews of topics such as the Standard Model, particle detectors, probability, and statistics. A booklet is available containing the Summary Tables and abbreviated versions of some of the other sections of this full Review. All tables, listings, and reviews (and errata) are also available on the Particle Data Group website: http: //pdg. lbl. gov.     

Accumulation of particles in time-dependent thermocapillary flow in a liquid bridge. Modeling of experiments

The study addresses the phenomenon of accumulation of rigid tracer particles suspended in a time-dependent thermocapillary flow in a liquid bridge. We report the results of the three-dimensional numerical modeling of recent experiments [1,2] in a non-isothermal liquid column. Exact physical properties of both liquids and particles are used for the modeling. Two liquids are investigated: sodium nitrate (NaNO₃) and n-decane (C₁₀H₂₂). The particles are modeled as perfect spheres suspended in already well developed time-dependent thermocapillary flow. The particle dynamics is described by the Maxey-Riley equation. The results of our simulations are in excellent agreement with the experimental observations. For the first time we reproduced numerically formation of the particle accumulation structure (PAS) both under gravity and under weightlessness conditions. Our analysis confirms the experimental observations that the existence of PAS depends on the strength of the flow field, on the ratio between liquid and particle density, and on the particle size.     

On Some Processes of Quantum Gravitation (Radiative Corrections)

Within the framework of the quantum theory of interactions of nonpoint (smeared out) particles [1], the radiative corrections due to gravitational interactions are examined. The gravitational masses are calculated for a photon, a graviton, and some other particles together with the renormalized gravitational interaction constant K(m) depending on the interacting particle mass m. The approximate character of the equivalence principle is confirmed.     

Measurement of chemical potentials of systems with strong excluded volume interactions by computing the density of states

At high densities and low temperatures, the conventional Widom test particle method to compute the chemical potential of a system of particles with excluded volume interactions fails owing to bad statistics. A way to circumvent this problem is the use of expanded ensemble simulation techniques or thermodynamic integration. In this article, we will describe an alternative method to compute the chemical potential which is conceptually much easier, by computing the density of states of systems of N and N + 1 particles directly; and by performing a test particle simulation at very high temperature. The advantage of our technique is that the densities of states of the N and N + 1 particle system are computed in an ensemble in which particles can pass each other, resulting in a more efficient sampling. We will demonstrate our method not only for single particles but also for chain molecules with intramolecular interactions. By using an infinite temperature expansion and an extension of the density of states to very high energies, we will show that it is also possible to compute the chemical potential without having to compute the density of states for the N + 1 particle system.     

Soft Dynamics simulation. 1. Normal approach of two deformable particles in a viscous fluid and optimal-approach strategy

Discrete simulation methods are efficient tools to investigate the behaviors of complex fluids such as dry granular materials or dilute suspensions of hard particles. By contrast, materials made of soft and/or concentrated units (emulsions, foams, vesicles, dense suspensions) can exhibit both significant elastic particle deflections (Hertz-like response) and strong viscous forces (squeezed liquid). We point out that the gap between two particles is then not determined solely by the positions of their centers, but rather exhibits its own dynamics. We provide the first ingredients of a new discrete numerical method, named Soft Dynamics, to simulate the combined dynamics of particles and contacts. As an illustration, we present the results for the approach of two particles. We recover the scaling behaviors expected in three limits: the Stokes limit for very large gaps, the Poiseuille-lubricated limit for small gaps and even smaller surface deflections, and the Hertz limit for significant surface deflections. We find that for each gap value, an optimal force achieves the fastest approach velocity. The principle of larger-scale simulations with this new method is provided. They will consitute a promising tool for investigating the collective behaviors of many complex materials.     

Effects of interaction of nanoparticles in a highly anisotropic ferrimagnet

This paper reports on a study of the field and temperature dependences of the parameters of the particle magnetic interaction in a densely packed system of nanocrystals of the highly anisotropic hexagonal ferrite BaFe₁₂O₁₉ with the particles distributed in diameter within the range 10—100 nm and having volumes satisfying the criterion of “small Stoner—Wohlfarth particles.” It is shown that the resultant particle interaction in the temperature range 300 K≤T≤640 K has a negative sign, whereas for T>640 K, it is positive. The maximum values of the parameter Δm allow classification of the interaction as moderate in strength. The temperature dependences of the interaction parameters are found to correlate with manifestation of the size and surface effects in the system, which are characteristic of small particles (transition to the superparamagnetic state, “surface” anisotropy, and reduced exchange interaction in a structurally defective near-surface layer of particles).     

Pair and singlet diffusion in a Lennard-Jones liquid

Inspired by the experimental work of Richter et al. [1], we have investigated the dynamic behaviour of pairs of particles of a Lennard-Jones liquid for four different thermodynamic states. The short and long time behaviour of dynamic correlation functions of pairs are studied and directly compared with that of single particle correlation functions. Only those relating to singlet and pair diffusion are treated.

A list of the correlation functions considered in this work is presented in the table of the Appendix.

The main results are the following:

• (i)the mean square displacement of pairs of particles departs only slightly from that of single particles (doubled);

• (ii)for long times, the mean square displacement of pairs is significantly reduced when pairs from only the first coordination shell are considered;

• (iii)in the short time region the mean square displacment of pairs resembles that of single atoms, however that of pairs being initially closer than the mean separation of particles in the liquid exhibits a small tendency to lie above the doubled one of single particles.

Our results, obtained by extensive molecular dynamics (MD) calculations, are in quantitative disagreement with results generated by other authors [3, 4], and the findings of Posch et al. [6] could not be confirmed in detail by our work.     

Exact solution of two-species ballistic annihilation with general pair-reaction probability

The reaction processA + B → ∅ is modeled for ballistic reactants on an infinite line with particle velocitiesυ _A =c andυ _B = -c and initially segregated conditions, i.e., allA particles to the left and allB particles to the right of the origin. Previous models of ballistic annihilation have particles that always react on contact, i.e., pair-reaction probabilityp = 1. The evolutions of such systems are wholly determined by the initial distributions of particles and therefore do not have a stochastic dynamics. However, in this paper the generalization is made to p< 1, allowing particles to pass through each other without necessarily reacting. In this way, theA andB particle domains overlap to form a fluctuating, finitesized reaction zone where the product ∅ is created. Fluctuations are also included in the currents ofA andB particles entering the overlap region, thereby inducing a stochastic motion of the reaction zone as a whole. These two types of fluctuations, in the reactions and particle currents, are characterised by theintrinsic reaction rate, seen in a single system, and theextrinsic reaction rate, seen in an average over many systems. The intrinsic and extrinsic behaviors are examined and compared to the case of isotropically diffusing reactants     

Generalized Theory for the Simultaneous Measurement of Particle Size and Velocity using laser doppler and laser two-focus methods

Essentially the laser two-focus (L2F) and laser Doppler anemometer (LDA) are time-of-flight anemometers. The velocity of particles in the micrometre range is determined within the limits of an optically fixed measurement distance. For this only one optoelectronic receiver for the detection of the scattered light is required [1, 2]. With an arrangement of two receiving optics, positioned under an off-axis angle φ and an elevation angle ± ψ to the optical axis of the measurement system, a resolution of three-dimensional structures can be achieved and with regard to spherical particles it is possible to determine the dimensions by means of the temporal or phase shift of the signals in the receiving optics. A particle size-dependent distance inside the measurement volume can be fixed, which has to be passed in order to change the signal from one receiving optics to the other. An LDA with an arrangement of two receiving optics for particle sizing is known as a phase Doppler anemometer (PDA); an L2F designed with two receiving optics can be termed a pulse-displacement two-focus anemometer (P2F). The physical analysis of the two methods with respect to a temporal signal displacement in the receivers yielded new results.     

Interlaced Particle Systems and Tilings of the Aztec Diamond

Motivated by the problem of domino tilings of the Aztec diamond, a weighted particle system is defined on N lines, with line j containing j particles. The particles are restricted to lattice points from 0 to N, and particles on successive lines are subject to an interlacing constraint. It is shown that this particle system is exactly solvable, to the extent that not only can the partition function be computed exactly, but so too can the marginal distributions. These results in turn are used to give new derivations within the particle picture of a number of known fundamental properties of the tiling problem, for example that the number of distinct configurations is 2 ^N(N+1)/2, and that there is a limit to the GUE minor process, which we show at the level of the joint PDFs. It is shown too that the study of tilings of the half Aztec diamond—not known from earlier literature—also leads to an interlaced particle system, now with successive lines 2n−1 and 2n (n=1,…,N/2−1) having n particles. Its exact solution allows for an analysis of the half Aztec diamond tilings analogous to that given for the Aztec diamond tilings.     

Magnetic dipole fields at interstitial sites in disordered binary alloys of ferromagnetic crystals

The dipole fields at interstitial sites occupied by positive particles (μ ⁺,p) in f.c.c. and b.c.c. diluted binary alloys of ferromagnetic metals are investigated. The characteristic shapes of the distribution functions of the dipole fields is shown to depend on the interstitial site occupied by the positive particle, and they may give information on the formation of diatomic complexes of the light interstitials with substitutional impurity atoms as well as on the degree of short-range order of the alloy.     

Finite-size particles in turbulent channel flow: quadrant analysis and acceleration statistics

The interaction between finite-size particles and turbulent channel flow in the absence of gravity is studied by direct numerical simulations (DNS). The study is motivated by DNS observations of a turbulent channel flow with high-density, pointwise particles, that cluster in regions of high streamwise root-mean-square (RMS) acceleration close to the wall, contrary to what is observed in homogeneous isotropic turbulence. The aim of the present study is to explore if this is still the case when size effects are taken into account in the DNS. Based on the analysis of the velocity and acceleration statistics, the present DNS shows that, close to the wall, particles with ρ_p/ρ_f ranging from 2 to 4 are surrounded by regions with low streamwise RMS velocity but high streamwise RMS acceleration. According to the normalised particle acceleration probability density functions (PDFs), size effects become important in the near-wall region. As particle inertia increases, the normalised PDFs of particle acceleration tend to a Gaussian distribution. The tails of the normalised PDFs of the fluid conditioned by the presence of particles are higher than that of the unconditioned fluid close to the wall.     

Transport as a consequence of state-dependent diffusion

Overdamped particles subject to a drift in a force field with sinusoidal space dependence and also a sinusoidally modulated space-dependent diffusion, with the same period as the drift, experience a net driving force. The resulting current depends on the amplitude of the modulation of the diffusion and is a periodic function of the phase difference between the sinusoidal drift and the sinusoidal modulation of the diffusion. For small modulation amplitudes a particle subject to state-dependent noise behaves the same way as a particle subject to thermal noise but with a drift which, in addition to the sinusoidal term, contains a net force term [M. Büttiker,Z. Phys. B 68:161 (1987)]. A specific example of this behavior [N. G. van Kampen,IBM J. Res. Dev. 32:107 (1988); R. Landauer,J. Stat. Phys. 53:233 (1988).] is the motion of overdamped particles in a ring subject to a nonuniform temperature field. When the drift and the temperature, which are periodic with a period equal to the ring circumference, are not in phase a noise-induced circulating current results.This paper will appear in a forthcoming issue of theJournal of Statistical Physics.