Notice

This paper is concerned with the transport of solid particles in a turbulent fluid flowing through a tank. The movement of the particles is approximated by one-dimensional random walks in both discrete and continuous time. General formulae are derived for the probabilities of sedimentation and exit of the particles, and detailed results obtained for the case where their movement is strictly non-negative. It is shown how Wald's identity can be used to find approximate expressions for these probabilities. Finally, formulae are found for the distribution of the sediment on the bottom of the tank.     

Supersymmetry, transfinite neural networks, hyperbolic manifolds, quantum gravity and the Higgs

Connections between the transfinite extension of neural networks and hyperbolic manifolds are pointed out. The found relations are subsequently used to elucidate a certain quantum gravity equation that is reinterpreted via Spinor varieties. Finally it is shown how a definite picture of ^(∞) spacetime may be obtained from a transfinite extension of a certain four dimensional hyperbolic manifold based on Coxeter 120-cell polytop and how this helps in understanding the need of postulating the existence of not only one but three Higgs particles.     

Quantum-Classical Wigner-Liouville Equation

We consider a quantum system that is partitioned into a subsystem and a bath. Starting from the Wigner transform of the von Neumann equation for the quantum-mechanical density matrix of the entire system, the quantum-classical Wigner-Liouville equation is obtained in the limit where the masses M of the bath particles are large as compared with the masses m of the subsystem particles. The structure of this equation is discussed and it is shown how the abstract operator form of the quantum-classical Liouville equation is obtained by taking the inverse Wigner transform on the subsystem. Solutions in terms of classical trajectory segments and quantum transition or momentum jumps are described. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 6, pp. 749–756, June, 2005.     

Continuity for the asymptotic shape in the frog model with random initial configurations

We consider the so-called frog model with random initial configurations, which is described by the following evolution mechanism of simple random walks on the multidimensional cubic lattice: Some particles are randomly assigned to any site of the multidimensional cubic lattice. Initially, only particles at the origin are active and they independently perform simple random walks. The other particles are sleeping and do not move at first. When sleeping particles are hit by an active particle, they become active and start doing independent simple random walks. An interest of this model is how initial configurations affect the asymptotic shape of the set of all sites visited by active particles up to a certain time. Thus, in this paper, we prove continuity for the asymptotic shape in the law of the initial configuration.     

Reasons why the current CERN experiment should discover at least one new spin zero elementary particle and probably several others

Within a general theory, various scenarios are discussed to answer the burning question of if and how many new elementary particles will the new CERN super hadron collider discover. The answer with a very high probability is that at least one spin zero particle will be discovered. In addition there is a reasonable possibility that up to five Higgs particles will be found.     

On dispersion of settling particles from an elevated source in an open-channel flow

Longitudinal dispersion of suspended particles with settling velocity in a turbulent shear flow over a rough-bed surface is investigated numerically when the settling particles are released from an elevated continuous line-source. A combined scheme of central and four-point upwind differences is used to solve the steady turbulent convection–diffusion equation and the alternating direction implicit (ADI) method is adopted for the unsteady equation. It is shown how the mixing of settling particles is influenced by the ‘log-wake law’ velocity and the corresponding eddy diffusivity when the initial distribution of concentration is regarded as a line-source. The concentration profiles for the steady-state conditions agree well with the existing experimental data and some other numerical results when the settling velocity is zero. The behaviours of iso-concentration lines in the vertical plane for different releasing heights are studied in terms of the relative importance of convection, eddy diffusion and settling velocity.     

Meta-control of an interacting-particle algorithm for global optimization

A common issue for stochastic global optimization algorithms is how to set the parameters of the sampling distribution (e.g. temperature, mutation/cross-over rates, selection rate, etc.) so that the samplings converge to the optimum effectively and efficiently. We consider an interacting-particle algorithm and develop a meta-control methodology which analytically guides the inverse temperature parameter of the algorithm to achieve desired performance characteristics (e.g. quality of the final outcome, algorithm running time, etc.). The main aspect of our meta-control methodology is to formulate an optimal control problem where the fractional change in the inverse temperature parameter is the control variable. The objectives of the optimal control problem are set according to the desired behavior of the interacting-particle algorithm. The control problem considers particles’ average behavior, rather than treating the behavior of individual particles. The solution to the control problem provides feedback on the inverse temperature parameter of the algorithm.     

Fermionic approach to soliton equations

In this paper we exploit the algebraic structure of the soliton equations and find solutions in terms of fermion particles. We show how determinants arise naturally in the fermionic approach to soliton equations. We write the τ-function for charged free fermions in terms of determinants. Examples of how to get soliton, rational and dromion solutions from τ-functions for the various soliton equations are given.     

Solution of soliton equations in terms of neutral fermion particles

In this paper we exploit the algebraic structure of the soliton equations and find solutions in terms of neutral free fermion particles. We show how pfaffians arise naturally in the fermionic approach to soliton equations. We write the τ-function for neutral free fermions in terms of pfaffians. Examples of how to get soliton, rational and dromion solutions from τ-functions for the various soliton equations are given.     

On the final evolution of the n-body problem

The Newtonian n-body problem is studied. The main result gives the asymptotic properties of the distances between particles as time (t) approaches infinity. It is shown how particles can separate into subsystems between which the distances are asymptotic to constant multiples of time, and within which the mutual distances are at most of the order $\text{t}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$. Each subsystem behaves asymptotically like a pure n-body problem in the sense that an energy and an angular momentum relationship are asymptotically satisfied. These techniques allow some three-body results found by Birkhoff and by Sundman to be extended to the n-body problem. Finally, some n-body results are derived for a motion which expands faster than time.     

Quantum game approach for capacity allocation decisions under strategic reasoning

From a common point of view, quantum mechanics, psychology, and decision science disciplines try to predict how unruly systems (atomic particles, human behaviors, and decision makers’ choices) might behave in the future. Effective predicting outcome of a capacity allocation game under various allocation policies requires a profound understanding as how strategic reasoning of decision makers contributes to the financial gain of players. A quantum game framework is employed in the current study to investigate how performance of allocation policies is affected when buyers strategize over order quantities. The results show that the degree of being manipulative for allocation mechanisms is not identical and adopting adaptive quantum method is the most effective approach to secure the highest fill rate and profit when it is practiced under a reasonable range of entanglement levels.

     

A probabilistic model associated with the pressureless gas dynamics

Using a method of stochastic perturbation of a Langevin system associated with the non-viscous Burgers equation we introduce a system of PDE that can be considered as a regularization of the pressureless gas dynamics describing sticky particles. By means of this regularization we describe how starting from smooth data a δ-singularity arises in the component of density. Namely, we find the asymptotics of solution at the point of the singularity formation as the parameter of stochastic perturbation tends to zero. Then we introduce a generalized solution in the sense of free particles (FP-solution) as a special limit of the solution to the regularized system. This solution corresponds to a medium consisting of non-interacting particles. The FP-solution is a bridging step to constructing solutions to the Riemann problem for the pressureless gas dynamics describing sticky particles. We analyze the difference in the behavior of discontinuous solutions for these two models and the relations between them. In our framework we obtain a unique entropy solution to the Riemann problem in 1D case.     

Cross-Diffusion Systems with Excluded-Volume Effects and Asymptotic Gradient Flow Structures

In this paper, we discuss the analysis of a cross-diffusion PDE system for a mixture of hard spheres, which was derived in Bruna and Chapman (J Chem Phys 137:204116-1–204116-16, 2012a) from a stochastic system of interacting Brownian particles using the method of matched asymptotic expansions. The resulting cross-diffusion system is valid in the limit of small volume fraction of particles. While the system has a gradient flow structure in the symmetric case of all particles having the same size and diffusivity, this is not valid in general. We discuss local stability and global existence for the symmetric case using the gradient flow structure and entropy variable techniques. For the general case, we introduce the concept of an asymptotic gradient flow structure and show how it can be used to study the behavior close to equilibrium. Finally, we illustrate the behavior of the model with various numerical simulations.     

Singularities for dynamical arrest in disordered systems

It will be shown how attraction and repulsion among interacting particles can lead to rich dynamical scenarios including glass transitions, gel transitions, and glass-glass transitions. The transition diagram is explained by the topologically stable singularities of the A_ℓ hierarchy. These singularities arise from the microscopic equations of motion and allow for asymptotic solutions. Close to glass-glass transitions asymptotic solutions are identified in results from computer simulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new path-integral representation for the many-particle Green function of the relativistic particles

Starting from the second quantized functional integral representation, the field path-integral representation for the total many-particle Green function for relativistic and nonrelativistic point-like charged Bose and Fermi particles in (3+1) or in (2+1) interacting via Maxwell or Chern-Simon fields is constructed and shown to be only an integral over the trajectories of the particles. The effective action depends on the coordinates and velocities of the particles, and is nonlocal in time due to causal interactions between the particles. In a static (nonrelativistic) approximation, the action is local in time and leads to expressions for the Hamiltonian for Coulomb interaction in (3+1), and for anyon interaction in (2+1) dimensions. This path integral representation automatically includes the usual connection between spin and statistics for the cases of an infinite flat space and trivial topology for the manifold of the charged fields. Our results are generalized in the presence of an external magnetic field. It is shown how to take into account the contribution of the vacuum polarization effects within the framework of the approach.D. N. Lebedev Institute of Physics, Moscow. Department of Physics, Middle East Technical University Ankara, Turkey. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 2, pp. 328–338, May, 1995.     

Form factors in the <Emphasis Type="Italic">N</Emphasis>=4 maximally supersymmetric Yang–Mills theory,soft theorems,and integrability

We discuss the universal soft behavior of form factors in the N=4 maximally supersymmetric Yang–Mills theory in the limit where the momentum of one of the particles tends to zero. We present details of how the tree-level form factors of this theory are related to eigenfunctions of a gl(4|4) integrable spin chain.     

Numerical studies to propose a ghost particle removed SPH (GR-SPH) method

This paper aims to propose a new modified SPH method with novel treatments at the boundaries. Although SPH methods decrease contradictions due to grid distortions compared to traditional mesh-based methods, the penetration of particles through boundaries and the consistency problem make the simulation of problems with definite boundaries a concern. The use of ghost boundary particles and the insertion of artificial forces at the boundaries are the most popular boundary consistency treatments proposed thus far. The use of artificial forces causes the mixing of molecular and finite theories, which can violate the conservation of momentum. This paper shows how the use of ghost boundary particles can violate the continuity equation in problems with non-zero velocity divergence. This study proposes a novel ghost particle removed SPH (GR-SPH) method that discards all ghost particles and artificial forces at the boundaries. Liner layers and liner particles have been defined inside the domain instead of ghost boundary particles in such a way that the so-called violations can partially be remedied. Based on the continuity equation and kernel function unity specification, a novel truncation correction factor has been defined for density renormalization to override the consistency problem at the boundaries. In addition, a new method is proposed to detect the particles near complex wall boundaries and evaluate the normal distance from boundaries. Finally, some benchmark problems have been solved to show the capabilities of the new modified SPH method for the prediction of both particle location and pressure distribution with acceptable accuracy. The GR-SPH method facilitates programming, with fewer particles contributing to the computations. Comparison of its outcomes with published results shows that the new treatments executed at the boundaries are effective.     

Cosserat Parameter Identification within the Frame of the Discrete Element Method

One of the main challenges using the Discrete Element Method is that there is no direct compliance to the well known continuum parameters such as elastic moduli. In this article we show how homogenization procedures using representative volume elements composed of discrete particles lead to Cosserat continua. Simulating a shear test with discrete elements it becomes obvious, that the evolving microstructure is mainly composed of contact chains that form triangles and quadrilaterals. For these contact chains we set up contact energies in normal and shear directions and combine those to derive the effective energy of the material. By comparison of this energy to a Cosserat energy we can derive formulas for the Lamé and Cosserat parameters. They are now only dependent on the interaction energies and radii of the particles. To show the validity of our assumptions and derivations we present some discrete element simulations of shear tests. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)