Complex evolution of a multi-particle system

This paper studies a discrete dynamical system of interacting particles that evolve by interacting among them. The computational model is an abstraction of the natural world, and real systems can range from the huge cosmological scale down to the scale of biological cell, or even molecules. Different conditions for the system evolution are tested. The emerging patterns are analysed by means of fractal dimension and entropy measures. It is observed that the population of particles evolves towards geometrical objects with a fractal nature. Moreover, the time signature of the entropy can be interpreted at the light of complex dynamical systems.

RELATIVISTIC 3-PARTICLE DYNAMICAL EQUATIONS .1. THEORETICAL DEVELOPMENT

Starting from the two-particle Bethe-Salpeter equation in the ladder approximation and integrating over the time component of momentum, we rederive three-dimensional scattering integral equations satisfying constraints of relativistic unitarity and convariance, first derived by Weinberg and by Blankenbecler and Sugar. These two-particle equations are shown to be related by a transformation of variables. Hence we show how to perform and relate identical dynamical calculation using these two equations. Similarly, starting from the Bethe-Salpeter-Faddeev equation for the three-particle system and integrating over the time component of momentum, we derive several three-dimensional three-particle scattering equations satisfying constraints of relativistic unitarity and convariance. We relate two of these three-particle equations by a transformation of variables as in the two-particle case. The three-particle equations we derive are very practical and suitable for performing relativistic scattering calculations. (C) 1994 Academic Press, Inc.

An interplanetary mission to Neptune system: Gravitational capture and maneuvers around triton

The gravitational capture was initially used to understand the capture of planetary satellites. However, in the 90's decade, this phenomenon was applied in spacecraft trajectories. Belbruno and Miller studied missions in the Earth-Moon system that uses this technique to save fuel during the insertion of the spacecraft in its final orbit around the Moon. Using a parameter defined as twice the two-body energy of the planet-particle system, Yamakawa also studied the gravitational capture in the Earth-Moon system. In the present paper, this technique is used to study a mission that goes to the Neptune system and perform a gravitational capture in the satellite Triton. The results show direct and retrograde trajectories, for different values of the initial conditions.

Thermal relaxation for particle systems in interaction with several bosonic heat reservoirs

The present thesis is concerned with the study of a quantum physical system composed of a small particle system (such as a spin chain) and several quantized massless boson fields (as photon gasses or phonon fields) at positive temperature. The setup serves as a simplified model for matter in interaction with thermal "radiation" from different sources. Hereby, questions concerning the dynamical and thermodynamic properties of particle-boson configurations far from thermal equilibrium are in the center of interest. We study a specific situation where the particle system is brought in contact with the boson systems (occasionally referred to as heat reservoirs) where the reservoirs are prepared close to thermal equilibrium states, each at a different temperature. We analyze the interacting time evolution of such an initial configuration and we show thermal relaxation of the system into a stationary state, i.e., we prove the existence of a time invariant state which is the unique limit state of the considered initial configurations evolving in time. As long as the reservoirs have been prepared at different temperatures, this stationary state features thermodynamic characteristics as stationary energy fluxes and a positive entropy production rate which distinguishes it from being a thermal equilibrium at any temperature. Therefore, we refer to it as non-equilibrium stationary state or simply NESS. The physical setup is phrased mathematically in the language of C*-algebras. The thesis gives an extended review of the application of operator algebraic theories to quantum statistical mechanics and introduces in detail the mathematical objects to describe matter in interaction with radiation. The C*-theory is adapted to the concrete setup. The algebraic description of the system is lifted into a Hilbert space framework. The appropriate Hilbert space representation is given by a bosonic Fock space over a suitable L2-space. The first part of the present work is concluded by the derivation of a spectral theory which connects the dynamical and thermodynamic features with spectral properties of a suitable generator, say K, of the time evolution in this Hilbert space setting. That way, the question about thermal relaxation becomes a spectral problem. The operator K is of Pauli-Fierz type. The spectral analysis of the generator K follows. This task is the core part of the work and it employs various kinds of functional analytic techniques. The operator K results from a perturbation of an operator L0 which describes the non-interacting particle-boson system. All spectral considerations are done in a perturbative regime, i.e., we assume that the strength of the coupling is sufficiently small. The extraction of dynamical features of the system from properties of K requires, in particular, the knowledge about the spectrum of K in the nearest vicinity of eigenvalues of the unperturbed operator L0. Since convergent Neumann series expansions only qualify to study the perturbed spectrum in the neighborhood of the unperturbed one on a scale of order of the coupling strength we need to apply a more refined tool, the Feshbach map. This technique allows the analysis of the spectrum on a smaller scale by transferring the analysis to a spectral subspace. The need of spectral information on arbitrary scales requires an iteration of the Feshbach map. This procedure leads to an operator-theoretic renormalization group. The reader is introduced to the Feshbach technique and the renormalization procedure based on it is discussed in full detail. Further, it is explained how the spectral information is extracted from the renormalization group flow. The present dissertation is an extension of two kinds of a recent research contribution by Jakšić and Pillet to a similar physical setup. Firstly, we consider the more delicate situation of bosonic heat reservoirs instead of fermionic ones, and secondly, the system can be studied uniformly for small reservoir temperatures. The adaption of the Feshbach map-based renormalization procedure by Bach, Chen, Fröhlich, and Sigal to concrete spectral problems in quantum statistical mechanics is a further novelty of this work.

Convergence results for stochastic particle systems with social interaction

We consider stochastic individual-based models for social behaviour of groups of animals. In these models the trajectory of each animal is given by a stochastic differential equation with interaction. The social interaction is contained in the drift term of the SDE. We consider a global aggregation force and a short-range repulsion force. The repulsion range and strength gets rescaled with the number of animals N. We show that for N tending to infinity stochastic fluctuations disappear and a smoothed version of the empirical process converges uniformly towards the solution of a nonlinear, nonlocal partial differential equation of advection-reaction-diffusion type. The rescaling of the repulsion in the individual-based model implies that the corresponding term in the limit equation is local while the aggregation term is non-local. Moreover, we discuss the effect of a predator on the system and derive an analogous convergence result. The predator acts as an repulsive force. Different laws of motion for the predator are considered.

Stationarity of multivariate particle systems

A particle system is a family of i.i.d. stochastic processes with values translated by Poisson points. We obtain conditions that ensure the stationarity in time of the particle system in RdRd and in some cases provide a full characterisation of the stationarity property. In particular, a full characterisation of stationary multivariate Brown–Resnick processes is given.

Aspects of Markov Chains and Particle Systems

Thesis (Ph.D.)--University of Washington, 2016-06

Contours in Motion

My practice-led research explores and maps workflows for generating experimental creative work involving inertia based motion capture technology. Motion capture has often been used as a way to bridge animation and dance resulting in abstracted visuals outcomes. In early works this process was largely done by rotoscoping, reference footage and mechanical forms of motion capture. With the evolution of technology, optical and inertial forms of motion capture are now more accessible and able to accurately capture a larger range of complex movements. The creative work titled “Contours in Motion” was the first in a series of studies on captured motion data used to generating experimental visual forms that reverberate in space and time. With the source or ‘seed’ comes from using an Xsens MVN - Inertial Motion Capture system to capture spontaneous dance movements, with the visual generation conducted through a customised dynamics simulation. The aim of the creative work was to diverge way from a standard practice of using particle system and/or a simple re-targeting of the motion data to drive a 3d character as a means to produce abstracted visual forms. To facilitate this divergence a virtual dynamic object was tether to a selection of data points from a captured performance. The proprieties of the dynamic object were then adjusted to balance the influences from the human movement data with the influence of computer based randomization. The resulting outcome was a visual form that surpassed simple data visualization to project the intent of the performer’s movements into a visual shape itself. The reported outcomes from this investigation have contributed to a larger study on the use of motion capture in the generative arts, furthering the understanding of and generating theories on practice.

Dynamics after a sweep through a quantum critical point

The coherent quantum evolution of a one-dimensional many-particle system after slowly sweeping the Hamiltonian through a critical point is studied using a generalized quantum Ising model containing both integrable and nonintegrable regimes. It is known from previous work that universal power laws of the sweep rate appear in such quantities as the mean number of excitations created by the sweep. Several other phenomena are found that are not reflected by such averages: there are two different scaling behaviors of the entanglement entropy and a relaxation that is power law in time rather than exponential. The final state of evolution after the quench is not characterized by any effective temperature, and the Loschmidt echo converges algebraically for long times, with cusplike singularities in the integrable case that are dynamically broadened by nonintegrable perturbations.

Jauheiden erottuminen

Partikkelisysteemien segregaatio eli erottuminen on ilmiö, jossa tasalaatuisen jauheseoksen komponenteilla on taipumus erota toisistaan. Jauheen erottumistaipumus riippuu partikkelien ominaisuuksista, ympäröivistä olosuhteista ja partikkelien välisistä vuorovaikutuksista. Segregaatiomekanismeja on esitetty kirjallisuudessa valtava määrä ja pienetkin erot partikkelien välisissä ominaisuuksissa ja vuorovaikutuksissa voivat johtaa täysin eri segregaatiomekanismeihin. Segregaatioilmiö on lääketeollisuuden näkökulmasta hyvin keskeinen, eikä sitä tunneta vielä riittävän hyvin, jotta siltä osattaisiin systemaattisesti välttyä. Nykyinen segregaatiotutkimus perustuu suurelta osin yrityksen ja erehdyksen kautta tapahtuvaan oppimiseen. Todellisen segregaatioilmiön ymmärtämiseen tarvittaisiin innovatiivisia tutkimusmenetelmiä. Kokeellisen osan tarkoituksena oli kehittää ja perustestata menetelmä, jolla voidaan tutkia erilaisten partikkelisysteemien erottumiskäyttäytymistä, ja käyttää tätä menetelmää farmaseuttisten rae- ja pellettiseosten segregaation tutkimiseen. Tavoitteena oli todistaa kehitetyn Babel-laitteen toimintaperiaatteen soveltuvuus partikkelisysteemien erottumiskäyttäytymisen tutkimiseen, mutta suoritetut kokeet olivat lähinnä menetelmän ja laitteen testausta. Ongelmiksi muodostuivat Babel-laitteen asettamat rajoitukset, partikkelien sähköistyminen ja partikkelien väliset vuorovaikutukset. Käytetyt suoraviivaiset lähestymistavat eivät riittäneet segregaation aiheuttamiseen Babel-laitteella. Vertikaalisen ravistelun seurauksena syntynyt konvektiopyörre esti segregaation syntymisen. Johtopäätöksenä voidaan sanoa, että Babel-laite mittaa hyvin ja toistettavasti sekä se kykenee erottamaan erikokoiset partikkelit ja erilaiset kokojakaumat toisistaan. Laitteen kehittämistavoitteena olisi saada segregaatio paremmin näkyviin jauheseoksissa ravistelun seurauksena. Tällöin voitaisiin tehdä päätelmiä jauheseoksen erottumistaipumuksesta ja systeemissä vallitsevista erottumismekanismeista. Laitteen ja menetelmän jatkokehittäminen voisi tuottaa hyödyllistä lisätietoa, mikä edesauttaisi segregaation ymmärtämistä ilmiönä entistä paremmin

Dependence of collision frequency on distance between two hard-sphere particles estimated by computer simulation

A Monte Carlo simulation is performed to study the dependence of collision frequency on interparticle distance for a system composed of two hard-sphere particles. The simulation quantitatively shows that the collision frequency drops down sharply as the distance between two particles increases. This characteristic provides a useful evidence for the collision-reaction dynamics of aggregation process for the two-particle system described in the other reference.

Neutron-neutron scattering parameters from the ^3He(d,t)2p and ^3H(d,^3He)2n reactions

The energy spectra of tritons and Helium-3 nuclei from the reactions ³He(d,t)2p, ³H(d,³He)2n, ³He(d,³He)pn, and ³H(d,t)pn were measured between 6° and 20° at a bombarding energy of 10.9 MeV. An upper limit of 5 μb/sr. was obtained for producing a bound di-neutron at 6° and 7.5°. The ³He(d,t)2p and ³H(d,³He)2n data, together with previous measurements at higher energies, have been used to investigate whether one can unambiguously extract information on the two-nucleon system from these three-body final state reactions. As an aid to these theoretical investigations, Born approximation calculations were made employing realistic nucleon-nucleon potentials and an antisymmetrized final state wave function for the five-particle system. These calculations reproduce many of the features observed in the experimental data and indicate that the role of exchange processes cannot be ignored. The results show that previous attempts to obtain information on the neutron-neutron scattering length from the ³H(d,³He)2n reaction may have seriously overestimated the precision that could be attained.