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.

Reverse engineering in many-body quantum physics: Correspondence between many-body systems and effective single-particle equations

The mapping, exact or approximate, of a many-body problem onto an effective single-body problem is one of the most widely used conceptual and computational tools of physics. Here, we propose and investigate the inverse map of effective approximate single-particle equations onto the corresponding many-particle system. This approach allows us to understand which interacting system a given single-particle approximation is actually describing, and how far this is from the original physical many-body system. We illustrate the resulting reverse engineering process by means of the Kohn-Sham equations of density-functional theory. In this application, our procedure sheds light on the nonlocality of the density-potential mapping of density-functional theory, and on the self-interaction error inherent in approximate density functionals.

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

Magnetic properties of Fe(3)O(4) nanoparticles coated with oleic and dodecanoic acids

Magnetic nanoparticles (NP) of magnetite (Fe(3)O(4)) coated with oleic acid (OA) and dodecanoic acid (DA) were synthesized and investigated through transmission electron microscopy (TEM), magnetization M, and ac magnetic susceptibility measurements. The OA coated samples were produced with different magnetic concentrations (78%, 76%, and 65%) and the DA sample with 63% of Fe(3)O(4). Images from TEM indicate that the NP have a nearly spherical geometry and mean diameter similar to 5.5 nm. Magnetization measurements, performed in zero-field cooled (ZFC) and field cooled processes under different external magnetic fields H, exhibited a maximum at a given temperature T(B) in the ZFC curves, which depends on the NP coating (OA or DA), magnetite concentration, and H. The temperature T(B) decreases monotonically with increasing H and, for a given H, the increase in the magnetite concentration results in an increase in T(B). The observed behavior is related to the dipolar interaction between NP, which seems to be an important mechanism in all samples studied. This is supported by the results of the ac magnetic susceptibility chi(ac) measurements, where the temperature in which chi' peaks for different frequencies follows the Vogel-Fulcher model, a feature commonly found in systems with dipolar interactions. Curves of H versus T(B)/T(B) (H=0) for samples with different coatings and magnetite concentrations collapse into a universal curve, indicating that the qualitative magnetic behavior of the samples may be described by the NP themselves, instead of the coating or the strength of the dipolar interaction. Below T(B), M versus H curves show a coercive field (H(C)) that increases monotonically with decreasing temperature. The saturation magnetization (M(S)) follows the Bloch's law and values of M(S) at room temperature as high as 78 emu/g were estimated, a result corresponding to similar to 80% of the bulk value. The overlap of M/M(S) versus H/T curves for a given sample and the low H(C) at high temperatures suggest superparamagnetic behavior in all samples studied. The overlap of M/M(S) versus H curves at constant temperature for different samples indicates that the NP magnetization behavior is preserved, independently of the coating and magnetite concentration. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3311611]

Collision probabilities in the rarefaction fan of asymmetric exclusion processes

We consider the one-dimensional asymmetric simple exclusion process (ASEP) in which particles jump to the right at rate p is an element of (1/2, 1.] and to the left at rate 1 - p, interacting by exclusion. In the initial state there is a finite region such that to the left of this region all sites are occupied and to the right of it all sites are empty. Under this initial state, the hydrodynamical limit of the process converges to the rarefaction fan of the associated Burgers equation. In particular suppose that the initial state has first-class particles to the left of the origin, second-class particles at sites 0 and I, and holes to the right of site I. We show that the probability that the two second-class particles eventually collide is (1 + p)/(3p), where a collision occurs when one of the particles attempts to jump over the other. This also corresponds to the probability that two ASEP processes. started from appropriate initial states and coupled using the so-called ""basic coupling,"" eventually reach the same state. We give various other results about the behaviour of second-class particles in the ASEP. In the totally asymmetric case (p = 1) we explain a further representation in terms of a multi-type particle system, and also use the collision result to derive the probability of coexistence of both clusters in a two-type version of the corner growth model.

One-dimensional point interaction with three complex parameters

For a pair of non-Hermitian Hamiltonian H and its Hermitian adjoint H(dagger), there are situations in which their eigenfunctions form a biorthogonal system. We illustrate such a situation by means of a one-particle system with a one-dimensional point interaction in the form of the Fermi pseudo-potential. The interaction consists of three terms with three strength parameters g(i) (i = 1, 2 and 3), which are all complex. This complex point interaction is neither Hermitian nor PT-invariant in general. The S-matrix for the transmission reflection problem constructed with H (or with H(dagger)) in the usual manner is not unitary, but it conforms to the pseudo-unitarity that we define. The pseudounitarity is closely related to the biorthogonality of the eigenfunctions. The eigenvalue spectrum of H with the complex interaction is generally complex but there are cases where the spectrum is real. In such a case H and H(dagger) form a pseudo-Hermitian pair.

3D-partikkelit aloittelijoille : tarkastelussa 3D-Kalevala-projekti

Opinnäytteeni teososana on 3D-Kalevala projektinimellä tunnetun animaation partikkeliefektit. Tarkastelen tarkemmin elokuvan Lumi-, Kylä-, Paja- ja Luola-kohtauksia. 3D-Kalevala on tietokoneella tehty animaatio, joka kertoo Suomen kansalliseepoksen Kalevalan päähenkilöstä Väinämöisestä. Elokuvassa vanha Väinämöinen muistelee nuoruutensa tapahtumia. 3D-Kalevala-projekti käynnistettiin vuonna 2003, mutta sen alkuperäiset tekijät eivät saaneet sitä valmiiksi, ja projektin teko keskeytettiin vuonna 2005. Vuoden 2006 keväällä projektiin perustettiin uusi kahden verkkoviestinnän opiskelijan projektiryhmä, jonka tehtävänä oli saada elokuva valmiiksi vuoden 2007 kevääseen mennessä. Kun aloitimme projektin tekemisen, olin kolmiulotteisessa mallinnuksessa aloittelija. Tästä johtuen opinnäytteen kirjallinen osa on opas aloittelijoille partikkelien maailmaan. Selvitän raportissani miten elokuvan partikkeliefektit rakennettiin, mitä niiden tekeminen minulta vaati ja miten ne loppujen lopuksi mielestäni onnistuivat. Elokuvan efektit tehtiin 3D Studio Max-ohjelman versiolla 6.0, ja tämän takia kirjoitankin efektien rakentamisesta kyseisen ohjelman keinoin. Projektin suuruuden vuoksi molemmat tekijät pääsivät tekemään monenlaisia töitä, mutta päävastuualueet olivat selvät. Minun osani oli tehdä elokuvaan efektejä. Partikkeliefektit ovat proseduraalisia efektejä, joiden avulla on mahdollista tehdä aidon näköisiä luonnonilmiöitä, kuten tulta, savua, kipinöitä ja veden roiskeita. Koska partikkeliefektit mallintavat reaalimaailman ilmiöitä, on tekijän hyvä olla kiinnostunut selvittämään ilmiöiden käyttäytymistä luonnossa. Raportoin myös projektin aikana huomaamistani hyvistä tavoista opiskella itsenäisesti partikkelien rakentamiseen käytettyjä tekniikoita. On hyvä lukea 3D Studio Maxin tasokasta käyttöohjesovellusta, tutustua Internetissä löytyviin 3D-aiheisiin foorumeihin, käydä aiheesta tutoriaaleja läpi sekä tutustua ohjelman ominaisuuksiin kokeilemalla ja tutkimalla itsenäisesti. Elokuvan efektit onnistuivat mielestäni kiitettävästi ottaen huomioon lähtötasoni. Löysin tapoja kehittää itseäni ja helppoja keinoja toteuttaa realistista jälkeä efektien rakentamisessa. Toivon, että raportistani olisi jollekin 3D-partikkeliefekteistä kiinnostuneelle hyötyä.

Flokkulaation seuraaminen partikkelikokoanalysaattorilla

Tässä työssä tutkittiin kahden erilaisen partikkelikokoanalysaattorin, PSyA:n ja PIA:n soveltuvuutta flokkuloinnin online-seurantaan. Kummallekin menetelmälle määritettiin raja-arvot, kuten lietteen maksimisakeus. Lisäksi tutkittiin flokkulanttiannostuksen, sekoitusnopeuden, sekoitusajan ja lietteen kiintoainepitoisuuden vaikutusta flokkikokojakaumaan. Kirjallisuusosassa tarkasteltiin kolloidisen suspension ominaispiirteitä, koaguloinnin ja flokkuloinnin teoriaa, flokkulaation kokeellista tutkimista sekä prosessin jatkuvatoimiseen seurantaan soveltuvia laitteita. Lisäksi esitettiin taustaa hydrometallurgisesta prosessista, johon työ liittyy. Flokkauskokeissa käytettiin jätevettä, jonka koostumus vastasi metalliteollisuuden peittausjätevesien tyypillistä koostumusta. Tutkittava jätevesimäärä käsiteltiin ensin kalkkimaidolla, jonka jälkeen saostunut kiintoaine flokattiin synteettisellä polymeeriflokkulantilla. Lietteen keskimääräinen kiintoainepitoisuus oli n. 10 g/l. Esikokeiden perusteella PSyA:lla voitiin mitata ilman laimennusta, mutta PIA:lla tuloksia ei saatu ilman laimentamista kiintoainepitoisuuteen n. 2,5 g/l. Kokeiden aikana havaittiin, että flokit muodostuivat erittäin nopeasti. Flokkien hajoaminen alkoi pian sen jälkeen, kun flokkulantin annostelu lopetettiin. Sekoitusnopeudella 40 r/min tai alle flokit alkoivat laskeutua astian pohjalle sekoituksesta huolimatta ja ne pysyivät pitempään koossa kuin suuremmilla sekoitusnopeuksilla. 5 - 10 minuutin kuluttua flokkulantin lisäämisestä saavutettiin tasapaino, jolloin flokkien kokojakauma ei enää muuttunut. Sekoitusnopeuksilla 80 r/min ja 120 r/min tasapainotilanteen koko-jakauma oli selvästi kapeampi kuin pienimmällä sekoitusnopeudella. Alkuperäisessä lietteessä flokit olivat suurempia kuin laimennetussa lietteessä. PSyA:lla jännepituusjakaumien määrittäminen oli varsin hidasta prosessissa tapahtuviin muutoksiin verrattuna, ja tuloksissa oli suurta hajontaa. PIA:lla saadut partikkelikokojakaumat sitä vastoin olivat johdonmukaisempia, vaikka suurimpien flokkien määrittäminen osoittautuikin epämääräiseksi. Menetelmän suurimmaksi puutteeksi todettiin soveltumattomuus sakeiden lietteiden analysointiin. Kumpikaan menetelmä ei ilman modifiointia sovellu tutkitun lietteen kaltaisten prosessilietteiden flokkuloinnin seurantaan.