Modeling opinion formation in the kinetic theory of active particles I: spontaneous trend

The kinetic theory of active particles is used to model the formation and evolution of opinions in a structured population. The spatial structure is modeled by a network whose nodes mimic the geographic distribution of individuals, while the functional subsystems present in each node group together elements sharing a common orientation. In this paper we introduce a model, based on nonlinear and nonlinearly additive interactions among individuals, subsystems and nodes, related to the spontaneous evolution of opinion concerning given specific issues. Numerical solutions in a model situation not related with real data show how the mutual interactions are able to drive the subsystems opinion toward the emergence of collective structures characterizing this kind of complex systems.     

Nonlinear martingale problems involving singular integrals

We study a class of nonlinear martingale problems in one dimension, that involve a singular integral of the density in the drift term, and are related to systems of particles with singular interactions. First, we prove existence and uniqueness of regular solutions of the associated nonlinear evolution equation. Then, we establish a suitable framework and conditions where the martingale problem is well posed. This extends the results of Bonami et al. (J. Funct. Anal. 165 (1999) 390) to a wide class of coefficients and initial conditions. Finally, we obtain our solution of the martingale problem as the chaotic limit of some systems of particles interacting through regular approximating kernels.     

Computer studies of planetary-type evolution

In this paper a new, computer approach to the study of the interactions of particles with differing masses is applied to the study of planetary type evolution. The formulation contains an inherent self-reorganization property in which particles self-stratify in accordance with their masses. Computer examples are described and discussed.     

Flocking of Multi-particle Swarm with Group Coupling Structure and Measurement Delay

We investigate the flocking conditions of a group coupling system with time delays, in which the communication between particles includes inter-group and intra-group interactions, and the time delay comes from the theory of moving object observation. As an effective model, we introduce a system of nonlinear functional differential equations to describe its dynamic evolution mechanism. By constructing two differential inequalities on velocity and velocity fluctuation from a continuity argument, and using the Lyapunov functional approach, we present some sufficient conditions for the existence of asymptotic flocking solutions to the coupling system, in which an upper bound of the delay allowed by the system is quantitatively given to ensure the emergence of flocking behavior. All results are novel and can be illustrated by using some specific numerical simulations.     

Mean‐Field Evolution of Fermionic Mixed States

In this paper we study the dynamics of fermionic mixed states in the mean‐field regime. We consider initial states that are close to quasi‐free states and prove that, under suitable assumptions on the initial data and on the many‐body interaction, the quantum evolution of such initial data is well approximated by a suitable quasi‐free state. In particular, we prove that the evolution of the reduced one‐particle density matrix converges, as the number of particles goes to infinity, to the solution of the time‐dependent Hartree‐Fock equation. Our result holds for all times and gives effective estimates on the rate of convergence of the many‐body dynamics towards the Hartree‐Fock evolution.© 2015 Wiley Periodicals, Inc.     

Numerical Study on the Influence of Particle Inertia in Particle-Driven Gravity Currents

We present two-dimensional numerical simulations of particle-driven gravity currents in a lock-exchange configuration. The fluid is described in an Eulerian framework, whereas the particles are tracked in a Lagrangian manner. The study is restricted to dilute suspensions, allowing to neglect particle-particle interactions. The particle forces considered are buoyancy and the Stokes drag. We study the influence of particle inertia on the flow evolution by performing simulations with different Stokes numbers. We also consider the case where particle inertia is neglected. Generally, we observe significant changes in the form and structure of the gravity current with increasing particle Stokes numbers. Particularly, the formation of Kelvin-Helmholtz vortices is more and more suppressed. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The influence of random interactions and decision heuristics on norm evolution in social networks

In this paper we explore the effect that random social interactions have on the emergence and evolution of social norms in a simulated population of agents. In our model agents observe the behaviour of others and update their norms based on these observations. An agent’s norm is influenced by both their own fixed social network plus a second random network that is composed of a subset of the remaining population. Random interactions are based on a weighted selection algorithm that uses an individual’s path distance on the network to determine their chance of meeting a stranger. This means that friends-of-friends are more likely to randomly interact with one another than agents with a higher degree of separation. We then contrast the cases where agents make highest utility based rational decisions about which norm to adopt versus using a Markov Decision process that associates a weight with the best choice. Finally we examine the effect that these random interactions have on the evolution of a more complex social norm as it propagates throughout the population. We discover that increasing the frequency and weighting of random interactions results in higher levels of norm convergence and in a quicker time when agents have the choice between two competing alternatives. This can be attributed to more information passing through the population thereby allowing for quicker convergence. When the norm is allowed to evolve we observe both global consensus formation and group splintering depending on the cognitive agent model used.     

Modeling and simulation of the motion of nanoparticles in cylindrical capillaries allowing particle‐to‐wall interactions

The process of transporting nanoparticles at the blood vessels level stumbles upon various physical and physiological obstacles; therefore, a Mathematical modeling will provide a valuable means through which to understand better this complexity. In this paper, we consider the motion of nanoparticles in capillaries having cylindrical shapes (i.e., tubes of finite size). Under the assumption that these particles have spherical shapes, the motion of these particles reduces to the motion of their centers. Under these conditions, we derive the mathematical model, to describe the motion of these centers, from the equilibrium of the gravitational force, the hemodynamic force and the van der Waals interaction forces. We distinguish between the interaction between the particles and the interaction between each particle and the walls of the tube. Assuming that the minimum distance between the particles is large compared with the maximum radius R of the particles and hence neglecting the interactions between the particles, we derive simpler models for each particle taking into account the particles‐to‐wall interactions. At an error of order O(R) or O(R³)(depending if the particles are 'near' or 'very near' to the walls), we show that the horizontal component of each particle's displacement is solution of a nonlinear integral equation that we can solve via the fixed point theory. The vertical components of the displacement are computable in a straightforward manner as soon as the horizontal components are estimated. Finally, we support this theory with several numerical tests. Copyright © 2016 John Wiley & Sons, Ltd.     

The Lie algebra structure and controllability of spin systems

In this paper, we study the controllability properties and the Lie algebra structure of networks of particles with spin immersed in an electro-magnetic field. We relate the Lie algebra structure to the properties of a graph whose nodes represent the particles and an edge connects two nodes if and only if the interaction between the two corresponding particles is active. For networks with different gyromagnetic ratios, we provide a necessary and sufficient condition of controllability in terms of the properties of the above-mentioned graph and describe the Lie algebra structure in every case. For these systems all the controllability notions, including the possibility of driving the evolution operator and/or the state, are equivalent. For general networks (with possibly equal gyromagnetic ratios), we give a sufficient condition of controllability. A general form of interaction among the particles is assumed which includes both Ising and Heisenberg models as special cases. Assuming Heisenberg interaction we provide an analysis of low-dimensional cases (number of particles less than or equal to three) which includes necessary and sufficient controllability conditions as well as a study of their Lie algebra structure. This also provides an example of quantum mechanical systems where controllability of the state is verified while controllability of the evolution operator is not.     

Superdense dark matter clumps from superheavy particles

We describe some specific but reasonable conditions for the formation of superdense clumps (or minihalos) of dark matter. Such clumps can be produced by several mechanisms, most notably by spiky features in the spectrum of density perturbations. Being produced very early during the radiation-dominated epoch, these clumps evolve as isolated objects. They do not belong to hierarchical structures for a long time after production and are therefore not destroyed by tidal interactions during the formation of larger structures. If the clumps are constituted of superheavy dark matter particles, then the evolution of their central part can lead to a “gravithermal catastrophe,” increasing the central density and thus the annihilation signal. As a result, annihilations of superheavy neutralinos in dense clumps may lead to observable fluxes of annihilation products in the form of ultrahigh-energy particles.     

Hydrodynamic Capture and Release of Passively Driven Particles by Active Particles Under Hele-Shaw Flows

The transport of active and passive particles plays central roles in diverse biological phenomena and engineering applications. In this paper, we present a theoretical investigation of a system consisting of an active particle and a passive particle in a confined micro-fluidic flow. The introduction of an external flow is found to induce the capture of the passive particle by the active particle via long-range hydrodynamic interactions among the particles. This hydrodynamic capture mechanism relies on an attracting stable equilibrium configuration formed by the particles, which occurs when the external flow intensity exceeds a certain threshold. We evaluate this threshold by studying the stability of the equilibrium configurations analytically and numerically. Furthermore, we study the dynamics of typical capture and non-capture events and characterize the basins of attraction of the equilibrium configurations. Our findings reveal a critical dependence of the hydrodynamic capture mechanism on the external flow intensity. Through adjusting the external flow intensity across the stability threshold, we demonstrate that the active particle can capture and release the passive particle in a controllable manner. Such a capture-and-release mechanism is desirable for biomedical applications such as the capture and release of therapeutic payloads by synthetic micro-swimmers in targeted drug delivery.     

一类带有非线性爆炸的粒子增长的数学模型

所建立的数学模型是由可数无穷多个彼此相互关联的非线性常微分方程所组成的自治系统,它刻划了在只有基本粒子和i-粒子(i≥1 ) 进行碰撞反应的系统里,粒子增长过程中密度随时间的变化规律.研究了这一自治系统解的存在性、唯一性、密度守恒以及解的渐近性质.     

Global well-posedness for an inhomogeneous LSW–model in unbounded domains

The LSW–theory of domain coarsening describes the evolution of the size distribution of a system of particles evolving by diffusional mass exchange to reduce their total surface area. We prove existence and uniqueness of solutions to an inhomogeneous extension of the LSW–model in unbounded domains. This model arises naturally as a homogenization limit of the underlying free boundary problem in the case of a system of particles for which the screening length (the effective range of particle interactions) is smaller than the system size. The crucial ingredients in the analysis are, first, to establish the screening property by showing that the corresponding Greens function decreases exponentially over the relevant distances. Second, we have to control the mass fluxes between different regions to prevent aggregation of mass in few particles which would result in blow–up of the solution.Mathematics Subject Classification (2000):82C26, 35Q72, 35D05Received: 16, December 2001     

Nonlinear limit for a system of diffusing particles which alternate between two states

We study a system of particles and the nonlinear McKean-Vlasov diffusion that is its limit for weak interactions. Each particle switches between two states, both with their own diffusion dynamics. There is interaction, in particular, in the rates of the switches. We show existence and uniqueness for the system of particles by stopping-time techniques. For the nonlinear martingale problem, we use a time-change that allows us to return to a strong pathwise representation, and then we use a contraction argument for an appropriate metric. Finally, we show propagation of chaos.     

Dynamics of Vortex Rossby Waves in Tropical Cyclones,Part 2: Nonlinear Time‐Dependent Asymptotic Analysis on a β‐Plane

Vortex Rossby waves in cyclones in the tropical atmosphere are believed to play a role in the observed eyewall replacement cycle, a phenomenon in which concentric rings of intense rainbands develop outside the wall of the cyclone eye, strengthen and then contract inward to replace the original eyewall. In this paper, we present a two‐dimensional configuration that represents the propagation of forced Rossby waves in a cyclonic vortex and use it to explore mechanisms by which critical layer interactions could contribute to the evolution of the secondary eyewall location. The equations studied include the nonlinear terms that describe wave‐mean‐flow interactions, as well as the terms arising from the latitudinal gradient of the Coriolis parameter. Asymptotic methods based on perturbation theory and weakly nonlinear analysis are used to obtain the solution as an expansion in powers of two small parameters that represent nonlinearity and the Coriolis effects. The asymptotic solutions obtained give us insight into the temporal evolution of the forced waves and their effects on the mean vortex. In particular, there is an inward displacement of the location of the critical radius with time which can be interpreted as part of the secondary eyewall cycle.     

Two stages of motion of anharmonic oscillators modeling fast particles in crystals

We consider the evolution of the spatial distribution of fast atomic particles as they pass through a crystal. At small penetration depths where the particles move without significant loss of coherence, the joint probability density function becomes smoother at the expense of the gradient of the anharmonic potential of the planar channel. We show that as a result of this smoothing, there arises a quasiequilibrium (quasistationary) state of the subsystem of fast particles. The further evolution of the particle distribution is caused by nonelastic scattering and is described by the kinetic equation. At this stage, the anharmonic character of particle oscillations between the channel walls results in a renormalization of the total phonon scattering cross section of particles, and the renormalization is determined by the fourth-order anharmonic interaction.     

Periodic solutions for the Landau-Lifshitz-Gilbert equation

Ferromagnetic materials tend to develop very complex magnetization patterns whose time evolution is modeled by the so-called Landau-Lifshitz-Gilbert equation (LLG). In this paper, we construct time-periodic solutions for LLG in the regime of soft and small ferromagnetic particles which satisfy a certain shape condition. Roughly speaking, it is assumed that the length of the particle is greater than its hight and its width. The approach is based on a perturbation argument and the spectral analysis of the corresponding linearized problem as well as the theory of sectorial operators.     

Observations on the nuclear interactions of cosmic ray pions and nucleons at energies ≳ 20 GeV

Detailed features of extremely collimated nuclear interactions induced by cosmic ray particles in carbon and brass (belonging to group I as classified in Part I of this series of papers) are presented. These extremely collimated nuclear interactions seem to be preferentially induced by pions rather than by nucleons; also the relative frequency of these seems to be less when brass is used as target compared to the case with carbon as target. The distribution of multiplicities of secondary particles emitted in the forward direction show certain regularities in the case of interactions induced by charged primaries. Observations on the γ-rays associated with these events give support to the interpretation that in these inelastic collisions pions are produced in pairs in the forward direction with low transverse momentum. It is suggested that such a low energy di-pion system could be the same as found in the so-called ABC effect.