309,151 research outputs found
Flows of singular vector fields and applications to fluid and kinetic equations
No full textSeveral physical phenomena arising in fluid dynamics and kinetic equations can be modeled by nonlinear transport PDE. Such quantities are the vorticity of a fluid, or the density of a collection of particles advected by a velocity field which is highly irregular. The theory of characteristics provides a link between this PDE and the ODE dX/dt=b(t,X(t,x)), where b is the velocity field. When b has Sobolev or BV regularity and bounded divergence, the theory of DiPerna-Lions and Ambrosio gives a good notion of solution to the ordinary differential equation using the concept of regular Lagrangian flow. Extending the results of Crippa-DeLellis, and more recently Bouchut-Crippa, we study Lagrangian flows associated to velocity fields with anisotropic regularity: those with gradient given by the singular integral of an L^1 function in some directions, and the singular integral of a measure in others. We exploit an anisotropic version of the previous arguments and estimate the difference quotients in this context, thereby gaining quantitative estimates in terms of the given regularity bounds. One then recovers well-posedness for the ordinary differential equation. This answers positively the question of existence of Lagrangian solutions to the Vlasov Poisson and Euler equations with L^1 data
A class of optimal control problems of McKean-Vlasov SDES
No full textLAUREA MAGISTRALEQuesta tesi studia una classe di problemi di controllo ottimo stocastico per equazioni differenziali stocastiche di tipo McKean-Vlasov. La differenza principale con i problemi standard di controllo ottimo stocastico consiste nel fatto che i coefficienti dell’equazione di stato e del funzionale guadagno dipendono anche dalla legge del processo di stato. I problemi di controllo di tipo McKean-Vlasov rappresentano un campo di ricerca che si è sviluppato molto recentemente grazie al legame con la teoria dei giochi a campo medio, un argomento introdotto solo pochi anni fa da J.-M. Lasry e P.-L. Lions. Sia i problemi di controllo di tipo McKean-Vlasov che i giochi a campo medio sono legati a giochi differenziali stocastici con un’infinità di giocatori, interagenti tra loro in modo simmetrico.
La classe di problemi di controllo di tipo McKean-Vlasov esplorata in questa tesi è caratterizzata dal fatto che i coefficienti dell’equazione di stato e del funzionale guadagno possono essere non limitati rispetto alla variabile di controllo. Inoltre, a differenza del classico caso lineare-quadratico, tali coefficienti possono essere non-lineari. Si tratta di una classe di problemi ancora inesplorata, infatti finora la letteratura si è concentrata sul caso di coefficienti limitati nella variabile di controllo.The present thesis focuses on a class of stochastic optimal control problems of McKean-Vlasov stochastic differential equations. The main difference from standard stochastic optimal control problems is that the coefficients of the state equation, as well as of the optimization functional, may depend on the law of the state process. McKean-Vlasov control problems have known a surge of interest with the emergence of mean field game theory, a pioneering subject introduced few years ago by J.-M. Lasry and P.-L. Lions. Both problems can be interpreted as searches for equilibria of stochastic differential games with a continuum of players, symmetrically interacting each other through the empirical distribution of the entire population.
In this thesis we study a particular class of McKean-Vlasov control problems, characterized by the fact that coefficients may be unbounded with respect to the control variable. Moreover, coefficients are not necessarily as in the classical linear-quadratic case, indeed they can be even non-linear. This is a still unexplored class of problems, in fact the literature on McKean-Vlasov control problems focuses on the standard case of coefficients bounded in the control variable
A parametric study of the numerical simulations of triggered VLF emissions
No full textThis work is concerned with the numerical modelling of VLF emissions triggered in the equatorial region of the Earth’s magnetosphere, using a well established 1D Vlasov Hybrid Simulation (VHS) code. Although this code reproduces observed ground based emissions well there is some uncertainty regarding the magnitude of simulation parameters such as saturation wave amplitude, cold plasma density, linear growth rate and simulation bandwidth. Concentrating on emissions triggered by pulses of VLF radio waves from the transmitter at Siple Station, Antarctica (L=4.2), these parameters, as well as triggering pulse length and amplitude, are systematically varied. This parametric study leads to an understanding of the physics of the triggering process and also of how the properties of these emissions, particularly their frequency time profile, depend upon these parameters. The main results are that weak power input tends to generate fallers, intermediate power input gives stable risers and strong growth rates give fallers, hooks or oscillating tones. The main factor determining the frequency sweep rate - of either sign - turns out to be the cold plasma density, lower densities giving larger sweep rates
Equilibrium and dynamics of collisionless current sheets
No full textIn this thesis examples of translationally invariant one-dimensional (1D) Vlasov-Maxwell (VM) equilibria are investigated. The 1D VM equilibrium equations are equivalent to the motion of
a pseudoparticle in a conservative pseudopotential, with the pseudopotential being proportional to one of the diagonal components of the plasma pressure tensor. A necessary condition on the pseudopotential (plasma pressure) to allow for force-free 1D VM equilibria is formulated. It is
shown that linear force-free 1D VM solutions correspond to the case where the pseudopotential is an attractive central potential. The pseudopotential for the force-free Harris sheet is found and a Fourier transform method is used to find the corresponding distribution function. The solution is extended to include a family of equilibria that describe the transition between the Harris sheet and the force-free Harris sheet. These equilibria are used in 2.5D particle-in-cell simulations of
magnetic reconnection. The structure of the diffusion region is compared for simulations starting from anti-parallel magnetic field configurations with different strengths of guide field and self-consistent linear and non-linear force-free magnetic fields. It is shown that gradients of off-diagonal
components of the electron pressure tensor are the dominant terms that give rise to the
reconnection electric field. The typical scale length of the electron pressure tensor components in the weak guide field case is of the order of the electron bounce widths in a field reversal. In the strong guide field case the scale length reduces to the electron Larmor radius in the guide magnetic field
Equilibrium and stability properties of collisionless current sheet models
No full textThe work in this thesis focuses primarily on equilibrium and stability properties of collisionless current sheet models, in particular of the force-free Harris sheet model.
A detailed investigation is carried out into the properties of the distribution function found by Harrison and Neukirch (Physical Review Letters 102, 135003, 2009) for the force-free Harris sheet, which is so far the only known nonlinear force-free Vlasov-Maxwell equilibrium. Exact conditions on the parameters of the distribution function are found, which show when it can be single or multi-peaked in two of the velocity space directions. This is important because it may have implications for the stability of the equilibrium.
One major aim of this thesis is to find new force-free equilibrium distribution functions. By using a new method which is different from that of Harrison and Neukirch, it is possible to find a complete family of distribution functions for the force-free Harris sheet, which includes the Harrison and Neukirch distribution function (Physical Review Letters 102, 135003, 2009). Each member of this family has a different dependence on the particle energy, although the dependence on the canonical momenta remains the same. Three detailed analytical examples are presented. Other possibilities for finding further collisionless force-free equilibrium distribution functions have been explored, but were unsuccessful.
The first linear stability analysis of the Harrison and Neukirch equilibrium distribution function is then carried out, concentrating on macroscopic instabilities, and considering two-dimensional perturbations only. The analysis is based on the technique of integration over unperturbed orbits. Similarly to the Harris sheet case (Nuovo Cimento, 23:115, 1962), this is only possible by using approximations to the exact orbits, which are unknown. Furthermore, the approximations for the Harris sheet case cannot be used for the force-free Harris sheet, and so new techniques have to be developed in order to make analytical progress. Full analytical expressions for the perturbed current density are derived but, for the sake of simplicity, only the long wavelength limit is investigated. The dependence of the stability on various equilibrium parameters is investigated
On the Nonlinear Triggering of VLF Emissions by Power Line Harmonic Radiation
No full textVLF ground data from Porojarvi in N. Finland has been presented. Spectrograms reveal frequent occurrence of power line harmonic radiation (PLHR), originating from the Finnish power system and from heavy industrial plant. This radiation is seen to penetrate the magnetosphere since numerous occurrences of PLHR triggered emissions are seen. Risers predominate but fallers and hooks are also observed. A well established 1D Vlasov simulation code has been used to simulate these emissions, using plausible magnetospheric data for a range of L values from L=4 to L=5.5. The code is able to reproduce risers fallers and hooks in close agreement with observations. The results shed considerable insight into the generation region structure of both risers and fallers
VLF emission triggering by a highly anisotropic electron plasma
No full textA recent paper by Bell et al (Bell et al,2000) reports observations from the POLAR spacecraft of highly anisotropic hot electron distribution functions in the equatorial region of the magnetosphere at L=3.4. The particle instrument HYDRA measures electron fluxes from 1-20 keV. VLF emissions triggered by pulses from Omega (Norway) are found to coincide with 'pancake' type electron distributions with average pitch angles >70 degrees, such distributions being effectively confined to the equatorial zone. We examine the linear and non linear wave particle interaction process between pancake distributions and CW ducted VLF signals. It is concluded that the pitch angle range 67-76 degrees dominates the interaction process, and that with in duct wave saturation amplitudes of 6pT strong non linear trapping occurs for these particles. It is difficult to avoid the impression that highly anisotropic pitch angle distributions don’t have a great effect on resonant particle dynamics. High anisotropy has raised the pitch angle of maximum non linear contribution from 61->72 degrees, and reduced particle non linearity somewhat, in that the onset of trapping occurs at 2pT rather than 1.6pT. Using this data a 1D Vlasov Hybrid Simulation (VHS) VLF code was run to numerically simulate risers triggered by a 1 s Omega pulse. The VHS algorithm defines a time varying phase space simulation box covering the trans-equatorial nonlinear trapping region and a segment of parallel velocity space centred on the local resonance velocity. The simulation particles have F defined as a constant on their trajectories by Liouville's theorem. At each time step F is interpolated from the particles onto the fixed phase space grid, allowing resonant particle current to be calculated. The VHS method is extremely efficient since at each step particles leaving the phase box are discarded, and fresh particles are embedded into the phase fluid where the latter flows into the phase box. Successful numerical triggering of emissions by Omega is shown, and examples of risers, fallers and hooks are shown. The integrated linear trans-equatorial amplification of ~10dB agreed well with figures calculated by Bell from HYDRA data. These successful simulations of Omega emissions with highly anisotropic distribution functions confirm that non linear trapping of cyclotron resonant electrons in the geomagnetic field is the root plasma physical mechanism behind the triggering of VLF emissions
Going Beyond Counting First Authors in Author Co-citation Analysis
No full textThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Neutral Vlasov kinetic theory of magnetized plasmas
No full textThe low-frequency limit of Maxwell equations is considered in the Maxwell-Vlasov system. This limit produces a neutral Vlasov system that captures essential features of plasma dynamics, while neglecting radiation effects. Euler-Poincar\'e reduction theory is used to show that the neutral Vlasov kinetic theory possesses a variational formulation in both Lagrangian and Eulerian coordinates. By construction, the model recovers all collisionless neutral models employed in plasma simulations. Then, comparisons between the neutral Vlasov system and hybrid kinetic-fluid models are presented in the linear regime
The Vlasov Bivector : A Parameter-Free Approach to Vlasov Kinematics
No full textPlasma kinetics, for both flat and curved spacetime, is conventionally performed on the mass shell, a 7--dimensional time-phase space with a Vlasov vector field, also known as the Liouville vector field. The choice of this time-phase space encodes the parameterisation of the underling 2nd order ordinary differential equations. By replacing the Vlasov vector on time-phase space with a bivector on an 8--dimensional sub-bundle of the tangent bundle, we create a parameterisation free version of Vlasov theory. This has a number of advantages, which include working for lightlike and ultra-relativistic particles, non metric connections, and metric-free and premetric theories. It also works for theories where no time-phase space can exist for topological topological reasons. An example of this is when we wish to consider all geodesics, including spacelike geodesics. We extend the particle density function to a 6--form on the subbundle of the tangent space, and define the transport equations, which correspond to the Vlasov equation. We then show how to define the corresponding 3--current on spacetime. We discuss the stress-energy tensor needed for the Einstein-Vlasov system. This theory can be generalised to create parameterisation invariant Vlasov theories for many 2nd order theories, on arbitrary manifolds. The relationship to sprays and semi-sprays is given and examples from Finsler geometry are also given
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