The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction, e.g. Coulomb. The equation was first suggested for description of plasma by Anatoly Vlasov in 1938 and later discussed by him in detail in a monograph. Difficulties of the standard kinetic approach First, Vlasov argues that the standard kinetic approach based on the Boltzmann equation has difficulties when applied to a description of the plasma with long-range Coulomb interaction. He mentions the following problems arising when applying the kinetic theory based on pair collisions to plasma dynamics:

Theory of pair collisions disagrees with the discovery by Rayleigh, Irving Langmuir and Lewi Tonks of natural vibrations in electron plasma.

Theory of pair collisions is formally not applicable to Coulomb interaction due to the divergence of the kinetic terms.

Theory of pair collisions cannot explain exp

A Framework for Numerical Simulations of Structure Formation

Claude Becker

In the absence of a full analytical treatment of nonlinear structure formation in the universe, numerical simulations provide the critical link between the properties of the underlying model and the features of the observed structures. Currently N-body simulations are the main tool to study structure growth. We explore an alternative framework for numerical simulations of structure formation. The underlying idea is to replace the long-range gravitational force in the Vlasov-Poisson system by a purely local interaction. To this end we trade the classical phase space distribution for its quantum mechanical counterpart, the Wigner distribution function. Its dynamical equation is equivalent to the Schrödinger equation and reduces to the Vlasov equation in the formal semi-classical limit. The proposed framework allows in principle to simulate systems with arbitrary phase space distributions and could for instance be beneficial for simulations of warm dark matter, where the velocity dispersion is important. We discuss several methods to obtain a set of wavefunctions whose Wigner distribution is close to a given initial phase space distribution function. An auxiliary gauge field is introduced to mediate the gravitational interaction, thereby obtaining a local Schrödinger-Maxwell system. We also use the ideas of lattice gauge theories to obtain a fully gauge-invariant discretization of the equations of motion. Their iterative solution was implemented in a three-dimensional simulation code. We discuss its computational complexity and memory requirements. Several testbed simulations were performed with this method. We compared the gravitational collapse of a Gaussian wavefunction with an independent numerical solution of the spherically symmetric Schrödinger-Newton system. The results were found to be in good agreement. Finally, a simple example of the growth of cosmic perturbations is investigated within our framework. We conclude by outlining various possible directions to optimize and develop our method.

EPFL2011

Vlasov description of the beam response to noise in the presence of wakefields in high-energy synchrotrons: beam transfer function, diffusion, and loss of Landau damping

Xavier Buffat, Sondre Vik Furuseth

Noise can have severe impacts on particle beams in high-energy synchrotrons. In particular, it has recently been discovered that noise combined with wakefields can cause a diffusion that leads to a loss of Landau damping after a latency. Such instabilities have been observed in the Large Hadron Collider. This paper, therefore, studies the beam response to noise in the presence of wakefields, within the framework of the Vlasov equation. First, a wakefield beam eigenmode transfer function (MTF) is derived, quantifying the amplitude of a wakefield eigenmode when excited by noise. Then, the MTFs of all the wakefield eigenmodes are combined to derive the beam transfer function (BTF) including the impact of wakefields. It is found to agree excellently with multi-particle tracking simulations. Finally, the MTFs are also used to derive the single-particle diffusion driven by the wakefield eigenmodes. This new Vlasov-based theory for the diffusion driven by noise-excited wakefields is found to be superior to an existing theory by comparing to multi-particle tracking simulations. Through sophisticated simulations that self-consistently model the evolution of the distribution and the stability diagram, the diffusion is found to lead to a loss of Landau damping after a latency. The most important technique to extend the latency and thereby mitigate these instabilities is to operate the synchrotron with a stability margin in detuning strength relative to the amount of detuning required to barely stabilize the beam with its initial distribution.

SPRINGER HEIDELBERG2022

A Framework for Numerical Simulations of Structure Formation

Claude Becker

EPFL2011