Vlasov Equation Research Articles (Page 1)

Context. The solar corona exhibits a striking temperature inversion, with plasma temperatures exceeding $10^6$ K above a much cooler chromosphere. How the coronal plasma reaches such extreme temperatures remains a fundamental open question in solar and plasma physics, known as the coronal heating problem. Aims. We investigate whether localized heating events, spatially distributed across the upper chromosphere and base of the transition region, combined with a collisionless corona, can self-consistently generate realistic temperature and density profiles without requiring direct energy deposition within the corona itself. Models. We develop a 3D kinetic model of a collisionless stellar atmosphere embedded in a uniform magnetic field, where heating occurs intermittently at the chromosphere–transition region interface. A surface coarse-graining procedure is introduced to capture the spatial intermittency of heating, leading to non-thermal boundary conditions for the Vlasov equation. We derive analytical expressions for the stationary distribution functions and compute the corresponding macroscopic profiles. Results. We show that spatially intermittent heating, when coarse-grained over a surface containing many localized events, produces suprathermal particle distributions and a temperature inversion via velocity filtration. The resulting density and temperature profiles feature a transition region followed by a hot corona, provided that heating events are spatially sparse, consistently with solar observations. This result holds independently of the specific statistical distribution of temperature increments. Importantly, no local heating is applied within the corona. Conclusions. The model demonstrates that spatial intermittency alone, i.e. a sparse distribution of heated regions at the chromospheric interface, is sufficient to explain the formation of the transition region and the high-temperature corona.

On the stability of the invariant probability measures of McKean–Vlasov equations

A dynamic domain semi-Lagrangian method for stochastic Vlasov equations

Traditional fluid models are unable to reproduce the kinetic dispersion relation of certain plasma waves and the onset of kinetic instabilities. Non-equilibrium fluid models (higher-order moment methods) permit to reproduce selected kinetic effects, but at a lower computational cost as compared to fully-kinetic models. In this work, we study the dispersion relation of longitudinal electrostatic electron modes. We show that the fourth-order maximum-entropy moment method is able to recover certain kinetic instabilities, although with a slightly increased region of stability if compared to the Vlasov equation. As in the kinetic-theory case, these instabilities appear when the gas is in a non-equilibrium state that is associated with a sufficiently bi-modal VDF. The maximum-entropy method is able to predict the classical two-stream instability within a single-fluid formulation, reproducing a reasonably accurate growth rate. The Vlasov-stability of maximum-entropy VDFs is also investigated via Nyquist diagrams and the Penrose criterion.

New fixed point approach based on certain topological properties is applied on the Vlasov equation to find classical non-negative solutions. This approach allows finding at least one solution and at least two solutions. It is shown how to apply this approach to the problem of finding classical non-negative solutions of the Vlasov equation.

Convergence of Splitting Methods on Rotating Grids for the Magnetized Vlasov Equation

The loss of Landau damping (LLD) can lead to beam quality degradation in single- and multi-bunch beams in synchrotrons. In the longitudinal plane, the LLD threshold is increased by reducing the beam-coupling impedance, applying controlled emittance blowup, or employing higher-harmonic rf systems. This work focused on the significant threshold modifications when moving from the single- to the double-harmonic rf system case, depending on the relative phase, the rf voltages, and harmonic ratios. Moreover, the impact of the impedance cutoff frequency is discussed. Special attention is given to the configuration where both rf systems are in phase bunch shorting mode (BSM) or in counterphase bunch lengthening mode at the bunch position. Although in BSM, an approximate analytical expression can still be derived, numerical solutions of the linearized Vlasov equation are required in most cases. For the first time, we demonstrate that nonmonotonic amplitude dependence of the synchrotron frequency does not always lead to a vanishing LLD threshold. In BSM, this nonmonotonicity also results in counterintuitive beam evolution after a phase perturbation. The theoretical and semianalytical predictions are validated by an unambiguous beam measurement technique in the Proton Synchrotron and Super Proton Synchrotron at CERN and supported by macroparticle tracking simulations.

Abstract In 1939, Oppenheimer and Snyder showed that the continued gravitational collapse of a self-gravitating matter distribution can result in the formation of a black hole, cf. Oppenheimer and Snyder (Phys Rev 56:455–459, 1939). In this paper, which has greatly influenced the evolution of ideas around the concept of a black hole, matter was modeled as dust, a fluid with pressure equal to zero. We prove that when the corresponding initial data are suitably approximated by data for a collisionless gas as modeled by the Vlasov equation, then a trapped surface forms before the corresponding solution to the Einstein–Vlasov system can develop a singularity and again a black hole arises. As opposed to the dust case the pressure does not vanish for such solutions. As a necessary starting point for the analysis, which is carried out in Painlevé–Gullstrand coordinates, we prove a local existence and uniqueness theorem for regular solutions together with a corresponding extension criterion. The latter result will also become useful when one perturbs dust solutions containing naked singularities in the Vlasov framework.

Euler–Maruyama scheme for delay-type stochastic McKean–Vlasov equations driven by fractional Brownian motion

The small mass limit is derived for a McKean–Vlasov equation subject to environmental noise with state-dependent friction. By applying the averaging approach to a non-autonomous stochastic slow-fast system with the microscopic and macroscopic scales, the convergence in distribution is obtained.

The Vlasov–Maxwell equations provide kinetic simulations of collisionless plasmas, but numerically solving them on classical computers is often impractical. This is due to the computational resource constraints imposed by the time evolution in the six-dimensional phase space, which requires broad spatial and temporal scales. The novelty of this study is to implement a quantum–classical hybrid Vlasov–Maxwell solver and the rigorous numerical scheme evaluation by numerical simulations. Specifically, the Vlasov solver implements the Hamiltonian simulation based on quantum singular value transformation, coupled with a classical Maxwell solver. We perform numerical simulation of a one-dimensional advection test and a one-spatial-dimension, one-velocity-dimension two-stream instability test on the Qiskit-Aer-GPU quantum circuit emulator with an A100 GPU. The computational complexity of our quantum algorithm can potentially be reduced from the classical $\mathcal{O}(N^6T^2/\epsilon )$ to $\mathcal{O}\left (\text{poly}(\log {N})\left (NT+T\log \left (T/\epsilon \right )\right )\right )$ for the $N$ grid system, simulation time $T$ and error tolerance $\epsilon$ in the limit where the number of queries is large enough and the error is small enough. Furthermore, the numerical analysis reveals that our quantum algorithm is robust under larger time steps compared with classical algorithms with the constraint of Courant–Friedrichs–Lewy condition.

The Poiseuille and thermal transpiration flows of a real gas confined between two parallel plates are investigated based on the Enskog–Vlasov kinetic equation under the diffuse reflection boundary condition. In contrast to an ideal gas, the gradients of pressure and normal stress components in the flow direction, as well as the density, are nonuniform in the direction normal to the plates. The net mass flows and the profiles of flow velocity and heat flow are obtained for various Knudsen numbers, molecular sizes, and magnitudes of the long-range intermolecular attractive force. Depending on the magnitude of the attractive force, the nondimensional net mass flow of the thermal transpiration flow exhibits a local minimum at an intermediate Knudsen number, analogous to the Knudsen minimum known for the Poiseuille flow. Both in the Poiseuille and thermal transpiration flows, the flow velocity and heat flow are enhanced due to the intermolecular attractive force, compared to the case of the Enskog equation for a purely repulsive hard-sphere dense gas. Unless the molecular diameter is sufficiently small compared to the distance between the plates, the enhancement is observed even at high Knudsen numbers, where the volume fraction of molecules is small. This occurs because, although the attractive force itself is small at high Knudsen numbers, its influence on molecules with small velocity components normal to the boundary remains significant. Due to the effect of intermolecular attractive force, the velocity distribution function of gas molecules becomes discontinuous not only on the boundary but also in the gas.

Abstract We present a proof showing that the weak error of a system of $n$ interacting stochastic particles approximating the solution of the McKean–Vlasov equation is $\mathcal{O}({n^{-1}})$. Our proof is based on the Kolmogorov backward equation for the particle system and bounds on the derivatives of its solution which we derive more generally using the variations of the stochastic particle system. The convergence rate is verified by numerical experiments which also indicate that the assumptions made here and in the literature can be relaxed.

The mechanisms by which rotation influences zonal flows (ZFs) in plasma are incompletely understood, presenting a significant challenge in the study of plasma dynamics. This research addresses this gap by investigating the role of non-inertial effects—specifically centrifugal and Coriolis forces—on Geodesic Acoustic Modes (GAMs) and ZFs in rotating tokamak plasmas. While previous studies have linked centrifugal convection to plasma toroidal rotation, they often overlook the Coriolis effects or inconsistently incorporate non-inertial terms into magneto-hydrodynamic (MHD) equations. In this work, we derive self-consistent drift-ordered two-fluid equations from the collisional Vlasov equation in a non-inertial frame, and we modify the Hermes cold ion code to simulate the impact of rotation on GAMs and ZFs. Our simulations reveal that toroidal rotation enhances ZF amplitude and GAM frequency, with Coriolis convection playing a critical role in GAM propagation and the global structure of ZFs. Analysis of simulation outcomes indicates that centrifugal drift drives parallel velocity growth, while Coriolis drift facilitates radial propagation of GAMs. This work may provide valuable insights into momentum transport and flow shear dynamics in tokamaks, with implications for turbulence suppression and confinement optimization.

Abstract The widely adopted initial force-balanced equilibrium prior to the occurrence of magnetic reconnection (MR) is the so-called Harris sheet model with antiparallel magnetic field. This study examines the tearing mode instability of the recently developed generalized Harris sheet equilibrium based on two-dimensional, resistive, isotropic, linear, and nonlinear MHD models. The generalized Harris equilibrium incorporating the nonthermal Kappa-like velocity distributions in the Vlasov–Maxwell equations may give rise to a family of plasma and magnetic field profiles with various current layer thicknesses which may be used to study various plasma instabilities and MR. It is shown that the linear growth rate is proportional to δ − 2 / 5 R m − 2 / 5 , where δ and R m are the current layer thickness and magnetic Reynolds number, respectively. The linear eigenmode solutions are used as the initial perturbations of the nonlinear MHD simulations to allow the full evolution of the tearing instability. It is shown that the cases with small R m may not be nonlinearly unstable, and the cases with relatively thinner current sheets and larger R m may grow faster with larger aspect ratios of magnetic islands. In the generalized kinetic Harris models, the thinning of current sheets may be achieved by the decrease of central temperature or the increase of the drift velocity of charged particles. The asymptotic magnetic field of the equilibrium profiles seems to play a minor role in the linear and nonlinear growth of resistive tearing mode instability.

Construction of Schrödinger, Pauli and Dirac equations from Vlasov equation in case of Lorenz gauge

The Thermionic Energy Converter (TEC) is an efficient static device that directly converts thermal energy into electrical energy through electron emission. In this study, we employ a fully kinetic particle-in-cell simulation method to accurately model the steady-state vacuum TEC current–voltage characteristics and electron transport properties. Our simulation results show excellent agreement with analytical solutions from the Child–Langmuir law and Langmuir space charge theory, verifying the accuracy of our approach. By analyzing electron phase space distributions and macroscopic quantities within the electrode gap under different operating modes (accelerating, flatband, and decelerating), we characterize the formation of virtual cathodes due to space charge effects and their impact on electron transport. Furthermore, we decompose the electron energy density into fluid kinetic and thermal components based on the moments of the Vlasov equation, revealing distinct energy conversion mechanisms during electron transport. The energy flux analysis demonstrates how electrons gain and lose energy during transport, with contributions from convection, pressure effects, and heat flow. These findings provide new insights into the fundamental physics governing vacuum TEC operation and suggest potential pathways for optimizing device performance.

A set of evolutionary equations is proposed for a self-consistent description of Langmuir wave dynamics in an inhomogeneous plasma. In contrast to the traditionally used set of equations consisting of the kinetic Boltzmann–Vlasov equation for the distribution function and the Poisson equation for the scalar potential of the wave electric field, we describe the latter with the help of a vector potential. This makes the set of equations an evolutionary type of system, which allows solving the initial problem for the distribution function and the wave field. The inclusion of non-resonant particles into consideration with the help of permittivity makes it possible to solve the kinetic equation only for resonant particles. This radically reduces the amount of numerical calculations and allows solving the problem on a personal computer, while the traditional approach in which the kinetic equation is solved for all particles requires much more computational resources. The results obtained include various cases of homogeneous and inhomogeneous plasma as well as the cases of a single wave and a wide spectrum.

In this paper, we focus on mean-field stochastic differential equations driven by G-Brownian motion (G-MFSDEs for short) with a drift coefficient satisfying the local one-sided Lipschitz condition with respect to the state variable and the global Lipschitz condition with respect to the law. We are concerned with the well-posedness and the numerical approximation of the G-MFSDE. Probability uncertainty leads the resulting expectation usually to be the G-expectation, which means that we cannot apply the numerical approximation for McKean–Vlasov equations to G-MFSDEs directly. To numerically approximate the G-MFSDE, with the help of G-expectation theory, we use the sample average value to represent the law and establish the interacting particle system whose mean square limit is the G-MFSDE. After this, we introduce the modified stochastic theta method to approximate the interacting particle system and study its strong convergence and asymptotic mean square stability. Finally, we present an example to verify our theoretical results.

ABSTRACTIt is argued that the Vlasov equation cannot fully account for collisionless shocks since it conserves entropy, while a shock does not. A rigorous mathematical theory of collisionless shocks could require working at the Klimontovich level.