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The paper focuses on the kinetic modeling of the vacuum arc initiation at the pulse-periodic plasma emission from the cathode surface. A comparative analysis is given to the pulse-periodic and continuous plasma emission from the cathode. An in-depth study concerns the mechanism of the anomalous acceleration of ions and cathode plasma expansion at a short-time emission current switch over, matching the plasma emission and its interruption. [ABSTRACT FROM AUTHOR]

Simulation of full wave, without paraxial approximation, high-resolution solution of wave equations in frequency domain in the electron cyclotron resonance (ECR) frequency range for realistic Tokamak plasma parameters became possible by using recently formulated hybrid iterative algorithm [Svidzinski et al., Phys. Plasmas 25, 082509 (2018)] for numerically solving discretized wave equations. This approach combines time evolution and iterative relaxation techniques into iteration cycles. This algorithm is implemented in 2D code FullWave, solving wave equations in Tokamaks in cold and hot plasma models, and it has been tested in 3D full wave iterative RF beams simulation tool, which is presently being developed to model 3D ECRH RF beams in fusion devices using dynamic grid adaptation. The results of 2D full wave modeling, assuming specified toroidal mode number, of ECRH RF beams in DIII-D plasma, performed in the cold and hot plasma models for outboard and top launch scenarios using FullWave are presented. Nonlocal hot plasma response model, based on accurate numerical solution of linearized Vlasov equation, is used to model beam propagation and absorption in the 2nd electron cyclotron harmonic region. Demonstration of capability of the hybrid iterative algorithm to model ECRH RF beams in 3D is made by simulating a substantial part of realistic beam in DIII-D, launched from outboard side of the machine. All relevant physics of RF beam propagation, most of which is not captured in paraxial approximation, such as beam's divergence, interference between the X and O modes in the beam, X-O mode conversion, beam splitting into the X and O mode beams, transformation of beam's cross section, and absorption at the 2nd electron cyclotron harmonic, is captured in the simulations. A numerical technique to find an optimal beam polarization at the launcher to launch a nearly pure X or O mode beam in plasma is developed and tested. [ABSTRACT FROM AUTHOR]

We develop a Hamiltonian theory for the nonlinear resonant interactions between energetic particles and nonlinear frequency chirping waves in the weakly inhomogeneous magnetic field. A canonical transformation is constructed to separate the fast and slowly varying scales, and the Hamiltonian of the resonant particle is transformed to the local resonance reference frames. The Vlasov equation of the local distribution function moving at the local resonance velocity is obtained using Liouville's theorem. The evolution for the slowly varying wave envelope is derived from the Ampère's law with both cold plasma and energetic particle currents. The Vlasov equation coupled with the wave envelope equation self-consistently describes the dynamics of the deeply trapped resonant particles and the slowly varying coherent wave envelope. The application of the theory to the frequency chirping chorus wave in magnetospheric plasmas is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we first derive the existence and uniqueness theorems for solutions to a class of generalized mean-field delay stochastic differential equations and mean-field anticipated backward stochastic differential equations (MFABSDEs). Then we study the stochastic maximum principle for generalized mean-field delay control problem. Since the state equation is distribution-depending, we define the adjoint equation as a MFABSDE in which all the derivatives of the coefficients are in Lions' sense. We also provide a sufficient condition for the optimality of the control. [ABSTRACT FROM AUTHOR]

This is a companion paper to the previous work [P. H. Yoon, Phys. Plasmas 31, 032309 (2024)] in which the nonlinear susceptibilities of weakly turbulent magnetized plasma are derived under a simplifying assumption of electrostatic interaction. The present paper extends the analysis to a general situation of electromagnetic interaction. The main novelty of the previous and present papers is that by employing the Bessel function addition theorem, the mathematical definitions for the susceptibilities are substantially simplified, a procedure that has not been discussed in the existing literature. In the present paper, a full set of Maxwell's equations are considered in conjunction with the nonlinear Vlasov equation, which is solved by a perturbative method. The result is a fully general nonlinear susceptibility, given in tensorial form, which is applicable for weakly turbulent magnetized plasmas. [ABSTRACT FROM AUTHOR]

We study the Schwinger pair creation of scalar charged particles by a homogeneous electric field in an expanding universe in the quantum kinetic approach. We introduce an adiabatic vacuum for the scalar field based on the Wentzel–Kramers–Brillouin solution to the mode equation in conformal time and apply the formalism of Bogolyubov coefficients to derive a system of quantum Vlasov equations for three real kinetic functions. Compared to the analogous system of equations previously reported in the literature, the new one has two advantages. First, its solutions exhibit a faster decrease at large momenta which makes it more suitable for numerical computations. Second, it predicts no particle creation in the case of conformally coupled massless scalar field in the vanishing electric field, i.e., it respects the conformal symmetry of the system. We identify the ultraviolet divergences in the electric current and energy–momentum tensor of produced particles and introduce the corresponding counterterms in order to cancel them. [ABSTRACT FROM AUTHOR]

Our critical analysis of Hutchinson's work, expressed in our recent article in the Annalen der Physik, which goes beyond mutual misunderstandings and misrepresentations, is also maintained in light of his Comment. The main reason for the limited yield in structural description is the author's adherence to the BGK method and the associated lack of a nonlinear dispersion relation (NDR). Even with an additional asymptotic constraint, as a supposed replacement for the NDR, the author's theory remains inferior to the pseudo‐potential method. [ABSTRACT FROM AUTHOR]

We propose an infinite dimensional generating function method for finding the analytical solution of the so-called chemical diffusion master equation (CDME) for creation and mutual annihilation chemical reactions. CDMEs model by means of an infinite system of coupled Fokker–Planck equations the probabilistic evolution of chemical reaction kinetics associated with spatial diffusion of individual particles; here, we focus an creation and mutual annihilation chemical reactions combined with Brownian diffusion of the single particles. Using our method we are able to link certain finite dimensional projections of the solution of the CDME to the solution of a single linear fourth order partial differential equation containing as many variables as the dimension of the aforementioned projection space. Our technique extends the one presented in Lanconelli [J. Math. Anal. Appl. 526, 127352 (2023)] and Lanconelli et al. [arXiv:2302.10700 [math.PR] (2023)] which allowed for an explicit representation for the solution of birth-death type CDMEs. [ABSTRACT FROM AUTHOR]

In this paper, we study three numerical schemes for the McKean-Vlasov equation {dXt=b(t,Xt,μt)dt+σ(t,Xt,μt)dBt,∀t∈[0,T],μt is the probability distribution of Xt, { d X t = b (t , X t , μ t) d t + σ (t , X t , μ t) d B t , ∀ t ∈ [ 0 , T ] , μ t  is the probability distribution of   X t , $ \{ \begin{array}{l}dX_t=b{(t,X_t,\mu_t)}dt+\sigma{(t,X_t,\mu_t)}dB_t,\\ \forall t\in\lbrack 0,T\rbrack,\mu_t\text{is the probability distribution of }X_t,\end{array} $ where X0 : (Ω, F, ℙ) → (ℝd, B(ℝd)) is a known random variable. Under the assumption on the Lipschitz continuity of the coefficients b and σ, our first result proves the convergence rate of the particle method with respect to the Wasserstein distance, which extends previous work [M. Bossy and D. Talay, Math. Comput. 66 (1997) 157–192.] established in a one-dimensional setting. In the second part, we present and analyse two quantization-based schemes, including the recursive quantization scheme (deterministic scheme) in the Vlasov setting, and the hybrid particle-quantization scheme (random scheme inspired by the K-means clustering). Two simulations are presented at the end of this paper: Burgers equation introduced in [M. Bossy and D. Talay, Math. Comput. 66 (1997) 157–192.] and the network of FitzHugh- Nagumo neurons (see [J. Baladron, D. Fasoli, O. Faugeras and J. Touboul, J. Math. Neurosci. 2 (2012) 1–50.] and [M. Bossy, O. Faugeras and D. Talay, J. Math. Neurosci. 5 (2015) 1–23.]) in dimension 3. [ABSTRACT FROM AUTHOR]

In this paper, we present a numerical approach to solve the McKean–Vlasov equations, which are distribution-dependent stochastic differential equations, under some non-globally Lipschitz conditions for both the drift and diffusion coefficients. We establish a propagation of chaos result, based on which the McKean–Vlasov equation is approximated by an interacting particle system. A truncated Euler scheme is then proposed for the interacting particle system allowing for a Khasminskii- type condition on the coefficients. To reduce the computational cost, the random batch approximation proposed in [S. Jin, L. Li and J. Liu, J. Comput. Phys. 400 (2020) 108877.] is extended to the interacting particle system where the interaction could take place in the diffusion term. An almost half order of convergence is proved in Lp sense. Numerical tests are performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

We consider the fully-coupled McKean--Vlasov equation with multi-time-scale potentials, and all the coefficients depend on the distributions of both the slow component and the fast motion. By studying the smoothness of the solution of the Poisson equation on Wasserstein space, we derive the asymptotic limit as well as the quantitative error estimate of the convergence for the slow process. An extra homogenized drift term containing derivative in the measure argument of the solution of the Poisson equation appears in the limit, which seems to be new and is unique for systems involving the fast distribution. [ABSTRACT FROM AUTHOR]

We investigate the influence of a self-propelling, out-of-equilibrium active particle on generalized elastic systems, including flexible and semi-flexible polymers, fluid membranes, and fluctuating interfaces, while accounting for long-ranged hydrodynamic effects. We derive the fractional Langevin equation governing the dynamics of the active particle, as well as that of any other passive particle (or probe) bound to the elastic system. This equation analytically demonstrates how the active particle dynamics is influenced by the interplay of both the non-equilibrium force and of the viscoelastic environment. Our study explores the diffusional behavior emerging for both the active particle and a distant probe. The active particle undergoes three different surprising and counter-intuitive regimes identified by the distinct dynamical time-scales: a pseudo-ballistic initial phase, a drastic decrease in the mobility, and an asymptotic subdiffusive regime. [ABSTRACT FROM AUTHOR]

In classical texts equations for gravitation and electromagnetic fields are proposed without deriving their right-hand sides [1–4]. In this paper, we derive the right-hand sides and analyze the energy–momentum tensor in the framework of Vlasov–Maxwell–Einstein equations and Milne–McCrea models. New models of accelerated expansion of the Universe without Einstein's lambda are proposed. [ABSTRACT FROM AUTHOR]

Efficient solution of the Vlasov equation, which can be up to six-dimensional, is key to the simulation of many difficult problems in plasma physics. The discontinuous Petrov-Galerkin (DPG) finite element methodology provides a framework for the development of stable (in the sense of Ladyzhenskaya–Babuška–Brezzi conditions) finite element formulations, with built-in mechanisms for adaptivity. While DPG has been studied extensively in the context of steady-state problems and to a lesser extent with space-time discretizations of transient problems, relatively little attention has been paid to time-marching approaches. In the present work, we study a first application of time-marching DPG to the Vlasov equation, using backward Euler for a Vlasov-Poisson discretization. We demonstrate adaptive mesh refinement for two problems: the two-stream instability problem, and a cold diode problem. We believe the present work is novel both in its application of unstructured adaptive mesh refinement (as opposed to block-structured adaptivity, which has been studied previously) in the context of Vlasov-Poisson, as well as in its application of DPG to the Vlasov-Poisson system. We also discuss extensive additions to the Camellia library in support of both the present formulation as well as extensions to higher dimensions, Maxwell equations, and space-time formulations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a conservative low rank tensor method to approximate nonlinear Vlasov solutions. The low rank approach is based on our earlier work [W. Guo and J.-M. Qiu, A Low Rank Tensor Representation of Linear Transport and Nonlinear Vlasov Solutions and Their Associated Flow Maps, preprint, https://arxiv.org/abs/2106.08834, 2021]. It takes advantage of the fact that the differential operators in the Vlasov equation are tensor friendly, based on which we propose to dynamically and adaptively build up low rank solution basis by adding new basis functions from discretization of the differential equation, and removing basis from a singular value decomposition (SVD)-type truncation procedure. For the discretization, we adopt a high order finite difference spatial discretization together with a second order strong stability preserving multistep time discretization. While the SVD truncation will remove the redundancy in representing the high dimensional Vlasov solution, it will destroy the conservation properties of the associated full conservative scheme. In this paper, we develop a conservative truncation procedure with conservation of mass, momentum, and kinetic energy densities. The conservative truncation is achieved by an orthogonal projection onto a subspace spanned by 1, v, and v2 in the velocity space associated with a weighted inner product. Then the algorithm performs a weighted SVD truncation of the remainder, which involves a scaling, followed by the standard SVD truncation and rescaling back. The algorithm is further developed in high dimensions with hierarchical Tucker tensor decomposition of high dimensional Vlasov solutions, overcoming the curse of dimensionality. An extensive set of nonlinear Vlasov examples are performed to show the effectiveness and conservation property of proposed conservative low rank approach. Comparison is performed against the nonconservative low rank tensor approach on conservation history of mass, momentum, and energy. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of the relativistic Vlasov–Klein–Gordon system in space dimension three. The dynamical model describes a collisionless ensemble of classical particles coupled with a Klein–Gordon field. By combining the Fourier method and smoothing effect of low velocity particles, we prove a higher regularity of the field, which is widely used for the study of Vlasov equation. [ABSTRACT FROM AUTHOR]

An open problem of the derivation of the relativistic Vlasov equation for systems of charged particles moving with velocities up to the speed of light and creating the electromagnetic field in accordance with the full set of the Maxwell equations is considered. Moreover, the method of derivation is illustrated on the non-relativistic kinetic model. Independent derivation of the relativistic hydrodynamics is also demonstrated. The key role of these derivations of the hydrodynamic and kinetic equations includes the explicit operator of averaging on the physically infinitesimal volume suggested by L.S. Kuzmenkov. [ABSTRACT FROM AUTHOR]

An analytical solution of the Fredholm linear integral equation of the second kind for the one-particle distribution function of a liquid near a solid molecularly smooth surface is obtained. The one-particle distribution function describes the change in local density for all distances of particles from the surface. The solution is obtained in the particular case of a molecular system of solid spheres. The kernel and right-hand side of the Fredholm equation are calculated in the Percus–Yevick approximation. The interaction of particles with the surface is treated in elastic approximation. [ABSTRACT FROM AUTHOR]

Equations of gravitation in the form of Vlasov–Poisson relativistic equations with Lambda term are derived from the classical principle of least action. Hamilton–Jacobi consequences are used to obtain cosmological solutions. The properties of Lagrange points are investigated. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for coupled system of Vlasov and non-Newtonian fluid equations. We establish local well-posedness of the strong solutions, provided that the initial data are regular enough. [ABSTRACT FROM AUTHOR]

We consider solutions to the massless Vlasov equation on the domain of outer communications of the Schwarschild black hole. By adapting the r p -weighted energy method of Dafermos and Rodnianski, used extensively in order to study wave equations, we prove superpolynomial decay for a non-degenerate energy flux of the Vlasov field f through a well-chosen foliation. An essential step of this methodology consists in proving a non-degenerate integrated local energy decay. For this, we take in particular advantage of the redshift effect near the event horizon. The trapping at the photon sphere requires, however, to lose an ϵ of integrability in the velocity variable. Pointwise decay estimates on the velocity average of f are then obtained by functional inequalities, adapted to the study of Vlasov fields, which allow us to deal with the lack of a conservation law for the radial derivative. [ABSTRACT FROM AUTHOR]

In near-collisionless plasmas, which are ubiquitous in astrophysics, entropy production relies on fully-nonlinear processes such as turbulence and reconnection, which lead to particle acceleration. Mechanisms for turbulent reconnection include multiple magnetic flux ropes interacting to generate thin current sheets which undergo reconnection, leading to mixing and magnetic merging and growth of coherent structures, unstable reconnection current layers that fragment and turbulent reconnection outflows. All of these processes act across, and encompass, multiple reconnection sites. We use Magnetospheric Multi Scale four-point satellite observations to characterize the magnetic field line topology within a single reconnection current layer. We examine magnetopause reconnection where the spacecraft encounter the Electron Diffusion Region (EDR). We find fluctuating magnetic field with topology identical to that found for dynamically evolving vortices in hydrodynamic turbulence. The turbulence is supported by an electron-magnetohydrodynamic (EMHD) flow in which the magnetic field is effectively frozen into the electron fluid. Accelerated electrons are found in the EDR edge where we identify a departure from this turbulent topology, towards two-dimensional sheet-like structures. This is consistent with a scenario in which sub-ion scale turbulence can suppress electron acceleration within the EDR which would otherwise be possible in the electric field at the X-line. [ABSTRACT FROM AUTHOR]

We study a one-dimensional system of equations for a discrete gas model (the McKean system). The McKean system is the Boltzmann kinetic equation of a model one-dimensional gas consisting of two groups of particles. Under certain conditions on a singularity variety, the system passes the Painlevé test. In addition, the kinetic system admits a reduction to a system of ordinary differential equations for which the Painlevé test is performed and it becomes possible to find a solution. [ABSTRACT FROM AUTHOR]

Multi-scale modeling of expanding plasmas is crucial for understanding the dynamics and evolution of various astrophysical plasma systems such as the solar and stellar winds. In this context, the Expanding Box Model (EBM) provides a valuable framework to mimic plasma expansion in a non-inertial reference frame, co-moving with the expansion but in a box with a fixed volume, which is especially useful for numerical simulations. Here, fundamentally based on the Vlasov equation for magnetized plasmas and the EBM formalism for coordinates transformations, for the first time, we develop a first principles description of radially expanding plasmas in the EB frame. From this approach, we aim to fill the gap between simulations and theory at microscopic scales to model plasma expansion at the kinetic level. Our results show that expansion introduces non-trivial changes in the Vlasov equation (in the EB frame), especially affecting its conservative form through non-inertial forces purely related to the expansion. In order to test the consistency of the equations, we also provide integral moments of the modified Vlasov equation, obtaining the related expanding moments (i.e., continuity, momentum, and energy equations). Comparing our results with the literature, we obtain the same fluids equations (ideal-MHD), but starting from a first principles approach. We also obtained the tensorial form of the energy/pressure equation in the EB frame. These results show the consistency between the kinetic and MHD descriptions. Thus, the expanding Vlasov kinetic theory provides a novel framework to explore plasma physics at both micro and macroscopic scales in complex astrophysical scenarios. [ABSTRACT FROM AUTHOR]

Consider the Vlasov–Poisson–Landau system with Coulomb potential in the weakly collisional regime on a 3-torus, i.e. \begin{align*} \partial _t F(t,x,v) + v_i \partial _{x_i} F(t,x,v) + E_i(t,x) \partial _{v_i} F(t,x,v) = \nu Q(F,F)(t,x,v),\\ E(t,x) = \nabla \Delta ^{-1} (\int _{\mathbb R^3} F(t,x,v)\, \mathrm {d} v - {{\int }\llap {-}}_{\mathbb T^3} \int _{\mathbb R^3} F(t,x,v)\, \mathrm {d} v \, \mathrm {d} x), \end{align*} with \nu \ll 1. We prove that for \epsilon >0 sufficiently small (but independent of \nu), initial data which are O(\epsilon \nu ^{1/3})-Sobolev space perturbations from the global Maxwellians lead to global-in-time solutions which converge to the global Maxwellians as t\to \infty. The solutions exhibit uniform-in-\nu Landau damping and enhanced dissipation. Our main result is analogous to an earlier result of Bedrossian for the Vlasov–Poisson–Fokker–Planck equation with the same threshold. However, unlike in the Fokker–Planck case, the linear operator cannot be inverted explicitly due to the complexity of the Landau collision operator. For this reason, we develop an energy-based framework, which combines Guo's weighted energy method with the hypocoercive energy method and the commuting vector field method. The proof also relies on pointwise resolvent estimates for the linearized density equation. [ABSTRACT FROM AUTHOR]

The main aim of this work is to study the asymptotic behavior for multi-scale McKean–Vlasov stochastic dynamical systems. Firstly, we obtain a central limit type theorem, i.e. the deviation between the slow component X ε and the solution X ¯ of the averaged equation converges weakly to a limiting process. More precisely, X ε - X ¯ ε converges weakly in C ([ 0 , T ] , R n) to the solution of certain distribution dependent stochastic differential equation, which involves an extra explicit stochastic integral term. Secondly, in order to estimate the probability of deviations away from the limiting process, we further investigate the Freidlin–Wentzell's large deviation principle for multi-scale McKean–Vlasov stochastic system when the small-noise regime parameter δ → 0 and the time scale parameter ε (δ) satisfies ε (δ) / δ → 0 . The main techniques are based on the Poisson equation for central limit type theorem and the weak convergence approach for large deviation principle. [ABSTRACT FROM AUTHOR]

Recent critical remarks, published in this journal, about the present author's analysis of electron and ion holes and their stability are addressed and shown to be misunderstandings and misrepresentations. [ABSTRACT FROM AUTHOR]

Ultrafast laser pulses interacting with plasmas can give rise to a rich spectrum of physical phenomena, which have been extensively studied both theoretically and experimentally. Less work has been devoted to the study of polarized plasmas, where the electron spin may play an important role. In this short review, we illustrate the use of phase-space methods to model and simulate spin-polarized plasmas. This approach is based on the Wigner representation of quantum mechanics, and its classical counterpart, the Vlasov equation, which are generalized to include the spin degrees of freedom. Our approach is illustrated through the study of the stimulated Raman scattering of a circularly polarized electromagnetic wave interacting with a dense electron plasma. [ABSTRACT FROM AUTHOR]

In this paper, we study exponentially fast mixing induced/enhanced by gravity and stochastic boundary in the kinetic theory of Vlasov equations. We consider the Vlasov equations with and without a vertical magnetic field inside a horizontally-periodic 3D half-space equipped with a non-isothermal diffusive reflection boundary condition of bounded continuous boundary temperature at the bottom. We construct both stationary solutions and global-in-time dynamical solutions in L ∞. We prove that moments of a dynamical fluctuation around the steady solutions decay exponentially fast in L ∞. As a key of this proof, we establish a uniform bound of so-called residual measures independently of the bouncing number of stochastic characteristics, by constructing a continuous stationary outgoing boundary flux which is strictly positive almost everywhere. [ABSTRACT FROM AUTHOR]

We propose an extended hydrodynamical model for plasmas, based on the moments of the electron distribution function which satisfies the Fokker–Planck–Landau (FPL) transport equation. The equations for the moments can be obtained by multiplying the FPL equation by the corresponding weight functions and integrating over the velocity space. The moments are decomposed in their convective and non–convective parts and closure relations for the fluxes and production terms can be obtained by using the maximum entropy distribution function, which depends on Lagrangian multipliers. These latter can be expressed in terms of the state variables by imposing the constraints that the maximum entropy distribution function reproduces the moments chosen as state variables. In particular, we will concentrate on the 13-moment system. As a first application, we treat the case of the relaxation towards equilibrium of a homogeneous plasma with a temperature anisotropy, showing that the results are in good agreement with those obtained by means of the Kogan solution of the kinetic equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of joint parameter estimation for drift and diffusion coefficients of a stochastic McKean–Vlasov equation and for the associated system of interacting particles. The analysis is provided in a general framework, as both coefficients depend on the solution and on the law of the solution itself. Starting from discrete observations of the interacting particle system over a fixed interval [ 0 , T ] , we propose a contrast function based on a pseudo likelihood approach. We show that the associated estimator is consistent when the discretization step (Δ n) and the number of particles (N) satisfy Δ n → 0 and N → ∞ , and asymptotically normal when additionally the condition Δ n N → 0 holds. [ABSTRACT FROM AUTHOR]

The main aim of this work is to investigate small noise limiting behavior of multi-scale McKean-Vlasov stochastic dynamical systems, where we allow the coefficients depend on the distributions of both slow and fast components. Firstly, the strong convergence in the functional law of large numbers is established by the time discretization scheme. Secondly, in order to characterize the probability of deviations away from the averaged limit, we prove the large deviation principle by the weak convergence approach for McKean-Vlasov equations. [ABSTRACT FROM AUTHOR]

We provide the rigorous derivation of the wave kinetic equation from the cubic nonlinear Schrödinger (NLS) equation at the kinetic timescale, under a particular scaling law that describes the limiting process. This solves a main conjecture in the theory of wave turbulence, i.e. the kinetic theory of nonlinear wave systems. Our result is the wave analog of Lanford's theorem on the derivation of the Boltzmann kinetic equation from particle systems, where in both cases one takes the thermodynamic limit as the size of the system diverges to infinity, and as the interaction strength of waves/radius of particles vanishes to 0, according to a particular scaling law (Boltzmann-Grad in the particle case). More precisely, in dimensions d ≥ 3 , we consider the (NLS) equation in a large box of size L with a weak nonlinearity of strength α . In the limit L → ∞ and α → 0 , under the scaling law α ∼ L − 1 , we show that the long-time behavior of (NLS) is statistically described by the wave kinetic equation, with well justified approximation, up to times that are O (1) (i.e. independent of L and α ) multiples of the kinetic timescale T kin ∼ α − 2 . This is the first result of its kind for any nonlinear dispersive system. [ABSTRACT FROM AUTHOR]

We analyse the problem of online parameter estimation for a stochastic McKean–Vlasov equation, and the associated system of weakly interacting particles. We propose an online estimator for the parameters of the McKean–Vlasov SDE, or the interacting particle system, which is based on a continuous-time stochastic gradient ascent scheme with respect to the asymptotic log-likelihood of the interacting particle system. We characterise the asymptotic behaviour of this estimator in the limit as t → ∞ , and also in the joint limit as t → ∞ and N → ∞. In these two cases, we obtain almost sure or L 1 convergence to the stationary points of a limiting contrast function, under suitable conditions which guarantee ergodicity and uniform-in-time propagation of chaos. We also establish, under the additional condition of global strong concavity, L 2 convergence to the unique maximiser of the asymptotic log-likelihood of the McKean–Vlasov SDE, with an asymptotic convergence rate which depends on the learning rate, the number of observations, and the dimension of the non-linear process. Our theoretical results are supported by two numerical examples, a linear mean field model and a stochastic opinion dynamics model. [ABSTRACT FROM AUTHOR]

We study the local-in-time well-posedness for an interface that separates an anisotropic plasma from a vacuum. The plasma flow is governed by the ideal Chew–Goldberger–Low (CGL) equations, which are the simplest collisionless fluid model with anisotropic pressure. The vacuum magnetic and electric fields are supposed to satisfy the pre-Maxwell equations. The plasma and vacuum magnetic fields are tangential to the interface. This represents a nonlinear hyperbolic-elliptic coupled problem with a characteristic free boundary. By a suitable symmetrization of the linearized CGL equations, we reduce the linearized free boundary problem to a problem analogous to that in isotropic magnetohydrodynamics (MHD). This enables us to prove the local existence and uniqueness of solutions to the nonlinear free boundary problem under the same non-collinearity condition for the plasma and vacuum magnetic fields on the initial interface required by Secchi and Trakhinin (Nonlinearity 27:105–169, 2014) in isotropic MHD. [ABSTRACT FROM AUTHOR]

We investigate the kinetic model of the relativistic Vlasov–Maxwell–Chern–Simons system, which originates from gauge theory. This system can be seen as an electromagnetic fields (i.e., Maxwell–Chern–Simons fields) perturbation for the classical Vlasov equation. By virtue of a nondecreasing function and an iteration method, the uniqueness and existence of the global solutions for the 1.5D case are obtained. [ABSTRACT FROM AUTHOR]

We provide a rigorous mathematical framework to establish the hydrodynamic limit of the Vlasov model introduced in Takata and Noguchi (J. Stat. Phys. 172:880-903, 2018) by Noguchi and Takata in order to describe phase transition of fluids by kinetic equations. We prove that, when the scale parameter tends to 0, this model converges to a nonlocal Cahn-Hilliard equation with degenerate mobility. For our analysis, we introduce apropriate forms of the short and long range potentials which allow us to derive Helmhotlz free energy estimates. Several compactness properties follow from the energy, the energy dissipation and kinetic averaging lemmas. In particular we prove a new weak compactness bound on the flux. [ABSTRACT FROM AUTHOR]

We estimated the residual entropy of Ice Ih by the recently developed simulation protocol, namely, the combination of the replica-exchange Wang–Landau algorithm and multicanonical replica-exchange method. We employed a model with the nearest neighbor interactions on the three-dimensional hexagonal lattice, which satisfied the ice rules in the ground state. The results showed that our estimate of the residual entropy is in accordance with various previous results. In this article, we not only give our latest estimate of the residual entropy of Ice Ih but also discuss the importance of the uniformity of a random number generator in Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

Microkinetic modeling has drawn increasing attention for quantitatively analyzing catalytic networks in recent decades, in which the speed and stability of the solver play a crucial role. However, for the multi-step complex systems with a wide variation of rate constants, the often encountered stiff problem leads to the low success rate and high computational cost in the numerical solution. Here, we report a new efficient sensitivity-supervised interlock algorithm (SSIA), which enables us to solve the steady state of heterogeneous catalytic systems in the microkinetic modeling with a 100% success rate. In SSIA, we introduce the coverage sensitivity of surface intermediates to monitor the low-precision time-integration of ordinary differential equations, through which a quasi-steady-state is located. Further optimized by the high-precision damped Newton's method, this quasi-steady-state can converge with a low computational cost. Besides, to simulate the large differences (usually by orders of magnitude) among the practical coverages of different intermediates, we propose the initial coverages in SSIA to be generated in exponential space, which allows a larger and more realistic search scope. On examining three representative catalytic models, we demonstrate that SSIA is superior in both speed and robustness compared with its traditional counterparts. This efficient algorithm can be promisingly applied in existing microkinetic solvers to achieve large-scale modeling of stiff catalytic networks. [ABSTRACT FROM AUTHOR]

This paper outlines a variety of possible applications of the Vlasov equation and its generalization, i.e., the Klimontovich equation, in various areas of many-body physics. In particular, these equations are shown to be used in relativistic plasma physics, the theory of semi-classical Bloch electrons, and the metriplectic description of dissipative processes. [ABSTRACT FROM AUTHOR]

Particle‐based simulations of the Vlasov equation typically require a large number of particles, which leads to a high‐dimensional system of ordinary differential equations. Solving such systems is computationally very expensive, especially when simulations for many different values of input parameters are desired. In this work, we compare several model reduction techniques and demonstrate their applicability to numerical simulations of the Vlasov equation. The necessity of symplectic model reduction algorithms is illustrated with a simple numerical experiment. [ABSTRACT FROM AUTHOR]

This article investigates the propagation of chaos property for weakly interacting mild solutions to semilinear stochastic partial differential equations whose coefficients might not satisfy Lipschitz conditions. Furthermore, we establish existence and uniqueness results for mild solutions to SPDEs with distribution dependent coefficients, so-called McKean–Vlasov SPDEs. [ABSTRACT FROM AUTHOR]

We re-examine the combined semi-classical and mean-field limit in the N-body fermionic Schrödinger equation with pure state initial data using the Husimi measure framework. The Husimi measure equation involves three residue types: kinetic, semiclassical, and mean-field. The main result of this paper is to provide better estimates for the kinetic and mean-field residue than those in Chen et al. (J Stat Phys 182(2):1–41, , 2021). Especially, the estimate for the mean-field residue is shown to be smaller than the semiclassical residue by a mixed-norm estimate of the two-particle reduced density matrix factorization. Our analysis also updates the oscillation estimate parts in the residual term estimates appeared in Chen et al. (J Stat Phys 182(2):1–41, , 2021). [ABSTRACT FROM AUTHOR]

The study of microscopic effects on the dispersion of surface magnetoplasmon polaritons is important. We use the collisionless Vlasov equation and Maxwell's equations to evaluate the dielectric tensor for evaluating the dispersion relations of surface magnetoplasmon polaritons. We treat the case in the Voigt geometry assuming a semi-infinite dielectric medium. The direction of the magnetic field is considered parallel to the surface and perpendicular to the propagation vector k. The analysis shows the influence of additional microscopic kinetic effects. Standard Drude model results are retrieved in the absence of these effects. [ABSTRACT FROM AUTHOR]

In this paper, we study the hydrodynamic limit for the fluid–particle model, which consists of the inhomogeneous incompressible Navier–Stokes equations coupled with the Vlasov equation through a drag force in a bounded domain of ℝ 2 with a homogeneous Dirichlet boundary condition on the fluid velocity field and Maxwell boundary condition on the kinetic distribution function. The proof relies on the relative entropy argument, which extends the work of El Ghani [Asymptotic analysis for a Vlasov–Navier–Stokes system in a bounded domain, J. Hyperbolic Differ. Equ. 7 (2010) 191–210] to inhomogeneous incompressible Navier–Stokes–Vlasov equations. [ABSTRACT FROM AUTHOR]

We investigate the regularising effect of certain perturbations by noise in singular interacting particle systems under the mean field scaling. In particular, we show that the addition of a suitably irregular path can regularise these dynamics and we recover the McKean–Vlasov limit under very broad assumptions on the interaction kernel; only requiring it to be controlled in a possibly distributional Besov space. In the particle system we include two sources of randomness, a common noise path Z which regularises the dynamics and a family of idiosyncratic noises, which we only assume to converge in mean field scaling to a representative noise in the McKean–Vlasov equation. [ABSTRACT FROM AUTHOR]

This work is concerned with the existence of optimal controllers for the Bolza optimal control problem governed by the nonlinear Fokker-Planck equation in L¹(Rd) with control input in the drift term. The solution to the control state system is a weak (mild) solution obtained from a vanishing viscosity approximation scheme. One obtains in particular the existence for the stochastic optimal control problem governed by McKean-Vlasov SDEs. For this problem, one proves the existence of a stochastic Markov optimal controller in feedback form. [ABSTRACT FROM AUTHOR]

In this work we design an explicit feedback controller u = Φ(ρ) for the exact controllability problem to the nonlinear Fokker-Planck equation ρt - Δβ(ρ) + div(uρ) = 0 in (0, T) x O, ρ(0) = ρ0, ρ(T) = ρ1, with reflecting boundary conditions (∇β(ρ) - uρ) ⋅ n = 0 on ∂O. Here, O ⊂ Rd, 1 ≤ d, is a bounded, open set with smooth boundary ∂O and β is a smooth monotonically increasing function. If ρ0 and ρ1 are probability densities, one obtains in particular the exact controllability for McKean-Vlasov stochastic differential equations on O with reflecting (impenetrable) barrier. [ABSTRACT FROM AUTHOR]

We investigate the semiclassical limit from the semirelativistic Hartree--Fock equation describing the time evolution of a system of fermions in the mean-field regime with a relativistic dispersion law and interacting through a singular potential of the form ..., and ..., with the convention ... . For mixed states, we show convergence in the Schatten norms with an explicit rate towards the Weyl transform of a solution to the relativistic Vlasov equation with singular potentials, thus generalizing [E. Dietler, S. Rademacher, and B. Schlein, J. Stat. Phys., 172 (2018), pp. 398--433], where the case of smooth potentials has been treated. Moreover, we provide new results on the well-posedness theory of the relativistic Vlasov equations with singular interactions. [ABSTRACT FROM AUTHOR]

Based on the dispersion chain of the Vlasov equations, the paper considers the construction of a new chain of equations of quantum mechanics of high kinematical values. The proposed approach can be applied to consideration of classical and quantum systems with radiation. A number of theorems are proved on the form of extensions of the Hamilton operators, Lagrange functions, Hamilton–Jacobi equations, and Maxwell equations to the case of a generalized phase space. In some special cases of lower dimensions, the dispersion chain of quantum mechanics is reduced to quantum mechanics in phase space (the Wigner function) and the de Broglie–Bohm «pilot wave» theory. An example of solving the Schrödinger equation of the second rank (for the phase space) is analyzed, which, in contrast to the Wigner function, gives a positive distribution density function. [ABSTRACT FROM AUTHOR]