Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations

Nonlocal Lotka–Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Is it possible to describe the dynamics of the limiting concentration points and of the weights of the Dirac masses? What is the long time asymptotics of these Dirac masses? Can several Dirac masses co-exist?

We will explain how these questions relate to the so-called “constrained Hamilton–Jacobi equation” and how a form of canonical equation can be established. This equation has been established assuming smoothness. Here we build a framework where smooth solutions exist and thus the full theory can be developed rigorously. We also show that our form of canonical equation comes with a kind of Lyapunov functional.

Numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation.

Our motivation comes from population adaptive evolution a branch of mathematical ecology which models Darwinian evolution.     

Symmetry and Classification of the Dirac–Fock Equation

We consider the properties of the Dirac–Fock equation with differential operators of the first-order symmetry. For a relativistic particle in an electromagnetic field, we describe the covariant properties of the Dirac equation in an arbitrary Riemannian space V₄ with the signature (?1,?1,?1, 1). We present a general form of the differential operator with a first-order symmetry and characterize the pair of such commuting operators. We list the spaces where the free Dirac equation admits at least one differential operator with a first-order symmetry. We perform a symmetry classification of electromagnetic field tensors and construct complete sets of symmetry operators.     

The explicit solutions to the nonlinear Dirac equation and Dirac-Klein-Gordon equation

Abstract In [3] Dias and Figueira have reported that the square of the solution for the nonlinear Dirac equation satisfies the linear wave equation in one space dimension. So the aim of this paper is to proceed with their work and to clarify a structure of the nonlinear Dirac equation. The explicit solutions to the nonlinear Dirac equation and Dirac-Klein-Gordon equation are obtained. Keywords: Nonlinear Dirac equation, Dirac-Klein-Gordon equation, Pauli matrix Mathematics Subject Classification (2000): 35C05, 35L45     

Mean field limits for nonlinear spatially extended Hawkes processes with exponential memory kernels

We consider spatially extended systems of interacting nonlinear Hawkes processes modeling large systems of neurons placed in ${\mathbb{R}}^d$ and study the associated mean field limits. As the total number of neurons tends to infinity, we prove that the evolution of a typical neuron, attached to a given spatial position, can be described by a nonlinear limit differential equation driven by a Poisson random measure. The limit process is described by a neural field equation. As a consequence, we provide a rigorous derivation of the neural field equation based on a thorough mean field analysis.     

The hydrodynamic limit for a system with interactions prescribed by Ginzburg-Landau type random Hamiltonian

Summary As a microscopic model we consider a system of interacting continuum like spin field overR ^d . Its evolution law is determined by the Ginzburg-Landau type random Hamiltonian and the total spin of the system is preserved by this evolution. We show that the spin field converges, under the hydrodynamic space-time scalling, to a deterministic limit which is a solution of a certain nonlinear diffusion equation. This equation describes the time evolution of the macroscopic field. The hydrodynamic scaling has an effect of the homogenization on the system at the same time.     

Vortex dynamics for the nonlinear wave equation

Vortex dynamics for the nonlinear wave equation is a typical model of the “particle and field” theories of classical physics. The formal derivation of the dynamical law was done by J.Neu. He also made an interesting connection between vortex dynamics and the Dirac theory of electrons. Here we give a rigorous mathematical proof of this natural dynamical law. © 1999 John Wiley & Sons, Inc.     

Neutral Fermion Having Electric and Magnetic Moments in an External Electromagnetic Field

We show that in 2+1 dimensions, the Dirac equation for a neutral fermion possessing electric and magnetic dipole moments in an external electromagnetic field reduces to the Dirac equation for a charged fermion in a external field characterized by a certain 3-pseudo-vector potential. The effective charge of the neutral fermion is determined by its dipole moments. The effects of coupling electric and magnetic moments of the neutral fermion to the external electromagnetic field seem to be inseparable in physical experiments of any type. We find an exact solution of the Dirac equation for a massive neutral fermion with electric and magnetic dipole moments in a external plane-wave electromagnetic field. We derive expressions for the fermionic vacuum current induced by neutral fermions in the presence of external electromagnetic fields.     

Dirac equation in a 5-dimensional Kaluza-Klein theory

Dirac equation is discussed in 5-dimensional space time having topology M ⁴ ×T ¹, whereM ⁴ and T ¹ both are curved. It is shown that 4-dimensional fermion can be obtained from 5-dimensional fermion, as a result of compactification of extra dimension. It is found that the realistic 4-dimensional fermions are possible in higher modes earlier than those in lower modes during the course of expansion of 4-dimensional universe. 4-dimensional Dirac equation, obtained from 5-dimensional Dirac equation after compactification, is solved for an arbitrary mode for superheavy as well as light (realistic) fermions. Time-dependence of polarization vector and magnetization density, as a result of Gordon-decomposition of the current vector for 4-dimensional spin-½ field (with arbitrary mode), is exhibited.     

WAVES IN FERROMAGNETIC MEDIA

ABSTRACT

It is shown that small perturbations of equilibrium states in ferromagnetic media give rise to standing and traveling waves that are stable for long times. The evolution of the wave profiles is governed by semilinear heat equations. The mathematical model underlying these results consists of the Landau–Lifshitz equation for the magnetization vector and Maxwell's equations for the electromagnetic field variables. The model belongs to a general class of hyperbolic equations for vector-valued functions, whose asymptotic properties are analyzed rigorously. The results are illustrated with numerical examples.     

Weakly nonlinear Schrödinger equation with random initial data

It is common practice to approximate a weakly nonlinear wave equation through a kinetic transport equation, thus raising the issue of controlling the validity of the kinetic limit for a suitable choice of the random initial data. While for the general case a proof of the kinetic limit remains open, we report on first progress. As wave equation we consider the nonlinear Schrödinger equation discretized on a hypercubic lattice. Since this is a Hamiltonian system, a natural choice of random initial data is distributing them according to the corresponding Gibbs measure with a chemical potential chosen so that the Gibbs field has exponential mixing. The solution ψ _t (x) of the nonlinear Schrödinger equation yields then a stochastic process stationary in x∈? ^d and t∈?. If λ denotes the strength of the nonlinearity, we prove that the space-time covariance of ψ _t (x) has a limit as λ→0 for t=λ ^?2 τ, with τ fixed and |τ| sufficiently small. The limit agrees with the prediction from kinetic theory.     

Sur quelques limites de la physique des particules chargées vers la (magnéto)hydrodynamiqueOn some limits in charged particle physics towards (magneto)hydrodynamic equations

We discuss the connection between different scalings limits of the quantum-relativistic Dirac–Maxwell system. In particular we give rigorous results for the quasi-neutral/non-relativistic limit of the Vlasov–Maxwell system: we obtain a magneto-hydro-dynamic system when we consider the magnetic field as a non-relativistic effect and we obtain the Euler equation when we see it as a relativistic effect. A mathematical key is the modulated energy method. To cite this article: Y. Brenier et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 239–244.     

Dirac γ-equation, classical gauge fields and Clifford algebra

An equation, we call Dirac γ-equation, is introduced with the help of the mathematical tools connected with the Clifford algebra. This equation can be considered as a generalization of the Dirac equation for the electron. Some features of Dirac γ-equation are investigated (plane waves, currents, canonical forms). Furthermore, on the basis of local gauge in variance regarding unitary group, a system of equations is introduced consisting of Dirac γ-equation and the Yang-Mills or Maxwell equations. This system of equations describes a Dirac’s field interacting with the Yang-Mills or Maxwell gauge field. Characteristics of this system of equations are studied for various gauge groups and the liaison between the new and the standard constructions of classical gauge fields is discussed. This paper is supported by the Russian Foundation for Basic Research, grant 95-10-00433a.     

Zero Modes of the Dirac Operator in Vortex-Type Gauge Fields

We consider the D4 Euclidean Dirac equation in a SU(2) gauge field with the form of intersecting central vortices and obtain its solutions corresponding to normalizable zero modes. In the context of the problem of constructing the mechanism of chiral symmetry breaking based on the model with randomly distributed central vortices, we study a possibility for splitting fermionic zero modes in the case of a superposition of many vortices with Pontryagin indices of different signs.     

Diracs Equation in 1+1 D from a Classical Random Walk

This paper is an investigation of the class of real classical Markov processes without a birth process that will generate the Dirac equation in 1+1 dimensions. The Markov process is assumed to evolve in an extra (ordinal) time dimension. The derivation requires that occupation by the Dirac particle of a space-time lattice site is encoded in a 4 state classical probability vector. Disregarding the state occupancy, the resulting Markov process is an homogeneous and almost isotropic binary random walk in Dirac space and Dirac time (including Dirac time reversals). It then emerges that the Dirac wavefunction can be identified with a polarization induced by the walk on the Dirac space-time lattice. The model predicts that QM observation must happen in ordinal time and that wavefunction collapse is due not to a dynamical discontinuity, but to selection of a particular ordinal time. Consequently, the model is more relativistically equitable in its treatment of space and time in that the observer is attributed no special ability to specify the Dirac time of observation.     

Distribution Functions in Quantum Mechanics and Wigner Functions

We formulate and solve the problem of finding a distribution function F(r,p,t) such that calculating statistical averages leads to the same local values of the number of particles, the momentum, and the energy as those in quantum mechanics. The method is based on the quantum mechanical definition of the probability density not limited by the number of particles in the system. The obtained distribution function coincides with the Wigner function only for spatially homogeneous systems. We obtain the chain of Bogoliubov equations, the Liouville equation for quantum distribution functions with an arbitrary number of particles in the system, the quantum kinetic equation with a self-consistent electromagnetic field, and the general expression for the dielectric permittivity tensor of the electron component of the plasma. In addition to the known physical effects that determine the dispersion of longitudinal and transverse waves in plasma, the latter tensor contains a contribution from the exchange Coulomb correlations significant for dense systems.     

High‐energy and smoothness asymptotic expansion of the scattering amplitude for the Dirac equation and application

We obtain an explicit formula for the diagonal singularities of the scattering amplitude for the Dirac equation with short‐range electromagnetic potentials. Using this expansion we uniquely reconstruct an electric potential and magnetic field from the high‐energy limit of the scattering amplitude. Moreover, supposing that the electric potential and magnetic field are asymptotic sums of homogeneous terms we give the unique reconstruction procedure for these asymptotics from the scattering amplitude, known for some energy E. Furthermore, we prove that the set of the averaged scattering solutions to the Dirac equation is dense in the set of all solutions to the Dirac equation that are in L²(Ω), where Ω is any connected bounded open set in with smooth boundary, and we show that if we know an electric potential and a magnetic field for , then the scattering amplitude, given for some energy E, uniquely determines these electric potential and magnetic field everywhere in . Combining this uniqueness result with the reconstruction procedure for the asymptotics of the electric potential and the magnetic field we show that the scattering amplitude, known for some E, uniquely determines a electric potential and a magnetic field, that are asymptotic sums of homogeneous terms, which converges to the electric potential and the magnetic field respectively. Moreover, we discuss the symmetries of the kernel of the scattering matrix, which follow from the parity, charge‐conjugation and time‐reversal transformations for the Dirac operator. Copyright © 2014 John Wiley & Sons, Ltd.     

Quantum electronic transport in graphene: A kinetic and fluid‐dynamic approach

We derive a fluid‐dynamic model for electron transport near a Dirac point in graphene. Starting from a kinetic model, based on spinorial Wigner functions, the derivation of the fluid model is based on the minimum entropy principle, which is exploited to close the moment system deduced from the Wigner equation. To this aim we make two main approximations: the usual semiclassical approximation (??1) and a new one, namely, the ‘strongly mixed state’ approximation, which allow to compute the closure explicitly. Particular solutions of the fluid‐dynamic equations are discussed which are of physical interest because of their connection with the Klein paradox phenomenon. Copyright © 2010 John Wiley & Sons, Ltd.     

Macroscaling Limit Theorems for Filtered Spatiotemporal Random Fields

This article addresses the problem of defining a general scaling setting in which Gaussian and non-Gaussian limit distributions of linear random fields can be obtained. The linear random fields considered are defined by the convolution of a Green kernel, satisfying suitable scaling conditions, with a non-linear transformation of a Gaussian centered homogeneous random field. The results derived cover the weak-dependence and strong-dependence cases for such Gaussian random fields. Extension to more general random initial conditions defined, for example, in terms of non-linear transformations of χ²-random fields, is also discussed. For an example, we consider the random fractional diffusion equation. The vectorial version of the limit theorems derived is also formulated, including the limit distribution of the parabolically rescaled solution to the Burgers equation in the cases of weakly and strongly dependent initial potentials.