物理代写|电磁学代写electromagnetism代考|ELEC3104

物理代写|电磁学代写electromagnetism代考|Coupled Approximate Models

When considering the Vlasov-Maxwell model, in many cases, the interactions between particles are mainly electrostatic; the self-consistent magnetic field is negligible. Furthermore, particles have velocities that are much smaller than $c$ : they obey the non-relativistic dynamic. So, one reverts to the position-velocity description of phase space $(x, v) \in \mathbb{R}_x^3 \times \mathbb{R}_v^3$; in addition. in the Lorentz force. the term $v \times \boldsymbol{B}$ is negligible before $\boldsymbol{E}$, unless there is a strong external magnetic field (as in tokamaks, for instance). One replaces the Maxwell’s equations with an electric quasi-static model; and the magnetic part (1.114)-(1.115) is irrelevant. The electric part (1.112)-(1.113) is rephrased as $\boldsymbol{E}=-\operatorname{grad} \phi$ and $-\Delta \phi=\varrho / \varepsilon_0$. Thus, we arrive at the Vlasov-Poisson system:
\begin{aligned} \frac{\partial f}{\partial t}+v \cdot \nabla_x f-\frac{q}{m} \nabla_x \phi \cdot \nabla_v f &=0 ; \ -\Delta_x \phi &=\frac{\varrho}{\varepsilon_0}, \end{aligned}
with $\varrho$ given by (1.82). Also, there exist intermediate models such as VlasovDarwin, which couples Eq. (1.87) with the model of Sect. 1.4.4 (see, for instance, $[7,36])$

物理代写|电磁学代写electromagnetism代考|Standard Differential Operators

Let us begin by recalling the definitions of the four operators grad, div, $\Delta$ and curl, which we use throughout this book.

Let $E_n$ be a finite-dimensional Euclidean space of dimension $n$, endowed with the scalar product ‘, and let $A_n$ be an affine space over $E_n$. Furthermore, let $U$ be an open subset of $A_n$. Respectively introduce a scalar field on $U, f: U \rightarrow \mathbb{R}$, and a vector field on $U, f: U \rightarrow E_n$.

Assume that $f$ is differentiable at $M \in U$, and let $D f(M)$ be its differential at $M$. Then, the gradient of $f$ at $M$ is defined by
$$\operatorname{grad} f(M) \cdot v:=D f(M) \bullet v, \quad \forall v \in E_n .$$
Provided that $f$ is differentiable on $U$, the vector field $M \mapsto \operatorname{grad} f(M)$ is called the gradient of $f$ on $U$. The operator, grad, is called the gradient operator.

Assume that $f$ is differentiable at $M \in U$, then the divergence of $f$ at $M$ is définéd by
$$\operatorname{div} f(M):=\operatorname{tr}(D f(M)),$$
where tr denotes the trace of a linear operator. Provided that $f$ is differentiable on $U$, the scalar field $M \mapsto \operatorname{div} f(M)$ is called the divergence of $f$ on $U$. The operator, div, is called the divergence operator.

Assume that $f$ is twice differentiable at $M \in U$, then the Laplacian of $f$ at $M$ is defined by
$$\Delta f(M):=\operatorname{div}(\operatorname{grad} f)(M)$$

物理代写|电磁学代写electromagnetism代考|Coupled Approximate Models

$$\frac{\partial f}{\partial t}+v \cdot \nabla_x f-\frac{q}{m} \nabla_x \phi \cdot \nabla_v f=0 ;-\Delta_x \phi=\frac{\varrho}{\varepsilon_0},$$

物理代写|电磁学代写electromagnetism代考|Standard Differential Operators

$$\operatorname{div} f(M):=\operatorname{tr}(D f(M))$$

$$\Delta f(M):=\operatorname{div}(\operatorname{grad} f)(M)$$

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