Surface plasmon resonance

The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range (for example, Coulomb) interaction. The equation was first suggested for description of plasma by Anatoly Vlasov in 1938 (see also ) 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:

1. Theory of pair collisions disagrees with the discovery by Rayleigh, Irving Langmuir and Lewi Tonks of natural vibrations in electron plasma.
2. Theory of pair collisions is formally not applicable to Coulomb interaction due to the divergence of the kinetic terms.
3. Theory of pair collisions cannot explain experiments by Harrison Merrill and Harold Webb on anomalous electron scattering in gaseous plasma.

Vlasov suggests that these difficulties originate from the long-range character of Coulomb interaction. He starts with the collisionless Boltzmann equation (sometimes anachronistically called the Vlasov equation in this context), in generalized coordinates:

${\frac {\operatorname {d} f({\bar {q}},{\bar {p}},t)}{\operatorname {d} t}}=0,$

explicitly a PDE:

${\frac {\partial f}{\partial t}}+{\frac {\operatorname {d} {\bar {q}}}{\operatorname {d} t}}\cdot {\frac {\partial f}{\partial {\bar {q}}}}+{\frac {\operatorname {d} {\bar {p}}}{\operatorname {d} t}}\cdot {\frac {\partial f}{\partial {\bar {p}}}}=0,$

and adapted it to the case of a plasma, leading to the systems of equations shown below.

The Vlasov–Maxwell system of equations (cgs units)

Instead of collision-based kinetic description for interaction of charged particles in plasma, Vlasov utilizes a self-consistent collective field created by the charged plasma particles. Such a description uses distribution functions $f_{e}({\vec {r}},{\vec {p}},t)$ and $f_{i}({\vec {r}},{\vec {p}},t)$ for electrons and (positive) plasma ions. The distribution function $f_{\alpha }({\vec {r}},{\vec {p}},t)$ for species $\alpha$ describes the number of particles of the species $\alpha$ having approximately the momentum ${\vec {p}}$ near the position ${\vec {r}}$ at time $t$. Instead of the Boltzmann equation, the following system of equations was proposed for description of charged components of plasma (electrons and positive ions):

${\frac {\partial f_{e}}{\partial t}}+{\vec {v}}_{e}\cdot \nabla f_{e}-e{\Bigl (}{\vec {E}}+{\frac {1}{c}}({\vec {v}}\times {\vec {B}}){\Bigr )}\cdot {\frac {\partial f_{e}}{\partial {\vec {p}}}}=0$

${\frac {\partial f_{i}}{\partial t}}+{\vec {v}}_{i}\cdot \nabla f_{i}+e{\Bigl (}{\vec {E}}+{\frac {1}{c}}({\vec {v}}\times {\vec {B}}){\Bigr )}\cdot {\frac {\partial f_{i}}{\partial {\vec {p}}}}=0$

$\nabla \times {\vec {B}}={\frac {4\pi {\vec {j}}}{c}}+{\frac {1}{c}}{\frac {\partial {\vec {E}}}{\partial t}},\quad \nabla \times {\vec {E}}=-{\frac {1}{c}}{\frac {\partial {\vec {B}}}{\partial t}}$

$\nabla \cdot {\vec {E}}=4\pi \rho ,\quad \nabla \cdot {\vec {B}}=0$

$\rho =e\int (f_{i}-f_{e})d^{3}p,\quad {\vec {j}}=e\int (f_{i}-f_{e}){\vec {v}}d^{3}p,\quad {\vec {v}}_{\alpha }={\frac {{\vec {p}}/m_{\alpha }}{(1+p^{2}/(m_{\alpha }c)^{2})^{1/2}}}$

Here $e$ is the electron charge, $c$ is the speed of light, $m_{\alpha }$ the mass of the electron and ion respectively, ${\vec {E}}({\vec {r}},t)$ and ${\vec {B}}({\vec {r}},t)$ represent collective self-consistent electromagnetic field created in the point ${\vec {r}}$ at time moment $t$ by all plasma particles. The essential difference of this system of equations from equations for particles in an external electromagnetic field is that the self-consistent electromagnetic field depends in a complex way on the distribution functions of electrons and ions $f_{e}({\vec {r}},{\vec {p}},t)$ and $f_{i}({\vec {r}},{\vec {p}},t)$.

The Vlasov–Poisson equation

The Vlasov–Poisson equations are an approximation of the Vlasov–Maxwell equations in the nonrelativistic zero-magnetic field limit:

${\frac {\partial f_{\alpha }}{\partial t}}+{\vec {v}}\cdot {\frac {\partial f_{\alpha }}{\partial {\vec {x}}}}+{\frac {q_{\alpha }{\vec {E}}}{m_{\alpha }}}\cdot {\frac {\partial f_{\alpha }}{\partial {\vec {v}}}}=0,$

and Poisson’s equation for self-consistent electric field:

$\varepsilon \nabla ^{2}\phi +\rho =0.$

Here $q_{\alpha }$ is the particle’s electric charge, $m_{\alpha }$ is the particle’s mass, ${\vec {E}}({\vec {x}},t)$ is the self-consistent electric field, $\phi ({\vec {x}},t)$ the self-consistent electric potential and $\rho$ is the electric charge density.

Vlasov–Poisson equations are used to describe various phenomena in plasma, in particular Landau damping and the distributions in a double layer plasma, where they are necessarily strongly non-Maxwellian, and therefore inaccessible to fluid models.

Moment equations

In fluid descriptions of plasmas (see plasma modeling and magnetohydrodynamics (MHD)) one does not consider the velocity distribution. This is achieved by replacing $f({\vec {r}},{\vec {v}},t)$ with plasma moments such as number density, $n$, mean velocity, $\mathbf {u}$ and pressure, $\mathbf {p}$ . They are named plasma moments because the nth moment of $f$ can be found by integrating $v^{n}f$ over velocity. These variables are only functions of position and time, which means that some information is lost. In multifluid theory, the different particle species are treated as different fluids with different pressures, densities and flow velocities. The equations governing the plasma moments are called the moment or fluid equations.

Below the two most used moment equations are presented (in SI units). Deriving the moment equations from the Vlasov equation requires no assumptions about the distribution function.

Continuity equation

The continuity equation describes how the density changes with time. It can be found by integration of the Vlasov equation over the entire velocity space.

$\int {\frac {\mathrm {d} }{\mathrm {d} t}}fd^{3}v=\int \left({\frac {\partial }{\partial t}}f+({\vec {v}}\cdot \nabla _{r})f+\nabla _{v}\cdot ({\vec {a}}f)\right)d^{3}v=0$

After some calculations, one ends up with

${\frac {\partial }{\partial t}}n+\nabla \cdot (n\mathbf {u} )=0$.

The particle density $n$, and the average velocity $\mathbf {u}$, are zeroth and first order moments:

$n=\int fd^{3}v$

$\mathbf {u} ={\frac {\int {\vec {v}}fd^{3}v}{\int fd^{3}v}}={\frac {1}{n}}\int {\vec {v}}fd^{3}v$

Momentum equation

The rate of change of momentum of a particle is given by the Lorentz equation:

$m{\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} t}}=q({\vec {E}}+{\vec {v}}\times {\vec {B}})$

By using this equation and the Vlasov Equation, the momentum equation for each fluid becomes

$mn{\frac {\mathrm {d} }{\mathrm {d} t}}\mathbf {u} =-\nabla \cdot \mathbf {p} +qn{\vec {E}}+qn\mathbf {u} \times {\vec {B}}$,

where $\mathbf {p}$ is the pressure tensor. The total time derivative is

${\frac {\mathrm {d} }{\mathrm {d} t}}={\frac {\partial }{\partial t}}+\mathbf {u} \cdot \nabla$.

The pressure tensor is defined as the mass density times the covariance matrix of the velocity:

$p_{ij}=m\int (v_{i}-u_{i})(v_{j}-u_{j})fd^{3}v$.

The frozen-in approximation

Before you choose any particular company it is vital to understand in full how the different plans can vary. There is no other better method than to create a message board so that people can relax and "chill" on your website and check out your articles more. You should read the HostGator review, even before registering with a web hosting company. but Hostgator in addition considers the surroundings. You can even use a Hostgator reseller coupon for unlimited web hosting at HostGator! Most of individuals by no means go for yearly subscription and choose month to month subscription. Several users commented that this was the deciding factor in picking HostGator but in any case there is a 45 day Money Back Guarantee and there is no contract so you can cancel at any time. GatorBill is able to send you an email notice about the new invoice. In certain cases a dedicated server can offer less overhead and a bigger revenue in investments. With the plan come a Free Billing Executive, Free sellers account and Free Hosting Templates.



This is one of the only things that require you to spend a little money to make money. Just go make an account, get a paypal account, and start selling. To go one step beyond just affiliating products and services is to create your own and sell it through your blog. Not great if you really enjoy trying out all the themes. Talking in real time having a real person causes it to be personal helping me personally to sort out how to proceed. The first step I took was search for a discount code, as I did with HostGator. Using a HostGator coupon is a beneficial method to get started. As long as the necessities are able to preserve the horizontal functionality of your site, you would pretty much be fine. As for ideal MHD, the plasma can be considered as tied to the magnetic field lines when certain conditions are fulfilled. One often says that the magnetic field lines are frozen into the plasma. The frozen-in conditions can be derived from Vlasov equation.

We introduce the scales $T$, $L$ and $V$ for time, distance and speed respectively. They represent magnitudes of the different parameters which give large changes in $f$. By large we mean that

${\frac {\partial f}{\partial t}}T\sim f\quad \left|{\frac {\partial f}{\partial {\vec {r}}}}\right|L\sim f\quad \left|{\frac {\partial f}{\partial {\vec {v}}}}\right|V\sim f.$

We then write

$t^{\prime }={\frac {t}{T}}\quad {\vec {r}}^{\prime }={\frac {\vec {r}}{L}}\quad {\vec {v}}^{\prime }={\frac {\vec {v}}{V}}.$

Vlasov equation can now be written

${\frac {1}{T}}{\frac {\partial f}{\partial t^{\prime }}}+{\frac {V}{L}}{\vec {v}}^{\prime }\cdot {\frac {\partial f}{\partial {\vec {r}}^{\prime }}}+{\frac {q}{mV}}({\vec {E}}+V{\vec {v}}^{\prime }\times {\vec {B}})\cdot {\frac {\partial f}{\partial {\vec {v}}^{\prime }}}=0.$

So far no approximations have been done. To be able to proceed we set $V=R\omega _{g}$, where $\omega _{g}=qB/m$ is the gyro frequency and R is the gyroradius. By dividing by $\omega _{g}$, we get

${\frac {1}{\omega _{g}T}}{\frac {\partial f}{\partial t^{\prime }}}+{\frac {R}{L}}{\vec {v}}^{\prime }\cdot {\frac {\partial f}{\partial {\vec {r}}^{\prime }}}+\left({\frac {\vec {E}}{VB}}+{\vec {v}}^{\prime }\times {\frac {\vec {B}}{B}}\right)\cdot {\frac {\partial f}{\partial {\vec {v}}^{\prime }}}=0$

If $1/\omega _{g}<<T$ and $R<<L$, the two first terms will be much less than one since $\partial f/\partial t^{\prime }\sim 1$, $v^{\prime }\lesssim 1$ and $\partial f/\partial {\vec {r}}^{\prime }\sim 1$ due to the definitions of $T$, $L$ and $V$ above. Since the last term is of the order of one, we can neglect the two first terms and write

$\left({\frac {\vec {E}}{VB}}+{\vec {v}}^{\prime }\times {\frac {\vec {B}}{B}}\right)\cdot {\frac {\partial f}{\partial {\vec {v}}^{\prime }}}\approx 0\Rightarrow ({\vec {E}}+{\vec {v}}\times {\vec {B}})\cdot {\frac {\partial f}{\partial {\vec {v}}}}\approx 0$

This equation can be decomposed into a field aligned and a perpendicular part:

${\vec {E}}_{||}{\frac {\partial f}{\partial {\vec {v}}_{||}}}+({\vec {E}}_{\perp }+{\vec {v}}\times {\vec {B}})\cdot {\frac {\partial f}{\partial {\vec {v}}_{\perp }}}\approx 0$

The next step is to write ${\vec {v}}={\vec {v}}_{0}+\Delta {\vec {v}}$, where

${\vec {v}}_{0}\times {\vec {B}}=-{\vec {E}}_{\perp }$

It will soon be clear why this is done. With this substitution, we get

${\vec {E}}_{||}{\frac {\partial f}{\partial {\vec {v}}_{||}}}+(\Delta {\vec {v}}_{\perp }\times {\vec {B}})\cdot {\frac {\partial f}{\partial {\vec {v}}_{\perp }}}\approx 0$

If the parallel electric field is small,

$(\Delta {\vec {v}}_{\perp }\times {\vec {B}})\cdot {\frac {\partial f}{\partial {\vec {v}}_{\perp }}}\approx 0$

This equation means that the distribution is gyrotropic. The mean velocity of a gyrotropic distribution is zero. Hence, ${\vec {v}}_{0}$ is identical with the mean velocity, $\mathbf {u}$, and we have

${\vec {E}}+\mathbf {u} \times {\vec {B}}\approx 0$

To summarize, the gyro period and the gyro radius must be much smaller than the typical times and lengths which give large changes in the distribution function. The gyro radius is often estimated by replacing $V$ with the thermal velocity or the Alfvén velocity. In the latter case $R$ is often called the inertial length. The frozen-in conditions must be evaluated for each particle species separately. Because electrons have much smaller gyro period and gyro radius than ions, the frozen-in conditions will more often be satisfied.

See also

• A. A. Vlasov, Many-Particle Theory and Its Application to Plasma, Gordon and Breach, 1961.
• List of plasma (physics) articles

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.