Vlasov equation
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The Vlasov equation is a
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
describing time evolution of the distribution function of
plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral), a green translucent silica mineral * Quark–gluon plasma, a state of matter in quantum chromodynamics Biology * Blood pla ...
consisting of
charged particles In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary particle, ...
with long-range interaction, e.g.
Coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...
. The equation was first suggested for description of plasma by
Anatoly Vlasov Anatoly Aleksandrovich Vlasov (russian: Анато́лий Алекса́ндрович Вла́сов; – 22 December 1975) was a Russian, later Soviet, theoretical physicist prominent in the fields of statistical mechanics, kinetics, and esp ...
in 1938 and later discussed by him in detail in a monograph.


Difficulties of the standard kinetic approach

First, Vlasov argues that the standard
kinetic Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory of gases, Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to i ...
approach based on the
Boltzmann equation The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
has difficulties when applied to a description of the plasma with long-range
Coulomb interaction Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
. He mentions the following problems arising when applying the kinetic theory based on pair collisions to plasma dynamics: # Theory of pair collisions disagrees with the discovery by Rayleigh,
Irving Langmuir Irving Langmuir (; January 31, 1881 – August 16, 1957) was an American chemist, physicist, and engineer. He was awarded the Nobel Prize in Chemistry in 1932 for his work in surface chemistry. Langmuir's most famous publication is the 1919 art ...
and
Lewi Tonks Lewi Tonks (1897–1971) was an American quantum physicist noted for his discovery (with Marvin D. Girardeau) of the Tonks–Girardeau gas. Tonks was employed by General Electric for most of his working life, researching microwaves and ferromagne ...
of natural vibrations in electron plasma. # Theory of pair collisions is formally not applicable to Coulomb interaction due to the divergence of the kinetic terms. # 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 called the Vlasov equation, anachronistically in this context), in
generalized coordinates In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
: \frac f(\mathbf r,\mathbf p,t) = 0, explicitly a PDE: \frac + \frac \cdot \frac + \frac \cdot \frac = 0, and adapted it to the case of a plasma, leading to the systems of equations shown below. Here is a general distribution function of particles with
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
at
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
and given
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
. Note that the term \frac is the force acting on the particle.


The Vlasov–Maxwell system of equations (Gaussian 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(\mathbf ,\mathbf ,t) and f_i(\mathbf ,\mathbf ,t) for
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
s and (positive) plasma
ion An ion () is an atom or molecule with a net electrical charge. The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conve ...
s. The distribution function f_(\mathbf ,\mathbf ,t) for species describes the number of particles of the species having approximately the
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
\mathbf near the
position Position often refers to: * Position (geometry), the spatial location (rather than orientation) of an entity * Position, a job or occupation Position may also refer to: Games and recreation * Position (poker), location relative to the dealer * ...
\mathbf at time . Instead of the Boltzmann equation, the following system of equations was proposed for description of charged components of plasma (electrons and positive ions): \begin \frac + \mathbf _e\cdot\nabla f_e &- e\left(\mathbf +\frac\times\mathbf \right)\cdot\frac = 0 \\ \frac + \mathbf _i\cdot\nabla f_i &+ Z_i e\left(\mathbf +\frac\times\mathbf \right)\cdot\frac = 0 \\ \nabla\times\mathbf &=\frac+\frac\frac \\ \nabla\times\mathbf &=-\frac\frac \\ \nabla\cdot\mathbf &=4\pi\rho \\ \nabla\cdot\mathbf &=0 \\ \end \rho = e \int (Z_i f_i-f_e) \mathrm^3p,\quad \mathbf = e\int(Z_i f_i \mathbf_i - f_e \mathbf_e) \mathrm^3p,\quad \mathbf _\alpha = \frac Here is the
elementary charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
(e>0), is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, is the mass of the ion, \mathbf (\mathbf ,t) and \mathbf (\mathbf , t) represent collective self-consistent electromagnetic field created in the point \mathbf at time moment 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(\mathbf ,\mathbf ,t) and f_i(\mathbf ,\mathbf ,t).


The Vlasov–Poisson equation

The Vlasov–Poisson equations are an approximation of the Vlasov–Maxwell equations in the non-relativistic zero-magnetic field limit: \frac + \mathbf _ \cdot \frac+ \frac \cdot \frac = 0, and
Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
for self-consistent electric field: \nabla^2 \phi + \rho = 0. Here is the particle's electric charge, is the particle's mass, \mathbf (\mathbf ,t) is the self-consistent
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
, \phi(\mathbf , t) the self-consistent
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
and is the
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
density. Vlasov–Poisson equations are used to describe various phenomena in plasma, in particular
Landau damping In physics, Landau damping, named after its discoverer,Landau, L. "On the vibration of the electronic plasma". ''JETP'' 16 (1946), 574. English translation in ''J. Phys. (USSR)'' 10 (1946), 25. Reproduced in Collected papers of L.D. Landau, edited a ...
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 Plasma modeling refers to solving equations of motion that describe the state of a plasma. It is generally coupled with Maxwell's equations for electromagnetic fields or Poisson's equation for electrostatic fields. There are several main types of p ...
and
magnetohydrodynamics Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydro­magnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magneto­fluids include plasmas, liquid metals, ...
(MHD)) one does not consider the velocity distribution. This is achieved by replacing f(\mathbf r,\mathbf v,t) with plasma moments such as
number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric number ...
,
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
and pressure . They are named plasma moments because the -th 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 The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
). 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^3v = \int \left(\fracf + (\mathbf \cdot\nabla_r)f +(\mathbf \cdot \nabla_v) f\right) \mathrm^3v=0 After some calculations, one ends up with \fracn + \nabla\cdot (n\mathbf)=0. The number density , and the momentum density , are zeroth and first order moments: n = \int f \mathrmv n \mathbf u = \int \mathbf v f \mathrm^3v


Momentum equation

The rate of change of momentum of a particle is given by the Lorentz equation: m\frac=q(\mathbf + \mathbf \times \mathbf ) By using this equation and the Vlasov Equation, the momentum equation for each fluid becomes mn\frac\mathbf= -\nabla\cdot \mathcal + qn\mathbf + qn\mathbf\times \mathbf , where \mathcal is the pressure tensor. The
material derivative In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material der ...
is \frac = \frac + \mathbf u \cdot \nabla. The pressure tensor is defined as the particle mass times the
covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...
of the velocity: p_ = m \int (v_i - u_i) (v_j - u_j)f \mathrm^3v.


The frozen-in approximation

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 , , and for time, distance and speed respectively. They represent magnitudes of the different parameters which give large changes in f. By large we mean that \fracT \sim f \quad \left, \frac\ L \sim f \quad\left, \frac\ V\sim f. We then write t' = \frac, \quad \mathbf r'=\frac, \quad \mathbf v' = \frac. Vlasov equation can now be written \frac \frac + \frac \mathbf v' \cdot \frac + \frac (\mathbf E + V \mathbf v' \times \mathbf B) \cdot \frac = 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 is the
gyroradius The gyroradius (also known as radius of gyration, Larmor radius or cyclotron radius) is the radius of the circular motion of a charged particle in the presence of a uniform magnetic field. In SI units, the non-relativistic gyroradius is given by :r_ ...
. By dividing by , we get \frac\frac + \frac \mathbf v' \cdot \frac + \left(\frac + \mathbf v'\times\frac\right) \cdot \frac = 0 If 1/\omega_g \ll T and R \ll L, the two first terms will be much less than f since \partial f/\partial t' \sim f, v' \lesssim 1 and \partial f / \partial \mathbf r' \sim f due to the definitions of , , and above. Since the last term is of the order of f, we can neglect the two first terms and write \left(\frac +\mathbf v' \times \frac\right)\cdot\frac \approx 0 \Rightarrow (\mathbf E + \mathbf v \times\mathbf B)\cdot\frac \approx 0 This equation can be decomposed into a field aligned and a perpendicular part: \mathbf E_\parallel \frac + (\mathbf E_\perp + \mathbf v \times \mathbf B) \cdot \frac \approx 0 The next step is to write \mathbf v = \mathbf v_0 + \Delta\mathbf v, where \mathbf v_0 \times\mathbf B = -\mathbf E_\perp It will soon be clear why this is done. With this substitution, we get \mathbf E_\parallel\frac+ (\Delta\mathbf v_\perp \times\mathbf B) \cdot \frac \approx 0 If the parallel electric field is small, (\Delta \mathbf v_\perp \times \mathbf B) \cdot \frac\approx 0 This equation means that the distribution is gyrotropic. The mean velocity of a gyrotropic distribution is zero. Hence, \mathbf v_0 is identical with the mean velocity, , and we have \mathbf E + \mathbf u \times \mathbf 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 with the
thermal velocity Thermal velocity or thermal speed is a typical velocity of the thermal motion of particles that make up a gas, liquid, etc. Thus, indirectly, thermal velocity is a measure of temperature. Technically speaking, it is a measure of the width of the pea ...
or the Alfvén velocity. In the latter case 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

*
List of plasma physics articles This is a list of plasma physics topics. A * Ablation * Abradable coating * Abraham–Lorentz force * Absorption band * Accretion disk * Active galactic nucleus * Adiabatic invariant * ADITYA (tokamak) * Aeronomy * Afterglow plasma * Air ...
* Fokker–Planck equation


References


Further reading

* {{Statistical mechanics topics Statistical mechanics Non-equilibrium thermodynamics Plasma physics Equations Transport phenomena Moment (physics)