II' Plasma Physics Fundamentals

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II. Plasma Physics Fundamentals

• 4. The Particle Picture
• 5. The Kinetic Theory
• 6.

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5. The Kinetic Theory

• 5.1 The Distribution Function
• 5.2 The Kinetic Equations
• 5.3 Relation to Macroscopic Quantities

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5.1 The Distribution Function

• 5.1.1 The Concept of Distribution Function
• 5.1.2 The Maxwellian Distribution

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5.1.1 The Concept of Distribution Function

• General distribution function ff(r,v,t)
• Meaning the number of particles per m3 at the
  position r, time t and velocity between v and
  vdv is f(r,v,t) dv, where dv dvx dvy dvz
• The density is then found as

• If the distribution is normalized as

then f represents a probability distribution
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5.1.2 The Maxwellian Distribution

• The maxwellian distribution is defined as

where

• The known result

yields
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The Maxwellian Distribution (II)

• The root mean square velocity for a maxwellian is

recall

• The average of the velocity magnitude vv is

• In one direction

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The Maxwellian Distribution (III)

• The distribution w.r.t. the magnitude of v

• For a Maxwellian

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5.2 The Kinetic Equations

• 5.2.1 The Boltzmann Equation
• 5.2.2 The Vlasov Equation
• 5.2.3 The Collisional Effects

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5.2.1 The Boltzmann Equation

• A distribution function ff(r,v,t) satisfies the
  Boltzmann equation

• The r.h.s. of the Boltzmann equation is simply
  the expansion of d f(r,v,t)/dt
• The Boltzmann equation states that in absence of
  collisions df/dt0

vx

tDt
Motion of a group of particles with constant
density in the phase space
t
x
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5.2.2 The Vlasov Equation

• In general, for sufficiently hot plasmas, the
  effect of collisions are less and less important
• For electromagnetic forces acting on the
  particles and no collisions the Boltzmann
  equation becomes

that is called the Vlasov equation

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5.2.3 The Collisional Effects

• The Vlasov equation does not account for
  collisions

• Short-range collisions like charged particles
  with neutrals can be described by a Boltzmann
  collision operator in the Boltzmann equation
• For long-range collisions, like Coulomb
  collisions, a statistical approach yields the
  Fokker-Planck collision term
• The Boltzmann equation with the Fokker-Planck
  collision term is simply named the Fokker-Planck
  equation.

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5.3 Relation to Macroscopic Quantities

• 5.3.1 The Moments of the Distribution Function
• 5.3.2 Derivation of the Fluid Equations

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5.3.1 The Moments of the Distribution Function

• Notation define

• If AA(v) the average of the function A for a
  distribution function ff(r,v,t) is defined as

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The Moments of the Distribution Function (II)

• General distribution function ff(r,v,t)
• The density is defined as the 0th order moment
  and was found to be

• The mass density can be then defined as

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The Moments of the Distribution Function (III)

• The 1st order moment is the average velocity or
  fluid velocity is defined as

• The momentum density can be then defined as

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The Moments of the Distribution Function (IV)

• Higher moments are found by diadic products with
  v
• The 2nd order moment gives the stress tensor
  (tensor of second order)

• In the frame of the moving fluid the velocity is
  wv-u. In this case the stress tensor becomes the
  pressure tensor

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5.3.2 Derivation of the Fluid Equations

• Boltzmann equation written for the Lorentz force

• Integrate in velocity space

• From the definition of density

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Derivation of the Fluid Equations (II)

• Since the gradient operator is independent from
  v

• Through integration by parts it can be shown that

• If there are no ionizations or recombination the
  collisional term will not cause any change in the
  number of particles (no particle sources or
  sinks) therefore

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Derivation of the Fluid Equations (III)

• The integrated Boltzmann equation then becomes

that is known as equation of continuity

• In general moments of the Boltzmann equation are
  taken by multiplying the equation by a vector
  function gg(v) and then integrating in the
  velocity space
• In the case of the continuity equation g1
• For gmv the fluid equation of motion, or
  momentum equation can be obtained

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Derivation of the Fluid Equations (IV)

• Integrate the Boltzmann equation in velocity
  space with gmv

• The first term is

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Derivation of the Fluid Equations (V)

• Further simplifications yield the final fluid
  equation of motion

where u is the fluid average velocity, P is the
stress tensor and Pcoll is the rate of momentum
change due to collisions

• Integrating the Boltzmann equation in velocity
  space with g½mvv the energy equation is obtained