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In classical gas theory, as taught in statistical mechanics classes, we learn that the statistical distribution function for each particle is described by the Boltzmann equation:

$$ \frac{\partial f}{\partial t}+\frac{\mathbf p}{m}\cdot\nabla f+\mathbf F\cdot\frac{\partial f}{\partial\mathbf p}=\left(\frac{df}{dt}\right)_{\rm coll} $$

In fusion plasma physics, rather than applying the Boltzmann equation directly, the Vlasov equation is often used alongside Maxwell's equations:

$$ \frac{\partial f_e}{\partial t} + \mathbf {v}_e\cdot\nabla f_{e}- e\left(\mathbf {E}+\frac{\mathbf {v_e}}{c}\times\mathbf {B}\right)\cdot\frac{\partial f_e}{\partial\mathbf {p}} = 0 \\ \frac{\partial f_i}{\partial t} + \mathbf {v}_i\cdot\nabla f_{i}+ Z_i e\left(\mathbf {E}+\frac{\mathbf {v_i}}{c}\times\mathbf {B}\right)\cdot\frac{\partial f_i}{\partial\mathbf {p}} = 0 $$

Here, I noticed that the collision term $\left( \frac{df}{dt} \right)_{\text{coll}}$ is missing. I thought this might be because long-range Coulomb interactions dominate, making collisions negligible.

However, it seems to me that this might not be a good approximation since electrons and ions have opposite charges and attract each other. It appears that the collision term $\left( \frac{df}{dt} \right)_{\text{coll}}$ could have significant effects and might include more profound physics.

1. What are the actual approximations that lead to the Vlasov equation?

2. Is there a scheme that incorporates collision effects in fusion plasma? If so, what is it, and what nontrivial physics does it produce?

K.R.Park
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2 Answers2

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it seems to me that this might not be a good approximation since electrons and ions have opposite charges and attract each other

Well I think this is your problem here: One only counts the interaction as a collision if the mean free path of the plasma is much larger than the interaction distance. Since this isn't true for (highly?) ionized plasmas one normally considers, it is a good approximation.

As far as I'm aware, the validity of the Vlasov equation depends on two things:

  1. mean particle distance is smaller than the Debye length
  2. the plasma frequency is much greater than the collisional frequency

(possibly arguable a third condition: the E&M fields are quasi-static to be able to ignore the time-dependence of the fields). Again, it is because of #2 that the collisional term can be treated as 0.

For more details, see Colonna's Boltzmann and Vlasov equations in plasma physics (IOP link).


As aside,

rather than applying the Boltzmann equation directly, the Vlasov equation is often used alongside Maxwell's equations:

this is confusing to me. The Vlasov equation is a specific case of the Boltzmann equation: the force is the Lorentz force and collisions are ignored. The literature often refers to the Vlasov equation as the collision-less Boltzmann equation further linking the two.

Kyle Kanos
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Collisionless plasma is a rather widely applied approximation in plasma physics, although obviously should not be applied to all the situations (just like any other approximation).

As per Wikipedia article Plasma (physics): Ideal plasma:

Collisionlessness: The electron plasma frequency (measuring plasma oscillations of the electrons) is much larger than the electron–neutral collision frequency. When this condition is valid, electrostatic interactions dominate over the processes of ordinary gas kinetics. Such plasmas are called collisionless.[29]

The reference is to Klimontivich's review Physics of collisionless plasma

The article on Vlasov equation also clearly states that Vlasov's approach is for collisionless plasma:

In plasma physics, the Vlasov equation is a differential equation describing time evolution of the distribution function of collisionless plasma consisting of charged particles with long-range interaction, such as the Coulomb interaction.

Roger V.
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