# Highly collisional yet low resistivity plasmas

In summary: If the mean free path is low and the field is low then how much KE can be gained before a collision. If you can find those two values then you can find the velocity gained and hence the KE lost per interaction. Your equation is macroscopic whilst you need the microscopic model to give you the number you want.
In approximations for the applicability of ideal MHD to plasmas, it states that plasmas are considered highly collisional to permit the assumption that the plasma (i.e. electrons) follow a Maxwellian velocity distribution. Although is not resistivity based on collisions in the plasma? If there is high collisionality, how can this result in collisions that somehow have low resistivity?

I would expect the resistivity to depend on the number of free electrons as much as anything. Frequent collisions would ensure that there were plenty of parallel paths for the current can take. But would the collisions necessarily transfer much net energy to thermal once equilibrium was established?

sophiecentaur said:
I would expect the resistivity to depend on the number of free electrons as much as anything. Frequent collisions would ensure that there were plenty of parallel paths for the current can take. But would the collisions necessarily transfer much net energy to thermal once equilibrium was established?

The number density of free electrons is certainly important in determining the rate of collisions. When considering momentum loss in plasmas, the rate is given in the first attachment. The actual magnitude for the energy loss would be calculated by multiplying the loss rate by the initial kinetic energy. Now if considering any collision in general (i.e. momentum loss rate), the ratio between the energy loss rate and momentum loss rate is given by the second attachment for thermal plasmas. If we're considering electron-electron collision to dominate (i.e. ##m_1 = m_2 = m_e##), then the two rates are equivalent. Thus in ideal MHD, why is it permissible to consider these plasmas to be highly collisional yet of low resistively if energy exchange does occur between individual particles during collisions?

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The actual magnitude for the energy loss would be calculated by multiplying the loss rate by the initial kinetic energy.
In a dense plasma, the KE, acquired from the overall field would be tiny? (I'm trying to look for reasons why the loss would be much less than you suggest.)

I have no idea about such extreme conditions but what would be the Resistivity that you refer to above? Wouldn't it all be a pretty High Impedance phenomenon? Out perhaps the interactions wouldn't be losing much energy from the electrons - so not actual 'collisions'?

sophiecentaur said:
I have no idea about such extreme conditions but what would be the Resistivity that you refer to above? Wouldn't it all be a pretty High Impedance phenomenon? Out perhaps the interactions wouldn't be losing much energy from the electrons - so not actual 'collisions'?

For the impedance to be negligible in MHD, we're considering the case: ## \eta \bf\vec{J} \ll \bf\vec{E} + \bf\vec{v} \times \bf\vec{B} ##, where ##\eta## is resistivity. Well that's what I'm trying to understand (and try to demonstrate in the above attachments): if collisions do take place, why would the collisions result in negligible energy loss?

For the impedance to be negligible in MHD, we're considering the case: ## \eta \bf\vec{J} \ll \bf\vec{E} + \bf\vec{v} \times \bf\vec{B} ##, where ##\eta## is resistivity. Well that's what I'm trying to understand (and try to demonstrate in the above attachments): if collisions do take place, why would the collisions result in negligible energy loss?

If the mean free path is low and the field is low then how much KE can be gained before a collision. If you can find those two values then you can find the velocity gained and hence the KE lost per interaction. Your equation is macroscopic whilst you need the microscopic model to give you the number you want.

## 1. What are highly collisional yet low resistivity plasmas?

Highly collisional yet low resistivity plasmas refer to plasma systems that have a high number of collisions between particles, but still maintain a low level of electrical resistivity. This means that while the particles in the plasma are constantly colliding with each other, the resistance to the flow of electrical current is still relatively low.

## 2. How is a plasma considered highly collisional?

A plasma is considered highly collisional when the mean free path of the particles is small compared to the size of the system. This means that the particles are colliding frequently, leading to a high collision rate.

## 3. Why is low resistivity important in plasma systems?

Low resistivity is important in plasma systems because it allows for the efficient flow of electrical current. This is necessary for many applications of plasma, such as in plasma processing for materials and in fusion reactors.

## 4. What factors can affect the collision rate and resistivity in plasmas?

The collision rate and resistivity in plasmas can be affected by various factors, such as the density and temperature of the plasma, the strength of the magnetic field, and the type of particles in the plasma.

## 5. How can highly collisional yet low resistivity plasmas be created and controlled?

Highly collisional yet low resistivity plasmas can be created and controlled through various methods, including the use of strong magnetic fields, adjusting the plasma density and temperature, and applying external forces such as radio frequency waves. These techniques can help to increase the collision rate and decrease the resistivity in plasmas.

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