Krook collision, also known as the Krook model or the Krook collision operator, is a simplified model used to describe the collisional processes in a plasma. It is named after the physicist Marcel Krook, who introduced this model in the field of kinetic theory.

In a plasma, collisions between charged particles play a crucial role in determining the plasma's behaviour. The Krook collision model provides a way to approximate these collisions by incorporating the effects of binary (two-particle) interactions in the kinetic equations.

The Krook collision model is often used in situations where the assumption of complete equilibrium or detailed knowledge of the collision processes is not necessary. It is particularly useful for studying plasma transport phenomena and relaxation processes.

The key idea behind the Krook collision model is to introduce a relaxation term in the kinetic equations that allows the distribution function of particles to approach a Maxwellian distribution, which represents a state of thermal equilibrium. The relaxation term is proportional to the difference between the current distribution function and the Maxwellian distribution, with a relaxation time parameter governing the rate of relaxation.

Distribution function

As distribution function is so important in kinetic theory. The number of particles within position (x, y, z) at time 't' with velocity components vx and vx+dvx, vy and vy+dvy, vz and vz + dvz is

The density is then given by

If f is normalized to unity i.e

In a volume element where velocity coordinates are very large, the number of representative points is relatively small since in any microscopic system there must be few particles with very large velocities. Physical consideration required therefore that f(arrow r, v, t) must tend to zero as velocity becomes infinite.

However, in the absence of external pressures, a plasma originally in an inhomogeneous condition eventually achieves an equilibrium state as a result of interactions between the particles. The distribution function doesn't depend on r in this homogenous state.

Number density is given by

Average velocity

The fundamental equation which f(r,v,t) has to satisfy is the Boltzmann equation.

where F = force acting on particles

(∂f/∂t) is time rate of change of f due to collisions.

Krook collision term

In a sufficiently hot plasma, collisions can be neglected. If the fundamental force F entirely electromagnetic, then Boltzmann equation takes the form,

when there are collisions with central atoms, the collision term in 1 can be approximated as

where fo = Maxwellian equilibrium distribution function, which depends on the plasma parameters such as temperature and particle densities.

𝜏 = constant collision time.

The Krook collision operator modifies the kinetic equations by including this relaxation term. It accounts for the effects of collisions and leads to a relaxation of the distribution function towards thermal equilibrium.

The use of the Krook collision model allows for a simplified treatment of the collisional processes in a plasma, making it easier to study the dynamics and transport phenomena in various plasma systems. However, it is important to note that the Krook collision model is an approximation and may not capture all the details and complexities of actual collision processes in a plasma.

This note is a part of the Physics Repository.

Explore

posts on Naya and its networks

Collections