Boltzmann Equation

Setup

Phase Space: -Space and -Space

Consider a closed system of particles in a region . Each particle has a position and a momentum , where .

Furthermore, assume that the particles are identical and indistinguishable, and they follow the canonical equation of motion:

Now, we define the phase space, (to be more precise, the -space) as the -dimensional space of position and momentum, i.e., .

The state of the system at time is represented as a set of points in -space, i.e., .

Instead of tracking individual particles (of which there could be more than ), which should track a single function that has all the information of the system.

We can also use the -space, which is the -dimensional space of all the positions and momenta of the particles, i.e., . In -space, the state of the system at time is represented as a single point . For simplicity, we may denote , as -space.

Klimontovich Distribution Function

In the hypothetical case that we know the exact position and momentum of each particle at each time , we can define the Klimontovich distribution function as:

The integral of over a region in -space gives the number of particles in that region at time :

One-Particle Distribution Function

Ensemble Average

Given a set of macroscopic physical quantities such as temperature, pressure, etc., the ensemble is a collection of virtual systems (called microstates) that share the same macroscopic physical quantities, but differ in the microscopic details, i.e., the exact position and momentum of each particle.

Each microstate can be represented as an ordered set of position and momentum of each particle, which correponds to a point in -space, i.e., . This point is called the representative point of the microstate in -space.

At each point, we can assign the likelihood of the system being in that microstate, which gives us a probability density function on -space. For example, if the temperature is high, the system is more likely to be spread in the momentum space, rather than near the origin of them momentum space. We call this assignment of likelihood the probability density such that at time , the probability of a region in -space is given by

which is the probability that systems in can be observed at time , given the macroscopic quantities.

Now, we can define the ensemble average of a microscopic quantity (a quantity that depends on the microscopic details of the system) as:

Conservation of Probability Density and Liouville’s Theorem

Since any microstate just keeps evolving, and no microstate can be created or destroyed,

where is the outflow of probability from at time . Since the probability of should be conserved, we have , which gives us the conservation of probability density:

Also,

by the equation of motion,

which gives us Liouville’s theorem:

One-Particle Distribution Function

Now, the Klimontovich distribution function is a function on the -space, given a microstate and its trajectory . In another perspective, we can also view as a function on -space, given a point in -space:

which gives us the freedom to move around for a fixed .

Now we can take the ensemble average of :

Physically, this means that we take the -space density to be moving with time

Note that the and in are replaced by the and in , which are independent variables in the integral, and they are not functions of :

Here, if we define as follows:

the sum can be rewritten as

Since the particles are identical and indistinguishable, we have for any ,

Often it is more useful to work with instead of , so we define the one-particle distribution function as

Boltzmann Equation

Reduced One-Particle Equation

Now, we also want to consider the time evolution of ,

where .

This is important because if we know , we know the particle number density, the average velocity and energy density at each point in . Notice that the time dependence of comes from , so finding the time evolution of necessitates finding the time evolution of .

From the previous discussion, we know that the probability density is locally conserved in -space:

here, we integrate both sides over , with :

here, we separate the term with from the rest of the terms:

Let us calculate the second term first:

where and .

The boundary consists of points where at least one of the particles (except the first particle) is at the boundary of or has infinite momentum. The spatial boundary of is essentially a wall which particles cannot move through, so the outflow of probability from is zero. Also, the probability of particles having infinite momentum is zero, so the outflow of probability from the boundary of momentum space is also zero. Therefore, we have

which gives us

and hence

Since the integral is taken over , we can take and as constants in the integral, which gives us

In summary,

This is the reduced one-particle equation.

If we take ,

and the integral can be viewed as probability current density in -space:

where is the -th component of . This gives us

where .

Under coordinate transformation ,

where .

Boltzmann Equation

Now, the probability current density can be decomposed into two parts: , which is the current density without collision/ interactions (i.e. if we assume and are independent of the other particles, such as only considering external forces), and , which is the current density due to collision/ interactions (collisions change the momenta of the particles, which brings in/ takes out ):

if we substitute this into the reduced one-particle equation, we have

If we multiply both sides by , we have

and letting RHS be , we have the conservative Boltzmann equation:

In a more familar form, we have

Equivalently, momentum is replaced by velocity:

Boltzmann Equation in Spherical Coordinates

Setup

We take the spatial coordinates to be spherical coordinates , with basis vectors , , and . The velocity direction is parameterized by the angular variables , where is the -axis, is the -axis, and is the -axis:

For convinience, we also write the velocity direction in terms of .

Now, from the following expression of the Boltzmann equation,

We know that , where is the solid angle element in the velocity space, and .

If we write out the summation explicitly, we have, for ,

Let us then calculate each term.

term: since is independent of , we have

term: here, is independent of , so we have

term: here, and are independent of , so we have

term: here, all the spatial coordinates are independent of , so we have

term: here, all the spatial coordinates are independent of , so we have

Which gives us a more simplified form of the Boltzmann equation in spherical coordinates:

To further proceed, we need to calculate the time derivatives , , , , and .

Calculating Time Derivatives

We consider that the particle moves at some constant speed , then we have

Now, we can also write the velocity in the position space with basis :

Comparing the components,

To re-write the LHS of the Boltzmann equation, we can only consider collisionless situation, where the particles are moving in a straight line:

the calculation here is a little convoluted, because we need to track change in both the component and basis of the velocity:

refer here for the detailed calculation of the above.

Then we have

Now compute the time derivatives of the scalar coefficients:

Hence

Since the motion is collisionless, the direction vector is constant along the trajectory, so

Because , , and are linearly independent, each coefficient must vanish:

From the first equation, assuming ,

Substitute

to obtain

Now use the second equation:

Substituting and , the first two terms cancel:

Thus

which gives

Substituting , we obtain

Finally, the third equation serves as a consistency check:

So the result is consistent.

Therefore, in the collisionless case,

But the variable we use is :

Together with

these are the characteristic equations for free streaming in spherical coordinates.

Remember the Boltzmann equation in spherical coordinates:

Substituting the above time derivatives, we have

We can take some factors out of the derivatives:

Dividing both sides by , and replacing for better readability, we have

which is the conservative form of the Boltzmann equation in spherical coordinates.