Solving the Boltzmann Equation in the Relaxation-Time Approximation

Hello! In our previous lesson, we grappled with the profound conceptual paradoxes of Loschmidt and Zermelo, ultimately resolving them by understanding the H-theorem and the second law as statistical, rather than absolute, mechanical laws. This established the Boltzmann equation as a sound basis for describing a system's approach to equilibrium.

Our focus now shifts from these foundational questions to a more practical one: how do we actually solve the Boltzmann equation to describe systems that are held slightly out of equilibrium by external influences? The full Boltzmann equation, with its intricate collision integral, is a notoriously difficult non-linear integro-differential equation. In this lesson, we will develop a powerful and widely used method to make it tractable: the relaxation-time approximation (RTA). Our goal is to understand this approximation and use it to find the non-equilibrium distribution function for a system under a small external perturbation, which is the crucial first step in calculating any transport property.

1. The Challenge of the Collision Integral

Let's recall the full Boltzmann transport equation (BTE):

\frac{\partial f}{\partial t} + v \cdot \nabla_{r} f + F \cdot \nabla_{p} f = (\frac{\partial f}{\partial t})_{coll}

The terms on the left-hand side represent the "streaming" of the distribution function $f (r, p, t)$ through phase space. As particles move, their contribution to $f$ flows from one point in phase space to another. The term $v \cdot \nabla_{r} f$ describes the change in $f$ at a point due to particles physically streaming in and out of the spatial volume element.

This diagram illustrates the core idea of the streaming term $\mathbf{v} \cdot \nabla_{\mathbf{r}} f$. The net change in particle density within the slab of thickness $dx$ depends on the difference between the flux entering at $x$ and the flux exiting at $x+dx$. The Boltzmann equation generalizes this concept to the full six-dimensional phase space.

The right-hand side, the collision integral, accounts for the abrupt changes in momentum due to scattering, knocking particles into and out of the state $(r, p)$ . As we saw when deriving the H-theorem, this term is a complicated integral over all possible scattering partners and outcomes, making the BTE formidable to solve directly.

2. The Relaxation-Time Approximation (RTA)

To make progress, we introduce a phenomenological but highly effective model for the collision term. The core idea is that collisions tend to restore the system to a state of local equilibrium. If the distribution function $f$ deviates from the local equilibrium distribution $f_{0}$ , collisions will drive it back towards $f_{0}$ over a characteristic timescale, $τ$ .

The following video provides an excellent conceptual introduction to this idea, known as the relaxation-time approximation (RTA), and discusses the conditions under which it can be rigorously justified.

Near-equilibrium Transport Lecture 7: Boltzmann Transport Equation

This segment from a lecture by Mark Lundstrom on the nanohub YouTube channel clearly explains the physical reasoning behind the relaxation-time approximation and its mathematical form.

Please watch the portion from this explanation. Pay close attention to how the complex collision term is replaced by a simple expression describing relaxation towards equilibrium, and note the physical conditions (near-equilibrium, elastic or isotropic scattering) required for its justification.

As the video explained, the RTA replaces the complicated collision integral with a much simpler expression:

(\frac{\partial f}{\partial t})_{coll} \approx - \frac{f - f _{0}}{τ} = - \frac{δ f}{τ}

Here, $δ f = f - f_{0}$ is the small deviation from the local equilibrium distribution $f_{0}$ , and $τ$ is the relaxation time. This approximation essentially states that the rate of change of $f$ due to collisions is proportional to how far $f$ is from equilibrium, and it acts to reduce this deviation.

This is a powerful simplification, but it's important to recognize its limitations. The RTA is, as the name suggests, an approximation. A key drawback, which you can explore in the following reading, is that this simple form does not strictly conserve collisional invariants (particle number, momentum, and energy) on its own, as the true collision integral does.

Contents

This short section from Daniel Arovas's lecture notes highlights a key theoretical flaw of the simple RTA.

Please read the subsection "Failure of the relaxation time approximation," found here. The author shows that if you only consider the RTA collision term, even the total particle number appears to decay, which is unphysical. This motivates more sophisticated models (like the Lorentz model or the full linearized collision operator) which we will not cover here but are important in advanced transport theory.

Despite this imperfection, the RTA provides invaluable physical insight and correct qualitative (and often semi-quantitative) results for a wide range of transport phenomena.

3. Solving the BTE for Small Perturbations

With the RTA, the Boltzmann equation becomes a linear integro-differential equation:

\frac{\partial f}{\partial t} + v \cdot \nabla_{r} f + F \cdot \nabla_{p} f = - \frac{f - f _{0}}{τ}

We can solve this for systems that are only slightly perturbed from equilibrium. This "diffusion approximation" or perturbation method is clearly laid out in the next video.

17. Solutions to Boltzmann Equation: Diffusion Laws

This video from an MIT OpenCourseWare lecture by Prof. Gang Chen details the perturbation method for solving the BTE under the RTA.

Please watch the segment from this derivation. The key idea is to substitute f = f_0 + g (where g is our \delta f) into the BTE and neglect terms that are second-order in the small perturbation g.

Following the procedure from the video, we write $f = f_{0} + δ f$ . Since $δ f$ is assumed to be small, any terms involving derivatives of $δ f$ on the left-hand side are considered second-order small and are neglected. This yields:

\frac{\partial f _{0}}{\partial t} + v \cdot \nabla_{r} f_{0} + F \cdot \nabla_{p} f_{0} \approx - \frac{δ f}{τ}

This is no longer a differential equation for the unknown $δ f$ ; it is an algebraic equation! We can solve for the deviation $δ f$ directly:

δ f \approx - τ (\frac{\partial f _{0}}{\partial t} + v \cdot \nabla_{r} f_{0} + F \cdot \nabla_{p} f_{0})

This is the general solution for the perturbation to the distribution function in the relaxation-time approximation. The deviation from equilibrium, $δ f$ , is proportional to the relaxation time $τ$ and the "driving terms" in the parentheses, which represent how quickly the local equilibrium distribution $f_{0}$ is changing in time and space due to the external perturbations.

For a steady-state situation ( $\partial f_{0} / \partial t = 0$ ), this simplifies to:

δ f = - τ (v \cdot \nabla_{r} f_{0} + F \cdot \nabla_{p} f_{0})

The problem of finding the non-equilibrium distribution now reduces to evaluating the gradients of the known equilibrium distribution $f_{0}$ .

4. Application: System with an Oscillating External Force

Let's apply this method to a concrete example: a spatially uniform system of particles (e.g., electrons in a metal) subjected to a weak, uniform, oscillating external force, such as from an AC electric field. This is a classic problem that leads to the Drude model of AC conductivity.

The setup is:

Spatially uniform: $\nabla_{r} f_{0} = 0$
Oscillating force: $F (t) = F e^{- iω t}$
We seek a steady-state response, so the distribution function will also oscillate at the same frequency: $f (t) = f_{0} + δ f (ω) e^{- iω t}$ . This implies $\frac{\partial f}{\partial t} = - iω δ f (t)$ .

The BTE in the RTA becomes:

\frac{\partial f}{\partial t} + F (t) \cdot \nabla_{p} f = - \frac{δ f ( t )}{τ}

Substituting $f = f_{0} + δ f (t)$ and linearizing (i.e., using $f \approx f_{0}$ in the force term):

- iω δ f (t) + F (t) \cdot \nabla_{p} f_{0} = - \frac{δ f ( t )}{τ}

We can now solve this algebraic equation for $δ f (t)$ .

Contents

This section from Arovas's notes provides the concise derivation for this exact problem.

Please read the subsection "Oscillating external force," found here. Follow the steps to solve the algebraic equation for \delta f and see how it leads directly to an expression for the frequency-dependent conductivity \sigma(\omega).

As the derivation shows, rearranging the terms gives:

δ f (t) (\frac{1}{τ} - iω) = - F (t) \cdot \nabla_{p} f_{0}

Solving for $δ f (t)$ yields:

δ f (t) = - \frac{τ}{1 - iω τ} (F (t) \cdot \nabla_{p} f_{0})

Using the chain rule, $\nabla_{p} f_{0} = \frac{\partial f _{0}}{\partial E} \nabla_{p} E = \frac{\partial f _{0}}{\partial E} v$ , we arrive at the final expression for the deviation:

δ f (t) = - \frac{τ}{1 - iω τ} (F (t) \cdot v) \frac{\partial f _{0}}{\partial E}

This result is elegant and powerful. It shows that the non-equilibrium part of the distribution function depends on the strength of the driving force, the particle's velocity, and its energy. Crucially, the response is modified by the complex factor $1/ (1 - iω τ)$ . This factor shows that for low frequencies ( $ω τ ≪ 1$ ), the response is in phase with the force, while for high frequencies ( $ω τ ≫ 1$ ), the response is reduced in magnitude and phase-shifted by 90 degrees. This is directly analogous to the frequency-dependent impedance in an R-L circuit, a familiar concept from your background in electronics.

Conclusion

In this lesson, we have developed a method to tame the formidable Boltzmann equation. By replacing the exact collision integral with the phenomenological relaxation-time approximation, we transformed the problem into a solvable one for systems near equilibrium.

Key Takeaways:

The Relaxation-Time Approximation (RTA) models the effect of collisions as a process that drives the distribution function $f$ back towards a local equilibrium $f_{0}$ with a characteristic time $τ$ . Mathematically, $(\partial f / \partial t)_{coll} = - (f - f_{0}) / τ$ .
By assuming the deviation from equilibrium $δ f = f - f_{0}$ is small, we can linearize the BTE, turning it into an algebraic equation for $δ f$ .
The general solution for the deviation is $δ f \approx - τ \times (streaming terms acting on f_{0})$ . The deviation is proportional to the relaxation time and the strength of the perturbation (e.g., external forces or spatial gradients).
We applied this method to a system with an oscillating external force, deriving the frequency-dependent deviation $δ f (ω)$ , which captures the essential physics of AC transport.

We now have the key tool needed to analyze transport phenomena. We have a method for finding the non-equilibrium part of the distribution function, $δ f$ . In our next lesson, we will put this solution to work. We will see how to calculate macroscopic transport coefficients, such as thermal conductivity and viscosity, by computing averages (or moments) using this perturbed distribution function.

Recommended resources

1. The Challenge of the Collision Integral

2. The Relaxation-Time Approximation (RTA)

3. Solving the BTE for Small Perturbations

4. Application: System with an Oscillating External Force

Conclusion