Particle transport C: Monte Carlo methods versus moment equations

This is the concluding of three articles introducing the basic concepts of the Monte Carlo method for radiation transport calculations. The demonstration calculations discussed in the previous articles give a basis for comparing the relative benefits and drawbacks of the Monte Carlo and transport equation approaches. We have seen that both strategies are based on averages over random distributions of particles whose statistical properties are known. Both methods give the same result in the limit of a large number of model particles. The main difference is that the averaging process is performed at the beginning of the transport equation calculation but at the end of the Monte Carlo solution.


  • In the transport equation approach to the two-dimensional random walk, the idea is to seek average quantities n or J and to find relationships between them (like Fick’s first and second laws). These relationships are accurate when there are large numbers of particles. To illustrate the meaning of large, note that the number of electrons in one cubic micrometer of aluminum equals 3 × 10^15. When averages are taken over such large numbers, the transport equations are effectively deterministic.
  • In the Monte Carlo method, the idea is to follow individual particles based on a knowledge of their interaction mechanisms. A practical computer simulation may involve millions of model particles, orders of magnitude below the actual particle number. Therefore, each model particle represents the average behavior of a large group of actual particles. In contrast to transport equations, the accuracy of Monte Carlo calculations is dominated by statistic variations.

An additional benefit of transport equations is that they often have closed-form solutions that lead to scaling relationships like Eq. 22 of the previous article. We could extract an approximation to the relationship from Monte Carlo results, although at the expense of some labor.

Despite the apparently favorable features of the transport equations, Monte Carlo is the primary tool for electron/photon transport. Let’s understand why. One advantage is apparent comparing the relative effort in the demonstration solutions — the Monte Carlo calculation is much easier to understand. A clear definition of physical properties of particle collisions was combined with a few simple rules. The only derivation required was that for the mean free path. The entire physical model was contained in a few lines of code. In contrast, the transport model required considerable insight and the derivation of several equations. In addition, it was necessary to introduce additional results like the divergence theorem. Most of us feel more comfortable staying close to the physics with a minimum of intervening mathematical constructions. This attitude represents good strategy, not laziness. Less abstraction means less chance for error. A computer calculation that closely adheres to the physics is called a simulation. Program managers and funding agents have a warm feeling for simulations.

Beyond the emotional appeal, there is an over-riding practical reason to apply Monte Carlo to electron/photon transport in matter. Transport equations become untenable when the interaction physics becomes complex. For example, consider the following scenario for a demonstration calculation:

In 20% of collisions, a particle splits into two particles with velocity 0.5v0 and 0.2v0. The two particles are emitted at a random angles separated by 60°. Each secondary particle has its own cross section for interaction with the background obstacles.

It would be relatively easy to modify the code of the first article to represent this history and even more complex ones. On the other hand, it would be require consider effort and theoretical insight to modify a transport equation. As a second example, suppose the medium were not uniform but had inclusions with different cross sections and with dimensions less than ?. In this case, the derivation of Fick’s first law is invalid. A much more complex relationship would be needed. Again, it would relatively simple to incorporate such a change in a Monte Carlo model. Although these scenarios may sound arbitrary, they are precisely the type of processes that occur in electron/photon showers.

In summary, the goal in collective physics is to describe behavior of huge numbers of particles. We have discussed two approaches:


  • Monte Carlo method. Define a large but reasonable set of model particles, where each model particle represents the behavior of a group of real particles with similar properties. Propagate the model particles as single particles using known physics and probabilities of interactions. Then, take averages to infer the group behavior.
  • Transport equation method. Define macroscopic quantities, averages over particle distributions. Derive and solve differential equations that describe the behavior of the macroscopic quantities.

The choice of method depends on the nature of the particles and their interaction mechanisms. Often, practical calculations usually use a combination of the two approaches. For example, consider the three types of calculations required for the design of X-ray devices (supported in our Xenos package):

  • Radiation transport in matter. Photons may be treated with the Monte Carlo technique, but mixed methods are necessary for electrons and positrons. In addition to discrete events (hard interactions) like Compton scattering, energetic electrons in matter undergo small angle scattering and energy loss with a vast number of background electrons (soft interactions). It would be impossible to model each interaction individually. Instead, averages based on transport calculations are used.
  • Heat transfer. Here, particles are the energy transferred from one atom to an adjacent one. Because the interaction model is simple and the mean-free-path is extremely small, transport equations are clearly the best choice.
  • Electric and magnetic fields. The standard approach is through the Maxwell equations. They are transport type equations, derived by taking averages over a large number of charges. On the other other hand, we employ Monte-Carlo-type methods to treat contributions to fields from high-current electron beams.


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