Is there any availability shown on how OpenMc discusses/solved the Boltzmann transport equation (mathematically in details) using Monte-carlo approach?

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Is there any availability shown on how OpenMc discusses/solved the Boltzmann transport equation (mathematically, step by step in details) using Monte-carlo approach? I have been going through the online OpenMc documentation but couldn’t come across the mathematical solution/discussion on how the openmc module carry out it mathematical calculations in solving the Boltmaann neutron transport yet, only that of algorithm is available which is not what i want. Pls where can i get it?
Thank you.

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Hi @IbrahimDgreat , I am not quite sure what do mean by mathematically step by step cause OpenMC is based on monte carlo algorithm which works by simulating the exact scenarios inside a reactor core adapting stochastic processes in general [ Unlike deterministic methods that discretize the neutron transport equation (NTE) and solve it over elements or meshes]. Monte Carlo doesn’t solve for an exact solution but instead estimates it by simulating individual particle histories. These histories provide a stochastic solution that converges to the actual answer as the number of histories increases.

To give you an analogy, let’s say you’re trying to determine the probability of getting heads in a coin toss. Mathematically, you’d calculate this by counting the favorable outcomes (heads) out of the total possible outcomes (heads and tails), resulting in a probability of 0.5. However, you can also reach this same result by running a simulated experiment where you toss the coin 10,000 times and count the proportion of heads. This probabilistic approach is similar to how Monte Carlo simulations operate: rather than solving equations directly, they simulate outcomes and average them to find a solution.

As I said earlier that I am not quite sure what did you mean by mathematically, step by step in details i am gonna leave two options for you here:

  1. Monte carlo (there are some mathematical proof and benchmark that monte carlo works and it’s true for which you can look up Dr. Romano’s PhD thesis or Carter and Cashwell from the 1970s. It’s a bit dated but the fundamentals haven’t changed much, so it’s still a good resource: https://www.osti.gov/biblio/4167844
  2. Deterministic methods (maybe this is what you want [there are Sn,Pn,CPM,MOC algorithms ]
    Computational Methods of Neutron Transport by prof E.E. Lewis is a good book for that)

Thanks for your question. Welcome to the fun world of monte carlo