KAHIBARO
39.1. Monte Carlo Random Numbers
Table of Contents
Random sampling
Monte Carlo simulations in GATE rely on random sampling to model the inherently probabilistic nature of particle transport and detection. Every event in your simulation, from the emission of a particle to its interaction in a detector, is decided using pseudo-random numbers.
In practice, GATE calls the Geant4 random number generator repeatedly to draw numbers that are uniformly distributed in the range $[0,1)$. These uniform values are then mathematically transformed to sample from the physical probability distributions that describe your problem. For example, the distance to the next interaction, the scattering angle, or the deposited energy are all drawn from known probability density functions.
You will encounter random sampling in many parts of a GATE simulation. Source definitions use random numbers to decide when each particle is emitted, where it starts, its direction, and its energy, based on the distributions you specify. Physics processes use random sampling to decide whether an interaction occurs in a given step, which interaction type takes place, and the kinematics of the outgoing particles. Digitizers and actors that apply blurring or noise, such as energy or timing resolution, also use random sampling to transform the ideal physical signal into a realistic detector response.
Although these numbers are called “random,” they are actually generated by deterministic algorithms. If you start the generator from the same initial state, called the seed, you get exactly the same sequence of numbers and therefore exactly the same Monte Carlo history. This property is crucial for debugging and reproducibility, because you can rerun the same simulation and obtain identical results when you use the same seed and configuration.
Random sampling is also central to how statistical uncertainty appears in Monte Carlo simulations. Each run represents one realization of many random choices. If you repeat the same simulation with different seeds, the mean results should be consistent, but individual tallies such as dose in a voxel or counts in a detector bin will fluctuate. These fluctuations decrease as you increase the number of simulated events, approximately with the well known square root law.
Important: Monte Carlo results contain statistical uncertainty. For a quantity estimated from $N$ independent events, the relative statistical uncertainty typically scales as
$$
\frac{\sigma}{\mu} \propto \frac{1}{\sqrt{N}}.
$$
Doubling the precision often requires roughly four times more simulated events.
When you design simulations, it is important to remember that every additional stochastic effect you include, for example finer physics detail or realistic detector blurring, adds more random sampling. This can increase noise in your output for a given number of events. You control this not by changing how random numbers are drawn, but by choosing appropriate physics, digitizer settings, and by running enough events for the precision you need.
Random engines
Behind every random number used in GATE is a random engine, a specific algorithm that generates a long deterministic sequence of numbers that appear random. GATE inherits its random number infrastructure from Geant4, which provides several pseudo-random generators such as Mersenne Twister variants and other well tested engines suitable for scientific Monte Carlo work.
A random engine maintains an internal state that completely determines the sequence of random numbers it will produce. When you initialize a simulation, GATE sets up this engine and its initial state, often using a seed value that you can control. From that moment, every random draw, whether for a source or a physics process, advances the engine state and yields the next number in the sequence.
Because GATE is built on Geant4, you normally do not interact directly with the random engine in simple simulations. However, you should understand two key ideas. First, different engines can have different properties such as period length and statistical quality. Geant4 defaults are chosen to be safe for most applications in medical physics, including imaging and dosimetry. Second, reproducibility depends both on the choice of engine and on its initial seed. Using the same engine and the same seed produces the same sequence of random numbers and thus the same Monte Carlo history, provided the simulation code and configuration are unchanged.
In multithreaded simulations the situation is more subtle. Each worker thread typically has its own random engine or a separate subsequence of a common engine. This is designed to keep results statistically independent between threads while avoiding unwanted correlations. For reproducible multithreaded runs, the random engines for each thread must be initialized in a controlled way. GATE follows the Geant4 strategy for thread level random number management, but you should be aware that changing the number of threads or the order of operations can change how many times each engine is called and therefore alter the exact sequence of random numbers consumed.
From the user perspective, the practical role of the random engine is to provide a consistent and high quality source of pseudo-random numbers that all parts of GATE can share. In later sections on random seeds, independent simulations, and reproducible research, you will see how controlling these engines through seeding and configuration lets you repeat, vary, and document your simulations in a scientifically robust way.
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