High Energy Physics Algorithms in GEANT4 I have been learning to use the GEANT4 software and I came across the following statement in the Users Guide, and I couldn't help being curious as to why this was? I'm primarily a mathematician, so I apologise in advance for any ignorance in HEP.

"...Geant4 can not offer a single modelling algorithm to cover the entire energy domain from zero to the TeV scale, for all known processes and particles."

 A: Let's be a bit more specific, shall we, and look at one example: a photon track. So a photon of energy $E_\gamma$ enters one of the small volume GEANT divided the whole detector into. GEANT takes into account several possible processes:


*

*Photo-electric effect: the photon is absorbed and an electron is ejected.

*Compton scattering: an inelastic collision between the photon and the electron, which result in an electron ejected and a photon with a different energy.

*Conversion to $e^+e^-$ pair


Each of them use very different semi-empirical formulae for the cross sections, the angular distributions of the final particles, the excitations level of the material, etc., all adjusted to experimental data. For example the photo-electric cross section is a series in powers of $1/E_\gamma$ [1, eqn (5.1)] whereas that of the Compton scattering has a logarithmic term and a rational approximant of order 3, both depending on $E_\gamma/m_ec^2$. [1, eqn (5.7)] I wrote semi-empirical because they are based on some theoretical considerations but they are definitively not a pure theoretical computations based entirely on QED. 
Now here are the absorption coefficients (which are proportional to the cross sections) for each process across the energy range, for the material lead (Pb). Credits goes to Joshua Hykes , Wikipedia.

As you can see, the relative importance of each process drastically change depending on the energy. So that's an illustration of different models being used across the energy range.
[1] Physics Reference Manual – Geant4
A: I think this applies not only to high energy physics, but to all sorts of computational physics programs.
Computational complexity is a very important thing when it comes to these programs, and certain algorithms are better for certain problems.
As an analogy, I'd say: 
If I'd want to simulate collisions between two or three particles, let's say, I'd be well off with a time driven dynamics simulation. (writing an event driven simulation would be overkill)
If I were to simulate collisions between millions of particles, on the other hand, it would be stupid to go with a time based simulation because of how long it'd take to run the program. 
As for another example, it'd be okay to model the dynamics a small bunch of particles individually, but when this number is very very large, it is almost always better to model them as a fluid. 
A: GEANT is a very extensive and HUGE Monte-Carlo (MC)-program. It typically tracks particles through detectors filled with all kind of materials and computes at each step the probability (via the cross sections) a certain number of interactions, scattering, decays and so forth. Via (at each step generated) random numbers according to the probability distributions given by the cross sections (MC) it will be  "determined" if a certain process will be happen (or better said simulated) or not. These cross sections are often only defined in certain energy or angle ranges and in the remaining energy or angle ranges represented by other formulas which provide in the given actual energy or angle range a better approximation. The program includes all processes with respect to the interaction of particles (especially ionizing particles but also non-ionizing particles) with matter. As you can certainly imagine, there are lot of different particles interacting in a different way with different types of material (of which one important parameter is $Z$ the atomic number, but it is not the only one), the particles having all kind of different energies, hundreds (if not thousand) of different parametrisations (of course all based on physics formulas) of these interactions can take place. There is no unique picture. Just an example: A low energy photon will experience "photo effect", at medium energy "compton effect" and at high energy mainly "pair creation", 3 rather different processes with their individual probability distributions. Nevertheless all three are based on electromagnetic interaction, so the processes can be derived from a common theory. Other particles only interact at low energy electromagnetically, but at higher energies also can have nuclear interaction. At the basis everything in GEANT is some kind of based of the standard model. However, another aspect plays an important role: how a certain type of force (there a 3) structures matter. The structure of the material the particle is interacting with plays a fundamental role. And nature allows so many different type of structures of the matter, which cannot be viewed in a unique picture. And GEANT  is supposed to deal with "all" types of structures typical materials can have. But finally, in certain parameter areas GEANT is not well presenting the real physics, in these parameter areas other simulation programs have to be used. 
You have to realize that on different scales (which represent different energies) different physics models apply, this is a kind of fundamental concept in physics. That is in particular true for the simulation of the physics implemented in GEANT.  
