What are some examples of other interactions we could implement between particles?
Particle systems are often used to model phenomena such as smoke, fire, and explosions. Objects in particle systems typically have certain attributes such as position, velocity, size, age, and lifetime. A dynamics model is used to describe how the attributes of particles change over time, and after a particle's age reaches the value of its lifetime, it is destroyed. Through particle systems, we can render physical phenomena whose global behavior is determined by many complex local interactions.
Would mixing paint (like digital painting systems that have realistic mixing between "oil paint" or other wet media) require a particle system to simulate?
Adding on to @drewkaul, I've been developing a video game in Unity and have been incredibly surprised at the versatility of particle systems. I have used them to render fire, smoke, snow and rain in real time. Furthermore, one can specific so many of the attributes of a particle system (velocities, lifetimes, color, image, color changes, pattern, number count) to virtually achieve any dynamic movement. Although somewhat taxing on computers as you increase the particle account, particle systems still provide some incredible effects and visuals!
Particle systems seem very useful in simulating interesting effects because we can use a large number of particles and each of them can be define by a set of physical force. I think some materials would need significantly more particles than others, like fluid. The large number of particle can be a challenge in computation.
Summary: Particle systems are very useful in simulating things like fire, moving water, smoke, light trails, galaxies, clouds, fog, or more fantastical things like magic. They can also be used to render things like hair or grass. Like mentioned in previous slides, they simplify a lot of physical properties in order to save on computation time but still get a good visual result.
The Material Point Method is a technique used for simulating continuum materials (these are materials modeled as continuous mass, not discrete particles). It is particularly good at handling snow dynamics due to variety of states snow can be in (icy, melting, etc.). Specifically, one can represent snow using an elasto-plastic constitutive model, treating snow as a continuous body rather than granular material.