I just started studying about rarefied gases and I came across the concepts of Molecular Dynamics (MD) and Direct Simulation Monte Carlo (DSMC). My question is: How are these two fields related to each other? Is MD the physical theory and DSMC just a mathematical method to study molecular dynamics?
The difference can be described as follow: In MD (basic theory), algorithm solve equations of motion in deterministic way. You can have collision, but all of them contribute in state of system. In DSMC, all part same as MD but instead of equation of motion a random collision sampling (randomly collide with particles in specific range). This part implicit relate this algorithm to Monte Carlo algorithm. You can see the this link for complete description of difference. MD is more fundamental than DSMC, because we enter randomness directly into algorithm. Of course, there are some doubt about if we can directly can find statistical properties from MD (see FPU paradox) without direct entrance of randomness or noise.
I am afraid I am not an expert on Discrete Simulation Monte Carlo, but I am familier with other collision based simulation techniques for fluids.
Molecular Dynamics has potentials defining the forces between particles. Newton's equations of motion are then used to update the particles' positions and velocities. The algorithm can be summarized as:
1) Calculate forces using physical potentials like Lennard-Jones and electrostatics
2) Update velocities using $$(v(t+\Delta t)=F(t)/m+v(t)$$
3) Update positions using $$(x(t+\Delta t)=v(t) \Delta t+x(t)$$
In contrast Discrete Simulation Monte Carlo has this algorithm.
1) Update velocities based on a collision related to kinetic gas theory.
2) Update positions using $$(x(t+\Delta t)=v(t) \Delta t+x(t)$$
So the only real difference is the collision based on kinetic gas theory. The first step of calculating the collision is to divide the simulation box into small boxes. Any particles which are in the same smaller box "collide" with each other. I am not familiar with the collision operators used in Discrete Simulation Monte Carlo, but based on other "collision" algorithms I've studied a simple example would be that all particles simply take the average velocity of the particles in the box. This would probably work to a degree, but probably violates the Fluctuation-Dissipation theorem and has other undesirable properties. Thus is practice the "collisions" are usually more complicated. The one thing they pretty much all have in common is that they conserve momentum (the total momentum of all the particles in the sub-box is the same before and after the collision). In fact, in the first Multi-Particle Collision Dynamics simulations made no effort to either have physical meaning nor obey the Fluctuation-Dissipation theorem. I assume the same is true of Discrete Simulation Monte Carlo, and the reason why the word "phenomelogical" appears in the Wikipedia article.