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I know that when deriving the hydrodynamic equations from Boltzmann's equation we take the first three moments along momentum space to get the conservations laws. Taking infinitely many moments would get us a system equivalent to the Boltzmann equation and to stop at three we use a state equation to close.

For the Particle-In-Cell method, I've seen the same thing done along the position and momentum space to get the equations of motion for $\mathbf{r}_{p}(t)$ and $\mathbf{p}_{p}(t)$ of equation $$ f \left( \mathbf{r}, \mathbf{p}, t \right) = \sum\limits_p S_{\mathbf{r}} \left(\mathbf{r} - \mathbf{r}_{p}(t) \right) S_{\mathbf{p}} \left(\mathbf{p} - \mathbf{p}_{p}(t) \right) $$ in this reference (starting in page 10). My question is what information am I still losing in the Vlasov equation even if I use a huge number of particles? For example, the equations of motion of the super-particles don't seem to reflect that the shape factor $S_{\mathbf{r}}$ should change if $S_{\mathbf{p}}$ is not a Dirac delta.

Any insight will be appreciated. Thank you.

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My question is what information am I still losing in the Vlasov equation even if I use a huge number of particles?

The particle-in-cell (PIC) uses a bunch of super particles to represent the dynamics of a system of discrete particles obeying a set of rules, e.g., physical laws like Maxwell's equations. While PIC simulations do have the potential to be nearly realistic, they are limited by a few factors.

Information Losses
This question asks how and what information is lost when running a PIC simulation, i.e., what causes irreversibility in the system? By irreversibility I simply mean a loss of information that may or may not prevent one from running the simulation in reverse to return to the initial conditions.

  • The first is how one updates at each time step. This requires some form of interpolation to a grid, from which the cell derives in the name. The interpolation is done in numerous different ways depending upon the problem or the writer of the code, but it is akin (somewhat loosely) to taking a spatial ensemble average. Any time one performs an ensemble average, information is lost.
  • Perhaps not too surprisingly, the shape of the super particles can also introduce irreversibility because it "smoothes out" the discreteness of the particles in an attempt to reduce numerical/simulation noise. These shape functions can be very critical for simulations with a low number of particles per cell.
  • Taking a velocity moment of a distribution function is a form of ensemble averaging, which again inserts a form of irreversibility into your system. Thus, if you use the results from the moment calculations to update the system at each time step, you are inserting irreversibility into the system.
  • If you use the Vlasov equation as one of the equations of motion to govern the sysem, you may assume a form for the velocity distribution fuction (as suggested by your equation). Any assumption of a continuous function is another way one inserts irreversibility into a system.
  • PIC simulations are inherently noisy but the level/magnitude of noise can be reduced by increasing the number of particles per cell and using different shape functions for the super particles. In either case, all methods used to mitigate the noise will introduce irreversibility into a system. The noise arises from some of the issues I raised above but that also introduces irreversibility into a system.

As an aside, a true Vlasov simulation should be as close to time-reversible as one can get in a kinetic simulation. The main issues here become rounding errors for small parameters and/or "bad" boundary conditions (e.g., truly open boundary conditions may cause irreversible losses but that may actually reflect nature, thus the quotes around the word bad).

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