User:
Problem of variable probability
Equation
So far we had assumed that the probability of interaction was independent of position. However the systems are usually not homogeneous and therefore it is necessary to consider that the probabilities depend on the position.
In this case it is necessary to define the probability in function of the position and to consider in the estimation of the free path the probabilities along the route. Both situations make it necessary to discretize the medium and move from cell to cell.
ID:(9102, 0)
Problem of discontinuous probability
Equation
The introduction of a grid means that average probabilities must be defined per cell. This in turn restricts the size of the grid, since it must be defined so that fluctuations in the probability function are not lost.
On the other hand, in a two- and three-dimensional system, the discontinuity in the probability function leads to the behavior of the particles being affected by the anisotropy of the discrete modeling of the probability function.
ID:(9105, 0)
Cell size problem
Equation
The main problem with the introduction of cells is that positions are no longer continuous but are reduced to discrete values in which the center of a cell is considered.
On the other hand, the introduction of more dimensions leads to the introduction of anisotropies since the centers of the cells involve directions and thus discrete angles. For example, in a two-dimensional system there are the possibilities of:
- four neighboring cells with angles of $\pi/2$
- six neighboring cells with angles of $\pi/3$
Any other geometry has the problem that the distances between centers are no longer the same which in itself makes it more difficult to work with constant time intervals and speeds.
ID:(9104, 0)