Am I correct that you want to program a certain curve that is perfectly reflective? (in a simulation of particles)

Dear Frederik,

I want to see the different possibilities of how to simulate reflective boundaries for the particle motion. In other words, what is the function that generates this behaviour that is included in the particle motion if this latter exceeds the boundaries of the simulation domain ?

Thank you

Ilhem

Well, mathematically it is quite simple. I = R + T, Incident intensity = Reflected Intensity + Transmitted Intensity.

In the case of boundaries, it matters at which energy level your particles are, and what height the boundary is. In essende your boundary is a barrier. Unless the barrier is infinitely high (and R = 100%), you should probably make a finite size barrier, with a finite height. So that some particles tunnel (Transmit), and some particles reflect. (You could even absorb some in the barrier). This way your box follows physics quite well. And if you take 'Free' particles, it shouldn't take too much numerical strength either.

What is the purpose of your simulation?

I assume that you are doing a time-stepping simulation such that during a time step the path of each particle is a straight line. Then there is a simple strategy: decompose your last velocity vector into components parallel and perpendicular to the normal vector to your reflecting (not reflected!) boundary. Then you obtain the velocity vector for the next step by reversing the parallel component while leaving the perpendicular one unchanged. If you use large time-steps which let the particles go far into the 'forbidden territory' you may insert a intermediate step which first brings the particle back to the surface.

An alternative method (which I like most) is to replace the boundary wall by a potential (e.g. proportional to the 6th power of the penetration depth. Then the forces guide the particle along a path which a bit away from the wall is a vey good approximation to elastic reflection.

I similar ideas are expressed in the last contribution of Frederik. Hope to have added a bit to the insight provided already.

There are some difficulties. Thus, if you have several boundaries, you must at each time step check whether you crossed any of them. However, if in one time step you crossed boundary B1, you should immediately afterwards *not* test for B1, as this can create nasty side effects, such as staying stuck on B1 forever. Similarly, if you have curved boundaries, you may want to be careful about orbits that get in and out of the forbidden region in one time step. I do not know of any general approach that works and believe thiese issues are best tackled on a case-to-case basis.

Dear Ilhem, if you are doing MC simulation then you can apply the condition "If the next location hit the reflective boundary" for the situation "if the next location goes out form the domain" i.e. you can use either white boundary condition or periodic boundary condition. Please find the attachment for details.

I would generally follow the approach first proposed here by Ulrich. With one obvious correction: not the parallel but perpendicular component of particle velocity should be reversed at the boundary. But when you consider non-zero temperature of the reflecting wall (why not?), then this picture need not to hold, i.e. the energy of a given particle may change during reflection. But this is another story and will not develop it further here.

Many years ago I was carrying similar simulations with particles living in a 3D rectangular prism. The problem I encountered was: what is the fate of a particle hitting the edge? And how to detect such a situation? Experimentally found answer was: check whether the particle after such 'reflection' still stays within the box. From purely geometric point of view it may seem that the probability of the mentioned event is negligible, precisely null. In practice, however, the numerical accuracy and unavoidable roundings make it happen quite frequently. In my case, one particle per roughly one thousand of them had to be 'forgotten' (eliminated from further computations). This was happening on a 8-bit machine, with 32 bit real numbers. Your mileage may vary substantially.

Hi,

Thanks to all of you to your valuable explanations and suggestions. Actually, I'm trying to simulate the motion of particles in 3D environment by using for the 2 horizontal axis (x and y ) a periodic boundaries and reflective boundaries for the vertical axis (z).

So is it possible to define a reflective boundaries conditions in this context? for example adding a (+-) random variable to the limits of the z-axis in order to fall inside the domain at the next simulation time?

Thank you in advance

Ilhem

@Marek,

please notice that my vector components were defined with respect to the normal vector to the boundary!

@IIhem

this is exactly the situation that I had in mind when writing my contribution. Your random variable adds an unnecessary complication. I guess your z-delimiting boundaries are planes? This makes things particularly simple since the normal vector than is constant and easy to define.