Particle Sphere

[Dardo Hameed]

Inquantum mechanics, a spherically symmetric potential is a system of which the potential only depends on the radial distance from the spherical center and a location in space. A particle in a spherically symmetric potential will behave accordingly to said potential and can therefore be used as an approximation, for example, of the electron in a hydrogen atom or of the formation of chemical bonds.[1]

In the general time-independent case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian of the following form: H ^ = p ^ 2 2 m 0 + V ( r ) \displaystyle \hat H=\frac \hat p^22m_0+V(r) Here, m 0 \displaystyle m_0 is the mass of the particle, p ^ \displaystyle \hat p is the momentum operator, and the potential V ( r ) \displaystyle V(r) depends only on the vector magnitude of the position vector, that is, the radial distance from the origin (hence the spherical symmetry of the problem).

To describe a particle in a spherically symmetric system, it is convenient to use spherical coordinates; denoted by r \displaystyle r , θ \displaystyle \theta and ϕ \displaystyle \phi . The time-independent Schrdinger equation for the system is then a separable, partial differential equation. This means solutions to the angular dimensions of the equation can be found independently of the radial dimension. This leaves an ordinary differential equation in terms only of the radius, r \displaystyle r , which determines the eigenstates for the particular potential, V ( r ) \displaystyle V(r) .

We first consider bound states, i.e. states which display the particle mostly inside the box (confined states). Those have an energy E \displaystyle E less than the potential outside the sphere, i.e., they have negative energy. Also worth noticing is that unlike Coulomb potential, featuring an infinite number of discrete bound states, the spherical square well has only a finite (if any) number because of its finite range.

First we transform the radial equation by a few successive substitutions to the generalized Laguerre differential equation, which has known solutions: the generalized Laguerre functions. Then we normalize the generalized Laguerre functions to unity. This normalization is with the usual volume element r2 dr.

Your settings looks fine. Maybe the sphere is using a sphere collider component instead of a mesh collider component? (it must use a mesh collider component if you want the inverted mesh normals to have any effect)

If your graphical card allows it, instancing would be more efficient than the particleSOP. The particleSOP is calculated on the CPU where with instancing there is just 1 SOP but being duplicated on the GPU with certain parameters (position / size / rotation / color etc).

Using CHOP or TOP data you can reference each particle separately, like growing the size of 1 particular particle out of all your particles etc.

So this covers the surface of a sphere with radius 1. If you want to have a different radius you can simply multiply x, y and z with a number to scale the whole sphere. If you vary this per pixel you can manipulate/deform the sphere super fast.

If you then multiply that top with something else, you can alter the radius like described above. Then I use the final null1 TOP as source of instanced particles, positioning them on the xyz I just computed.

The particles themselves are created by using an addSOP (Points/add1) to create a point, then converting it to a particle using a convertSOP (Points/convert1), this makes sure it gets rendered as a point.

Hope this all makes a bit of sense. Not sure how much of the math you understand, perhaps check out Sphere - Wikipedia and Spherical coordinate system - Wikipedia where they explain it way better than I ever can

But the problem occurs if particles are moving very quickly, because then the particle basically jumps to the other side of the sphere (P1 is the previous position, P2 is the current position, P' is how I think it should be resolved), instead of returning back to the previous position:

Now, I could try to return in the direction of the previous point, but since the sphere might also be moving, I am not sure if P1 is even a valid position (and if it will make sense). Also, it seems to be more computationally expensive - is this how I am supposed to do it, or not?

Cloth like things snapping to the wrong side of an obstacle and getting stuck there is not too uncommon. Even more common is fast moving objects overlapping way too much when the collision is detected.

A common solution is, on detecting a collision, sub divide the previous step until the collision is less severe and then resolve it. I think you will find trying to detect how deep the collision is to be difficult in your case, but if you could limit the top speed of spheres in your system you could binary split the frames in which collisions occur a fixed number of times and assume it will be good enough?

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My assumption was that when one of the particles passes through the wall of the sphere, they all render outside. I tried to rig up an inner sphere, with reversed normals and a mesh collider to trap the particles inside, as per a similar solution i found elsewhere. That worked to keep the particles inside, but I was seeing the same behavior. Figured I should ask the internet. Hope you can help

To solve for the eigenvalues of a particle on the surface of a sphere, you can use the Laplace-Beltrami operator and the associated eigenvalue equation. This equation involves finding the eigenvalues that satisfy the boundary conditions on the surface of the sphere.

The eigenvalues represent the energy levels of the particle on the surface of the sphere. These energy levels can provide valuable information about the behavior and properties of the particle on the sphere.

Yes, the eigenvalues can be calculated analytically using mathematical techniques such as separation of variables and spherical harmonics. However, for more complex systems, numerical methods may be necessary.

The eigenvalues of the particle on the surface of the sphere correspond to the quantum numbers of the system. These quantum numbers represent the different states or configurations of the particle on the sphere.

Yes, the study of particles on the surface of a sphere has applications in various fields such as quantum mechanics, geophysics, and materials science. Understanding the behavior of particles on curved surfaces can help in the development of new technologies and materials.

yes there is a fundamental limit - which is machine precision :-)

The obvious solution would be (a) coarse graining or (b) using another unit system such as cgs if you want to stick to small particles

>>As a more general question then, are there problems I should expect modelling the behaviour of small particles?

expect e.g. dominant cohesive/van der walls forces which makes it difficult to model

>>I assume one can overlay the LAMMPS VdW implementation over the granular particles

To my modest knowledge this is not possible since LAMMPS VdW is an atomistic model whereas for granular particles of mascroscopic size this should be some kind of "subgrid model". However, it would be possible to use the LAMMPS VdW model to developsuch a model.

But I am sure there are also models readily available in the literature

I am also interested in looking at similar systems to you (i.e. particles around the 1 m size) and so this issue of the particle size limit and combination of granular and vdW forces is also important to me. I am also interested in coupling the particles with fluid motion.

Now that I have moved to the latest version of LIGGGHTS and CFDEM, I have been hit with this particle size limit issue (it worked fine in the older versions). I haven't figured this one out yet. Looks like I will have to change unit systems.

However, I have managed to overlay the vdW forces (from LAMMPS) onto the granular atoms (LIGGGHTS). The hybrid/overlay command from LAMMPS works fine. I am still busy sorting out the material constants, so I haven't produced any usable data just yet. Also, my system is still pretty dilute as I am still setting up my model, but I have managed to get a stable model while including these forces. I am working on pushing up the concentrations at the moment. So it is possible.

In conventional granular methods, it is OK to model particles with a diameter in microns, but the model would run very slowly because very small timestep is required to achieve a computation stability.

>>My question is if MD could simulate the dynamics between atoms, why it could not simulate the

>>interaction/contact between the particles at micron scale???

there are groups which try to go in this direction, e.g. Marcus Bannerman (group of Prof. Pschel/Univ Erlangen)

But to my knowledge true MD for a full contact of larger particles is hard because the largest systems that can be simulated with MD at the moment have the size of Lincoln's nose on a penny

I am dealing with micro range particles.When I change the gravity value 9.81 to 981 to chage the unit from si to cgs,particles move with fast speed and leave the simulation domain.

can any one please sort it out.

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