Implementing a 3-Sphere Model for Molecular Dynamics Simulations

[quetzalcoatl9]

Hi everyone,

I'm currently implementing some molecular dynamics code. What I have so far is a system of particles interacting over cubic periodic boundaries..so when one particle leaves through one face of the cube, it pop back in through the other face. In reality however, the implementation simply replicates cubes and takes the minimum image. The fact that a boundary is introduced means that there is a cutoff potential...

To make a long story short, I am now implementing a model where the system would be implemented in a 3-sphere so that there would be no boundary considerations for interacting particles. If you consider the 2D analogy of particles sitting in a 2-sphere, they can all force interact with each other, without the need for a cutoff due to the boundary. This would be a better model. I wish to do this without embedding in 4-space.

While I understand the theoretical framework involved, I'm not so sure about how to handle the coordinate systems as an applied problem. I will use a spherical metric to calculate distance and angles, and I don't need the geodesic equation since I am doing a timestep integration to calculate the next position...but in what coordinates?

here's what i was thinking:

a) use x,y,z for coordinates
b) set 3 poles as 3 origins for a coordinate system, that cycle from (-a, a), they will just go round and round always having a coordinate in this range.
c) calculate distance between particles using the metric tensor, and apply force calculations to get acceleration
d) move the particles by Newtonian integration,
$$x = x_i + v_x dt + a_x (dt)(dt)$$

so, will keeping a standard coordinate system (but just cycling the coordinates without boundary) but using a spherical spatial metric work? any ideas?

 

[mathwonk]

a little less jargon and just maybe i can help.

 

[M98Ranger]

Hi there,

I think implementing a 3-sphere model for molecular dynamics simulations is a great idea! It definitely has the potential to improve the accuracy and realism of your simulations. Your approach of using a spherical metric to calculate distances and angles and then using Newtonian integration to move the particles sounds like a good plan. As for the coordinate system, I agree that using x,y,z coordinates and setting 3 poles as origins for the system is a logical choice. This will allow for easy calculation of distances and angles using the spherical metric. I would also suggest considering using polar coordinates (r, theta, phi) instead of Cartesian coordinates, as it may simplify some of the calculations and make them more intuitive in a spherical system. Overall, I think your approach is solid and I'm excited to see the results of your simulations using this 3-sphere model. Good luck with your implementation!