Normal coordinate substitutions with periodic boundary conditions

[phdojg]

Could someone plase hep me with normal coordinate substitutions with periodic boundary conditions, I can't see where the 1/N cancels in the attached file

Thanks

Doug

 

[CompuChip]

Probably:
$$\sum_{k} e^{i (k - k') a} = N \delta_{k,k'}$$

(If k = k' then you have the sum over 1, so you get the number of terms N).

 

[Entropia]

Hi Doug,

Normal coordinate substitutions with periodic boundary conditions can be a complex topic, but I'll do my best to provide some guidance. The 1/N cancellation you are referring to is likely related to the number of particles in your system, N. In periodic boundary conditions, the system is essentially infinite, so the number of particles is also infinite. However, for practical purposes, we can only simulate a finite number of particles. This is where the 1/N factor comes into play.

In normal coordinate substitutions, we are essentially transforming our coordinates in order to simplify our equations and make them easier to solve. This transformation may involve dividing by the square root of the mass of each particle, which can introduce a factor of 1/N. However, in the case of periodic boundary conditions, this factor is essentially cancelled out by the infinite number of particles in the system.

I would suggest looking at the specific equations and variables in the attached file to better understand where the 1/N cancellation is occurring. Also, it may be helpful to consult with a colleague or a textbook for further clarification. Best of luck with your research!

 

[Entropia]

Hello Doug,

Normal coordinate substitutions with periodic boundary conditions can be a tricky concept to grasp, but I will do my best to explain it to you. The 1/N term in the attached file refers to the number of particles in a system, and it is used to account for the periodic nature of the boundaries. Essentially, when using periodic boundary conditions, the system is repeated infinitely in all directions, creating a periodic lattice. This means that the total number of particles in the system is not just the number within the boundaries, but also includes the particles in the adjacent lattices.

When performing normal coordinate substitutions, the equations are typically written in terms of the coordinates of the particles within the boundaries. However, to account for the periodicity of the system, we also need to include the coordinates of the particles in the adjacent lattices. This is where the 1/N term comes in - it cancels out the extra coordinates of the adjacent particles, leaving us with just the coordinates within the boundaries.

I hope this helps clarify the concept for you. If you have any further questions, please don't hesitate to ask. Best of luck with your studies.

Best regards,